Properties

Label 5390.2.a.bk.1.1
Level $5390$
Weight $2$
Character 5390.1
Self dual yes
Analytic conductor $43.039$
Analytic rank $1$
Dimension $2$
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] = [5390,2,Mod(1,5390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5390, 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("5390.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.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: \(43.0393666895\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 5390.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.73205 q^{6} -1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.73205 q^{6} -1.00000 q^{8} +4.46410 q^{9} +1.00000 q^{10} +1.00000 q^{11} -2.73205 q^{12} +1.46410 q^{13} +2.73205 q^{15} +1.00000 q^{16} -3.46410 q^{17} -4.46410 q^{18} +2.73205 q^{19} -1.00000 q^{20} -1.00000 q^{22} +1.26795 q^{23} +2.73205 q^{24} +1.00000 q^{25} -1.46410 q^{26} -4.00000 q^{27} -4.73205 q^{29} -2.73205 q^{30} -8.92820 q^{31} -1.00000 q^{32} -2.73205 q^{33} +3.46410 q^{34} +4.46410 q^{36} +3.26795 q^{37} -2.73205 q^{38} -4.00000 q^{39} +1.00000 q^{40} +4.73205 q^{41} -4.92820 q^{43} +1.00000 q^{44} -4.46410 q^{45} -1.26795 q^{46} -2.73205 q^{48} -1.00000 q^{50} +9.46410 q^{51} +1.46410 q^{52} -4.73205 q^{53} +4.00000 q^{54} -1.00000 q^{55} -7.46410 q^{57} +4.73205 q^{58} +13.8564 q^{59} +2.73205 q^{60} -2.00000 q^{61} +8.92820 q^{62} +1.00000 q^{64} -1.46410 q^{65} +2.73205 q^{66} -10.9282 q^{67} -3.46410 q^{68} -3.46410 q^{69} +9.46410 q^{71} -4.46410 q^{72} -11.4641 q^{73} -3.26795 q^{74} -2.73205 q^{75} +2.73205 q^{76} +4.00000 q^{78} +6.73205 q^{79} -1.00000 q^{80} -2.46410 q^{81} -4.73205 q^{82} +4.39230 q^{83} +3.46410 q^{85} +4.92820 q^{86} +12.9282 q^{87} -1.00000 q^{88} +15.4641 q^{89} +4.46410 q^{90} +1.26795 q^{92} +24.3923 q^{93} -2.73205 q^{95} +2.73205 q^{96} -5.80385 q^{97} +4.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{15} + 2 q^{16} - 2 q^{18} + 2 q^{19} - 2 q^{20} - 2 q^{22} + 6 q^{23} + 2 q^{24} + 2 q^{25} + 4 q^{26} - 8 q^{27} - 6 q^{29} - 2 q^{30} - 4 q^{31} - 2 q^{32} - 2 q^{33} + 2 q^{36} + 10 q^{37} - 2 q^{38} - 8 q^{39} + 2 q^{40} + 6 q^{41} + 4 q^{43} + 2 q^{44} - 2 q^{45} - 6 q^{46} - 2 q^{48} - 2 q^{50} + 12 q^{51} - 4 q^{52} - 6 q^{53} + 8 q^{54} - 2 q^{55} - 8 q^{57} + 6 q^{58} + 2 q^{60} - 4 q^{61} + 4 q^{62} + 2 q^{64} + 4 q^{65} + 2 q^{66} - 8 q^{67} + 12 q^{71} - 2 q^{72} - 16 q^{73} - 10 q^{74} - 2 q^{75} + 2 q^{76} + 8 q^{78} + 10 q^{79} - 2 q^{80} + 2 q^{81} - 6 q^{82} - 12 q^{83} - 4 q^{86} + 12 q^{87} - 2 q^{88} + 24 q^{89} + 2 q^{90} + 6 q^{92} + 28 q^{93} - 2 q^{95} + 2 q^{96} - 22 q^{97} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.73205 −1.57735 −0.788675 0.614810i \(-0.789233\pi\)
−0.788675 + 0.614810i \(0.789233\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.73205 1.11536
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 4.46410 1.48803
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) −2.73205 −0.788675
\(13\) 1.46410 0.406069 0.203034 0.979172i \(-0.434920\pi\)
0.203034 + 0.979172i \(0.434920\pi\)
\(14\) 0 0
\(15\) 2.73205 0.705412
\(16\) 1.00000 0.250000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) −4.46410 −1.05220
\(19\) 2.73205 0.626775 0.313388 0.949625i \(-0.398536\pi\)
0.313388 + 0.949625i \(0.398536\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 1.26795 0.264386 0.132193 0.991224i \(-0.457798\pi\)
0.132193 + 0.991224i \(0.457798\pi\)
\(24\) 2.73205 0.557678
\(25\) 1.00000 0.200000
\(26\) −1.46410 −0.287134
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −4.73205 −0.878720 −0.439360 0.898311i \(-0.644795\pi\)
−0.439360 + 0.898311i \(0.644795\pi\)
\(30\) −2.73205 −0.498802
\(31\) −8.92820 −1.60355 −0.801776 0.597624i \(-0.796111\pi\)
−0.801776 + 0.597624i \(0.796111\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.73205 −0.475589
\(34\) 3.46410 0.594089
\(35\) 0 0
\(36\) 4.46410 0.744017
\(37\) 3.26795 0.537248 0.268624 0.963245i \(-0.413431\pi\)
0.268624 + 0.963245i \(0.413431\pi\)
\(38\) −2.73205 −0.443197
\(39\) −4.00000 −0.640513
\(40\) 1.00000 0.158114
\(41\) 4.73205 0.739022 0.369511 0.929226i \(-0.379525\pi\)
0.369511 + 0.929226i \(0.379525\pi\)
\(42\) 0 0
\(43\) −4.92820 −0.751544 −0.375772 0.926712i \(-0.622622\pi\)
−0.375772 + 0.926712i \(0.622622\pi\)
\(44\) 1.00000 0.150756
\(45\) −4.46410 −0.665469
\(46\) −1.26795 −0.186949
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.73205 −0.394338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 9.46410 1.32524
\(52\) 1.46410 0.203034
\(53\) −4.73205 −0.649997 −0.324999 0.945715i \(-0.605364\pi\)
−0.324999 + 0.945715i \(0.605364\pi\)
\(54\) 4.00000 0.544331
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −7.46410 −0.988644
\(58\) 4.73205 0.621349
