Properties

Label 2366.2.a.bb.1.3
Level $2366$
Weight $2$
Character 2366.1
Self dual yes
Analytic conductor $18.893$
Analytic rank $1$
Dimension $3$
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] = [2366,2,Mod(1,2366)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2366, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2366.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.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: \(18.8926051182\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 2366.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.69202 q^{3} +1.00000 q^{4} -2.55496 q^{5} +1.69202 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.137063 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.69202 q^{3} +1.00000 q^{4} -2.55496 q^{5} +1.69202 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.137063 q^{9} -2.55496 q^{10} -1.64310 q^{11} +1.69202 q^{12} -1.00000 q^{14} -4.32304 q^{15} +1.00000 q^{16} -4.34481 q^{17} -0.137063 q^{18} -1.08815 q^{19} -2.55496 q^{20} -1.69202 q^{21} -1.64310 q^{22} -2.04892 q^{23} +1.69202 q^{24} +1.52781 q^{25} -5.30798 q^{27} -1.00000 q^{28} +10.3666 q^{29} -4.32304 q^{30} -11.0489 q^{31} +1.00000 q^{32} -2.78017 q^{33} -4.34481 q^{34} +2.55496 q^{35} -0.137063 q^{36} -1.26875 q^{37} -1.08815 q^{38} -2.55496 q^{40} -4.76271 q^{41} -1.69202 q^{42} -1.30798 q^{43} -1.64310 q^{44} +0.350191 q^{45} -2.04892 q^{46} -7.24698 q^{47} +1.69202 q^{48} +1.00000 q^{49} +1.52781 q^{50} -7.35152 q^{51} +1.29590 q^{53} -5.30798 q^{54} +4.19806 q^{55} -1.00000 q^{56} -1.84117 q^{57} +10.3666 q^{58} +8.45473 q^{59} -4.32304 q^{60} +3.52111 q^{61} -11.0489 q^{62} +0.137063 q^{63} +1.00000 q^{64} -2.78017 q^{66} -6.57002 q^{67} -4.34481 q^{68} -3.46681 q^{69} +2.55496 q^{70} -15.4547 q^{71} -0.137063 q^{72} -5.18598 q^{73} -1.26875 q^{74} +2.58509 q^{75} -1.08815 q^{76} +1.64310 q^{77} +10.2295 q^{79} -2.55496 q^{80} -8.57002 q^{81} -4.76271 q^{82} +13.9705 q^{83} -1.69202 q^{84} +11.1008 q^{85} -1.30798 q^{86} +17.5405 q^{87} -1.64310 q^{88} -3.93900 q^{89} +0.350191 q^{90} -2.04892 q^{92} -18.6950 q^{93} -7.24698 q^{94} +2.78017 q^{95} +1.69202 q^{96} +12.0489 q^{97} +1.00000 q^{98} +0.225209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 8 q^{5} - 3 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} - 8 q^{5} - 3 q^{7} + 3 q^{8} + 5 q^{9} - 8 q^{10} - 9 q^{11} - 3 q^{14} + 7 q^{15} + 3 q^{16} + 10 q^{17} + 5 q^{18} - 7 q^{19} - 8 q^{20} - 9 q^{22} + 3 q^{23} + 11 q^{25} - 21 q^{27} - 3 q^{28} + 5 q^{29} + 7 q^{30} - 24 q^{31} + 3 q^{32} - 7 q^{33} + 10 q^{34} + 8 q^{35} + 5 q^{36} + 4 q^{37} - 7 q^{38} - 8 q^{40} + 3 q^{41} - 9 q^{43} - 9 q^{44} - 25 q^{45} + 3 q^{46} - 17 q^{47} + 3 q^{49} + 11 q^{50} - 21 q^{51} - 10 q^{53} - 21 q^{54} + 17 q^{55} - 3 q^{56} - 14 q^{57} + 5 q^{58} + 3 q^{59} + 7 q^{60} - 5 q^{61} - 24 q^{62} - 5 q^{63} + 3 q^{64} - 7 q^{66} + 5 q^{67} + 10 q^{68} - 7 q^{69} + 8 q^{70} - 24 q^{71} + 5 q^{72} - q^{73} + 4 q^{74} - 42 q^{75} - 7 q^{76} + 9 q^{77} + 10 q^{79} - 8 q^{80} - q^{81} + 3 q^{82} + 7 q^{83} - 29 q^{85} - 9 q^{86} - 7 q^{87} - 9 q^{88} - 2 q^{89} - 25 q^{90} + 3 q^{92} - 7 q^{93} - 17 q^{94} + 7 q^{95} + 27 q^{97} + 3 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.69202 0.976889 0.488445 0.872595i \(-0.337564\pi\)
