Properties

Label 6027.2.a.be.1.10
Level $6027$
Weight $2$
Character 6027.1
Self dual yes
Analytic conductor $48.126$
Analytic rank $0$
Dimension $10$
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: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 11x^{8} + 56x^{7} + 26x^{6} - 266x^{5} + 52x^{4} + 526x^{3} - 255x^{2} - 372x + 239 \) 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 1.10
Root \(2.80657\) 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)
 
\(f(q)\) \(=\) \(q+2.80657 q^{2} +1.00000 q^{3} +5.87681 q^{4} +4.11270 q^{5} +2.80657 q^{6} +10.8805 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.80657 q^{2} +1.00000 q^{3} +5.87681 q^{4} +4.11270 q^{5} +2.80657 q^{6} +10.8805 q^{8} +1.00000 q^{9} +11.5426 q^{10} -4.01832 q^{11} +5.87681 q^{12} +0.753718 q^{13} +4.11270 q^{15} +18.7833 q^{16} -2.86815 q^{17} +2.80657 q^{18} -8.05398 q^{19} +24.1696 q^{20} -11.2777 q^{22} -6.62187 q^{23} +10.8805 q^{24} +11.9143 q^{25} +2.11536 q^{26} +1.00000 q^{27} -3.55799 q^{29} +11.5426 q^{30} -1.91801 q^{31} +30.9555 q^{32} -4.01832 q^{33} -8.04966 q^{34} +5.87681 q^{36} +5.80800 q^{37} -22.6040 q^{38} +0.753718 q^{39} +44.7484 q^{40} +1.00000 q^{41} -2.59707 q^{43} -23.6149 q^{44} +4.11270 q^{45} -18.5847 q^{46} -1.45654 q^{47} +18.7833 q^{48} +33.4383 q^{50} -2.86815 q^{51} +4.42946 q^{52} +7.12681 q^{53} +2.80657 q^{54} -16.5262 q^{55} -8.05398 q^{57} -9.98574 q^{58} +2.95445 q^{59} +24.1696 q^{60} -5.68932 q^{61} -5.38302 q^{62} +49.3121 q^{64} +3.09982 q^{65} -11.2777 q^{66} +4.99671 q^{67} -16.8556 q^{68} -6.62187 q^{69} +5.96041 q^{71} +10.8805 q^{72} -14.7998 q^{73} +16.3005 q^{74} +11.9143 q^{75} -47.3317 q^{76} +2.11536 q^{78} +8.86055 q^{79} +77.2502 q^{80} +1.00000 q^{81} +2.80657 q^{82} -1.22105 q^{83} -11.7959 q^{85} -7.28885 q^{86} -3.55799 q^{87} -43.7215 q^{88} +6.89799 q^{89} +11.5426 q^{90} -38.9155 q^{92} -1.91801 q^{93} -4.08787 q^{94} -33.1236 q^{95} +30.9555 q^{96} -1.83652 q^{97} -4.01832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} + 10 q^{3} + 18 q^{4} + 6 q^{5} + 4 q^{6} + 12 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{2} + 10 q^{3} + 18 q^{4} + 6 q^{5} + 4 q^{6} + 12 q^{8} + 10 q^{9} + 2 q^{10} - 2 q^{11} + 18 q^{12} + 6 q^{15} + 14 q^{16} + 8 q^{17} + 4 q^{18} + 6 q^{19} + 20 q^{20} + 2 q^{22} + 12 q^{24} + 10 q^{25} + 16 q^{26} + 10 q^{27} + 16 q^{29} + 2 q^{30} + 2 q^{31} + 38 q^{32} - 2 q^{33} - 4 q^{34} + 18 q^{36} + 24 q^{37} - 26 q^{38} + 40 q^{40} + 10 q^{41} + 8 q^{43} - 8 q^{44} + 6 q^{45} + 4 q^{46} - 8 q^{47} + 14 q^{48} + 44 q^{50} + 8 q^{51} - 30 q^{52} + 24 q^{53} + 4 q^{54} + 6 q^{57} - 14 q^{58} + 6 q^{59} + 20 q^{60} - 14 q^{61} - 2 q^{62} + 86 q^{64} + 28 q^{65} + 2 q^{66} + 26 q^{67} - 6 q^{68} + 14 q^{71} + 12 q^{72} - 36 q^{73} + 18 q^{74} + 10 q^{75} - 32 q^{76} + 16 q^{78} + 20 q^{79} + 70 q^{80} + 10 q^{81} + 4 q^{82} + 40 q^{83} + 24 q^{85} - 36 q^{86} + 16 q^{87} + 14 q^{88} + 2 q^{89} + 2 q^{90} + 8 q^{92} + 2 q^{93} - 54 q^{94} - 24 q^{95} + 38 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.80657 1.98454 0.992271 0.124090i \(-0.0396011\pi\)
0.992271 + 0.124090i \(0.0396011\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.87681 2.93841
\(5\) 4.11270 1.83926 0.919628 0.392790i \(-0.128490\pi\)
0.919628 + 0.392790i \(0.128490\pi\)
\(6\) 2.80657 1.14578
\(7\) 0 0
\(8\) 10.8805 3.84685
\(9\) 1.00000 0.333333
\(10\) 11.5426 3.65008
\(11\) −4.01832 −1.21157 −0.605785 0.795628i \(-0.707141\pi\)
−0.605785 + 0.795628i \(0.707141\pi\)
\(12\) 5.87681 1.69649
\(13\) 0.753718 0.209044 0.104522 0.994523i \(-0.466669\pi\)
0.104522 + 0.994523i \(0.466669\pi\)
\(14\) 0 0
\(15\) 4.11270 1.06190
\(16\) 18.7833 4.69583
\(17\) −2.86815 −0.695629 −0.347815 0.937563i \(-0.613076\pi\)
−0.347815 + 0.937563i \(0.613076\pi\)
\(18\) 2.80657 0.661514
\(19\) −8.05398 −1.84771 −0.923855 0.382744i \(-0.874979\pi\)
−0.923855 + 0.382744i \(0.874979\pi\)
\(20\) 24.1696 5.40448
\(21\) 0 0
\(22\) −11.2777 −2.40441
\(23\) −6.62187 −1.38075 −0.690377 0.723449i \(-0.742555\pi\)
−0.690377 + 0.723449i \(0.742555\pi\)
\(24\) 10.8805 2.22098
\(25\) 11.9143 2.38286
\(26\) 2.11536 0.414856
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.55799 −0.660702 −0.330351 0.943858i \(-0.607167\pi\)
−0.330351 + 0.943858i \(0.607167\pi\)
\(30\) 11.5426 2.10738
\(31\) −1.91801 −0.344485 −0.172242 0.985055i \(-0.555101\pi\)
−0.172242 + 0.985055i \(0.555101\pi\)
\(32\) 30.9555 5.47222
\(33\) −4.01832 −0.699500
\(34\) −8.04966 −1.38051
\(35\) 0 0
\(36\) 5.87681 0.979469
\(37\) 5.80800 0.954829 0.477414 0.878678i \(-0.341574\pi\)
0.477414 + 0.878678i \(0.341574\pi\)
\(38\) −22.6040 −3.66686
\(39\) 0.753718 0.120691
\(40\) 44.7484 7.07534
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −2.59707 −0.396049 −0.198025 0.980197i \(-0.563453\pi\)
−0.198025 + 0.980197i \(0.563453\pi\)
\(44\) −23.6149 −3.56009
\(45\) 4.11270 0.613085
