Properties

Label 6027.2.a.bf.1.9
Level $6027$
Weight $2$
Character 6027.1
Self dual yes
Analytic conductor $48.126$
Analytic rank $1$
Dimension $12$
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] = [6027,2,Mod(1,6027)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, 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("6027.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6027.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: \(48.1258372982\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 15 x^{10} + 30 x^{9} + 74 x^{8} - 149 x^{7} - 140 x^{6} + 278 x^{5} + 126 x^{4} - 211 x^{3} - 64 x^{2} + 53 x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 861)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-1.05133\) of defining polynomial
Character \(\chi\) \(=\) 6027.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.05133 q^{2} -1.00000 q^{3} -0.894704 q^{4} -4.26226 q^{5} -1.05133 q^{6} -3.04329 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.05133 q^{2} -1.00000 q^{3} -0.894704 q^{4} -4.26226 q^{5} -1.05133 q^{6} -3.04329 q^{8} +1.00000 q^{9} -4.48105 q^{10} -2.24822 q^{11} +0.894704 q^{12} +0.346093 q^{13} +4.26226 q^{15} -1.41010 q^{16} +4.66895 q^{17} +1.05133 q^{18} -2.00040 q^{19} +3.81346 q^{20} -2.36363 q^{22} +2.47117 q^{23} +3.04329 q^{24} +13.1669 q^{25} +0.363858 q^{26} -1.00000 q^{27} +6.18926 q^{29} +4.48105 q^{30} -3.71258 q^{31} +4.60410 q^{32} +2.24822 q^{33} +4.90861 q^{34} -0.894704 q^{36} -4.73147 q^{37} -2.10309 q^{38} -0.346093 q^{39} +12.9713 q^{40} +1.00000 q^{41} -0.374747 q^{43} +2.01149 q^{44} -4.26226 q^{45} +2.59802 q^{46} -3.92338 q^{47} +1.41010 q^{48} +13.8427 q^{50} -4.66895 q^{51} -0.309650 q^{52} +10.4390 q^{53} -1.05133 q^{54} +9.58251 q^{55} +2.00040 q^{57} +6.50696 q^{58} -7.37689 q^{59} -3.81346 q^{60} -11.7053 q^{61} -3.90315 q^{62} +7.66063 q^{64} -1.47514 q^{65} +2.36363 q^{66} +14.2025 q^{67} -4.17732 q^{68} -2.47117 q^{69} +1.60184 q^{71} -3.04329 q^{72} +15.1789 q^{73} -4.97434 q^{74} -13.1669 q^{75} +1.78977 q^{76} -0.363858 q^{78} +10.1626 q^{79} +6.01021 q^{80} +1.00000 q^{81} +1.05133 q^{82} +12.3018 q^{83} -19.9003 q^{85} -0.393984 q^{86} -6.18926 q^{87} +6.84199 q^{88} -7.79875 q^{89} -4.48105 q^{90} -2.21097 q^{92} +3.71258 q^{93} -4.12477 q^{94} +8.52624 q^{95} -4.60410 q^{96} -12.3944 q^{97} -2.24822 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 12 q^{3} + 10 q^{4} - 12 q^{5} + 2 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - 12 q^{3} + 10 q^{4} - 12 q^{5} + 2 q^{6} + 12 q^{9} - 11 q^{10} + 10 q^{11} - 10 q^{12} - 15 q^{13} + 12 q^{15} + 14 q^{16} - 8 q^{17} - 2 q^{18} - 2 q^{19} - 16 q^{20} - 7 q^{22} + 5 q^{23} + 20 q^{25} - 12 q^{27} + 20 q^{29} + 11 q^{30} - 10 q^{31} + 3 q^{32} - 10 q^{33} + 23 q^{34} + 10 q^{36} - 17 q^{37} - 6 q^{38} + 15 q^{39} - 39 q^{40} + 12 q^{41} + 12 q^{43} + 20 q^{44} - 12 q^{45} - 36 q^{46} - 34 q^{47} - 14 q^{48} + 59 q^{50} + 8 q^{51} - 26 q^{52} + 6 q^{53} + 2 q^{54} + q^{55} + 2 q^{57} - 11 q^{58} - 27 q^{59} + 16 q^{60} - 22 q^{61} + 45 q^{62} + 26 q^{64} + 7 q^{66} - 26 q^{67} - 33 q^{68} - 5 q^{69} + 50 q^{71} - 21 q^{73} - 35 q^{74} - 20 q^{75} + 24 q^{76} - 10 q^{79} - 22 q^{80} + 12 q^{81} - 2 q^{82} - 8 q^{83} + 8 q^{85} - 17 q^{86} - 20 q^{87} - 46 q^{88} - 11 q^{89} - 11 q^{90} + 63 q^{92} + 10 q^{93} - 10 q^{94} + 35 q^{95} - 3 q^{96} - 32 q^{97} + 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.05133 0.743403 0.371702 0.928352i \(-0.378774\pi\)
0.371702 + 0.928352i \(0.378774\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.894704 −0.447352
\(5\) −4.26226 −1.90614 −0.953071 0.302748i \(-0.902096\pi\)
−0.953071 + 0.302748i \(0.902096\pi\)
\(6\) −1.05133 −0.429204
\(7\) 0 0
\(8\) −3.04329 −1.07597
\(9\) 1.00000 0.333333
\(10\) −4.48105 −1.41703
\(11\) −2.24822 −0.677865 −0.338932 0.940811i \(-0.610066\pi\)
−0.338932 + 0.940811i \(0.610066\pi\)
\(12\) 0.894704 0.258279
\(13\) 0.346093 0.0959888 0.0479944 0.998848i \(-0.484717\pi\)
0.0479944 + 0.998848i \(0.484717\pi\)
\(14\) 0 0
\(15\) 4.26226 1.10051
\(16\) −1.41010 −0.352525
\(17\) 4.66895 1.13239 0.566193 0.824273i \(-0.308416\pi\)
0.566193 + 0.824273i \(0.308416\pi\)
\(18\) 1.05133 0.247801
\(19\) −2.00040 −0.458924 −0.229462 0.973318i \(-0.573697\pi\)
−0.229462 + 0.973318i \(0.573697\pi\)
\(20\) 3.81346 0.852716
\(21\) 0 0
\(22\) −2.36363 −0.503927
\(23\) 2.47117 0.515275 0.257638 0.966242i \(-0.417056\pi\)
0.257638 + 0.966242i \(0.417056\pi\)
\(24\) 3.04329 0.621209
\(25\) 13.1669 2.63338
\(26\) 0.363858 0.0713584
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.18926 1.14932 0.574658 0.818394i \(-0.305135\pi\)
0.574658 + 0.818394i \(0.305135\pi\)
\(30\) 4.48105 0.818123
\(31\) −3.71258 −0.666799 −0.333400 0.942786i \(-0.608196\pi\)
−0.333400 + 0.942786i \(0.608196\pi\)
\(32\) 4.60410 0.813898
\(33\) 2.24822 0.391365
\(34\) 4.90861 0.841819
\(35\) 0 0
\(36\) −0.894704 −0.149117
\(37\) −4.73147 −0.777848 −0.388924 0.921270i \(-0.627153\pi\)
−0.388924 + 0.921270i \(0.627153\pi\)