0.488445 + 0.872595i \(0.337564\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.55496 −1.14261 −0.571306 0.820737i \(-0.693563\pi\)
−0.571306 + 0.820737i \(0.693563\pi\)
\(6\) 1.69202 0.690765
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −0.137063 −0.0456878
\(10\) −2.55496 −0.807949
\(11\) −1.64310 −0.495415 −0.247707 0.968835i \(-0.579677\pi\)
−0.247707 + 0.968835i \(0.579677\pi\)
\(12\) 1.69202 0.488445
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −4.32304 −1.11621
\(16\) 1.00000 0.250000
\(17\) −4.34481 −1.05377 −0.526886 0.849936i \(-0.676641\pi\)
−0.526886 + 0.849936i \(0.676641\pi\)
\(18\) −0.137063 −0.0323061
\(19\) −1.08815 −0.249638 −0.124819 0.992180i \(-0.539835\pi\)
−0.124819 + 0.992180i \(0.539835\pi\)
\(20\) −2.55496 −0.571306
\(21\) −1.69202 −0.369229
\(22\) −1.64310 −0.350311
\(23\) −2.04892 −0.427229 −0.213614 0.976918i \(-0.568524\pi\)
−0.213614 + 0.976918i \(0.568524\pi\)
\(24\) 1.69202 0.345382
\(25\) 1.52781 0.305562
\(26\) 0 0
\(27\) −5.30798 −1.02152
\(28\) −1.00000 −0.188982
\(29\) 10.3666 1.92503 0.962513 0.271235i \(-0.0874320\pi\)
0.962513 + 0.271235i \(0.0874320\pi\)
\(30\) −4.32304 −0.789276
\(31\) −11.0489 −1.98444 −0.992222 0.124480i \(-0.960274\pi\)
−0.992222 + 0.124480i \(0.960274\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.78017 −0.483965
\(34\) −4.34481 −0.745130
\(35\) 2.55496 0.431867
\(36\) −0.137063 −0.0228439
\(37\) −1.26875 −0.208581 −0.104291 0.994547i \(-0.533257\pi\)
−0.104291 + 0.994547i \(0.533257\pi\)
\(38\) −1.08815 −0.176521
\(39\) 0 0
\(40\) −2.55496 −0.403974
\(41\) −4.76271 −0.743810 −0.371905 0.928271i \(-0.621295\pi\)
−0.371905 + 0.928271i \(0.621295\pi\)
\(42\) −1.69202 −0.261085
\(43\) −1.30798 −0.199465 −0.0997324 0.995014i \(-0.531799\pi\)
−0.0997324 + 0.995014i \(0.531799\pi\)
\(44\) −1.64310 −0.247707
\(45\) 0.350191 0.0522034
\(46\) −2.04892 −0.302096
\(47\) −7.24698 −1.05708 −0.528540 0.848908i \(-0.677260\pi\)
−0.528540 + 0.848908i \(0.677260\pi\)
\(48\) 1.69202 0.244222
\(49\) 1.00000 0.142857
\(50\) 1.52781 0.216065
\(51\) −7.35152 −1.02942
\(52\) 0 0
\(53\) 1.29590 0.178005 0.0890026 0.996031i \(-0.471632\pi\)
0.0890026 + 0.996031i \(0.471632\pi\)
\(54\) −5.30798 −0.722324
\(55\) 4.19806 0.566067
\(56\) −1.00000 −0.133631
\(57\) −1.84117 −0.243868
\(58\) 10.3666 1.36120
\(59\) 8.45473 1.10071 0.550356 0.834930i \(-0.314492\pi\)
0.550356 + 0.834930i \(0.314492\pi\)
\(60\) −4.32304 −0.558103
\(61\) 3.52111 0.450831 0.225416 0.974263i \(-0.427626\pi\)
0.225416 + 0.974263i \(0.427626\pi\)
\(62\) −11.0489 −1.40321
\(63\) 0.137063 0.0172684
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.78017 −0.342215
\(67\) −6.57002 −0.802656 −0.401328 0.915934i \(-0.631451\pi\)
−0.401328 + 0.915934i \(0.631451\pi\)
\(68\) −4.34481 −0.526886
\(69\) −3.46681 −0.417355
\(70\) 2.55496 0.305376