\(46\) −18.5847 −2.74017
\(47\) −1.45654 −0.212458 −0.106229 0.994342i \(-0.533878\pi\)
−0.106229 + 0.994342i \(0.533878\pi\)
\(48\) 18.7833 2.71114
\(49\) 0 0
\(50\) 33.4383 4.72889
\(51\) −2.86815 −0.401622
\(52\) 4.42946 0.614255
\(53\) 7.12681 0.978943 0.489471 0.872019i \(-0.337190\pi\)
0.489471 + 0.872019i \(0.337190\pi\)
\(54\) 2.80657 0.381925
\(55\) −16.5262 −2.22839
\(56\) 0 0
\(57\) −8.05398 −1.06678
\(58\) −9.98574 −1.31119
\(59\) 2.95445 0.384637 0.192318 0.981333i \(-0.438399\pi\)
0.192318 + 0.981333i \(0.438399\pi\)
\(60\) 24.1696 3.12028
\(61\) −5.68932 −0.728443 −0.364222 0.931312i \(-0.618665\pi\)
−0.364222 + 0.931312i \(0.618665\pi\)
\(62\) −5.38302 −0.683644
\(63\) 0 0
\(64\) 49.3121 6.16402
\(65\) 3.09982 0.384485
\(66\) −11.2777 −1.38819
\(67\) 4.99671 0.610446 0.305223 0.952281i \(-0.401269\pi\)
0.305223 + 0.952281i \(0.401269\pi\)
\(68\) −16.8556 −2.04404
\(69\) −6.62187 −0.797179
\(70\) 0 0
\(71\) 5.96041 0.707370 0.353685 0.935365i \(-0.384928\pi\)
0.353685 + 0.935365i \(0.384928\pi\)
\(72\) 10.8805 1.28228
\(73\) −14.7998 −1.73218 −0.866092 0.499885i \(-0.833375\pi\)
−0.866092 + 0.499885i \(0.833375\pi\)
\(74\) 16.3005 1.89490
\(75\) 11.9143 1.37575
\(76\) −47.3317 −5.42932
\(77\) 0 0
\(78\) 2.11536 0.239517
\(79\) 8.86055 0.996890 0.498445 0.866921i \(-0.333905\pi\)
0.498445 + 0.866921i \(0.333905\pi\)
\(80\) 77.2502 8.63683
\(81\) 1.00000 0.111111
\(82\) 2.80657 0.309933
\(83\) −1.22105 −0.134028 −0.0670141 0.997752i \(-0.521347\pi\)
−0.0670141 + 0.997752i \(0.521347\pi\)
\(84\) 0 0
\(85\) −11.7959 −1.27944
\(86\) −7.28885 −0.785977
\(87\) −3.55799 −0.381457
\(88\) −43.7215 −4.66073
\(89\) 6.89799 0.731186 0.365593 0.930775i \(-0.380866\pi\)
0.365593 + 0.930775i \(0.380866\pi\)
\(90\) 11.5426 1.21669
\(91\) 0 0
\(92\) −38.9155 −4.05722
\(93\) −1.91801 −0.198888
\(94\) −4.08787 −0.421632
\(95\) −33.1236 −3.39841
\(96\) 30.9555 3.15939
\(97\) −1.83652 −0.186471 −0.0932353 0.995644i \(-0.529721\pi\)
−0.0932353 + 0.995644i \(0.529721\pi\)
\(98\) 0 0
\(99\) −4.01832 −0.403857
\(100\) 70.0182 7.00182
\(101\) −16.4538 −1.63722 −0.818609 0.574351i \(-0.805255\pi\)
−0.818609 + 0.574351i \(0.805255\pi\)
\(102\) −8.04966 −0.797035
\(103\) 7.19377 0.708823 0.354412 0.935090i \(-0.384681\pi\)
0.354412 + 0.935090i \(0.384681\pi\)
\(104\) 8.20085 0.804160
\(105\) 0 0
\(106\) 20.0019 1.94275
\(107\) 8.56687 0.828191 0.414095 0.910234i \(-0.364098\pi\)
0.414095 + 0.910234i \(0.364098\pi\)
\(108\) 5.87681 0.565497
\(109\) 13.9372 1.33494 0.667469 0.744638i \(-0.267378\pi\)
0.667469 + 0.744638i \(0.267378\pi\)
\(110\) −46.3818 −4.42233
\(111\) 5.80800 0.551271
\(112\) 0 0
\(113\) −10.8482 −1.02051 −0.510256 0.860022i \(-0.670449\pi\)
−0.510256 + 0.860022i \(0.670449\pi\)
\(114\) −22.6040 −2.11706
\(115\) −27.2338 −2.53956
\(116\) −20.9097 −1.94141
\(117\) 0.753718 0.0696812
\(118\) 8.29186 0.763327
\(119\) 0 0
\(120\) 44.7484 4.08495
\(121\) 5.14693 0.467903
\(122\) −15.9675 −1.44563
\(123\) 1.00000 0.0901670
\(124\) −11.2718 −1.01224
\(125\) 28.4365 2.54344
\(126\) 0 0
\(127\) −13.7606 −1.22106 −0.610530 0.791993i \(-0.709043\pi\)
−0.610530 + 0.791993i \(0.709043\pi\)
\(128\) 76.4867 6.76053
\(129\) −2.59707 −0.228659
\(130\) 8.69984 0.763027
\(131\) 7.79788 0.681304 0.340652 0.940190i \(-0.389352\pi\)
0.340652 + 0.940190i \(0.389352\pi\)
\(132\) −23.6149 −2.05542
\(133\) 0 0
\(134\) 14.0236 1.21145
\(135\) 4.11270 0.353965
\(136\) −31.2070 −2.67598
\(137\) 9.01085 0.769849 0.384924 0.922948i \(-0.374227\pi\)
0.384924 + 0.922948i \(0.374227\pi\)
\(138\) −18.5847 −1.58204
\(139\) −14.5224 −1.23177 −0.615885 0.787836i \(-0.711202\pi\)
−0.615885 + 0.787836i \(0.711202\pi\)
\(140\) 0 0
\(141\) −1.45654 −0.122663
\(142\) 16.7283 1.40381
\(143\) −3.02868 −0.253271
\(144\) 18.7833 1.56528
\(145\) −14.6330 −1.21520
\(146\) −41.5366 −3.43759
\(147\) 0 0
\(148\) 34.1325 2.80568
\(149\) 5.01078 0.410499 0.205249 0.978710i \(-0.434199\pi\)
0.205249 + 0.978710i \(0.434199\pi\)
\(150\) 33.4383 2.73023
\(151\) −13.6609 −1.11171 −0.555853 0.831280i \(-0.687608\pi\)
−0.555853 + 0.831280i \(0.687608\pi\)
\(152\) −87.6316 −7.10786
\(153\) −2.86815 −0.231876
\(154\) 0 0
\(155\) −7.88820 −0.633596
\(156\) 4.42946 0.354641
\(157\) −2.36196 −0.188505 −0.0942526 0.995548i \(-0.530046\pi\)
−0.0942526 + 0.995548i \(0.530046\pi\)
\(158\) 24.8677 1.97837
\(159\) 7.12681 0.565193
\(160\) 127.311 10.0648
\(161\) 0 0
\(162\) 2.80657 0.220505
\(163\) 0.595417 0.0466367 0.0233183 0.999728i \(-0.492577\pi\)
0.0233183 + 0.999728i \(0.492577\pi\)
\(164\) 5.87681 0.458902
\(165\) −16.5262 −1.28656
\(166\) −3.42697 −0.265985
\(167\) 3.69545 0.285963 0.142981 0.989725i \(-0.454331\pi\)
0.142981 + 0.989725i \(0.454331\pi\)
\(168\) 0 0
\(169\) −12.4319 −0.956301
\(170\) −33.1059 −2.53910
\(171\) −8.05398 −0.615903
\(172\) −15.2625 −1.16375
\(173\) −8.93647 −0.679427 −0.339714 0.940529i \(-0.610330\pi\)