\(59\) 13.8564 1.80395 0.901975 0.431788i \(-0.142117\pi\)
0.901975 + 0.431788i \(0.142117\pi\)
\(60\) 2.73205 0.352706
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.92820 1.13388
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.46410 −0.181599
\(66\) 2.73205 0.336292
\(67\) −10.9282 −1.33509 −0.667546 0.744568i \(-0.732655\pi\)
−0.667546 + 0.744568i \(0.732655\pi\)
\(68\) −3.46410 −0.420084
\(69\) −3.46410 −0.417029
\(70\) 0 0
\(71\) 9.46410 1.12318 0.561591 0.827415i \(-0.310189\pi\)
0.561591 + 0.827415i \(0.310189\pi\)
\(72\) −4.46410 −0.526099
\(73\) −11.4641 −1.34177 −0.670886 0.741561i \(-0.734086\pi\)
−0.670886 + 0.741561i \(0.734086\pi\)
\(74\) −3.26795 −0.379891
\(75\) −2.73205 −0.315470
\(76\) 2.73205 0.313388
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 6.73205 0.757415 0.378707 0.925516i \(-0.376369\pi\)
0.378707 + 0.925516i \(0.376369\pi\)
\(80\) −1.00000 −0.111803
\(81\) −2.46410 −0.273789
\(82\) −4.73205 −0.522568
\(83\) 4.39230 0.482118 0.241059 0.970510i \(-0.422505\pi\)
0.241059 + 0.970510i \(0.422505\pi\)
\(84\) 0 0
\(85\) 3.46410 0.375735
\(86\) 4.92820 0.531422
\(87\) 12.9282 1.38605
\(88\) −1.00000 −0.106600
\(89\) 15.4641 1.63919 0.819596 0.572942i \(-0.194198\pi\)
0.819596 + 0.572942i \(0.194198\pi\)
\(90\) 4.46410 0.470558
\(91\) 0 0
\(92\) 1.26795 0.132193
\(93\) 24.3923 2.52936
\(94\) 0 0
\(95\) −2.73205 −0.280302
\(96\) 2.73205 0.278839
\(97\) −5.80385 −0.589291 −0.294646 0.955607i \(-0.595202\pi\)
−0.294646 + 0.955607i \(0.595202\pi\)
\(98\) 0 0
\(99\) 4.46410 0.448659
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −9.46410 −0.937086
\(103\) −10.5359 −1.03813 −0.519066 0.854734i \(-0.673720\pi\)
−0.519066 + 0.854734i \(0.673720\pi\)
\(104\) −1.46410 −0.143567
\(105\) 0 0
\(106\) 4.73205 0.459617
\(107\) 0.928203 0.0897328 0.0448664 0.998993i \(-0.485714\pi\)
0.0448664 + 0.998993i \(0.485714\pi\)
\(108\) −4.00000 −0.384900
\(109\) −1.80385 −0.172777 −0.0863886 0.996262i \(-0.527533\pi\)
−0.0863886 + 0.996262i \(0.527533\pi\)
\(110\) 1.00000 0.0953463
\(111\) −8.92820 −0.847428
\(112\) 0 0
\(113\) 12.9282 1.21618 0.608092 0.793867i \(-0.291935\pi\)
0.608092 + 0.793867i \(0.291935\pi\)
\(114\) 7.46410 0.699077
\(115\) −1.26795 −0.118237
\(116\) −4.73205 −0.439360
\(117\) 6.53590 0.604244
\(118\) −13.8564 −1.27559
\(119\) 0 0
\(120\) −2.73205 −0.249401
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) −12.9282 −1.16570
\(124\) −8.92820 −0.801776
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 9.85641 0.874615 0.437307 0.899312i \(-0.355932\pi\)
0.437307 + 0.899312i \(0.355932\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 13.4641 1.18545
\(130\) 1.46410 0.128410
\(131\) 17.6603 1.54298 0.771492 0.636239i \(-0.219511\pi\)
0.771492 + 0.636239i \(0.219511\pi\)
\(132\) −2.73205 −0.237795
\(133\) 0 0
\(134\) 10.9282 0.944053
\(135\) 4.00000 0.344265
\(136\) 3.46410 0.297044
\(137\) 7.85641 0.671218 0.335609 0.942001i \(-0.391058\pi\)
0.335609 + 0.942001i \(0.391058\pi\)
\(138\) 3.46410 0.294884
\(139\) −4.19615 −0.355913 −0.177957 0.984038i \(-0.556949\pi\)
−0.177957 + 0.984038i \(0.556949\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −9.46410 −0.794210
\(143\) 1.46410 0.122434
\(144\) 4.46410 0.372008
\(145\) 4.73205 0.392975
\(146\) 11.4641 0.948776
\(147\) 0 0
\(148\) 3.26795 0.268624
\(149\) −7.26795 −0.595414 −0.297707 0.954657i \(-0.596222\pi\)
−0.297707 + 0.954657i \(0.596222\pi\)
\(150\) 2.73205 0.223071
\(151\) −7.80385 −0.635068 −0.317534 0.948247i \(-0.602855\pi\)
−0.317534 + 0.948247i \(0.602855\pi\)
\(152\) −2.73205 −0.221599
\(153\) −15.4641 −1.25020
\(154\) 0 0
\(155\) 8.92820 0.717131
\(156\) −4.00000 −0.320256
\(157\) −6.39230 −0.510161 −0.255081 0.966920i \(-0.582102\pi\)
−0.255081 + 0.966920i \(0.582102\pi\)
\(158\) −6.73205 −0.535573
\(159\) 12.9282 1.02527
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 2.46410 0.193598
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 4.73205 0.369511
\(165\) 2.73205 0.212690
\(166\) −4.39230 −0.340909
\(167\) 13.8564 1.07224 0.536120 0.844141i \(-0.319889\pi\)
0.536120 + 0.844141i \(0.319889\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) −3.46410 −0.265684
\(171\) 12.1962 0.932663
\(172\) −4.92820 −0.375772
\(173\) −0.928203 −0.0705700 −0.0352850 0.999377i \(-0.511234\pi\)
−0.0352850 + 0.999377i \(0.511234\pi\)
\(174\) −12.9282 −0.980085
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −37.8564 −2.84546
\(178\) −15.4641 −1.15908
\(179\) −19.8564 −1.48414 −0.742069 0.670324i \(-0.766155\pi\)
−0.742069 + 0.670324i \(0.766155\pi\)
\(180\) −4.46410 −0.332734
\(181\) 11.8564 0.881280 0.440640 0.897684i \(-0.354752\pi\)
0.440640 + 0.897684i \(0.354752\pi\)
\(182\) 0 0
\(183\) 5.46410 0.403918
\(184\) −1.26795 −0.0934745
\(185\) −3.26795 −0.240264
\(186\) −24.3923 −1.78853