\(38\) −2.10309 −0.341166
\(39\) −0.346093 −0.0554192
\(40\) 12.9713 2.05094
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −0.374747 −0.0571485 −0.0285742 0.999592i \(-0.509097\pi\)
−0.0285742 + 0.999592i \(0.509097\pi\)
\(44\) 2.01149 0.303244
\(45\) −4.26226 −0.635380
\(46\) 2.59802 0.383057
\(47\) −3.92338 −0.572284 −0.286142 0.958187i \(-0.592373\pi\)
−0.286142 + 0.958187i \(0.592373\pi\)
\(48\) 1.41010 0.203530
\(49\) 0 0
\(50\) 13.8427 1.95766
\(51\) −4.66895 −0.653783
\(52\) −0.309650 −0.0429408
\(53\) 10.4390 1.43391 0.716953 0.697121i \(-0.245536\pi\)
0.716953 + 0.697121i \(0.245536\pi\)
\(54\) −1.05133 −0.143068
\(55\) 9.58251 1.29211
\(56\) 0 0
\(57\) 2.00040 0.264960
\(58\) 6.50696 0.854405
\(59\) −7.37689 −0.960389 −0.480194 0.877162i \(-0.659434\pi\)
−0.480194 + 0.877162i \(0.659434\pi\)
\(60\) −3.81346 −0.492316
\(61\) −11.7053 −1.49871 −0.749356 0.662168i \(-0.769637\pi\)
−0.749356 + 0.662168i \(0.769637\pi\)
\(62\) −3.90315 −0.495701
\(63\) 0 0
\(64\) 7.66063 0.957579
\(65\) −1.47514 −0.182968
\(66\) 2.36363 0.290942
\(67\) 14.2025 1.73511 0.867556 0.497339i \(-0.165689\pi\)
0.867556 + 0.497339i \(0.165689\pi\)
\(68\) −4.17732 −0.506575
\(69\) −2.47117 −0.297494
\(70\) 0 0
\(71\) 1.60184 0.190104 0.0950519 0.995472i \(-0.469698\pi\)
0.0950519 + 0.995472i \(0.469698\pi\)
\(72\) −3.04329 −0.358655
\(73\) 15.1789 1.77656 0.888279 0.459305i \(-0.151901\pi\)
0.888279 + 0.459305i \(0.151901\pi\)
\(74\) −4.97434 −0.578255
\(75\) −13.1669 −1.52038
\(76\) 1.78977 0.205301
\(77\) 0 0
\(78\) −0.363858 −0.0411988
\(79\) 10.1626 1.14338 0.571692 0.820468i \(-0.306287\pi\)
0.571692 + 0.820468i \(0.306287\pi\)
\(80\) 6.01021 0.671962
\(81\) 1.00000 0.111111
\(82\) 1.05133 0.116100
\(83\) 12.3018 1.35030 0.675150 0.737681i \(-0.264079\pi\)
0.675150 + 0.737681i \(0.264079\pi\)
\(84\) 0 0
\(85\) −19.9003 −2.15849
\(86\) −0.393984 −0.0424843
\(87\) −6.18926 −0.663558
\(88\) 6.84199 0.729359
\(89\) −7.79875 −0.826666 −0.413333 0.910580i \(-0.635635\pi\)
−0.413333 + 0.910580i \(0.635635\pi\)
\(90\) −4.48105 −0.472344
\(91\) 0 0
\(92\) −2.21097 −0.230509
\(93\) 3.71258 0.384977
\(94\) −4.12477 −0.425438
\(95\) 8.52624 0.874774
\(96\) −4.60410 −0.469904
\(97\) −12.3944 −1.25847 −0.629233 0.777217i \(-0.716631\pi\)
−0.629233 + 0.777217i \(0.716631\pi\)
\(98\) 0 0
\(99\) −2.24822 −0.225955
\(100\) −11.7805 −1.17805
\(101\) −9.64182 −0.959397 −0.479699 0.877433i \(-0.659254\pi\)
−0.479699 + 0.877433i \(0.659254\pi\)
\(102\) −4.90861 −0.486025
\(103\) 4.40291 0.433832 0.216916 0.976190i \(-0.430400\pi\)
0.216916 + 0.976190i \(0.430400\pi\)
\(104\) −1.05326 −0.103281
\(105\) 0 0
\(106\) 10.9748 1.06597
\(107\) 3.45652 0.334155 0.167077 0.985944i \(-0.446567\pi\)
0.167077 + 0.985944i \(0.446567\pi\)
\(108\) 0.894704 0.0860929
\(109\) −18.0191 −1.72592 −0.862959 0.505274i \(-0.831391\pi\)
−0.862959 + 0.505274i \(0.831391\pi\)
\(110\) 10.0744 0.960555
\(111\) 4.73147 0.449091
\(112\) 0 0
\(113\) 16.2569 1.52932 0.764659 0.644435i \(-0.222907\pi\)
0.764659 + 0.644435i \(0.222907\pi\)
\(114\) 2.10309 0.196972
\(115\) −10.5328 −0.982187
\(116\) −5.53755 −0.514149
\(117\) 0.346093 0.0319963
\(118\) −7.75555 −0.713956
\(119\) 0 0
\(120\) −12.9713 −1.18411
\(121\) −5.94550 −0.540500
\(122\) −12.3062 −1.11415
\(123\) −1.00000 −0.0901670
\(124\) 3.32166 0.298294
\(125\) −34.8094 −3.11344
\(126\) 0 0
\(127\) −2.48589 −0.220587 −0.110294 0.993899i \(-0.535179\pi\)
−0.110294 + 0.993899i \(0.535179\pi\)
\(128\) −1.15435 −0.102031
\(129\) 0.374747 0.0329947
\(130\) −1.55086 −0.136019
\(131\) 6.81857 0.595742 0.297871 0.954606i \(-0.403724\pi\)
0.297871 + 0.954606i \(0.403724\pi\)
\(132\) −2.01149 −0.175078
\(133\) 0 0
\(134\) 14.9315 1.28989
\(135\) 4.26226 0.366837
\(136\) −14.2090 −1.21841
\(137\) −3.50002 −0.299027 −0.149513 0.988760i \(-0.547771\pi\)
−0.149513 + 0.988760i \(0.547771\pi\)
\(138\) −2.59802 −0.221158
\(139\) −20.1733 −1.71108 −0.855540 0.517737i \(-0.826775\pi\)
−0.855540 + 0.517737i \(0.826775\pi\)
\(140\) 0 0
\(141\) 3.92338 0.330409
\(142\) 1.68407 0.141324
\(143\) −0.778093 −0.0650674
\(144\) −1.41010 −0.117508
\(145\) −26.3802 −2.19076
\(146\) 15.9581 1.32070
\(147\) 0 0
\(148\) 4.23326 0.347972
\(149\) −13.0123 −1.06601 −0.533005 0.846112i \(-0.678937\pi\)
−0.533005 + 0.846112i \(0.678937\pi\)
\(150\) −13.8427 −1.13026
\(151\) −10.8585 −0.883652 −0.441826 0.897101i \(-0.645669\pi\)
−0.441826 + 0.897101i \(0.645669\pi\)
\(152\) 6.08781 0.493787
\(153\) 4.66895 0.377462
\(154\) 0 0
\(155\) 15.8240 1.27101
\(156\) 0.309650 0.0247919
\(157\) −2.08189 −0.166153 −0.0830765 0.996543i \(-0.526475\pi\)
−0.0830765 + 0.996543i \(0.526475\pi\)
\(158\) 10.6843 0.849995
\(159\) −10.4390 −0.827866
\(160\) −19.6239 −1.55140
\(161\) 0 0
\(162\) 1.05133 0.0826003
\(163\) −13.1660 −1.03124 −0.515619 0.856818i \(-0.672438\pi\)
−0.515619 + 0.856818i \(0.672438\pi\)
\(164\) −0.894704 −0.0698646
\(165\) −9.58251 −0.745998
\(166\) 12.9333 1.00382
\(167\) 13.6237 1.05423 0.527117 0.849792i \(-0.323273\pi\)