−0.339714 + 0.940529i \(0.610330\pi\)
\(174\) −9.98574 −0.757017
\(175\) 0 0
\(176\) −75.4774 −5.68933
\(177\) 2.95445 0.222070
\(178\) 19.3597 1.45107
\(179\) −7.70618 −0.575987 −0.287994 0.957632i \(-0.592988\pi\)
−0.287994 + 0.957632i \(0.592988\pi\)
\(180\) 24.1696 1.80149
\(181\) −9.67962 −0.719480 −0.359740 0.933053i \(-0.617135\pi\)
−0.359740 + 0.933053i \(0.617135\pi\)
\(182\) 0 0
\(183\) −5.68932 −0.420567
\(184\) −72.0494 −5.31156
\(185\) 23.8866 1.75617
\(186\) −5.38302 −0.394702
\(187\) 11.5252 0.842804
\(188\) −8.55981 −0.624288
\(189\) 0 0
\(190\) −92.9636 −6.74429
\(191\) 20.9409 1.51523 0.757614 0.652702i \(-0.226365\pi\)
0.757614 + 0.652702i \(0.226365\pi\)
\(192\) 49.3121 3.55880
\(193\) 6.85788 0.493641 0.246821 0.969061i \(-0.420614\pi\)
0.246821 + 0.969061i \(0.420614\pi\)
\(194\) −5.15432 −0.370059
\(195\) 3.09982 0.221982
\(196\) 0 0
\(197\) 5.85575 0.417205 0.208603 0.978001i \(-0.433109\pi\)
0.208603 + 0.978001i \(0.433109\pi\)
\(198\) −11.2777 −0.801471
\(199\) −9.49177 −0.672854 −0.336427 0.941710i \(-0.609219\pi\)
−0.336427 + 0.941710i \(0.609219\pi\)
\(200\) 129.634 9.16652
\(201\) 4.99671 0.352441
\(202\) −46.1788 −3.24913
\(203\) 0 0
\(204\) −16.8556 −1.18013
\(205\) 4.11270 0.287244
\(206\) 20.1898 1.40669
\(207\) −6.62187 −0.460252
\(208\) 14.1573 0.981633
\(209\) 32.3635 2.23863
\(210\) 0 0
\(211\) −19.9337 −1.37229 −0.686146 0.727464i \(-0.740699\pi\)
−0.686146 + 0.727464i \(0.740699\pi\)
\(212\) 41.8829 2.87653
\(213\) 5.96041 0.408400
\(214\) 24.0435 1.64358
\(215\) −10.6810 −0.728436
\(216\) 10.8805 0.740327
\(217\) 0 0
\(218\) 39.1156 2.64924
\(219\) −14.7998 −1.00008
\(220\) −97.1212 −6.54791
\(221\) −2.16178 −0.145417
\(222\) 16.3005 1.09402
\(223\) 2.16735 0.145136 0.0725682 0.997363i \(-0.476880\pi\)
0.0725682 + 0.997363i \(0.476880\pi\)
\(224\) 0 0
\(225\) 11.9143 0.794288
\(226\) −30.4462 −2.02525
\(227\) −27.4094 −1.81923 −0.909613 0.415457i \(-0.863622\pi\)
−0.909613 + 0.415457i \(0.863622\pi\)
\(228\) −47.3317 −3.13462
\(229\) 12.3463 0.815867 0.407933 0.913012i \(-0.366250\pi\)
0.407933 + 0.913012i \(0.366250\pi\)
\(230\) −76.4334 −5.03987
\(231\) 0 0
\(232\) −38.7128 −2.54162
\(233\) −11.9621 −0.783664 −0.391832 0.920037i \(-0.628159\pi\)
−0.391832 + 0.920037i \(0.628159\pi\)
\(234\) 2.11536 0.138285
\(235\) −5.99031 −0.390765
\(236\) 17.3627 1.13022
\(237\) 8.86055 0.575555
\(238\) 0 0
\(239\) −11.4914 −0.743320 −0.371660 0.928369i \(-0.621211\pi\)
−0.371660 + 0.928369i \(0.621211\pi\)
\(240\) 77.2502 4.98648
\(241\) 24.4262 1.57343 0.786715 0.617317i \(-0.211780\pi\)
0.786715 + 0.617317i \(0.211780\pi\)
\(242\) 14.4452 0.928573
\(243\) 1.00000 0.0641500
\(244\) −33.4351 −2.14046
\(245\) 0 0
\(246\) 2.80657 0.178940
\(247\) −6.07043 −0.386252
\(248\) −20.8690 −1.32518
\(249\) −1.22105 −0.0773812
\(250\) 79.8090 5.04757
\(251\) −17.3618 −1.09587 −0.547935 0.836521i \(-0.684586\pi\)
−0.547935 + 0.836521i \(0.684586\pi\)
\(252\) 0 0
\(253\) 26.6088 1.67288
\(254\) −38.6201 −2.42324
\(255\) −11.7959 −0.738686
\(256\) 116.041 7.25254
\(257\) 15.9782 0.996693 0.498347 0.866978i \(-0.333941\pi\)
0.498347 + 0.866978i \(0.333941\pi\)
\(258\) −7.28885 −0.453784
\(259\) 0 0
\(260\) 18.2170 1.12977
\(261\) −3.55799 −0.220234
\(262\) 21.8853 1.35208
\(263\) −3.35156 −0.206666 −0.103333 0.994647i \(-0.532951\pi\)
−0.103333 + 0.994647i \(0.532951\pi\)
\(264\) −43.7215 −2.69087
\(265\) 29.3104 1.80053
\(266\) 0 0
\(267\) 6.89799 0.422150
\(268\) 29.3647 1.79374
\(269\) 8.96079 0.546349 0.273174 0.961965i \(-0.411926\pi\)
0.273174 + 0.961965i \(0.411926\pi\)
\(270\) 11.5426 0.702458
\(271\) −2.53831 −0.154191 −0.0770957 0.997024i \(-0.524565\pi\)
−0.0770957 + 0.997024i \(0.524565\pi\)
\(272\) −53.8734 −3.26656
\(273\) 0 0
\(274\) 25.2896 1.52780
\(275\) −47.8756 −2.88701
\(276\) −38.9155 −2.34244
\(277\) 19.0038 1.14183 0.570914 0.821010i \(-0.306589\pi\)
0.570914 + 0.821010i \(0.306589\pi\)
\(278\) −40.7580 −2.44450
\(279\) −1.91801 −0.114828
\(280\) 0 0
\(281\) −5.50424 −0.328356 −0.164178 0.986431i \(-0.552497\pi\)
−0.164178 + 0.986431i \(0.552497\pi\)
\(282\) −4.08787 −0.243429
\(283\) 25.6985 1.52762 0.763809 0.645443i \(-0.223327\pi\)
0.763809 + 0.645443i \(0.223327\pi\)
\(284\) 35.0282 2.07854
\(285\) −33.1236 −1.96207
\(286\) −8.50020 −0.502627
\(287\) 0 0
\(288\) 30.9555 1.82407
\(289\) −8.77369 −0.516100
\(290\) −41.0684 −2.41162
\(291\) −1.83652 −0.107659
\(292\) −86.9755 −5.08986
\(293\) −15.9943 −0.934395 −0.467197 0.884153i \(-0.654736\pi\)
−0.467197 + 0.884153i \(0.654736\pi\)
\(294\) 0 0
\(295\) 12.1508 0.707445
\(296\) 63.1941 3.67308
\(297\) −4.01832 −0.233167
\(298\) 14.0631 0.814652
\(299\) −4.99102 −0.288638
\(300\) 70.0182 4.04250
\(301\) 0 0
\(302\) −38.3402 −2.20623
\(303\) −16.4538 −0.945249
\(304\) −151.280 −8.67652
\(305\) −23.3985 −1.33979
\(306\) −8.04966 −0.460169
\(307\) 20.1627 1.15075 0.575373 0.817891i \(-0.304857\pi\)