\(187\) −3.46410 −0.253320
\(188\) 0 0
\(189\) 0 0
\(190\) 2.73205 0.198204
\(191\) 6.92820 0.501307 0.250654 0.968077i \(-0.419354\pi\)
0.250654 + 0.968077i \(0.419354\pi\)
\(192\) −2.73205 −0.197169
\(193\) −25.4641 −1.83295 −0.916473 0.400096i \(-0.868977\pi\)
−0.916473 + 0.400096i \(0.868977\pi\)
\(194\) 5.80385 0.416692
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) −22.3923 −1.59539 −0.797693 0.603064i \(-0.793946\pi\)
−0.797693 + 0.603064i \(0.793946\pi\)
\(198\) −4.46410 −0.317250
\(199\) 24.7846 1.75693 0.878467 0.477803i \(-0.158567\pi\)
0.878467 + 0.477803i \(0.158567\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 29.8564 2.10591
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 9.46410 0.662620
\(205\) −4.73205 −0.330501
\(206\) 10.5359 0.734071
\(207\) 5.66025 0.393415
\(208\) 1.46410 0.101517
\(209\) 2.73205 0.188980
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −4.73205 −0.324999
\(213\) −25.8564 −1.77165
\(214\) −0.928203 −0.0634507
\(215\) 4.92820 0.336101
\(216\) 4.00000 0.272166
\(217\) 0 0
\(218\) 1.80385 0.122172
\(219\) 31.3205 2.11644
\(220\) −1.00000 −0.0674200
\(221\) −5.07180 −0.341166
\(222\) 8.92820 0.599222
\(223\) 8.39230 0.561990 0.280995 0.959709i \(-0.409336\pi\)
0.280995 + 0.959709i \(0.409336\pi\)
\(224\) 0 0
\(225\) 4.46410 0.297607
\(226\) −12.9282 −0.859971
\(227\) −13.8564 −0.919682 −0.459841 0.888001i \(-0.652094\pi\)
−0.459841 + 0.888001i \(0.652094\pi\)
\(228\) −7.46410 −0.494322
\(229\) 13.4641 0.889733 0.444866 0.895597i \(-0.353251\pi\)
0.444866 + 0.895597i \(0.353251\pi\)
\(230\) 1.26795 0.0836061
\(231\) 0 0
\(232\) 4.73205 0.310674
\(233\) 7.85641 0.514690 0.257345 0.966320i \(-0.417152\pi\)
0.257345 + 0.966320i \(0.417152\pi\)
\(234\) −6.53590 −0.427265
\(235\) 0 0
\(236\) 13.8564 0.901975
\(237\) −18.3923 −1.19471
\(238\) 0 0
\(239\) −3.80385 −0.246050 −0.123025 0.992404i \(-0.539260\pi\)
−0.123025 + 0.992404i \(0.539260\pi\)
\(240\) 2.73205 0.176353
\(241\) 18.1962 1.17212 0.586059 0.810269i \(-0.300679\pi\)
0.586059 + 0.810269i \(0.300679\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 18.7321 1.20166
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 12.9282 0.824272
\(247\) 4.00000 0.254514
\(248\) 8.92820 0.566941
\(249\) −12.0000 −0.760469
\(250\) 1.00000 0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 1.26795 0.0797153
\(254\) −9.85641 −0.618446
\(255\) −9.46410 −0.592665
\(256\) 1.00000 0.0625000
\(257\) −22.9808 −1.43350 −0.716750 0.697330i \(-0.754371\pi\)
−0.716750 + 0.697330i \(0.754371\pi\)
\(258\) −13.4641 −0.838238
\(259\) 0 0
\(260\) −1.46410 −0.0907997
\(261\) −21.1244 −1.30756
\(262\) −17.6603 −1.09105
\(263\) 18.9282 1.16716 0.583582 0.812055i \(-0.301651\pi\)
0.583582 + 0.812055i \(0.301651\pi\)
\(264\) 2.73205 0.168146
\(265\) 4.73205 0.290688
\(266\) 0 0
\(267\) −42.2487 −2.58558
\(268\) −10.9282 −0.667546
\(269\) −4.39230 −0.267804 −0.133902 0.990995i \(-0.542751\pi\)
−0.133902 + 0.990995i \(0.542751\pi\)
\(270\) −4.00000 −0.243432
\(271\) −24.3923 −1.48173 −0.740863 0.671656i \(-0.765583\pi\)
−0.740863 + 0.671656i \(0.765583\pi\)
\(272\) −3.46410 −0.210042
\(273\) 0 0
\(274\) −7.85641 −0.474623
\(275\) 1.00000 0.0603023
\(276\) −3.46410 −0.208514
\(277\) 27.8564 1.67373 0.836865 0.547410i \(-0.184386\pi\)
0.836865 + 0.547410i \(0.184386\pi\)
\(278\) 4.19615 0.251668
\(279\) −39.8564 −2.38614
\(280\) 0 0
\(281\) −10.3923 −0.619953 −0.309976 0.950744i \(-0.600321\pi\)
−0.309976 + 0.950744i \(0.600321\pi\)
\(282\) 0 0
\(283\) 10.9282 0.649614 0.324807 0.945780i \(-0.394701\pi\)
0.324807 + 0.945780i \(0.394701\pi\)
\(284\) 9.46410 0.561591
\(285\) 7.46410 0.442135
\(286\) −1.46410 −0.0865741
\(287\) 0 0
\(288\) −4.46410 −0.263050
\(289\) −5.00000 −0.294118
\(290\) −4.73205 −0.277876
\(291\) 15.8564 0.929519
\(292\) −11.4641 −0.670886
\(293\) 2.53590 0.148149 0.0740744 0.997253i \(-0.476400\pi\)
0.0740744 + 0.997253i \(0.476400\pi\)
\(294\) 0 0
\(295\) −13.8564 −0.806751
\(296\) −3.26795 −0.189946
\(297\) −4.00000 −0.232104
\(298\) 7.26795 0.421021
\(299\) 1.85641 0.107359
\(300\) −2.73205 −0.157735
\(301\) 0 0
\(302\) 7.80385 0.449061
\(303\) −16.3923 −0.941713
\(304\) 2.73205 0.156694
\(305\) 2.00000 0.114520
\(306\) 15.4641 0.884024
\(307\) −31.3205 −1.78756 −0.893778 0.448510i \(-0.851955\pi\)
−0.893778 + 0.448510i \(0.851955\pi\)
\(308\) 0 0
\(309\) 28.7846 1.63750
\(310\) −8.92820 −0.507088
\(311\) −7.85641 −0.445496 −0.222748 0.974876i \(-0.571503\pi\)
−0.222748 + 0.974876i \(0.571503\pi\)
\(312\) 4.00000 0.226455
\(313\) −27.2679 −1.54128 −0.770638 0.637273i \(-0.780062\pi\)
−0.770638 + 0.637273i \(0.780062\pi\)
\(314\) 6.39230 0.360739
\(315\) 0 0
\(316\) 6.73205 0.378707
\(317\) −32.4449 −1.82229 −0.911143 0.412091i \(-0.864798\pi\)
−0.911143 + 0.412091i \(0.864798\pi\)
\(318\) −12.9282 −0.724978