0.527117 + 0.849792i \(0.323273\pi\)
\(168\) 0 0
\(169\) −12.8802 −0.990786
\(170\) −20.9218 −1.60463
\(171\) −2.00040 −0.152975
\(172\) 0.335288 0.0255655
\(173\) −11.4936 −0.873841 −0.436921 0.899500i \(-0.643931\pi\)
−0.436921 + 0.899500i \(0.643931\pi\)
\(174\) −6.50696 −0.493291
\(175\) 0 0
\(176\) 3.17021 0.238964
\(177\) 7.37689 0.554481
\(178\) −8.19906 −0.614546
\(179\) −22.3270 −1.66880 −0.834400 0.551159i \(-0.814186\pi\)
−0.834400 + 0.551159i \(0.814186\pi\)
\(180\) 3.81346 0.284239
\(181\) 0.572312 0.0425396 0.0212698 0.999774i \(-0.493229\pi\)
0.0212698 + 0.999774i \(0.493229\pi\)
\(182\) 0 0
\(183\) 11.7053 0.865282
\(184\) −7.52050 −0.554419
\(185\) 20.1668 1.48269
\(186\) 3.90315 0.286193
\(187\) −10.4968 −0.767604
\(188\) 3.51027 0.256012
\(189\) 0 0
\(190\) 8.96390 0.650310
\(191\) −0.378591 −0.0273939 −0.0136969 0.999906i \(-0.504360\pi\)
−0.0136969 + 0.999906i \(0.504360\pi\)
\(192\) −7.66063 −0.552858
\(193\) 9.61273 0.691939 0.345970 0.938246i \(-0.387550\pi\)
0.345970 + 0.938246i \(0.387550\pi\)
\(194\) −13.0307 −0.935547
\(195\) 1.47514 0.105637
\(196\) 0 0
\(197\) −10.3730 −0.739042 −0.369521 0.929222i \(-0.620478\pi\)
−0.369521 + 0.929222i \(0.620478\pi\)
\(198\) −2.36363 −0.167976
\(199\) 17.1983 1.21915 0.609576 0.792727i \(-0.291340\pi\)
0.609576 + 0.792727i \(0.291340\pi\)
\(200\) −40.0706 −2.83342
\(201\) −14.2025 −1.00177
\(202\) −10.1367 −0.713219
\(203\) 0 0
\(204\) 4.17732 0.292471
\(205\) −4.26226 −0.297689
\(206\) 4.62892 0.322512
\(207\) 2.47117 0.171758
\(208\) −0.488024 −0.0338384
\(209\) 4.49735 0.311088
\(210\) 0 0
\(211\) −2.94273 −0.202586 −0.101293 0.994857i \(-0.532298\pi\)
−0.101293 + 0.994857i \(0.532298\pi\)
\(212\) −9.33981 −0.641461
\(213\) −1.60184 −0.109757
\(214\) 3.63395 0.248412
\(215\) 1.59727 0.108933
\(216\) 3.04329 0.207070
\(217\) 0 0
\(218\) −18.9440 −1.28305
\(219\) −15.1789 −1.02570
\(220\) −8.57351 −0.578026
\(221\) 1.61589 0.108696
\(222\) 4.97434 0.333856
\(223\) 1.34407 0.0900058 0.0450029 0.998987i \(-0.485670\pi\)
0.0450029 + 0.998987i \(0.485670\pi\)
\(224\) 0 0
\(225\) 13.1669 0.877792
\(226\) 17.0914 1.13690
\(227\) 26.4743 1.75716 0.878579 0.477597i \(-0.158492\pi\)
0.878579 + 0.477597i \(0.158492\pi\)
\(228\) −1.78977 −0.118530
\(229\) −14.7753 −0.976378 −0.488189 0.872738i \(-0.662342\pi\)
−0.488189 + 0.872738i \(0.662342\pi\)
\(230\) −11.0734 −0.730161
\(231\) 0 0
\(232\) −18.8357 −1.23663
\(233\) −11.9956 −0.785860 −0.392930 0.919568i \(-0.628539\pi\)
−0.392930 + 0.919568i \(0.628539\pi\)
\(234\) 0.363858 0.0237861
\(235\) 16.7225 1.09086
\(236\) 6.60013 0.429632
\(237\) −10.1626 −0.660133
\(238\) 0 0
\(239\) 15.3398 0.992248 0.496124 0.868252i \(-0.334756\pi\)
0.496124 + 0.868252i \(0.334756\pi\)
\(240\) −6.01021 −0.387957
\(241\) 17.6355 1.13600 0.568000 0.823029i \(-0.307717\pi\)
0.568000 + 0.823029i \(0.307717\pi\)
\(242\) −6.25068 −0.401809
\(243\) −1.00000 −0.0641500
\(244\) 10.4728 0.670451
\(245\) 0 0
\(246\) −1.05133 −0.0670304
\(247\) −0.692325 −0.0440516
\(248\) 11.2985 0.717453
\(249\) −12.3018 −0.779596
\(250\) −36.5962 −2.31454
\(251\) 21.4380 1.35315 0.676576 0.736373i \(-0.263463\pi\)
0.676576 + 0.736373i \(0.263463\pi\)
\(252\) 0 0
\(253\) −5.55575 −0.349287
\(254\) −2.61350 −0.163985
\(255\) 19.9003 1.24620
\(256\) −16.5349 −1.03343
\(257\) 8.64434 0.539219 0.269610 0.962970i \(-0.413105\pi\)
0.269610 + 0.962970i \(0.413105\pi\)
\(258\) 0.393984 0.0245283
\(259\) 0 0
\(260\) 1.31981 0.0818512
\(261\) 6.18926 0.383105
\(262\) 7.16858 0.442876
\(263\) −15.6653 −0.965964 −0.482982 0.875630i \(-0.660446\pi\)
−0.482982 + 0.875630i \(0.660446\pi\)
\(264\) −6.84199 −0.421096
\(265\) −44.4937 −2.73323
\(266\) 0 0
\(267\) 7.79875 0.477276
\(268\) −12.7070 −0.776206
\(269\) 13.1787 0.803519 0.401760 0.915745i \(-0.368399\pi\)
0.401760 + 0.915745i \(0.368399\pi\)
\(270\) 4.48105 0.272708
\(271\) −7.35125 −0.446556 −0.223278 0.974755i \(-0.571676\pi\)
−0.223278 + 0.974755i \(0.571676\pi\)
\(272\) −6.58367 −0.399194
\(273\) 0 0
\(274\) −3.67968 −0.222297
\(275\) −29.6021 −1.78507
\(276\) 2.21097 0.133085
\(277\) −1.18852 −0.0714115 −0.0357057 0.999362i \(-0.511368\pi\)
−0.0357057 + 0.999362i \(0.511368\pi\)
\(278\) −21.2088 −1.27202
\(279\) −3.71258 −0.222266
\(280\) 0 0
\(281\) 6.99493 0.417283 0.208641 0.977992i \(-0.433096\pi\)
0.208641 + 0.977992i \(0.433096\pi\)
\(282\) 4.12477 0.245627
\(283\) −11.1116 −0.660518 −0.330259 0.943890i \(-0.607136\pi\)
−0.330259 + 0.943890i \(0.607136\pi\)
\(284\) −1.43318 −0.0850433
\(285\) −8.52624 −0.505051
\(286\) −0.818033 −0.0483713
\(287\) 0 0
\(288\) 4.60410 0.271299
\(289\) 4.79907 0.282298
\(290\) −27.7344 −1.62862
\(291\) 12.3944 0.726575
\(292\) −13.5806 −0.794746
\(293\) 11.5761 0.676283 0.338141 0.941095i \(-0.390202\pi\)
0.338141 + 0.941095i \(0.390202\pi\)
\(294\) 0 0
\(295\) 31.4422 1.83064
\(296\) 14.3992 0.836938
\(297\) 2.24822 0.130455
\(298\) −13.6802 −0.792475
\(299\) 0.855254 0.0494606
\(300\) 11.7805 0.680145
\(301\) 0 0
\(302\) −11.4159 −0.656909