\(71\) −15.4547 −1.83414 −0.917070 0.398727i \(-0.869452\pi\)
−0.917070 + 0.398727i \(0.869452\pi\)
\(72\) −0.137063 −0.0161531
\(73\) −5.18598 −0.606973 −0.303487 0.952836i \(-0.598151\pi\)
−0.303487 + 0.952836i \(0.598151\pi\)
\(74\) −1.26875 −0.147489
\(75\) 2.58509 0.298500
\(76\) −1.08815 −0.124819
\(77\) 1.64310 0.187249
\(78\) 0 0
\(79\) 10.2295 1.15091 0.575456 0.817833i \(-0.304825\pi\)
0.575456 + 0.817833i \(0.304825\pi\)
\(80\) −2.55496 −0.285653
\(81\) −8.57002 −0.952225
\(82\) −4.76271 −0.525953
\(83\) 13.9705 1.53346 0.766729 0.641971i \(-0.221883\pi\)
0.766729 + 0.641971i \(0.221883\pi\)
\(84\) −1.69202 −0.184615
\(85\) 11.1008 1.20405
\(86\) −1.30798 −0.141043
\(87\) 17.5405 1.88054
\(88\) −1.64310 −0.175155
\(89\) −3.93900 −0.417533 −0.208767 0.977965i \(-0.566945\pi\)
−0.208767 + 0.977965i \(0.566945\pi\)
\(90\) 0.350191 0.0369134
\(91\) 0 0
\(92\) −2.04892 −0.213614
\(93\) −18.6950 −1.93858
\(94\) −7.24698 −0.747469
\(95\) 2.78017 0.285239
\(96\) 1.69202 0.172691
\(97\) 12.0489 1.22338 0.611691 0.791097i \(-0.290490\pi\)
0.611691 + 0.791097i \(0.290490\pi\)
\(98\) 1.00000 0.101015
\(99\) 0.225209 0.0226344
\(100\) 1.52781 0.152781
\(101\) −1.54288 −0.153522 −0.0767610 0.997050i \(-0.524458\pi\)
−0.0767610 + 0.997050i \(0.524458\pi\)
\(102\) −7.35152 −0.727909
\(103\) −2.63773 −0.259903 −0.129951 0.991520i \(-0.541482\pi\)
−0.129951 + 0.991520i \(0.541482\pi\)
\(104\) 0 0
\(105\) 4.32304 0.421886
\(106\) 1.29590 0.125869
\(107\) 10.1564 0.981860 0.490930 0.871199i \(-0.336657\pi\)
0.490930 + 0.871199i \(0.336657\pi\)
\(108\) −5.30798 −0.510760
\(109\) −13.2567 −1.26976 −0.634879 0.772611i \(-0.718950\pi\)
−0.634879 + 0.772611i \(0.718950\pi\)
\(110\) 4.19806 0.400270
\(111\) −2.14675 −0.203761
\(112\) −1.00000 −0.0944911
\(113\) −9.86054 −0.927602 −0.463801 0.885939i \(-0.653515\pi\)
−0.463801 + 0.885939i \(0.653515\pi\)
\(114\) −1.84117 −0.172441
\(115\) 5.23490 0.488157
\(116\) 10.3666 0.962513
\(117\) 0 0
\(118\) 8.45473 0.778321
\(119\) 4.34481 0.398288
\(120\) −4.32304 −0.394638
\(121\) −8.30021 −0.754564
\(122\) 3.52111 0.318786
\(123\) −8.05861 −0.726620
\(124\) −11.0489 −0.992222
\(125\) 8.87130 0.793473
\(126\) 0.137063 0.0122106
\(127\) −16.7114 −1.48290 −0.741448 0.671010i \(-0.765861\pi\)
−0.741448 + 0.671010i \(0.765861\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.21313 −0.194855
\(130\) 0 0
\(131\) −5.85086 −0.511192 −0.255596 0.966784i \(-0.582272\pi\)
−0.255596 + 0.966784i \(0.582272\pi\)
\(132\) −2.78017 −0.241983
\(133\) 1.08815 0.0943542
\(134\) −6.57002 −0.567564
\(135\) 13.5617 1.16720
\(136\) −4.34481 −0.372565
\(137\) 20.7995 1.77703 0.888513 0.458852i \(-0.151739\pi\)
0.888513 + 0.458852i \(0.151739\pi\)
\(138\) −3.46681 −0.295115
\(139\) 4.79225 0.406473 0.203237 0.979130i \(-0.434854\pi\)
0.203237 + 0.979130i \(0.434854\pi\)
\(140\) 2.55496 0.215933
\(141\) −12.2620 −1.03265
\(142\) −15.4547 −1.29693
\(143\) 0 0
\(144\) −0.137063 −0.0114219
\(145\) −26.4862 −2.19956
\(146\) −5.18598 −0.429195
\(147\) 1.69202 0.139556
\(148\) −1.26875 −0.104291
\(149\) 15.3448 1.25710 0.628548 0.777771i \(-0.283650\pi\)