\(319\) −4.73205 −0.264944
\(320\) −1.00000 −0.0559017
\(321\) −2.53590 −0.141540
\(322\) 0 0
\(323\) −9.46410 −0.526597
\(324\) −2.46410 −0.136895
\(325\) 1.46410 0.0812137
\(326\) −8.00000 −0.443079
\(327\) 4.92820 0.272530
\(328\) −4.73205 −0.261284
\(329\) 0 0
\(330\) −2.73205 −0.150394
\(331\) −4.92820 −0.270879 −0.135439 0.990786i \(-0.543245\pi\)
−0.135439 + 0.990786i \(0.543245\pi\)
\(332\) 4.39230 0.241059
\(333\) 14.5885 0.799443
\(334\) −13.8564 −0.758189
\(335\) 10.9282 0.597072
\(336\) 0 0
\(337\) −30.7846 −1.67694 −0.838472 0.544944i \(-0.816551\pi\)
−0.838472 + 0.544944i \(0.816551\pi\)
\(338\) 10.8564 0.590511
\(339\) −35.3205 −1.91835
\(340\) 3.46410 0.187867
\(341\) −8.92820 −0.483489
\(342\) −12.1962 −0.659492
\(343\) 0 0
\(344\) 4.92820 0.265711
\(345\) 3.46410 0.186501
\(346\) 0.928203 0.0499005
\(347\) 7.85641 0.421754 0.210877 0.977513i \(-0.432368\pi\)
0.210877 + 0.977513i \(0.432368\pi\)
\(348\) 12.9282 0.693024
\(349\) −30.3923 −1.62686 −0.813431 0.581661i \(-0.802403\pi\)
−0.813431 + 0.581661i \(0.802403\pi\)
\(350\) 0 0
\(351\) −5.85641 −0.312592
\(352\) −1.00000 −0.0533002
\(353\) 32.4449 1.72687 0.863433 0.504464i \(-0.168310\pi\)
0.863433 + 0.504464i \(0.168310\pi\)
\(354\) 37.8564 2.01205
\(355\) −9.46410 −0.502302
\(356\) 15.4641 0.819596
\(357\) 0 0
\(358\) 19.8564 1.04944
\(359\) −1.26795 −0.0669198 −0.0334599 0.999440i \(-0.510653\pi\)
−0.0334599 + 0.999440i \(0.510653\pi\)
\(360\) 4.46410 0.235279
\(361\) −11.5359 −0.607153
\(362\) −11.8564 −0.623159
\(363\) −2.73205 −0.143395
\(364\) 0 0
\(365\) 11.4641 0.600059
\(366\) −5.46410 −0.285613
\(367\) −29.4641 −1.53801 −0.769007 0.639241i \(-0.779249\pi\)
−0.769007 + 0.639241i \(0.779249\pi\)
\(368\) 1.26795 0.0660964
\(369\) 21.1244 1.09969
\(370\) 3.26795 0.169893
\(371\) 0 0
\(372\) 24.3923 1.26468
\(373\) 30.3923 1.57365 0.786827 0.617174i \(-0.211722\pi\)
0.786827 + 0.617174i \(0.211722\pi\)
\(374\) 3.46410 0.179124
\(375\) 2.73205 0.141082
\(376\) 0 0
\(377\) −6.92820 −0.356821
\(378\) 0 0
\(379\) 23.7128 1.21805 0.609023 0.793153i \(-0.291562\pi\)
0.609023 + 0.793153i \(0.291562\pi\)
\(380\) −2.73205 −0.140151
\(381\) −26.9282 −1.37957
\(382\) −6.92820 −0.354478
\(383\) −16.3923 −0.837608 −0.418804 0.908077i \(-0.637551\pi\)
−0.418804 + 0.908077i \(0.637551\pi\)
\(384\) 2.73205 0.139419
\(385\) 0 0
\(386\) 25.4641 1.29609
\(387\) −22.0000 −1.11832
\(388\) −5.80385 −0.294646
\(389\) −24.2487 −1.22946 −0.614729 0.788738i \(-0.710735\pi\)
−0.614729 + 0.788738i \(0.710735\pi\)
\(390\) −4.00000 −0.202548
\(391\) −4.39230 −0.222128
\(392\) 0 0
\(393\) −48.2487 −2.43383
\(394\) 22.3923 1.12811
\(395\) −6.73205 −0.338726
\(396\) 4.46410 0.224330
\(397\) 38.3923 1.92685 0.963427 0.267970i \(-0.0863528\pi\)
0.963427 + 0.267970i \(0.0863528\pi\)
\(398\) −24.7846 −1.24234
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −25.1769 −1.25728 −0.628638 0.777698i \(-0.716387\pi\)
−0.628638 + 0.777698i \(0.716387\pi\)
\(402\) −29.8564 −1.48910
\(403\) −13.0718 −0.651153
\(404\) 6.00000 0.298511
\(405\) 2.46410 0.122442
\(406\) 0 0
\(407\) 3.26795 0.161986
\(408\) −9.46410 −0.468543
\(409\) −22.1962 −1.09753 −0.548765 0.835977i \(-0.684902\pi\)
−0.548765 + 0.835977i \(0.684902\pi\)
\(410\) 4.73205 0.233699
\(411\) −21.4641 −1.05875
\(412\) −10.5359 −0.519066
\(413\) 0 0
\(414\) −5.66025 −0.278186
\(415\) −4.39230 −0.215610
\(416\) −1.46410 −0.0717835
\(417\) 11.4641 0.561399
\(418\) −2.73205 −0.133629
\(419\) −32.7846 −1.60163 −0.800816 0.598910i \(-0.795601\pi\)
−0.800816 + 0.598910i \(0.795601\pi\)
\(420\) 0 0
\(421\) 10.7846 0.525610 0.262805 0.964849i \(-0.415352\pi\)
0.262805 + 0.964849i \(0.415352\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) 4.73205 0.229809
\(425\) −3.46410 −0.168034
\(426\) 25.8564 1.25275
\(427\) 0 0
\(428\) 0.928203 0.0448664
\(429\) −4.00000 −0.193122
\(430\) −4.92820 −0.237659
\(431\) 18.3397 0.883394 0.441697 0.897164i \(-0.354377\pi\)
0.441697 + 0.897164i \(0.354377\pi\)
\(432\) −4.00000 −0.192450
\(433\) −0.732051 −0.0351801 −0.0175901 0.999845i \(-0.505599\pi\)
−0.0175901 + 0.999845i \(0.505599\pi\)
\(434\) 0 0
\(435\) −12.9282 −0.619860
\(436\) −1.80385 −0.0863886
\(437\) 3.46410 0.165710
\(438\) −31.3205 −1.49655
\(439\) −36.3923 −1.73691 −0.868455 0.495768i \(-0.834887\pi\)
−0.868455 + 0.495768i \(0.834887\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 5.07180 0.241241
\(443\) 20.7846 0.987507 0.493753 0.869602i \(-0.335625\pi\)
0.493753 + 0.869602i \(0.335625\pi\)
\(444\) −8.92820 −0.423714
\(445\) −15.4641 −0.733069
\(446\) −8.39230 −0.397387
\(447\) 19.8564 0.939176
\(448\) 0 0
\(449\) −35.3205 −1.66688 −0.833439 0.552612i \(-0.813631\pi\)
−0.833439 + 0.552612i \(0.813631\pi\)
\(450\) −4.46410 −0.210440
\(451\) 4.73205 0.222824
\(452\) 12.9282 0.608092
\(453\) 21.3205 1.00172