\(303\) 9.64182 0.553908
\(304\) 2.82077 0.161782
\(305\) 49.8911 2.85676
\(306\) 4.90861 0.280606
\(307\) 15.4598 0.882336 0.441168 0.897424i \(-0.354564\pi\)
0.441168 + 0.897424i \(0.354564\pi\)
\(308\) 0 0
\(309\) −4.40291 −0.250473
\(310\) 16.6363 0.944876
\(311\) 16.4265 0.931462 0.465731 0.884926i \(-0.345791\pi\)
0.465731 + 0.884926i \(0.345791\pi\)
\(312\) 1.05326 0.0596291
\(313\) −29.9007 −1.69009 −0.845043 0.534699i \(-0.820425\pi\)
−0.845043 + 0.534699i \(0.820425\pi\)
\(314\) −2.18876 −0.123519
\(315\) 0 0
\(316\) −9.09253 −0.511495
\(317\) 4.91153 0.275859 0.137929 0.990442i \(-0.455955\pi\)
0.137929 + 0.990442i \(0.455955\pi\)
\(318\) −10.9748 −0.615438
\(319\) −13.9148 −0.779081
\(320\) −32.6516 −1.82528
\(321\) −3.45652 −0.192924
\(322\) 0 0
\(323\) −9.33978 −0.519679
\(324\) −0.894704 −0.0497058
\(325\) 4.55696 0.252774
\(326\) −13.8418 −0.766626
\(327\) 18.0191 0.996459
\(328\) −3.04329 −0.168038
\(329\) 0 0
\(330\) −10.0744 −0.554577
\(331\) 5.41473 0.297621 0.148810 0.988866i \(-0.452456\pi\)
0.148810 + 0.988866i \(0.452456\pi\)
\(332\) −11.0065 −0.604059
\(333\) −4.73147 −0.259283
\(334\) 14.3230 0.783722
\(335\) −60.5348 −3.30737
\(336\) 0 0
\(337\) 33.3160 1.81484 0.907420 0.420225i \(-0.138049\pi\)
0.907420 + 0.420225i \(0.138049\pi\)
\(338\) −13.5414 −0.736554
\(339\) −16.2569 −0.882953
\(340\) 17.8048 0.965603
\(341\) 8.34671 0.452000
\(342\) −2.10309 −0.113722
\(343\) 0 0
\(344\) 1.14047 0.0614898
\(345\) 10.5328 0.567066
\(346\) −12.0836 −0.649616
\(347\) 32.8297 1.76239 0.881195 0.472753i \(-0.156740\pi\)
0.881195 + 0.472753i \(0.156740\pi\)
\(348\) 5.53755 0.296844
\(349\) 8.42964 0.451228 0.225614 0.974217i \(-0.427561\pi\)
0.225614 + 0.974217i \(0.427561\pi\)
\(350\) 0 0
\(351\) −0.346093 −0.0184731
\(352\) −10.3510 −0.551713
\(353\) −19.6337 −1.04500 −0.522499 0.852640i \(-0.675000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(354\) 7.75555 0.412203
\(355\) −6.82748 −0.362365
\(356\) 6.97757 0.369810
\(357\) 0 0
\(358\) −23.4731 −1.24059
\(359\) 8.78107 0.463447 0.231724 0.972782i \(-0.425564\pi\)
0.231724 + 0.972782i \(0.425564\pi\)
\(360\) 12.9713 0.683648
\(361\) −14.9984 −0.789389
\(362\) 0.601689 0.0316241
\(363\) 5.94550 0.312058
\(364\) 0 0
\(365\) −64.6965 −3.38637
\(366\) 12.3062 0.643253
\(367\) −15.7089 −0.819999 −0.410000 0.912086i \(-0.634471\pi\)
−0.410000 + 0.912086i \(0.634471\pi\)
\(368\) −3.48460 −0.181647
\(369\) 1.00000 0.0520579
\(370\) 21.2019 1.10224
\(371\) 0 0
\(372\) −3.32166 −0.172220
\(373\) −13.7795 −0.713473 −0.356737 0.934205i \(-0.616111\pi\)
−0.356737 + 0.934205i \(0.616111\pi\)
\(374\) −11.0356 −0.570639
\(375\) 34.8094 1.79755
\(376\) 11.9400 0.615759
\(377\) 2.14206 0.110321
\(378\) 0 0
\(379\) −25.6908 −1.31965 −0.659824 0.751420i \(-0.729369\pi\)
−0.659824 + 0.751420i \(0.729369\pi\)
\(380\) −7.62846 −0.391332
\(381\) 2.48589 0.127356
\(382\) −0.398024 −0.0203647
\(383\) 24.8594 1.27026 0.635128 0.772407i \(-0.280947\pi\)
0.635128 + 0.772407i \(0.280947\pi\)
\(384\) 1.15435 0.0589075
\(385\) 0 0
\(386\) 10.1062 0.514390
\(387\) −0.374747 −0.0190495
\(388\) 11.0894 0.562977
\(389\) 9.84230 0.499025 0.249512 0.968372i \(-0.419730\pi\)
0.249512 + 0.968372i \(0.419730\pi\)
\(390\) 1.55086 0.0785307
\(391\) 11.5378 0.583490
\(392\) 0 0
\(393\) −6.81857 −0.343952
\(394\) −10.9054 −0.549406
\(395\) −43.3157 −2.17945
\(396\) 2.01149 0.101081
\(397\) −15.4375 −0.774788 −0.387394 0.921914i \(-0.626625\pi\)
−0.387394 + 0.921914i \(0.626625\pi\)
\(398\) 18.0811 0.906322
\(399\) 0 0
\(400\) −18.5666 −0.928329
\(401\) 29.8758 1.49192 0.745962 0.665988i \(-0.231990\pi\)
0.745962 + 0.665988i \(0.231990\pi\)
\(402\) −14.9315 −0.744717
\(403\) −1.28490 −0.0640053
\(404\) 8.62658 0.429188
\(405\) −4.26226 −0.211793
\(406\) 0 0
\(407\) 10.6374 0.527276
\(408\) 14.2090 0.703449
\(409\) 9.99523 0.494233 0.247116 0.968986i \(-0.420517\pi\)
0.247116 + 0.968986i \(0.420517\pi\)
\(410\) −4.48105 −0.221303
\(411\) 3.50002 0.172643
\(412\) −3.93930 −0.194076
\(413\) 0 0
\(414\) 2.59802 0.127686
\(415\) −52.4335 −2.57386
\(416\) 1.59345 0.0781251
\(417\) 20.1733 0.987893
\(418\) 4.72820 0.231264
\(419\) 30.0143 1.46629 0.733147 0.680070i \(-0.238050\pi\)
0.733147 + 0.680070i \(0.238050\pi\)
\(420\) 0 0
\(421\) −2.76729 −0.134870 −0.0674348 0.997724i \(-0.521481\pi\)
−0.0674348 + 0.997724i \(0.521481\pi\)
\(422\) −3.09379 −0.150603
\(423\) −3.92338 −0.190761
\(424\) −31.7689 −1.54283
\(425\) 61.4754 2.98200
\(426\) −1.68407 −0.0815933
\(427\) 0 0
\(428\) −3.09256 −0.149485
\(429\) 0.778093 0.0375667
\(430\) 1.67926 0.0809812
\(431\) −16.9439 −0.816160 −0.408080 0.912946i \(-0.633802\pi\)
−0.408080 + 0.912946i \(0.633802\pi\)
\(432\) 1.41010 0.0678434
\(433\) −0.950181 −0.0456628 −0.0228314 0.999739i \(-0.507268\pi\)
−0.0228314 + 0.999739i \(0.507268\pi\)
\(434\) 0 0
\(435\) 26.3802 1.26484
\(436\) 16.1218 0.772092
\(437\) −4.94334 −0.236472
\(438\) −15.9581 −0.762505
\(439\) −16.3634 −0.780984 −0.390492 0.920606i \(-0.627695\pi\)
−0.390492 + 0.920606i \(0.627695\pi\)