0.575373 + 0.817891i \(0.304857\pi\)
\(308\) 0 0
\(309\) 7.19377 0.409239
\(310\) −22.1388 −1.25740
\(311\) 23.6861 1.34312 0.671558 0.740952i \(-0.265626\pi\)
0.671558 + 0.740952i \(0.265626\pi\)
\(312\) 8.20085 0.464282
\(313\) 25.9892 1.46900 0.734499 0.678610i \(-0.237417\pi\)
0.734499 + 0.678610i \(0.237417\pi\)
\(314\) −6.62901 −0.374097
\(315\) 0 0
\(316\) 52.0718 2.92927
\(317\) 12.9378 0.726659 0.363329 0.931661i \(-0.381640\pi\)
0.363329 + 0.931661i \(0.381640\pi\)
\(318\) 20.0019 1.12165
\(319\) 14.2972 0.800487
\(320\) 202.806 11.3372
\(321\) 8.56687 0.478156
\(322\) 0 0
\(323\) 23.1000 1.28532
\(324\) 5.87681 0.326490
\(325\) 8.98003 0.498123
\(326\) 1.67108 0.0925524
\(327\) 13.9372 0.770727
\(328\) 10.8805 0.600777
\(329\) 0 0
\(330\) −46.3818 −2.55323
\(331\) −15.8948 −0.873655 −0.436827 0.899545i \(-0.643898\pi\)
−0.436827 + 0.899545i \(0.643898\pi\)
\(332\) −7.17591 −0.393829
\(333\) 5.80800 0.318276
\(334\) 10.3715 0.567505
\(335\) 20.5500 1.12277
\(336\) 0 0
\(337\) −7.73799 −0.421515 −0.210758 0.977538i \(-0.567593\pi\)
−0.210758 + 0.977538i \(0.567593\pi\)
\(338\) −34.8910 −1.89782
\(339\) −10.8482 −0.589193
\(340\) −69.3221 −3.75952
\(341\) 7.70719 0.417368
\(342\) −22.6040 −1.22229
\(343\) 0 0
\(344\) −28.2575 −1.52354
\(345\) −27.2338 −1.46622
\(346\) −25.0808 −1.34835
\(347\) −6.92049 −0.371511 −0.185756 0.982596i \(-0.559473\pi\)
−0.185756 + 0.982596i \(0.559473\pi\)
\(348\) −20.9097 −1.12087
\(349\) 33.9500 1.81730 0.908651 0.417556i \(-0.137113\pi\)
0.908651 + 0.417556i \(0.137113\pi\)
\(350\) 0 0
\(351\) 0.753718 0.0402305
\(352\) −124.389 −6.62998
\(353\) 19.8577 1.05692 0.528460 0.848958i \(-0.322770\pi\)
0.528460 + 0.848958i \(0.322770\pi\)
\(354\) 8.29186 0.440707
\(355\) 24.5134 1.30104
\(356\) 40.5382 2.14852
\(357\) 0 0
\(358\) −21.6279 −1.14307
\(359\) 5.95353 0.314216 0.157108 0.987581i \(-0.449783\pi\)
0.157108 + 0.987581i \(0.449783\pi\)
\(360\) 44.7484 2.35845
\(361\) 45.8666 2.41403
\(362\) −27.1665 −1.42784
\(363\) 5.14693 0.270144
\(364\) 0 0
\(365\) −60.8671 −3.18593
\(366\) −15.9675 −0.834633
\(367\) −29.4136 −1.53538 −0.767690 0.640822i \(-0.778594\pi\)
−0.767690 + 0.640822i \(0.778594\pi\)
\(368\) −124.381 −6.48379
\(369\) 1.00000 0.0520579
\(370\) 67.0392 3.48520
\(371\) 0 0
\(372\) −11.2718 −0.584415
\(373\) −1.46339 −0.0757715 −0.0378857 0.999282i \(-0.512062\pi\)
−0.0378857 + 0.999282i \(0.512062\pi\)
\(374\) 32.3462 1.67258
\(375\) 28.4365 1.46846
\(376\) −15.8479 −0.817294
\(377\) −2.68172 −0.138116
\(378\) 0 0
\(379\) −28.8794 −1.48343 −0.741717 0.670713i \(-0.765988\pi\)
−0.741717 + 0.670713i \(0.765988\pi\)
\(380\) −194.661 −9.98591
\(381\) −13.7606 −0.704979
\(382\) 58.7720 3.00704
\(383\) −9.31702 −0.476077 −0.238039 0.971256i \(-0.576505\pi\)
−0.238039 + 0.971256i \(0.576505\pi\)
\(384\) 76.4867 3.90320
\(385\) 0 0
\(386\) 19.2471 0.979652
\(387\) −2.59707 −0.132016
\(388\) −10.7929 −0.547926
\(389\) 14.1181 0.715816 0.357908 0.933757i \(-0.383490\pi\)
0.357908 + 0.933757i \(0.383490\pi\)
\(390\) 8.69984 0.440534
\(391\) 18.9925 0.960494
\(392\) 0 0
\(393\) 7.79788 0.393351
\(394\) 16.4346 0.827961
\(395\) 36.4408 1.83354
\(396\) −23.6149 −1.18670
\(397\) −32.9826 −1.65535 −0.827674 0.561209i \(-0.810336\pi\)
−0.827674 + 0.561209i \(0.810336\pi\)
\(398\) −26.6393 −1.33531
\(399\) 0 0
\(400\) 223.790 11.1895
\(401\) 20.1923 1.00836 0.504178 0.863600i \(-0.331796\pi\)
0.504178 + 0.863600i \(0.331796\pi\)
\(402\) 14.0236 0.699434
\(403\) −1.44564 −0.0720124
\(404\) −96.6962 −4.81081
\(405\) 4.11270 0.204362
\(406\) 0 0
\(407\) −23.3384 −1.15684
\(408\) −31.2070 −1.54498
\(409\) 8.05070 0.398082 0.199041 0.979991i \(-0.436217\pi\)
0.199041 + 0.979991i \(0.436217\pi\)
\(410\) 11.5426 0.570047
\(411\) 9.01085 0.444473
\(412\) 42.2764 2.08281
\(413\) 0 0
\(414\) −18.5847 −0.913388
\(415\) −5.02183 −0.246512
\(416\) 23.3317 1.14393
\(417\) −14.5224 −0.711163
\(418\) 90.8303 4.44265
\(419\) 14.8930 0.727570 0.363785 0.931483i \(-0.381484\pi\)
0.363785 + 0.931483i \(0.381484\pi\)
\(420\) 0 0
\(421\) 2.32962 0.113539 0.0567695 0.998387i \(-0.481920\pi\)
0.0567695 + 0.998387i \(0.481920\pi\)
\(422\) −55.9452 −2.72337
\(423\) −1.45654 −0.0708194
\(424\) 77.5435 3.76585
\(425\) −34.1721 −1.65759
\(426\) 16.7283 0.810488
\(427\) 0 0
\(428\) 50.3459 2.43356
\(429\) −3.02868 −0.146226
\(430\) −29.9769 −1.44561
\(431\) 11.9770 0.576914 0.288457 0.957493i \(-0.406858\pi\)
0.288457 + 0.957493i \(0.406858\pi\)
\(432\) 18.7833 0.903712
\(433\) 28.6259 1.37568 0.687838 0.725865i \(-0.258560\pi\)
0.687838 + 0.725865i \(0.258560\pi\)
\(434\) 0 0
\(435\) −14.6330 −0.701597
\(436\) 81.9061 3.92259
\(437\) 53.3324 2.55123
\(438\) −41.5366 −1.98469
\(439\) 14.2078 0.678102 0.339051 0.940768i \(-0.389894\pi\)
0.339051 + 0.940768i \(0.389894\pi\)
\(440\) −179.814 −8.57228
\(441\) 0 0
\(442\) −6.06717 −0.288586
\(443\) −9.29302 −0.441525 −0.220762 0.975328i \(-0.570855\pi\)