\(454\) 13.8564 0.650313
\(455\) 0 0
\(456\) 7.46410 0.349539
\(457\) −34.0000 −1.59045 −0.795226 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) −13.4641 −0.629136
\(459\) 13.8564 0.646762
\(460\) −1.26795 −0.0591184
\(461\) −29.3205 −1.36559 −0.682796 0.730609i \(-0.739236\pi\)
−0.682796 + 0.730609i \(0.739236\pi\)
\(462\) 0 0
\(463\) 12.9808 0.603267 0.301634 0.953424i \(-0.402468\pi\)
0.301634 + 0.953424i \(0.402468\pi\)
\(464\) −4.73205 −0.219680
\(465\) −24.3923 −1.13117
\(466\) −7.85641 −0.363941
\(467\) −17.6603 −0.817219 −0.408610 0.912709i \(-0.633986\pi\)
−0.408610 + 0.912709i \(0.633986\pi\)
\(468\) 6.53590 0.302122
\(469\) 0 0
\(470\) 0 0
\(471\) 17.4641 0.804703
\(472\) −13.8564 −0.637793
\(473\) −4.92820 −0.226599
\(474\) 18.3923 0.844787
\(475\) 2.73205 0.125355
\(476\) 0 0
\(477\) −21.1244 −0.967218
\(478\) 3.80385 0.173984
\(479\) −13.8564 −0.633115 −0.316558 0.948573i \(-0.602527\pi\)
−0.316558 + 0.948573i \(0.602527\pi\)
\(480\) −2.73205 −0.124700
\(481\) 4.78461 0.218159
\(482\) −18.1962 −0.828812
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 5.80385 0.263539
\(486\) −18.7321 −0.849703
\(487\) 4.87564 0.220937 0.110468 0.993880i \(-0.464765\pi\)
0.110468 + 0.993880i \(0.464765\pi\)
\(488\) 2.00000 0.0905357
\(489\) −21.8564 −0.988381
\(490\) 0 0
\(491\) 42.9282 1.93732 0.968661 0.248385i \(-0.0798999\pi\)
0.968661 + 0.248385i \(0.0798999\pi\)
\(492\) −12.9282 −0.582848
\(493\) 16.3923 0.738272
\(494\) −4.00000 −0.179969
\(495\) −4.46410 −0.200646
\(496\) −8.92820 −0.400888
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 2.00000 0.0895323 0.0447661 0.998997i \(-0.485746\pi\)
0.0447661 + 0.998997i \(0.485746\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −37.8564 −1.69130
\(502\) 0 0
\(503\) 18.9282 0.843967 0.421983 0.906604i \(-0.361334\pi\)
0.421983 + 0.906604i \(0.361334\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) −1.26795 −0.0563672
\(507\) 29.6603 1.31726
\(508\) 9.85641 0.437307
\(509\) −2.78461 −0.123426 −0.0617128 0.998094i \(-0.519656\pi\)
−0.0617128 + 0.998094i \(0.519656\pi\)
\(510\) 9.46410 0.419077
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −10.9282 −0.482492
\(514\) 22.9808 1.01364
\(515\) 10.5359 0.464267
\(516\) 13.4641 0.592724
\(517\) 0 0
\(518\) 0 0
\(519\) 2.53590 0.111314
\(520\) 1.46410 0.0642051
\(521\) −10.3923 −0.455295 −0.227648 0.973744i \(-0.573103\pi\)
−0.227648 + 0.973744i \(0.573103\pi\)
\(522\) 21.1244 0.924588
\(523\) −19.3205 −0.844827 −0.422413 0.906403i \(-0.638817\pi\)
−0.422413 + 0.906403i \(0.638817\pi\)
\(524\) 17.6603 0.771492
\(525\) 0 0
\(526\) −18.9282 −0.825309
\(527\) 30.9282 1.34725
\(528\) −2.73205 −0.118897
\(529\) −21.3923 −0.930100
\(530\) −4.73205 −0.205547
\(531\) 61.8564 2.68434
\(532\) 0 0
\(533\) 6.92820 0.300094
\(534\) 42.2487 1.82828
\(535\) −0.928203 −0.0401297
\(536\) 10.9282 0.472026
\(537\) 54.2487 2.34100
\(538\) 4.39230 0.189366
\(539\) 0 0
\(540\) 4.00000 0.172133
\(541\) 17.1244 0.736234 0.368117 0.929780i \(-0.380003\pi\)
0.368117 + 0.929780i \(0.380003\pi\)
\(542\) 24.3923 1.04774
\(543\) −32.3923 −1.39009
\(544\) 3.46410 0.148522
\(545\) 1.80385 0.0772683
\(546\) 0 0
\(547\) 4.78461 0.204575 0.102288 0.994755i \(-0.467384\pi\)
0.102288 + 0.994755i \(0.467384\pi\)
\(548\) 7.85641 0.335609
\(549\) −8.92820 −0.381046
\(550\) −1.00000 −0.0426401
\(551\) −12.9282 −0.550760
\(552\) 3.46410 0.147442
\(553\) 0 0
\(554\) −27.8564 −1.18351
\(555\) 8.92820 0.378981
\(556\) −4.19615 −0.177957
\(557\) −12.2487 −0.518995 −0.259497 0.965744i \(-0.583557\pi\)
−0.259497 + 0.965744i \(0.583557\pi\)
\(558\) 39.8564 1.68726
\(559\) −7.21539 −0.305178
\(560\) 0 0
\(561\) 9.46410 0.399575
\(562\) 10.3923 0.438373
\(563\) 20.7846 0.875967 0.437983 0.898983i \(-0.355693\pi\)
0.437983 + 0.898983i \(0.355693\pi\)
\(564\) 0 0
\(565\) −12.9282 −0.543894
\(566\) −10.9282 −0.459347
\(567\) 0 0
\(568\) −9.46410 −0.397105
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) −7.46410 −0.312637
\(571\) −20.3923 −0.853391 −0.426696 0.904395i \(-0.640322\pi\)
−0.426696 + 0.904395i \(0.640322\pi\)
\(572\) 1.46410 0.0612172
\(573\) −18.9282 −0.790737
\(574\) 0 0
\(575\) 1.26795 0.0528771
\(576\) 4.46410 0.186004
\(577\) −8.33975 −0.347188 −0.173594 0.984817i \(-0.555538\pi\)
−0.173594 + 0.984817i \(0.555538\pi\)
\(578\) 5.00000 0.207973
\(579\) 69.5692 2.89120
\(580\) 4.73205 0.196488
\(581\) 0 0
\(582\) −15.8564 −0.657269
\(583\) −4.73205 −0.195982
\(584\) 11.4641 0.474388
\(585\) −6.53590 −0.270226
\(586\) −2.53590 −0.104757
\(587\) 28.9808 1.19616 0.598082 0.801435i \(-0.295930\pi\)
0.598082 + 0.801435i \(0.295930\pi\)
\(588\) 0 0
\(589\) −24.3923 −1.00507
\(590\) 13.8564 0.570459
\(591\) 61.1769 2.51648
\(592\) 3.26795 0.134312
\(593\) −32.5359 −1.33609 −0.668045 0.744121i \(-0.732868\pi\)