\(440\) −29.1624 −1.39026
\(441\) 0 0
\(442\) 1.69883 0.0808052
\(443\) −13.1007 −0.622435 −0.311217 0.950339i \(-0.600737\pi\)
−0.311217 + 0.950339i \(0.600737\pi\)
\(444\) −4.23326 −0.200902
\(445\) 33.2403 1.57574
\(446\) 1.41307 0.0669106
\(447\) 13.0123 0.615461
\(448\) 0 0
\(449\) −10.2850 −0.485378 −0.242689 0.970104i \(-0.578030\pi\)
−0.242689 + 0.970104i \(0.578030\pi\)
\(450\) 13.8427 0.652553
\(451\) −2.24822 −0.105865
\(452\) −14.5451 −0.684144
\(453\) 10.8585 0.510177
\(454\) 27.8332 1.30628
\(455\) 0 0
\(456\) −6.08781 −0.285088
\(457\) −7.57107 −0.354160 −0.177080 0.984197i \(-0.556665\pi\)
−0.177080 + 0.984197i \(0.556665\pi\)
\(458\) −15.5337 −0.725843
\(459\) −4.66895 −0.217928
\(460\) 9.42372 0.439383
\(461\) −37.7253 −1.75704 −0.878520 0.477705i \(-0.841469\pi\)
−0.878520 + 0.477705i \(0.841469\pi\)
\(462\) 0 0
\(463\) −1.46827 −0.0682365 −0.0341182 0.999418i \(-0.510862\pi\)
−0.0341182 + 0.999418i \(0.510862\pi\)
\(464\) −8.72746 −0.405162
\(465\) −15.8240 −0.733820
\(466\) −12.6114 −0.584211
\(467\) 18.3223 0.847853 0.423927 0.905697i \(-0.360651\pi\)
0.423927 + 0.905697i \(0.360651\pi\)
\(468\) −0.309650 −0.0143136
\(469\) 0 0
\(470\) 17.5809 0.810945
\(471\) 2.08189 0.0959285
\(472\) 22.4500 1.03335
\(473\) 0.842516 0.0387389
\(474\) −10.6843 −0.490745
\(475\) −26.3391 −1.20852
\(476\) 0 0
\(477\) 10.4390 0.477969
\(478\) 16.1272 0.737640
\(479\) −6.47641 −0.295915 −0.147957 0.988994i \(-0.547270\pi\)
−0.147957 + 0.988994i \(0.547270\pi\)
\(480\) 19.6239 0.895704
\(481\) −1.63753 −0.0746647
\(482\) 18.5407 0.844505
\(483\) 0 0
\(484\) 5.31946 0.241794
\(485\) 52.8284 2.39881
\(486\) −1.05133 −0.0476893
\(487\) −0.574842 −0.0260486 −0.0130243 0.999915i \(-0.504146\pi\)
−0.0130243 + 0.999915i \(0.504146\pi\)
\(488\) 35.6227 1.61256
\(489\) 13.1660 0.595386
\(490\) 0 0
\(491\) −5.62113 −0.253678 −0.126839 0.991923i \(-0.540483\pi\)
−0.126839 + 0.991923i \(0.540483\pi\)
\(492\) 0.894704 0.0403364
\(493\) 28.8973 1.30147
\(494\) −0.727862 −0.0327481
\(495\) 9.58251 0.430702
\(496\) 5.23511 0.235063
\(497\) 0 0
\(498\) −12.9333 −0.579554
\(499\) 11.4910 0.514409 0.257205 0.966357i \(-0.417199\pi\)
0.257205 + 0.966357i \(0.417199\pi\)
\(500\) 31.1441 1.39280
\(501\) −13.6237 −0.608663
\(502\) 22.5384 1.00594
\(503\) −19.5016 −0.869534 −0.434767 0.900543i \(-0.643169\pi\)
−0.434767 + 0.900543i \(0.643169\pi\)
\(504\) 0 0
\(505\) 41.0960 1.82875
\(506\) −5.84093 −0.259661
\(507\) 12.8802 0.572031
\(508\) 2.22414 0.0986802
\(509\) −9.19847 −0.407715 −0.203858 0.979001i \(-0.565348\pi\)
−0.203858 + 0.979001i \(0.565348\pi\)
\(510\) 20.9218 0.926432
\(511\) 0 0
\(512\) −15.0749 −0.666223
\(513\) 2.00040 0.0883200
\(514\) 9.08806 0.400857
\(515\) −18.7664 −0.826945
\(516\) −0.335288 −0.0147602
\(517\) 8.82064 0.387931
\(518\) 0 0
\(519\) 11.4936 0.504513
\(520\) 4.48927 0.196868
\(521\) −10.7828 −0.472403 −0.236202 0.971704i \(-0.575903\pi\)
−0.236202 + 0.971704i \(0.575903\pi\)
\(522\) 6.50696 0.284802
\(523\) −2.56702 −0.112248 −0.0561240 0.998424i \(-0.517874\pi\)
−0.0561240 + 0.998424i \(0.517874\pi\)
\(524\) −6.10060 −0.266506
\(525\) 0 0
\(526\) −16.4694 −0.718100
\(527\) −17.3338 −0.755074
\(528\) −3.17021 −0.137966
\(529\) −16.8933 −0.734491
\(530\) −46.7776 −2.03189
\(531\) −7.37689 −0.320130
\(532\) 0 0
\(533\) 0.346093 0.0149909
\(534\) 8.19906 0.354808
\(535\) −14.7326 −0.636946
\(536\) −43.2224 −1.86692
\(537\) 22.3270 0.963482
\(538\) 13.8552 0.597339
\(539\) 0 0
\(540\) −3.81346 −0.164105
\(541\) 24.0028 1.03196 0.515980 0.856601i \(-0.327428\pi\)
0.515980 + 0.856601i \(0.327428\pi\)
\(542\) −7.72859 −0.331971
\(543\) −0.572312 −0.0245603
\(544\) 21.4963 0.921647
\(545\) 76.8022 3.28984
\(546\) 0 0
\(547\) 14.3624 0.614090 0.307045 0.951695i \(-0.400660\pi\)
0.307045 + 0.951695i \(0.400660\pi\)
\(548\) 3.13148 0.133770
\(549\) −11.7053 −0.499571
\(550\) −31.1216 −1.32703
\(551\) −12.3810 −0.527449
\(552\) 7.52050 0.320094
\(553\) 0 0
\(554\) −1.24953 −0.0530875
\(555\) −20.1668 −0.856031
\(556\) 18.0492 0.765455
\(557\) −25.2968 −1.07186 −0.535929 0.844263i \(-0.680039\pi\)
−0.535929 + 0.844263i \(0.680039\pi\)
\(558\) −3.90315 −0.165234
\(559\) −0.129697 −0.00548561
\(560\) 0 0
\(561\) 10.4968 0.443177
\(562\) 7.35399 0.310209
\(563\) −42.5322 −1.79252 −0.896260 0.443530i \(-0.853726\pi\)
−0.896260 + 0.443530i \(0.853726\pi\)
\(564\) −3.51027 −0.147809
\(565\) −69.2911 −2.91510
\(566\) −11.6820 −0.491031
\(567\) 0 0
\(568\) −4.87488 −0.204545
\(569\) −39.2312 −1.64466 −0.822329 0.569013i \(-0.807325\pi\)
−0.822329 + 0.569013i \(0.807325\pi\)
\(570\) −8.96390 −0.375457
\(571\) 12.0130 0.502728 0.251364 0.967893i \(-0.419121\pi\)
0.251364 + 0.967893i \(0.419121\pi\)
\(572\) 0.696162 0.0291080
\(573\) 0.378591 0.0158159
\(574\) 0 0
\(575\) 32.5376 1.35691
\(576\) 7.66063 0.319193
\(577\) −42.5206 −1.77016 −0.885078 0.465443i \(-0.845895\pi\)
−0.885078 + 0.465443i \(0.845895\pi\)
\(578\) 5.04541 0.209861
\(579\) −9.61273 −0.399491
\(580\) 23.6025 0.980040
\(581\) 0 0