−0.220762 + 0.975328i \(0.570855\pi\)
\(444\) 34.1325 1.61986
\(445\) 28.3694 1.34484
\(446\) 6.08281 0.288029
\(447\) 5.01078 0.237002
\(448\) 0 0
\(449\) −37.3303 −1.76173 −0.880864 0.473369i \(-0.843038\pi\)
−0.880864 + 0.473369i \(0.843038\pi\)
\(450\) 33.4383 1.57630
\(451\) −4.01832 −0.189216
\(452\) −63.7528 −2.99868
\(453\) −13.6609 −0.641844
\(454\) −76.9263 −3.61033
\(455\) 0 0
\(456\) −87.6316 −4.10372
\(457\) 12.5438 0.586776 0.293388 0.955993i \(-0.405217\pi\)
0.293388 + 0.955993i \(0.405217\pi\)
\(458\) 34.6507 1.61912
\(459\) −2.86815 −0.133874
\(460\) −160.048 −7.46227
\(461\) −16.7456 −0.779919 −0.389959 0.920832i \(-0.627511\pi\)
−0.389959 + 0.920832i \(0.627511\pi\)
\(462\) 0 0
\(463\) −16.4001 −0.762176 −0.381088 0.924539i \(-0.624451\pi\)
−0.381088 + 0.924539i \(0.624451\pi\)
\(464\) −66.8309 −3.10254
\(465\) −7.88820 −0.365807
\(466\) −33.5725 −1.55521
\(467\) 16.4957 0.763331 0.381666 0.924300i \(-0.375351\pi\)
0.381666 + 0.924300i \(0.375351\pi\)
\(468\) 4.42946 0.204752
\(469\) 0 0
\(470\) −16.8122 −0.775489
\(471\) −2.36196 −0.108834
\(472\) 32.1460 1.47964
\(473\) 10.4359 0.479842
\(474\) 24.8677 1.14221
\(475\) −95.9577 −4.40284
\(476\) 0 0
\(477\) 7.12681 0.326314
\(478\) −32.2515 −1.47515
\(479\) 13.3546 0.610185 0.305093 0.952323i \(-0.401313\pi\)
0.305093 + 0.952323i \(0.401313\pi\)
\(480\) 127.311 5.81092
\(481\) 4.37759 0.199601
\(482\) 68.5537 3.12254
\(483\) 0 0
\(484\) 30.2476 1.37489
\(485\) −7.55307 −0.342967
\(486\) 2.80657 0.127308
\(487\) 3.34242 0.151460 0.0757298 0.997128i \(-0.475871\pi\)
0.0757298 + 0.997128i \(0.475871\pi\)
\(488\) −61.9029 −2.80221
\(489\) 0.595417 0.0269257
\(490\) 0 0
\(491\) 21.1104 0.952699 0.476349 0.879256i \(-0.341960\pi\)
0.476349 + 0.879256i \(0.341960\pi\)
\(492\) 5.87681 0.264947
\(493\) 10.2049 0.459604
\(494\) −17.0371 −0.766533
\(495\) −16.5262 −0.742796
\(496\) −36.0266 −1.61764
\(497\) 0 0
\(498\) −3.42697 −0.153566
\(499\) −0.109003 −0.00487963 −0.00243981 0.999997i \(-0.500777\pi\)
−0.00243981 + 0.999997i \(0.500777\pi\)
\(500\) 167.116 7.47366
\(501\) 3.69545 0.165101
\(502\) −48.7271 −2.17480
\(503\) 3.20557 0.142930 0.0714648 0.997443i \(-0.477233\pi\)
0.0714648 + 0.997443i \(0.477233\pi\)
\(504\) 0 0
\(505\) −67.6698 −3.01127
\(506\) 74.6794 3.31990
\(507\) −12.4319 −0.552120
\(508\) −80.8687 −3.58797
\(509\) −13.9817 −0.619727 −0.309863 0.950781i \(-0.600283\pi\)
−0.309863 + 0.950781i \(0.600283\pi\)
\(510\) −33.1059 −1.46595
\(511\) 0 0
\(512\) 172.703 7.63245
\(513\) −8.05398 −0.355592
\(514\) 44.8439 1.97798
\(515\) 29.5858 1.30371
\(516\) −15.2625 −0.671894
\(517\) 5.85285 0.257408
\(518\) 0 0
\(519\) −8.93647 −0.392268
\(520\) 33.7277 1.47906
\(521\) −11.6932 −0.512288 −0.256144 0.966639i \(-0.582452\pi\)
−0.256144 + 0.966639i \(0.582452\pi\)
\(522\) −9.98574 −0.437064
\(523\) 11.6517 0.509492 0.254746 0.967008i \(-0.418008\pi\)
0.254746 + 0.967008i \(0.418008\pi\)
\(524\) 45.8267 2.00195
\(525\) 0 0
\(526\) −9.40637 −0.410137
\(527\) 5.50115 0.239634
\(528\) −75.4774 −3.28473
\(529\) 20.8491 0.906483
\(530\) 82.2617 3.57322
\(531\) 2.95445 0.128212
\(532\) 0 0
\(533\) 0.753718 0.0326471
\(534\) 19.3597 0.837775
\(535\) 35.2330 1.52325
\(536\) 54.3669 2.34829
\(537\) −7.70618 −0.332546
\(538\) 25.1490 1.08425
\(539\) 0 0
\(540\) 24.1696 1.04009
\(541\) 23.6914 1.01857 0.509285 0.860598i \(-0.329910\pi\)
0.509285 + 0.860598i \(0.329910\pi\)
\(542\) −7.12393 −0.305999
\(543\) −9.67962 −0.415392
\(544\) −88.7852 −3.80664
\(545\) 57.3194 2.45529
\(546\) 0 0
\(547\) 45.3071 1.93719 0.968597 0.248637i \(-0.0799826\pi\)
0.968597 + 0.248637i \(0.0799826\pi\)
\(548\) 52.9551 2.26213
\(549\) −5.68932 −0.242814
\(550\) −134.366 −5.72939
\(551\) 28.6560 1.22079
\(552\) −72.0494 −3.06663
\(553\) 0 0
\(554\) 53.3355 2.26601
\(555\) 23.8866 1.01393
\(556\) −85.3452 −3.61944
\(557\) 20.3869 0.863820 0.431910 0.901917i \(-0.357840\pi\)
0.431910 + 0.901917i \(0.357840\pi\)
\(558\) −5.38302 −0.227881
\(559\) −1.95746 −0.0827916
\(560\) 0 0
\(561\) 11.5252 0.486593
\(562\) −15.4480 −0.651635
\(563\) −25.8965 −1.09141 −0.545703 0.837978i \(-0.683737\pi\)
−0.545703 + 0.837978i \(0.683737\pi\)
\(564\) −8.55981 −0.360433
\(565\) −44.6154 −1.87698
\(566\) 72.1246 3.03162
\(567\) 0 0
\(568\) 64.8524 2.72115
\(569\) 17.4999 0.733632 0.366816 0.930293i \(-0.380448\pi\)
0.366816 + 0.930293i \(0.380448\pi\)
\(570\) −92.9636 −3.89382
\(571\) 23.4334 0.980656 0.490328 0.871538i \(-0.336877\pi\)
0.490328 + 0.871538i \(0.336877\pi\)
\(572\) −17.7990 −0.744214
\(573\) 20.9409 0.874818
\(574\) 0 0
\(575\) −78.8950 −3.29015
\(576\) 49.3121 2.05467
\(577\) 27.6328 1.15037 0.575184 0.818024i \(-0.304930\pi\)
0.575184 + 0.818024i \(0.304930\pi\)
\(578\) −24.6240 −1.02422
\(579\) 6.85788 0.285004
\(580\) −85.9952 −3.57076
\(581\) 0 0
\(582\) −5.15432 −0.213653
\(583\) −28.6378 −1.18606
\(584\) −161.029 −6.66345