−0.668045 + 0.744121i \(0.732868\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −7.26795 −0.297707
\(597\) −67.7128 −2.77130
\(598\) −1.85641 −0.0759141
\(599\) −32.1051 −1.31178 −0.655890 0.754857i \(-0.727706\pi\)
−0.655890 + 0.754857i \(0.727706\pi\)
\(600\) 2.73205 0.111536
\(601\) −28.4449 −1.16029 −0.580145 0.814513i \(-0.697004\pi\)
−0.580145 + 0.814513i \(0.697004\pi\)
\(602\) 0 0
\(603\) −48.7846 −1.98666
\(604\) −7.80385 −0.317534
\(605\) −1.00000 −0.0406558
\(606\) 16.3923 0.665892
\(607\) −34.7846 −1.41186 −0.705932 0.708280i \(-0.749472\pi\)
−0.705932 + 0.708280i \(0.749472\pi\)
\(608\) −2.73205 −0.110799
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) −15.4641 −0.625099
\(613\) 27.1769 1.09767 0.548833 0.835932i \(-0.315072\pi\)
0.548833 + 0.835932i \(0.315072\pi\)
\(614\) 31.3205 1.26399
\(615\) 12.9282 0.521315
\(616\) 0 0
\(617\) 8.53590 0.343642 0.171821 0.985128i \(-0.445035\pi\)
0.171821 + 0.985128i \(0.445035\pi\)
\(618\) −28.7846 −1.15789
\(619\) 10.9282 0.439242 0.219621 0.975585i \(-0.429518\pi\)
0.219621 + 0.975585i \(0.429518\pi\)
\(620\) 8.92820 0.358565
\(621\) −5.07180 −0.203524
\(622\) 7.85641 0.315013
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 27.2679 1.08985
\(627\) −7.46410 −0.298088
\(628\) −6.39230 −0.255081
\(629\) −11.3205 −0.451378
\(630\) 0 0
\(631\) −12.7846 −0.508947 −0.254474 0.967080i \(-0.581902\pi\)
−0.254474 + 0.967080i \(0.581902\pi\)
\(632\) −6.73205 −0.267787
\(633\) −21.8564 −0.868714
\(634\) 32.4449 1.28855
\(635\) −9.85641 −0.391140
\(636\) 12.9282 0.512637
\(637\) 0 0
\(638\) 4.73205 0.187344
\(639\) 42.2487 1.67133
\(640\) 1.00000 0.0395285
\(641\) 19.8564 0.784281 0.392140 0.919905i \(-0.371735\pi\)
0.392140 + 0.919905i \(0.371735\pi\)
\(642\) 2.53590 0.100084
\(643\) 14.7321 0.580975 0.290488 0.956879i \(-0.406182\pi\)
0.290488 + 0.956879i \(0.406182\pi\)
\(644\) 0 0
\(645\) −13.4641 −0.530148
\(646\) 9.46410 0.372360
\(647\) −37.1769 −1.46158 −0.730788 0.682605i \(-0.760847\pi\)
−0.730788 + 0.682605i \(0.760847\pi\)
\(648\) 2.46410 0.0967991
\(649\) 13.8564 0.543912
\(650\) −1.46410 −0.0574268
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) −9.80385 −0.383654 −0.191827 0.981429i \(-0.561441\pi\)
−0.191827 + 0.981429i \(0.561441\pi\)
\(654\) −4.92820 −0.192708
\(655\) −17.6603 −0.690043
\(656\) 4.73205 0.184756
\(657\) −51.1769 −1.99660
\(658\) 0 0
\(659\) 6.24871 0.243415 0.121708 0.992566i \(-0.461163\pi\)
0.121708 + 0.992566i \(0.461163\pi\)
\(660\) 2.73205 0.106345
\(661\) −3.85641 −0.149997 −0.0749984 0.997184i \(-0.523895\pi\)
−0.0749984 + 0.997184i \(0.523895\pi\)
\(662\) 4.92820 0.191540
\(663\) 13.8564 0.538138
\(664\) −4.39230 −0.170454
\(665\) 0 0
\(666\) −14.5885 −0.565291
\(667\) −6.00000 −0.232321
\(668\) 13.8564 0.536120
\(669\) −22.9282 −0.886456
\(670\) −10.9282 −0.422193
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) −20.1436 −0.776478 −0.388239 0.921559i \(-0.626917\pi\)
−0.388239 + 0.921559i \(0.626917\pi\)
\(674\) 30.7846 1.18578
\(675\) −4.00000 −0.153960
\(676\) −10.8564 −0.417554
\(677\) −19.8564 −0.763144 −0.381572 0.924339i \(-0.624617\pi\)
−0.381572 + 0.924339i \(0.624617\pi\)
\(678\) 35.3205 1.35648
\(679\) 0 0
\(680\) −3.46410 −0.132842
\(681\) 37.8564 1.45066
\(682\) 8.92820 0.341879
\(683\) 42.2487 1.61660 0.808301 0.588769i \(-0.200387\pi\)
0.808301 + 0.588769i \(0.200387\pi\)
\(684\) 12.1962 0.466332
\(685\) −7.85641 −0.300178
\(686\) 0 0
\(687\) −36.7846 −1.40342
\(688\) −4.92820 −0.187886
\(689\) −6.92820 −0.263944
\(690\) −3.46410 −0.131876
\(691\) −31.3205 −1.19149 −0.595744 0.803174i \(-0.703143\pi\)
−0.595744 + 0.803174i \(0.703143\pi\)
\(692\) −0.928203 −0.0352850
\(693\) 0 0
\(694\) −7.85641 −0.298225
\(695\) 4.19615 0.159169
\(696\) −12.9282 −0.490042
\(697\) −16.3923 −0.620903
\(698\) 30.3923 1.15037
\(699\) −21.4641 −0.811847
\(700\) 0 0
\(701\) 14.1962 0.536181 0.268091 0.963394i \(-0.413607\pi\)
0.268091 + 0.963394i \(0.413607\pi\)
\(702\) 5.85641 0.221036
\(703\) 8.92820 0.336734
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −32.4449 −1.22108
\(707\) 0 0
\(708\) −37.8564 −1.42273
\(709\) −2.39230 −0.0898449 −0.0449224 0.998990i \(-0.514304\pi\)
−0.0449224 + 0.998990i \(0.514304\pi\)
\(710\) 9.46410 0.355181
\(711\) 30.0526 1.12706
\(712\) −15.4641 −0.579542
\(713\) −11.3205 −0.423956
\(714\) 0 0
\(715\) −1.46410 −0.0547543
\(716\) −19.8564 −0.742069
\(717\) 10.3923 0.388108
\(718\) 1.26795 0.0473194
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) −4.46410 −0.166367
\(721\) 0 0
\(722\) 11.5359 0.429322
\(723\) −49.7128 −1.84884
\(724\) 11.8564 0.440640
\(725\) −4.73205 −0.175744
\(726\) 2.73205 0.101396
\(727\) 38.6410 1.43312 0.716558 0.697528i \(-0.245717\pi\)
0.716558 + 0.697528i \(0.245717\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) −11.4641 −0.424305