\(582\) 13.0307 0.540138
\(583\) −23.4692 −0.971994
\(584\) −46.1938 −1.91151
\(585\) −1.47514 −0.0609894
\(586\) 12.1703 0.502751
\(587\) −43.3776 −1.79038 −0.895192 0.445681i \(-0.852962\pi\)
−0.895192 + 0.445681i \(0.852962\pi\)
\(588\) 0 0
\(589\) 7.42666 0.306010
\(590\) 33.0562 1.36090
\(591\) 10.3730 0.426686
\(592\) 6.67183 0.274211
\(593\) −42.2748 −1.73602 −0.868008 0.496550i \(-0.834600\pi\)
−0.868008 + 0.496550i \(0.834600\pi\)
\(594\) 2.36363 0.0969807
\(595\) 0 0
\(596\) 11.6422 0.476881
\(597\) −17.1983 −0.703878
\(598\) 0.899155 0.0367692
\(599\) −28.0250 −1.14507 −0.572536 0.819879i \(-0.694040\pi\)
−0.572536 + 0.819879i \(0.694040\pi\)
\(600\) 40.0706 1.63588
\(601\) 1.15074 0.0469396 0.0234698 0.999725i \(-0.492529\pi\)
0.0234698 + 0.999725i \(0.492529\pi\)
\(602\) 0 0
\(603\) 14.2025 0.578371
\(604\) 9.71513 0.395303
\(605\) 25.3413 1.03027
\(606\) 10.1367 0.411777
\(607\) 42.1717 1.71170 0.855849 0.517226i \(-0.173035\pi\)
0.855849 + 0.517226i \(0.173035\pi\)
\(608\) −9.21006 −0.373517
\(609\) 0 0
\(610\) 52.4521 2.12372
\(611\) −1.35785 −0.0549329
\(612\) −4.17732 −0.168858
\(613\) −29.9745 −1.21066 −0.605330 0.795975i \(-0.706959\pi\)
−0.605330 + 0.795975i \(0.706959\pi\)
\(614\) 16.2533 0.655932
\(615\) 4.26226 0.171871
\(616\) 0 0
\(617\) 16.9897 0.683979 0.341989 0.939704i \(-0.388899\pi\)
0.341989 + 0.939704i \(0.388899\pi\)
\(618\) −4.62892 −0.186202
\(619\) −4.99009 −0.200569 −0.100284 0.994959i \(-0.531975\pi\)
−0.100284 + 0.994959i \(0.531975\pi\)
\(620\) −14.1578 −0.568590
\(621\) −2.47117 −0.0991648
\(622\) 17.2697 0.692452
\(623\) 0 0
\(624\) 0.488024 0.0195366
\(625\) 82.5322 3.30129
\(626\) −31.4355 −1.25641
\(627\) −4.49735 −0.179607
\(628\) 1.86268 0.0743289
\(629\) −22.0910 −0.880825
\(630\) 0 0
\(631\) −39.1594 −1.55891 −0.779456 0.626458i \(-0.784504\pi\)
−0.779456 + 0.626458i \(0.784504\pi\)
\(632\) −30.9278 −1.23024
\(633\) 2.94273 0.116963
\(634\) 5.16364 0.205074
\(635\) 10.5955 0.420471
\(636\) 9.33981 0.370347
\(637\) 0 0
\(638\) −14.6291 −0.579171
\(639\) 1.60184 0.0633680
\(640\) 4.92013 0.194485
\(641\) 35.5086 1.40251 0.701253 0.712912i \(-0.252624\pi\)
0.701253 + 0.712912i \(0.252624\pi\)
\(642\) −3.63395 −0.143421
\(643\) −28.2461 −1.11392 −0.556959 0.830540i \(-0.688032\pi\)
−0.556959 + 0.830540i \(0.688032\pi\)
\(644\) 0 0
\(645\) −1.59727 −0.0628925
\(646\) −9.81920 −0.386331
\(647\) −21.9398 −0.862544 −0.431272 0.902222i \(-0.641935\pi\)
−0.431272 + 0.902222i \(0.641935\pi\)
\(648\) −3.04329 −0.119552
\(649\) 16.5849 0.651014
\(650\) 4.79087 0.187913
\(651\) 0 0
\(652\) 11.7796 0.461326
\(653\) 45.6206 1.78527 0.892636 0.450778i \(-0.148853\pi\)
0.892636 + 0.450778i \(0.148853\pi\)
\(654\) 18.9440 0.740771
\(655\) −29.0625 −1.13557
\(656\) −1.41010 −0.0550551
\(657\) 15.1789 0.592186
\(658\) 0 0
\(659\) −9.26037 −0.360733 −0.180366 0.983599i \(-0.557728\pi\)
−0.180366 + 0.983599i \(0.557728\pi\)
\(660\) 8.57351 0.333723
\(661\) 8.79472 0.342075 0.171038 0.985265i \(-0.445288\pi\)
0.171038 + 0.985265i \(0.445288\pi\)
\(662\) 5.69267 0.221252
\(663\) −1.61589 −0.0627559
\(664\) −37.4380 −1.45288
\(665\) 0 0
\(666\) −4.97434 −0.192752
\(667\) 15.2947 0.592214
\(668\) −12.1892 −0.471614
\(669\) −1.34407 −0.0519649
\(670\) −63.6421 −2.45871
\(671\) 26.3161 1.01592
\(672\) 0 0
\(673\) −20.4719 −0.789135 −0.394568 0.918867i \(-0.629106\pi\)
−0.394568 + 0.918867i \(0.629106\pi\)
\(674\) 35.0262 1.34916
\(675\) −13.1669 −0.506793
\(676\) 11.5240 0.443230
\(677\) 8.68940 0.333961 0.166980 0.985960i \(-0.446598\pi\)
0.166980 + 0.985960i \(0.446598\pi\)
\(678\) −17.0914 −0.656390
\(679\) 0 0
\(680\) 60.5623 2.32246
\(681\) −26.4743 −1.01450
\(682\) 8.77515 0.336018
\(683\) −4.49190 −0.171878 −0.0859389 0.996300i \(-0.527389\pi\)
−0.0859389 + 0.996300i \(0.527389\pi\)
\(684\) 1.78977 0.0684335
\(685\) 14.9180 0.569987
\(686\) 0 0
\(687\) 14.7753 0.563712
\(688\) 0.528431 0.0201462
\(689\) 3.61286 0.137639
\(690\) 11.0734 0.421559
\(691\) −2.49897 −0.0950654 −0.0475327 0.998870i \(-0.515136\pi\)
−0.0475327 + 0.998870i \(0.515136\pi\)
\(692\) 10.2834 0.390915
\(693\) 0 0
\(694\) 34.5149 1.31017
\(695\) 85.9840 3.26156
\(696\) 18.8357 0.713966
\(697\) 4.66895 0.176849
\(698\) 8.86234 0.335445
\(699\) 11.9956 0.453717
\(700\) 0 0
\(701\) −18.1085 −0.683949 −0.341975 0.939709i \(-0.611096\pi\)
−0.341975 + 0.939709i \(0.611096\pi\)
\(702\) −0.363858 −0.0137329
\(703\) 9.46485 0.356973
\(704\) −17.2228 −0.649109
\(705\) −16.7225 −0.629805
\(706\) −20.6416 −0.776855
\(707\) 0 0
\(708\) −6.60013 −0.248048
\(709\) 9.04329 0.339628 0.169814 0.985476i \(-0.445683\pi\)
0.169814 + 0.985476i \(0.445683\pi\)
\(710\) −7.17794 −0.269383
\(711\) 10.1626 0.381128
\(712\) 23.7339 0.889464
\(713\) −9.17443 −0.343585
\(714\) 0 0
\(715\) 3.31644 0.124028
\(716\) 19.9761 0.746541
\(717\) −15.3398 −0.572875
\(718\) 9.23181 0.344528
\(719\) 27.2172 1.01503 0.507515 0.861643i \(-0.330564\pi\)
0.507515 + 0.861643i \(0.330564\pi\)
\(720\) 6.01021 0.223987
\(721\) 0 0