\(585\) 3.09982 0.128162
\(586\) −44.8889 −1.85435
\(587\) 3.95052 0.163056 0.0815278 0.996671i \(-0.474020\pi\)
0.0815278 + 0.996671i \(0.474020\pi\)
\(588\) 0 0
\(589\) 15.4476 0.636508
\(590\) 34.1019 1.40395
\(591\) 5.85575 0.240873
\(592\) 109.093 4.48371
\(593\) 30.9520 1.27105 0.635523 0.772082i \(-0.280785\pi\)
0.635523 + 0.772082i \(0.280785\pi\)
\(594\) −11.2777 −0.462729
\(595\) 0 0
\(596\) 29.4474 1.20621
\(597\) −9.49177 −0.388472
\(598\) −14.0076 −0.572814
\(599\) 0.0709266 0.00289798 0.00144899 0.999999i \(-0.499539\pi\)
0.00144899 + 0.999999i \(0.499539\pi\)
\(600\) 129.634 5.29229
\(601\) −24.8758 −1.01471 −0.507353 0.861738i \(-0.669376\pi\)
−0.507353 + 0.861738i \(0.669376\pi\)
\(602\) 0 0
\(603\) 4.99671 0.203482
\(604\) −80.2825 −3.26665
\(605\) 21.1678 0.860593
\(606\) −46.1788 −1.87589
\(607\) −14.5138 −0.589096 −0.294548 0.955637i \(-0.595169\pi\)
−0.294548 + 0.955637i \(0.595169\pi\)
\(608\) −249.315 −10.1111
\(609\) 0 0
\(610\) −65.6694 −2.65888
\(611\) −1.09782 −0.0444130
\(612\) −16.8556 −0.681347
\(613\) −22.0987 −0.892559 −0.446279 0.894894i \(-0.647251\pi\)
−0.446279 + 0.894894i \(0.647251\pi\)
\(614\) 56.5879 2.28370
\(615\) 4.11270 0.165840
\(616\) 0 0
\(617\) 13.7730 0.554482 0.277241 0.960800i \(-0.410580\pi\)
0.277241 + 0.960800i \(0.410580\pi\)
\(618\) 20.1898 0.812153
\(619\) 7.60818 0.305799 0.152899 0.988242i \(-0.451139\pi\)
0.152899 + 0.988242i \(0.451139\pi\)
\(620\) −46.3575 −1.86176
\(621\) −6.62187 −0.265726
\(622\) 66.4766 2.66547
\(623\) 0 0
\(624\) 14.1573 0.566746
\(625\) 57.3794 2.29518
\(626\) 72.9405 2.91529
\(627\) 32.3635 1.29247
\(628\) −13.8808 −0.553905
\(629\) −16.6582 −0.664207
\(630\) 0 0
\(631\) −19.7825 −0.787530 −0.393765 0.919211i \(-0.628828\pi\)
−0.393765 + 0.919211i \(0.628828\pi\)
\(632\) 96.4075 3.83489
\(633\) −19.9337 −0.792293
\(634\) 36.3108 1.44209
\(635\) −56.5934 −2.24584
\(636\) 41.8829 1.66077
\(637\) 0 0
\(638\) 40.1259 1.58860
\(639\) 5.96041 0.235790
\(640\) 314.567 12.4344
\(641\) −2.29745 −0.0907439 −0.0453719 0.998970i \(-0.514447\pi\)
−0.0453719 + 0.998970i \(0.514447\pi\)
\(642\) 24.0435 0.948921
\(643\) −32.2860 −1.27323 −0.636617 0.771180i \(-0.719667\pi\)
−0.636617 + 0.771180i \(0.719667\pi\)
\(644\) 0 0
\(645\) −10.6810 −0.420563
\(646\) 64.8318 2.55077
\(647\) −38.0027 −1.49404 −0.747019 0.664802i \(-0.768516\pi\)
−0.747019 + 0.664802i \(0.768516\pi\)
\(648\) 10.8805 0.427428
\(649\) −11.8719 −0.466014
\(650\) 25.2031 0.988545
\(651\) 0 0
\(652\) 3.49915 0.137037
\(653\) −5.33327 −0.208707 −0.104354 0.994540i \(-0.533277\pi\)
−0.104354 + 0.994540i \(0.533277\pi\)
\(654\) 39.1156 1.52954
\(655\) 32.0703 1.25309
\(656\) 18.7833 0.733365
\(657\) −14.7998 −0.577394
\(658\) 0 0
\(659\) 37.9072 1.47665 0.738327 0.674443i \(-0.235616\pi\)
0.738327 + 0.674443i \(0.235616\pi\)
\(660\) −97.1212 −3.78044
\(661\) −23.6532 −0.920004 −0.460002 0.887918i \(-0.652151\pi\)
−0.460002 + 0.887918i \(0.652151\pi\)
\(662\) −44.6097 −1.73380
\(663\) −2.16178 −0.0839565
\(664\) −13.2857 −0.515586
\(665\) 0 0
\(666\) 16.3005 0.631633
\(667\) 23.5605 0.912268
\(668\) 21.7175 0.840275
\(669\) 2.16735 0.0837946
\(670\) 57.6749 2.22818
\(671\) 22.8615 0.882560
\(672\) 0 0
\(673\) −37.8125 −1.45757 −0.728783 0.684745i \(-0.759914\pi\)
−0.728783 + 0.684745i \(0.759914\pi\)
\(674\) −21.7172 −0.836515
\(675\) 11.9143 0.458582
\(676\) −73.0600 −2.81000
\(677\) −46.9680 −1.80513 −0.902563 0.430557i \(-0.858317\pi\)
−0.902563 + 0.430557i \(0.858317\pi\)
\(678\) −30.4462 −1.16928
\(679\) 0 0
\(680\) −128.345 −4.92182
\(681\) −27.4094 −1.05033
\(682\) 21.6307 0.828283
\(683\) 6.20097 0.237274 0.118637 0.992938i \(-0.462148\pi\)
0.118637 + 0.992938i \(0.462148\pi\)
\(684\) −47.3317 −1.80977
\(685\) 37.0589 1.41595
\(686\) 0 0
\(687\) 12.3463 0.471041
\(688\) −48.7816 −1.85978
\(689\) 5.37160 0.204642
\(690\) −76.4334 −2.90977
\(691\) −2.33309 −0.0887548 −0.0443774 0.999015i \(-0.514130\pi\)
−0.0443774 + 0.999015i \(0.514130\pi\)
\(692\) −52.5180 −1.99643
\(693\) 0 0
\(694\) −19.4228 −0.737280
\(695\) −59.7262 −2.26554
\(696\) −38.7128 −1.46741
\(697\) −2.86815 −0.108639
\(698\) 95.2830 3.60651
\(699\) −11.9621 −0.452449
\(700\) 0 0
\(701\) 16.8503 0.636427 0.318213 0.948019i \(-0.396917\pi\)
0.318213 + 0.948019i \(0.396917\pi\)
\(702\) 2.11536 0.0798391
\(703\) −46.7775 −1.76425
\(704\) −198.152 −7.46814
\(705\) −5.99031 −0.225608
\(706\) 55.7320 2.09750
\(707\) 0 0
\(708\) 17.3627 0.652532
\(709\) −12.6657 −0.475670 −0.237835 0.971306i \(-0.576438\pi\)
−0.237835 + 0.971306i \(0.576438\pi\)
\(710\) 68.7984 2.58196
\(711\) 8.86055 0.332297
\(712\) 75.0538 2.81276
\(713\) 12.7008 0.475649
\(714\) 0 0
\(715\) −12.4561 −0.465831
\(716\) −45.2878 −1.69248
\(717\) −11.4914 −0.429156
\(718\) 16.7090 0.623574
\(719\) 21.7824 0.812348 0.406174 0.913796i \(-0.366863\pi\)
0.406174 + 0.913796i \(0.366863\pi\)
\(720\) 77.2502 2.87894
\(721\) 0 0