\(731\) 17.0718 0.631423
\(732\) 5.46410 0.201959
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 29.4641 1.08754
\(735\) 0 0
\(736\) −1.26795 −0.0467372
\(737\) −10.9282 −0.402546
\(738\) −21.1244 −0.777598
\(739\) 18.1436 0.667423 0.333711 0.942675i \(-0.391699\pi\)
0.333711 + 0.942675i \(0.391699\pi\)
\(740\) −3.26795 −0.120132
\(741\) −10.9282 −0.401458
\(742\) 0 0
\(743\) 3.71281 0.136210 0.0681049 0.997678i \(-0.478305\pi\)
0.0681049 + 0.997678i \(0.478305\pi\)
\(744\) −24.3923 −0.894265
\(745\) 7.26795 0.266277
\(746\) −30.3923 −1.11274
\(747\) 19.6077 0.717408
\(748\) −3.46410 −0.126660
\(749\) 0 0
\(750\) −2.73205 −0.0997604
\(751\) 2.24871 0.0820566 0.0410283 0.999158i \(-0.486937\pi\)
0.0410283 + 0.999158i \(0.486937\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 6.92820 0.252310
\(755\) 7.80385 0.284011
\(756\) 0 0
\(757\) 23.3731 0.849509 0.424754 0.905309i \(-0.360360\pi\)
0.424754 + 0.905309i \(0.360360\pi\)
\(758\) −23.7128 −0.861288
\(759\) −3.46410 −0.125739
\(760\) 2.73205 0.0991019
\(761\) −19.2679 −0.698463 −0.349231 0.937037i \(-0.613557\pi\)
−0.349231 + 0.937037i \(0.613557\pi\)
\(762\) 26.9282 0.975506
\(763\) 0 0
\(764\) 6.92820 0.250654
\(765\) 15.4641 0.559106
\(766\) 16.3923 0.592278
\(767\) 20.2872 0.732528
\(768\) −2.73205 −0.0985844
\(769\) 24.4449 0.881504 0.440752 0.897629i \(-0.354712\pi\)
0.440752 + 0.897629i \(0.354712\pi\)
\(770\) 0 0
\(771\) 62.7846 2.26113
\(772\) −25.4641 −0.916473
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 22.0000 0.790774
\(775\) −8.92820 −0.320711
\(776\) 5.80385 0.208346
\(777\) 0 0
\(778\) 24.2487 0.869358
\(779\) 12.9282 0.463201
\(780\) 4.00000 0.143223
\(781\) 9.46410 0.338652
\(782\) 4.39230 0.157069
\(783\) 18.9282 0.676439
\(784\) 0 0
\(785\) 6.39230 0.228151
\(786\) 48.2487 1.72097
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −22.3923 −0.797693
\(789\) −51.7128 −1.84102
\(790\) 6.73205 0.239516
\(791\) 0 0
\(792\) −4.46410 −0.158625
\(793\) −2.92820 −0.103984
\(794\) −38.3923 −1.36249
\(795\) −12.9282 −0.458516
\(796\) 24.7846 0.878467
\(797\) −6.67949 −0.236600 −0.118300 0.992978i \(-0.537744\pi\)
−0.118300 + 0.992978i \(0.537744\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 69.0333 2.43917
\(802\) 25.1769 0.889028
\(803\) −11.4641 −0.404559
\(804\) 29.8564 1.05295
\(805\) 0 0
\(806\) 13.0718 0.460434
\(807\) 12.0000 0.422420
\(808\) −6.00000 −0.211079
\(809\) 5.32051 0.187059 0.0935296 0.995617i \(-0.470185\pi\)
0.0935296 + 0.995617i \(0.470185\pi\)
\(810\) −2.46410 −0.0865797
\(811\) 4.58846 0.161123 0.0805613 0.996750i \(-0.474329\pi\)
0.0805613 + 0.996750i \(0.474329\pi\)
\(812\) 0 0
\(813\) 66.6410 2.33720
\(814\) −3.26795 −0.114542
\(815\) −8.00000 −0.280228
\(816\) 9.46410 0.331310
\(817\) −13.4641 −0.471049
\(818\) 22.1962 0.776070
\(819\) 0 0
\(820\) −4.73205 −0.165250
\(821\) 52.0526 1.81665 0.908323 0.418269i \(-0.137363\pi\)
0.908323 + 0.418269i \(0.137363\pi\)
\(822\) 21.4641 0.748647
\(823\) −24.1962 −0.843425 −0.421712 0.906730i \(-0.638571\pi\)
−0.421712 + 0.906730i \(0.638571\pi\)
\(824\) 10.5359 0.367035
\(825\) −2.73205 −0.0951178
\(826\) 0 0
\(827\) 53.5692 1.86278 0.931392 0.364017i \(-0.118595\pi\)
0.931392 + 0.364017i \(0.118595\pi\)
\(828\) 5.66025 0.196707
\(829\) −40.1051 −1.39291 −0.696454 0.717601i \(-0.745240\pi\)
−0.696454 + 0.717601i \(0.745240\pi\)
\(830\) 4.39230 0.152459
\(831\) −76.1051 −2.64006
\(832\) 1.46410 0.0507586
\(833\) 0 0
\(834\) −11.4641 −0.396969
\(835\) −13.8564 −0.479521
\(836\) 2.73205 0.0944900
\(837\) 35.7128 1.23442
\(838\) 32.7846 1.13253
\(839\) 38.7846 1.33899 0.669497 0.742815i \(-0.266510\pi\)
0.669497 + 0.742815i \(0.266510\pi\)
\(840\) 0 0
\(841\) −6.60770 −0.227852
\(842\) −10.7846 −0.371662
\(843\) 28.3923 0.977883
\(844\) 8.00000 0.275371
\(845\) 10.8564 0.373472
\(846\) 0 0
\(847\) 0 0
\(848\) −4.73205 −0.162499
\(849\) −29.8564 −1.02467
\(850\) 3.46410 0.118818
\(851\) 4.14359 0.142041
\(852\) −25.8564 −0.885826
\(853\) −20.9282 −0.716568 −0.358284 0.933613i \(-0.616638\pi\)
−0.358284 + 0.933613i \(0.616638\pi\)
\(854\) 0 0
\(855\) −12.1962 −0.417100
\(856\) −0.928203 −0.0317253
\(857\) 19.1769 0.655071 0.327535 0.944839i \(-0.393782\pi\)
0.327535 + 0.944839i \(0.393782\pi\)
\(858\) 4.00000 0.136558
\(859\) −40.7846 −1.39155 −0.695776 0.718258i \(-0.744940\pi\)
−0.695776 + 0.718258i \(0.744940\pi\)
\(860\) 4.92820 0.168050
\(861\) 0 0
\(862\) −18.3397 −0.624654
\(863\) 24.5885 0.837001 0.418500 0.908217i \(-0.362556\pi\)
0.418500 + 0.908217i \(0.362556\pi\)
\(864\) 4.00000 0.136083
\(865\) 0.928203 0.0315599
\(866\) 0.732051 0.0248761
\(867\) 13.6603 0.463927
\(868\) 0 0
\(869\) 6.73205 0.228369
\(870\) 12.9282 0.438307
\(871\) −16.0000 −0.542139
\(872\) 1.80385 0.0610860
\(873\) −25.9090 −0.876886
\(874\) −3.46410 −0.117175