\(722\) −15.7683 −0.586834
\(723\) −17.6355 −0.655870
\(724\) −0.512050 −0.0190302
\(725\) 81.4932 3.02658
\(726\) 6.25068 0.231985
\(727\) −23.1275 −0.857753 −0.428876 0.903363i \(-0.641090\pi\)
−0.428876 + 0.903363i \(0.641090\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −68.0174 −2.51744
\(731\) −1.74968 −0.0647141
\(732\) −10.4728 −0.387085
\(733\) 33.7507 1.24661 0.623305 0.781979i \(-0.285790\pi\)
0.623305 + 0.781979i \(0.285790\pi\)
\(734\) −16.5153 −0.609590
\(735\) 0 0
\(736\) 11.3775 0.419381
\(737\) −31.9304 −1.17617
\(738\) 1.05133 0.0387000
\(739\) 9.45395 0.347769 0.173885 0.984766i \(-0.444368\pi\)
0.173885 + 0.984766i \(0.444368\pi\)
\(740\) −18.0433 −0.663284
\(741\) 0.692325 0.0254332
\(742\) 0 0
\(743\) −10.9106 −0.400269 −0.200135 0.979768i \(-0.564138\pi\)
−0.200135 + 0.979768i \(0.564138\pi\)
\(744\) −11.2985 −0.414222
\(745\) 55.4619 2.03197
\(746\) −14.4868 −0.530398
\(747\) 12.3018 0.450100
\(748\) 9.39155 0.343389
\(749\) 0 0
\(750\) 36.5962 1.33630
\(751\) −41.5320 −1.51552 −0.757762 0.652531i \(-0.773707\pi\)
−0.757762 + 0.652531i \(0.773707\pi\)
\(752\) 5.53236 0.201744
\(753\) −21.4380 −0.781242
\(754\) 2.25201 0.0820133
\(755\) 46.2817 1.68437
\(756\) 0 0
\(757\) −17.7227 −0.644142 −0.322071 0.946716i \(-0.604379\pi\)
−0.322071 + 0.946716i \(0.604379\pi\)
\(758\) −27.0095 −0.981030
\(759\) 5.55575 0.201661
\(760\) −25.9478 −0.941227
\(761\) 0.980289 0.0355354 0.0177677 0.999842i \(-0.494344\pi\)
0.0177677 + 0.999842i \(0.494344\pi\)
\(762\) 2.61350 0.0946770
\(763\) 0 0
\(764\) 0.338726 0.0122547
\(765\) −19.9003 −0.719496
\(766\) 26.1355 0.944313
\(767\) −2.55309 −0.0921866
\(768\) 16.5349 0.596650
\(769\) 9.98688 0.360136 0.180068 0.983654i \(-0.442368\pi\)
0.180068 + 0.983654i \(0.442368\pi\)
\(770\) 0 0
\(771\) −8.64434 −0.311318
\(772\) −8.60054 −0.309540
\(773\) 4.24629 0.152729 0.0763643 0.997080i \(-0.475669\pi\)
0.0763643 + 0.997080i \(0.475669\pi\)
\(774\) −0.393984 −0.0141614
\(775\) −48.8831 −1.75593
\(776\) 37.7199 1.35407
\(777\) 0 0
\(778\) 10.3475 0.370976
\(779\) −2.00040 −0.0716719
\(780\) −1.31981 −0.0472568
\(781\) −3.60130 −0.128865
\(782\) 12.1300 0.433769
\(783\) −6.18926 −0.221186
\(784\) 0 0
\(785\) 8.87357 0.316711
\(786\) −7.16858 −0.255695
\(787\) 41.1788 1.46787 0.733933 0.679222i \(-0.237683\pi\)
0.733933 + 0.679222i \(0.237683\pi\)
\(788\) 9.28072 0.330612
\(789\) 15.6653 0.557699
\(790\) −45.5392 −1.62021
\(791\) 0 0
\(792\) 6.84199 0.243120
\(793\) −4.05112 −0.143860
\(794\) −16.2300 −0.575980
\(795\) 44.4937 1.57803
\(796\) −15.3873 −0.545390
\(797\) 15.2255 0.539316 0.269658 0.962956i \(-0.413089\pi\)
0.269658 + 0.962956i \(0.413089\pi\)
\(798\) 0 0
\(799\) −18.3181 −0.648047
\(800\) 60.6216 2.14330
\(801\) −7.79875 −0.275555
\(802\) 31.4093 1.10910
\(803\) −34.1256 −1.20427
\(804\) 12.7070 0.448143
\(805\) 0 0
\(806\) −1.35085 −0.0475817
\(807\) −13.1787 −0.463912
\(808\) 29.3429 1.03228
\(809\) −50.6646 −1.78127 −0.890637 0.454715i \(-0.849741\pi\)
−0.890637 + 0.454715i \(0.849741\pi\)
\(810\) −4.48105 −0.157448
\(811\) −8.70882 −0.305808 −0.152904 0.988241i \(-0.548863\pi\)
−0.152904 + 0.988241i \(0.548863\pi\)
\(812\) 0 0
\(813\) 7.35125 0.257819
\(814\) 11.1834 0.391979
\(815\) 56.1168 1.96569
\(816\) 6.58367 0.230475
\(817\) 0.749646 0.0262268
\(818\) 10.5083 0.367414
\(819\) 0 0
\(820\) 3.81346 0.133172
\(821\) −2.59866 −0.0906938 −0.0453469 0.998971i \(-0.514439\pi\)
−0.0453469 + 0.998971i \(0.514439\pi\)
\(822\) 3.67968 0.128343
\(823\) −44.7617 −1.56029 −0.780147 0.625596i \(-0.784856\pi\)
−0.780147 + 0.625596i \(0.784856\pi\)
\(824\) −13.3993 −0.466788
\(825\) 29.6021 1.03061
\(826\) 0 0
\(827\) 9.57720 0.333032 0.166516 0.986039i \(-0.446748\pi\)
0.166516 + 0.986039i \(0.446748\pi\)
\(828\) −2.21097 −0.0768364
\(829\) 30.0318 1.04305 0.521523 0.853237i \(-0.325364\pi\)
0.521523 + 0.853237i \(0.325364\pi\)
\(830\) −55.1250 −1.91342
\(831\) 1.18852 0.0412294
\(832\) 2.65129 0.0919168
\(833\) 0 0
\(834\) 21.2088 0.734402
\(835\) −58.0679 −2.00952
\(836\) −4.02380 −0.139166
\(837\) 3.71258 0.128326
\(838\) 31.5550 1.09005
\(839\) 0.824294 0.0284578 0.0142289 0.999899i \(-0.495471\pi\)
0.0142289 + 0.999899i \(0.495471\pi\)
\(840\) 0 0
\(841\) 9.30692 0.320928
\(842\) −2.90934 −0.100262
\(843\) −6.99493 −0.240918
\(844\) 2.63287 0.0906273
\(845\) 54.8989 1.88858
\(846\) −4.12477 −0.141813
\(847\) 0 0
\(848\) −14.7200 −0.505487
\(849\) 11.1116 0.381350
\(850\) 64.6310 2.21683
\(851\) −11.6923 −0.400806
\(852\) 1.43318 0.0490998
\(853\) −27.1552 −0.929777 −0.464889 0.885369i \(-0.653906\pi\)
−0.464889 + 0.885369i \(0.653906\pi\)
\(854\) 0 0
\(855\) 8.52624 0.291591
\(856\) −10.5192 −0.359539
\(857\) −14.1642 −0.483840 −0.241920 0.970296i \(-0.577777\pi\)
−0.241920 + 0.970296i \(0.577777\pi\)
\(858\) 0.818033 0.0279272
\(859\) 41.4434 1.41403 0.707015 0.707198i \(-0.250041\pi\)
0.707015 + 0.707198i \(0.250041\pi\)
\(860\) −1.42908 −0.0487314
\(861\) 0 0
\(862\) −17.8137 −0.606736
\(863\) 51.3552 1.74815 0.874076 0.485789i \(-0.161468\pi\)