\(722\) 128.728 4.79074
\(723\) 24.4262 0.908420
\(724\) −56.8853 −2.11413
\(725\) −42.3910 −1.57436
\(726\) 14.4452 0.536112
\(727\) −3.60601 −0.133739 −0.0668697 0.997762i \(-0.521301\pi\)
−0.0668697 + 0.997762i \(0.521301\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −170.827 −6.32261
\(731\) 7.44879 0.275504
\(732\) −33.4351 −1.23580
\(733\) 44.4399 1.64143 0.820713 0.571341i \(-0.193576\pi\)
0.820713 + 0.571341i \(0.193576\pi\)
\(734\) −82.5513 −3.04702
\(735\) 0 0
\(736\) −204.983 −7.55579
\(737\) −20.0784 −0.739598
\(738\) 2.80657 0.103311
\(739\) −23.7944 −0.875291 −0.437646 0.899148i \(-0.644188\pi\)
−0.437646 + 0.899148i \(0.644188\pi\)
\(740\) 140.377 5.16036
\(741\) −6.07043 −0.223003
\(742\) 0 0
\(743\) −49.0075 −1.79791 −0.898956 0.438038i \(-0.855673\pi\)
−0.898956 + 0.438038i \(0.855673\pi\)
\(744\) −20.8690 −0.765094
\(745\) 20.6078 0.755012
\(746\) −4.10710 −0.150372
\(747\) −1.22105 −0.0446761
\(748\) 67.7313 2.47650
\(749\) 0 0
\(750\) 79.8090 2.91421
\(751\) 1.45799 0.0532030 0.0266015 0.999646i \(-0.491531\pi\)
0.0266015 + 0.999646i \(0.491531\pi\)
\(752\) −27.3586 −0.997667
\(753\) −17.3618 −0.632700
\(754\) −7.52643 −0.274096
\(755\) −56.1831 −2.04471
\(756\) 0 0
\(757\) 34.3969 1.25018 0.625089 0.780554i \(-0.285063\pi\)
0.625089 + 0.780554i \(0.285063\pi\)
\(758\) −81.0518 −2.94394
\(759\) 26.6088 0.965839
\(760\) −360.403 −13.0732
\(761\) −39.2505 −1.42283 −0.711415 0.702772i \(-0.751945\pi\)
−0.711415 + 0.702772i \(0.751945\pi\)
\(762\) −38.6201 −1.39906
\(763\) 0 0
\(764\) 123.066 4.45236
\(765\) −11.7959 −0.426480
\(766\) −26.1488 −0.944796
\(767\) 2.22682 0.0804058
\(768\) 116.041 4.18726
\(769\) −47.5049 −1.71307 −0.856535 0.516089i \(-0.827387\pi\)
−0.856535 + 0.516089i \(0.827387\pi\)
\(770\) 0 0
\(771\) 15.9782 0.575441
\(772\) 40.3025 1.45052
\(773\) 18.5442 0.666989 0.333494 0.942752i \(-0.391772\pi\)
0.333494 + 0.942752i \(0.391772\pi\)
\(774\) −7.28885 −0.261992
\(775\) −22.8518 −0.820860
\(776\) −19.9823 −0.717324
\(777\) 0 0
\(778\) 39.6234 1.42057
\(779\) −8.05398 −0.288564
\(780\) 18.2170 0.652275
\(781\) −23.9509 −0.857029
\(782\) 53.3038 1.90614
\(783\) −3.55799 −0.127152
\(784\) 0 0
\(785\) −9.71406 −0.346710
\(786\) 21.8853 0.780621
\(787\) 38.9277 1.38762 0.693812 0.720156i \(-0.255930\pi\)
0.693812 + 0.720156i \(0.255930\pi\)
\(788\) 34.4132 1.22592
\(789\) −3.35156 −0.119319
\(790\) 102.274 3.63873
\(791\) 0 0
\(792\) −43.7215 −1.55358
\(793\) −4.28814 −0.152276
\(794\) −92.5678 −3.28511
\(795\) 29.3104 1.03953
\(796\) −55.7814 −1.97712
\(797\) 4.22206 0.149553 0.0747766 0.997200i \(-0.476176\pi\)
0.0747766 + 0.997200i \(0.476176\pi\)
\(798\) 0 0
\(799\) 4.17758 0.147792
\(800\) 368.814 13.0395
\(801\) 6.89799 0.243729
\(802\) 56.6711 2.00112
\(803\) 59.4703 2.09866
\(804\) 29.3647 1.03561
\(805\) 0 0
\(806\) −4.05728 −0.142912
\(807\) 8.96079 0.315435
\(808\) −179.027 −6.29813
\(809\) −18.3897 −0.646548 −0.323274 0.946305i \(-0.604784\pi\)
−0.323274 + 0.946305i \(0.604784\pi\)
\(810\) 11.5426 0.405565
\(811\) 17.8646 0.627311 0.313656 0.949537i \(-0.398446\pi\)
0.313656 + 0.949537i \(0.398446\pi\)
\(812\) 0 0
\(813\) −2.53831 −0.0890224
\(814\) −65.5008 −2.29580
\(815\) 2.44877 0.0857768
\(816\) −53.8734 −1.88595
\(817\) 20.9167 0.731784
\(818\) 22.5948 0.790010
\(819\) 0 0
\(820\) 24.1696 0.844038
\(821\) 22.7936 0.795502 0.397751 0.917493i \(-0.369791\pi\)
0.397751 + 0.917493i \(0.369791\pi\)
\(822\) 25.2896 0.882074
\(823\) 26.6506 0.928983 0.464491 0.885578i \(-0.346237\pi\)
0.464491 + 0.885578i \(0.346237\pi\)
\(824\) 78.2721 2.72674
\(825\) −47.8756 −1.66681
\(826\) 0 0
\(827\) −40.0514 −1.39272 −0.696361 0.717692i \(-0.745199\pi\)
−0.696361 + 0.717692i \(0.745199\pi\)
\(828\) −38.9155 −1.35241
\(829\) 29.9196 1.03915 0.519576 0.854424i \(-0.326090\pi\)
0.519576 + 0.854424i \(0.326090\pi\)
\(830\) −14.0941 −0.489214
\(831\) 19.0038 0.659235
\(832\) 37.1674 1.28855
\(833\) 0 0
\(834\) −40.7580 −1.41133
\(835\) 15.1983 0.525959
\(836\) 190.194 6.57800
\(837\) −1.91801 −0.0662961
\(838\) 41.7981 1.44389
\(839\) 46.0646 1.59033 0.795163 0.606396i \(-0.207385\pi\)
0.795163 + 0.606396i \(0.207385\pi\)
\(840\) 0 0
\(841\) −16.3407 −0.563472
\(842\) 6.53824 0.225323
\(843\) −5.50424 −0.189576
\(844\) −117.147 −4.03235
\(845\) −51.1287 −1.75888
\(846\) −4.08787 −0.140544
\(847\) 0 0
\(848\) 133.865 4.59695
\(849\) 25.6985 0.881970
\(850\) −95.9063 −3.28956
\(851\) −38.4598 −1.31838
\(852\) 35.0282 1.20005
\(853\) −6.91770 −0.236858 −0.118429 0.992963i \(-0.537786\pi\)
−0.118429 + 0.992963i \(0.537786\pi\)
\(854\) 0 0
\(855\) −33.1236 −1.13280
\(856\) 93.2121 3.18592
\(857\) −21.5052 −0.734603 −0.367302 0.930102i \(-0.619718\pi\)
−0.367302 + 0.930102i \(0.619718\pi\)
\(858\) −8.50020 −0.290192
\(859\) 18.6568 0.636562 0.318281 0.947996i \(-0.396894\pi\)
0.318281 + 0.947996i \(0.396894\pi\)
\(860\) −62.7701 −2.14044
\(861\) 0 0