\(875\) 0 0
\(876\) 31.3205 1.05822
\(877\) −23.1769 −0.782629 −0.391314 0.920257i \(-0.627979\pi\)
−0.391314 + 0.920257i \(0.627979\pi\)
\(878\) 36.3923 1.22818
\(879\) −6.92820 −0.233682
\(880\) −1.00000 −0.0337100
\(881\) −24.9282 −0.839853 −0.419926 0.907558i \(-0.637944\pi\)
−0.419926 + 0.907558i \(0.637944\pi\)
\(882\) 0 0
\(883\) −41.1769 −1.38571 −0.692857 0.721075i \(-0.743648\pi\)
−0.692857 + 0.721075i \(0.743648\pi\)
\(884\) −5.07180 −0.170583
\(885\) 37.8564 1.27253
\(886\) −20.7846 −0.698273
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 8.92820 0.299611
\(889\) 0 0
\(890\) 15.4641 0.518358
\(891\) −2.46410 −0.0825505
\(892\) 8.39230 0.280995
\(893\) 0 0
\(894\) −19.8564 −0.664098
\(895\) 19.8564 0.663726
\(896\) 0 0
\(897\) −5.07180 −0.169342
\(898\) 35.3205 1.17866
\(899\) 42.2487 1.40907
\(900\) 4.46410 0.148803
\(901\) 16.3923 0.546107
\(902\) −4.73205 −0.157560
\(903\) 0 0
\(904\) −12.9282 −0.429986
\(905\) −11.8564 −0.394120
\(906\) −21.3205 −0.708326
\(907\) −15.3205 −0.508709 −0.254355 0.967111i \(-0.581863\pi\)
−0.254355 + 0.967111i \(0.581863\pi\)
\(908\) −13.8564 −0.459841
\(909\) 26.7846 0.888389
\(910\) 0 0
\(911\) 14.5359 0.481596 0.240798 0.970575i \(-0.422591\pi\)
0.240798 + 0.970575i \(0.422591\pi\)
\(912\) −7.46410 −0.247161
\(913\) 4.39230 0.145364
\(914\) 34.0000 1.12462
\(915\) −5.46410 −0.180638
\(916\) 13.4641 0.444866
\(917\) 0 0
\(918\) −13.8564 −0.457330
\(919\) 24.9808 0.824039 0.412020 0.911175i \(-0.364824\pi\)
0.412020 + 0.911175i \(0.364824\pi\)
\(920\) 1.26795 0.0418030
\(921\) 85.5692 2.81960
\(922\) 29.3205 0.965620
\(923\) 13.8564 0.456089
\(924\) 0 0
\(925\) 3.26795 0.107450
\(926\) −12.9808 −0.426574
\(927\) −47.0333 −1.54478
\(928\) 4.73205 0.155337
\(929\) −26.1051 −0.856481 −0.428241 0.903665i \(-0.640866\pi\)
−0.428241 + 0.903665i \(0.640866\pi\)
\(930\) 24.3923 0.799855
\(931\) 0 0
\(932\) 7.85641 0.257345
\(933\) 21.4641 0.702703
\(934\) 17.6603 0.577861
\(935\) 3.46410 0.113288
\(936\) −6.53590 −0.213633
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) 74.4974 2.43113
\(940\) 0 0
\(941\) 35.5692 1.15952 0.579762 0.814786i \(-0.303146\pi\)
0.579762 + 0.814786i \(0.303146\pi\)
\(942\) −17.4641 −0.569011
\(943\) 6.00000 0.195387
\(944\) 13.8564 0.450988
\(945\) 0 0
\(946\) 4.92820 0.160230
\(947\) −25.1769 −0.818140 −0.409070 0.912503i \(-0.634147\pi\)
−0.409070 + 0.912503i \(0.634147\pi\)
\(948\) −18.3923 −0.597354
\(949\) −16.7846 −0.544851
\(950\) −2.73205 −0.0886394
\(951\) 88.6410 2.87438
\(952\) 0 0
\(953\) −11.3205 −0.366707 −0.183354 0.983047i \(-0.558695\pi\)
−0.183354 + 0.983047i \(0.558695\pi\)
\(954\) 21.1244 0.683926
\(955\) −6.92820 −0.224191
\(956\) −3.80385 −0.123025
\(957\) 12.9282 0.417909
\(958\) 13.8564 0.447680
\(959\) 0 0
\(960\) 2.73205 0.0881766
\(961\) 48.7128 1.57138
\(962\) −4.78461 −0.154262
\(963\) 4.14359 0.133525
\(964\) 18.1962 0.586059
\(965\) 25.4641 0.819718
\(966\) 0 0
\(967\) −26.1436 −0.840721 −0.420361 0.907357i \(-0.638096\pi\)
−0.420361 + 0.907357i \(0.638096\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 25.8564 0.830627
\(970\) −5.80385 −0.186350
\(971\) −49.8564 −1.59997 −0.799984 0.600021i \(-0.795159\pi\)
−0.799984 + 0.600021i \(0.795159\pi\)
\(972\) 18.7321 0.600831
\(973\) 0 0
\(974\) −4.87564 −0.156226
\(975\) −4.00000 −0.128103
\(976\) −2.00000 −0.0640184
\(977\) −0.928203 −0.0296959 −0.0148479 0.999890i \(-0.504726\pi\)
−0.0148479 + 0.999890i \(0.504726\pi\)
\(978\) 21.8564 0.698891
\(979\) 15.4641 0.494235
\(980\) 0 0
\(981\) −8.05256 −0.257098
\(982\) −42.9282 −1.36989
\(983\) 7.60770 0.242648 0.121324 0.992613i \(-0.461286\pi\)
0.121324 + 0.992613i \(0.461286\pi\)
\(984\) 12.9282 0.412136
\(985\) 22.3923 0.713478
\(986\) −16.3923 −0.522037
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) −6.24871 −0.198697
\(990\) 4.46410 0.141878
\(991\) −22.9282 −0.728338 −0.364169 0.931333i \(-0.618647\pi\)
−0.364169 + 0.931333i \(0.618647\pi\)
\(992\) 8.92820 0.283471
\(993\) 13.4641 0.427270
\(994\) 0 0
\(995\) −24.7846 −0.785725
\(996\) −12.0000 −0.380235
\(997\) 23.8564 0.755540 0.377770 0.925899i \(-0.376691\pi\)
0.377770 + 0.925899i \(0.376691\pi\)
\(998\) −2.00000 −0.0633089
\(999\) −13.0718 −0.413573
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 5390.2.a.bk.1.1 2
7.6 odd 2 770.2.a.h.1.2 2
21.20 even 2 6930.2.a.ca.1.1 2
28.27 even 2 6160.2.a.v.1.1 2
35.13 even 4 3850.2.c.s.1849.4 4
35.27 even 4 3850.2.c.s.1849.1 4
35.34 odd 2 3850.2.a.bm.1.1 2
77.76 even 2 8470.2.a.ce.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.h.1.2 2 7.6 odd 2
3850.2.a.bm.1.1 2 35.34 odd 2
3850.2.c.s.1849.1 4 35.27 even 4
3850.2.c.s.1849.4 4 35.13 even 4
5390.2.a.bk.1.1 2 1.1 even 1 trivial
6160.2.a.v.1.1 2 28.27 even 2
6930.2.a.ca.1.1 2 21.20 even 2
8470.2.a.ce.1.2 2 77.76 even 2