0.874076 + 0.485789i \(0.161468\pi\)
\(864\) −4.60410 −0.156635
\(865\) 48.9887 1.66567
\(866\) −0.998955 −0.0339459
\(867\) −4.79907 −0.162985
\(868\) 0 0
\(869\) −22.8478 −0.775059
\(870\) 27.7344 0.940283
\(871\) 4.91538 0.166551
\(872\) 54.8374 1.85703
\(873\) −12.3944 −0.419488
\(874\) −5.19709 −0.175794
\(875\) 0 0
\(876\) 13.5806 0.458847
\(877\) 27.3268 0.922761 0.461381 0.887202i \(-0.347354\pi\)
0.461381 + 0.887202i \(0.347354\pi\)
\(878\) −17.2034 −0.580586
\(879\) −11.5761 −0.390452
\(880\) −13.5123 −0.455499
\(881\) −32.1796 −1.08416 −0.542079 0.840328i \(-0.682363\pi\)
−0.542079 + 0.840328i \(0.682363\pi\)
\(882\) 0 0
\(883\) 35.3505 1.18964 0.594820 0.803859i \(-0.297223\pi\)
0.594820 + 0.803859i \(0.297223\pi\)
\(884\) −1.44574 −0.0486255
\(885\) −31.4422 −1.05692
\(886\) −13.7732 −0.462720
\(887\) 26.1857 0.879231 0.439616 0.898186i \(-0.355115\pi\)
0.439616 + 0.898186i \(0.355115\pi\)
\(888\) −14.3992 −0.483207
\(889\) 0 0
\(890\) 34.9466 1.17141
\(891\) −2.24822 −0.0753183
\(892\) −1.20255 −0.0402643
\(893\) 7.84835 0.262635
\(894\) 13.6802 0.457536
\(895\) 95.1636 3.18097
\(896\) 0 0
\(897\) −0.855254 −0.0285561
\(898\) −10.8129 −0.360832
\(899\) −22.9781 −0.766363
\(900\) −11.7805 −0.392682
\(901\) 48.7391 1.62374
\(902\) −2.36363 −0.0787001
\(903\) 0 0
\(904\) −49.4744 −1.64549
\(905\) −2.43934 −0.0810865
\(906\) 11.4159 0.379267
\(907\) 17.6397 0.585718 0.292859 0.956156i \(-0.405393\pi\)
0.292859 + 0.956156i \(0.405393\pi\)
\(908\) −23.6866 −0.786068
\(909\) −9.64182 −0.319799
\(910\) 0 0
\(911\) 56.8015 1.88192 0.940958 0.338523i \(-0.109927\pi\)
0.940958 + 0.338523i \(0.109927\pi\)
\(912\) −2.82077 −0.0934049
\(913\) −27.6572 −0.915320
\(914\) −7.95969 −0.263283
\(915\) −49.8911 −1.64935
\(916\) 13.2195 0.436785
\(917\) 0 0
\(918\) −4.90861 −0.162008
\(919\) −56.8023 −1.87373 −0.936867 0.349686i \(-0.886288\pi\)
−0.936867 + 0.349686i \(0.886288\pi\)
\(920\) 32.0543 1.05680
\(921\) −15.4598 −0.509417
\(922\) −39.6617 −1.30619
\(923\) 0.554386 0.0182478
\(924\) 0 0
\(925\) −62.2986 −2.04837
\(926\) −1.54364 −0.0507272
\(927\) 4.40291 0.144611
\(928\) 28.4960 0.935426
\(929\) 16.4404 0.539391 0.269696 0.962946i \(-0.413077\pi\)
0.269696 + 0.962946i \(0.413077\pi\)
\(930\) −16.6363 −0.545524
\(931\) 0 0
\(932\) 10.7325 0.351556
\(933\) −16.4265 −0.537780
\(934\) 19.2628 0.630297
\(935\) 44.7402 1.46316
\(936\) −1.05326 −0.0344269
\(937\) 46.5074 1.51933 0.759665 0.650314i \(-0.225363\pi\)
0.759665 + 0.650314i \(0.225363\pi\)
\(938\) 0 0
\(939\) 29.9007 0.975771
\(940\) −14.9617 −0.487996
\(941\) −17.9670 −0.585709 −0.292854 0.956157i \(-0.594605\pi\)
−0.292854 + 0.956157i \(0.594605\pi\)
\(942\) 2.18876 0.0713136
\(943\) 2.47117 0.0804725
\(944\) 10.4021 0.338561
\(945\) 0 0
\(946\) 0.885763 0.0287986
\(947\) −26.6907 −0.867332 −0.433666 0.901074i \(-0.642780\pi\)
−0.433666 + 0.901074i \(0.642780\pi\)
\(948\) 9.09253 0.295312
\(949\) 5.25331 0.170530
\(950\) −27.6911 −0.898417
\(951\) −4.91153 −0.159267
\(952\) 0 0
\(953\) 44.3116 1.43539 0.717697 0.696355i \(-0.245196\pi\)
0.717697 + 0.696355i \(0.245196\pi\)
\(954\) 10.9748 0.355323
\(955\) 1.61365 0.0522166
\(956\) −13.7246 −0.443884
\(957\) 13.9148 0.449803
\(958\) −6.80885 −0.219984
\(959\) 0 0
\(960\) 32.6516 1.05383
\(961\) −17.2167 −0.555379
\(962\) −1.72158 −0.0555060
\(963\) 3.45652 0.111385
\(964\) −15.7785 −0.508191
\(965\) −40.9720 −1.31893
\(966\) 0 0
\(967\) −45.6508 −1.46803 −0.734016 0.679132i \(-0.762356\pi\)
−0.734016 + 0.679132i \(0.762356\pi\)
\(968\) 18.0939 0.581559
\(969\) 9.33978 0.300037
\(970\) 55.5401 1.78329
\(971\) 44.2849 1.42117 0.710584 0.703612i \(-0.248431\pi\)
0.710584 + 0.703612i \(0.248431\pi\)
\(972\) 0.894704 0.0286976
\(973\) 0 0
\(974\) −0.604349 −0.0193646
\(975\) −4.55696 −0.145939
\(976\) 16.5056 0.528333
\(977\) 2.79496 0.0894186 0.0447093 0.999000i \(-0.485764\pi\)
0.0447093 + 0.999000i \(0.485764\pi\)
\(978\) 13.8418 0.442612
\(979\) 17.5333 0.560367
\(980\) 0 0
\(981\) −18.0191 −0.575306
\(982\) −5.90967 −0.188585
\(983\) 59.3041 1.89151 0.945753 0.324887i \(-0.105326\pi\)
0.945753 + 0.324887i \(0.105326\pi\)
\(984\) 3.04329 0.0970166
\(985\) 44.2122 1.40872
\(986\) 30.3806 0.967517
\(987\) 0 0
\(988\) 0.619425 0.0197065
\(989\) −0.926066 −0.0294472
\(990\) 10.0744 0.320185
\(991\) 3.98616 0.126625 0.0633123 0.997994i \(-0.479834\pi\)
0.0633123 + 0.997994i \(0.479834\pi\)
\(992\) −17.0931 −0.542707
\(993\) −5.41473 −0.171831
\(994\) 0 0
\(995\) −73.3035 −2.32388
\(996\) 11.0065 0.348753
\(997\) −14.6032 −0.462488 −0.231244 0.972896i \(-0.574280\pi\)
−0.231244 + 0.972896i \(0.574280\pi\)
\(998\) 12.0809 0.382413
\(999\) 4.73147 0.149697
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 6027.2.a.bf.1.9 12
7.3 odd 6 861.2.i.e.247.4 24
7.5 odd 6 861.2.i.e.739.4 yes 24
7.6 odd 2 6027.2.a.bg.1.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
861.2.i.e.247.4 24 7.3 odd 6
861.2.i.e.739.4 yes 24 7.5 odd 6
6027.2.a.bf.1.9 12 1.1 even 1 trivial
6027.2.a.bg.1.9 12 7.6 odd 2