\(862\) 33.6144 1.14491
\(863\) 31.7858 1.08200 0.541001 0.841022i \(-0.318046\pi\)
0.541001 + 0.841022i \(0.318046\pi\)
\(864\) 30.9555 1.05313
\(865\) −36.7530 −1.24964
\(866\) 80.3406 2.73009
\(867\) −8.77369 −0.297970
\(868\) 0 0
\(869\) −35.6046 −1.20780
\(870\) −41.0684 −1.39235
\(871\) 3.76611 0.127610
\(872\) 151.644 5.13531
\(873\) −1.83652 −0.0621569
\(874\) 149.681 5.06303
\(875\) 0 0
\(876\) −86.9755 −2.93863
\(877\) −19.4115 −0.655480 −0.327740 0.944768i \(-0.606287\pi\)
−0.327740 + 0.944768i \(0.606287\pi\)
\(878\) 39.8752 1.34572
\(879\) −15.9943 −0.539473
\(880\) −310.416 −10.4641
\(881\) 28.9178 0.974265 0.487133 0.873328i \(-0.338043\pi\)
0.487133 + 0.873328i \(0.338043\pi\)
\(882\) 0 0
\(883\) 43.0790 1.44972 0.724862 0.688894i \(-0.241903\pi\)
0.724862 + 0.688894i \(0.241903\pi\)
\(884\) −12.7044 −0.427294
\(885\) 12.1508 0.408444
\(886\) −26.0815 −0.876225
\(887\) −38.8059 −1.30297 −0.651487 0.758660i \(-0.725854\pi\)
−0.651487 + 0.758660i \(0.725854\pi\)
\(888\) 63.1941 2.12066
\(889\) 0 0
\(890\) 79.6206 2.66889
\(891\) −4.01832 −0.134619
\(892\) 12.7371 0.426470
\(893\) 11.7309 0.392561
\(894\) 14.0631 0.470340
\(895\) −31.6932 −1.05939
\(896\) 0 0
\(897\) −4.99102 −0.166645
\(898\) −104.770 −3.49622
\(899\) 6.82426 0.227602
\(900\) 70.0182 2.33394
\(901\) −20.4408 −0.680981
\(902\) −11.2777 −0.375506
\(903\) 0 0
\(904\) −118.034 −3.92576
\(905\) −39.8094 −1.32331
\(906\) −38.3402 −1.27377
\(907\) 10.6255 0.352815 0.176407 0.984317i \(-0.443552\pi\)
0.176407 + 0.984317i \(0.443552\pi\)
\(908\) −161.080 −5.34562
\(909\) −16.4538 −0.545740
\(910\) 0 0
\(911\) 15.7405 0.521505 0.260753 0.965406i \(-0.416029\pi\)
0.260753 + 0.965406i \(0.416029\pi\)
\(912\) −151.280 −5.00939
\(913\) 4.90659 0.162385
\(914\) 35.2051 1.16448
\(915\) −23.3985 −0.773530
\(916\) 72.5569 2.39735
\(917\) 0 0
\(918\) −8.04966 −0.265678
\(919\) 55.3915 1.82720 0.913599 0.406617i \(-0.133292\pi\)
0.913599 + 0.406617i \(0.133292\pi\)
\(920\) −296.318 −9.76931
\(921\) 20.1627 0.664384
\(922\) −46.9975 −1.54778
\(923\) 4.49247 0.147871
\(924\) 0 0
\(925\) 69.1983 2.27523
\(926\) −46.0279 −1.51257
\(927\) 7.19377 0.236274
\(928\) −110.140 −3.61551
\(929\) 59.4411 1.95020 0.975099 0.221769i \(-0.0711831\pi\)
0.975099 + 0.221769i \(0.0711831\pi\)
\(930\) −22.1388 −0.725959
\(931\) 0 0
\(932\) −70.2991 −2.30272
\(933\) 23.6861 0.775448
\(934\) 46.2963 1.51486
\(935\) 47.3996 1.55013
\(936\) 8.20085 0.268053
\(937\) 12.7418 0.416258 0.208129 0.978101i \(-0.433263\pi\)
0.208129 + 0.978101i \(0.433263\pi\)
\(938\) 0 0
\(939\) 25.9892 0.848127
\(940\) −35.2040 −1.14823
\(941\) −53.0284 −1.72868 −0.864338 0.502912i \(-0.832262\pi\)
−0.864338 + 0.502912i \(0.832262\pi\)
\(942\) −6.62901 −0.215985
\(943\) −6.62187 −0.215638
\(944\) 55.4943 1.80619
\(945\) 0 0
\(946\) 29.2890 0.952266
\(947\) 33.9701 1.10388 0.551940 0.833884i \(-0.313888\pi\)
0.551940 + 0.833884i \(0.313888\pi\)
\(948\) 52.0718 1.69121
\(949\) −11.1549 −0.362102
\(950\) −269.312 −8.73762
\(951\) 12.9378 0.419537
\(952\) 0 0
\(953\) 20.9729 0.679378 0.339689 0.940538i \(-0.389678\pi\)
0.339689 + 0.940538i \(0.389678\pi\)
\(954\) 20.0019 0.647584
\(955\) 86.1236 2.78689
\(956\) −67.5331 −2.18418
\(957\) 14.2972 0.462162
\(958\) 37.4804 1.21094
\(959\) 0 0
\(960\) 202.806 6.54554
\(961\) −27.3212 −0.881330
\(962\) 12.2860 0.396116
\(963\) 8.56687 0.276064
\(964\) 143.548 4.62338
\(965\) 28.2044 0.907933
\(966\) 0 0
\(967\) −31.4462 −1.01124 −0.505621 0.862756i \(-0.668737\pi\)
−0.505621 + 0.862756i \(0.668737\pi\)
\(968\) 56.0013 1.79995
\(969\) 23.1000 0.742080
\(970\) −21.1982 −0.680633
\(971\) 38.8183 1.24574 0.622869 0.782326i \(-0.285967\pi\)
0.622869 + 0.782326i \(0.285967\pi\)
\(972\) 5.87681 0.188499
\(973\) 0 0
\(974\) 9.38073 0.300578
\(975\) 8.98003 0.287591
\(976\) −106.864 −3.42064
\(977\) 34.6293 1.10789 0.553945 0.832554i \(-0.313122\pi\)
0.553945 + 0.832554i \(0.313122\pi\)
\(978\) 1.67108 0.0534352
\(979\) −27.7184 −0.885883
\(980\) 0 0
\(981\) 13.9372 0.444979
\(982\) 59.2477 1.89067
\(983\) 52.6484 1.67922 0.839612 0.543187i \(-0.182783\pi\)
0.839612 + 0.543187i \(0.182783\pi\)
\(984\) 10.8805 0.346859
\(985\) 24.0830 0.767347
\(986\) 28.6406 0.912104
\(987\) 0 0
\(988\) −35.6748 −1.13497
\(989\) 17.1974 0.546847
\(990\) −46.3818 −1.47411
\(991\) 17.1185 0.543788 0.271894 0.962327i \(-0.412350\pi\)
0.271894 + 0.962327i \(0.412350\pi\)
\(992\) −59.3730 −1.88510
\(993\) −15.8948 −0.504405
\(994\) 0 0
\(995\) −39.0368 −1.23755
\(996\) −7.17591 −0.227377
\(997\) −23.0058 −0.728601 −0.364301 0.931281i \(-0.618692\pi\)
−0.364301 + 0.931281i \(0.618692\pi\)
\(998\) −0.305923 −0.00968383
\(999\) 5.80800 0.183757
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.be.1.10 yes 10
7.6 odd 2 6027.2.a.bd.1.10 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6027.2.a.bd.1.10 10 7.6 odd 2
6027.2.a.be.1.10 yes 10 1.1 even 1 trivial