0.628548 + 0.777771i \(0.283650\pi\)
\(150\) 2.58509 0.211072
\(151\) 0.521106 0.0424070 0.0212035 0.999775i \(-0.493250\pi\)
0.0212035 + 0.999775i \(0.493250\pi\)
\(152\) −1.08815 −0.0882603
\(153\) 0.595515 0.0481445
\(154\) 1.64310 0.132405
\(155\) 28.2295 2.26745
\(156\) 0 0
\(157\) −23.0911 −1.84287 −0.921436 0.388529i \(-0.872983\pi\)
−0.921436 + 0.388529i \(0.872983\pi\)
\(158\) 10.2295 0.813817
\(159\) 2.19269 0.173891
\(160\) −2.55496 −0.201987
\(161\) 2.04892 0.161477
\(162\) −8.57002 −0.673325
\(163\) 22.9028 1.79388 0.896941 0.442150i \(-0.145784\pi\)
0.896941 + 0.442150i \(0.145784\pi\)
\(164\) −4.76271 −0.371905
\(165\) 7.10321 0.552984
\(166\) 13.9705 1.08432
\(167\) −8.05861 −0.623594 −0.311797 0.950149i \(-0.600931\pi\)
−0.311797 + 0.950149i \(0.600931\pi\)
\(168\) −1.69202 −0.130542
\(169\) 0 0
\(170\) 11.1008 0.851394
\(171\) 0.149145 0.0114054
\(172\) −1.30798 −0.0997324
\(173\) 9.82669 0.747110 0.373555 0.927608i \(-0.378139\pi\)
0.373555 + 0.927608i \(0.378139\pi\)
\(174\) 17.5405 1.32974
\(175\) −1.52781 −0.115492
\(176\) −1.64310 −0.123854
\(177\) 14.3056 1.07527
\(178\) −3.93900 −0.295241
\(179\) −19.2392 −1.43801 −0.719003 0.695007i \(-0.755401\pi\)
−0.719003 + 0.695007i \(0.755401\pi\)
\(180\) 0.350191 0.0261017
\(181\) 9.57971 0.712054 0.356027 0.934476i \(-0.384131\pi\)
0.356027 + 0.934476i \(0.384131\pi\)
\(182\) 0 0
\(183\) 5.95779 0.440412
\(184\) −2.04892 −0.151048
\(185\) 3.24160 0.238327
\(186\) −18.6950 −1.37078
\(187\) 7.13898 0.522054
\(188\) −7.24698 −0.528540
\(189\) 5.30798 0.386099
\(190\) 2.78017 0.201695
\(191\) 1.51142 0.109362 0.0546812 0.998504i \(-0.482586\pi\)
0.0546812 + 0.998504i \(0.482586\pi\)
\(192\) 1.69202 0.122111
\(193\) −14.6310 −1.05316 −0.526582 0.850124i \(-0.676527\pi\)
−0.526582 + 0.850124i \(0.676527\pi\)
\(194\) 12.0489 0.865062
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 22.4480 1.59936 0.799678 0.600429i \(-0.205004\pi\)
0.799678 + 0.600429i \(0.205004\pi\)
\(198\) 0.225209 0.0160049
\(199\) −8.32975 −0.590480 −0.295240 0.955423i \(-0.595400\pi\)
−0.295240 + 0.955423i \(0.595400\pi\)
\(200\) 1.52781 0.108033
\(201\) −11.1166 −0.784106
\(202\) −1.54288 −0.108556
\(203\) −10.3666 −0.727592
\(204\) −7.35152 −0.514709
\(205\) 12.1685 0.849886
\(206\) −2.63773 −0.183779
\(207\) 0.280831 0.0195191
\(208\) 0 0
\(209\) 1.78794 0.123674
\(210\) 4.32304 0.298318
\(211\) 10.4426 0.718902 0.359451 0.933164i \(-0.382964\pi\)
0.359451 + 0.933164i \(0.382964\pi\)
\(212\) 1.29590 0.0890026
\(213\) −26.1497 −1.79175
\(214\) 10.1564 0.694280
\(215\) 3.34183 0.227911
\(216\) −5.30798 −0.361162
\(217\) 11.0489 0.750049
\(218\) −13.2567 −0.897855
\(219\) −8.77479 −0.592945
\(220\) 4.19806 0.283033
\(221\) 0 0
\(222\) −2.14675 −0.144081
\(223\) 15.9312 1.06683 0.533417 0.845852i \(-0.320908\pi\)
0.533417 + 0.845852i \(0.320908\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −0.209407 −0.0139605
\(226\) −9.86054 −0.655914
\(227\) −5.53079 −0.367092 −0.183546 0.983011i \(-0.558758\pi\)
−0.183546 + 0.983011i \(0.558758\pi\)
\(228\) −1.84117 −0.121934
\(229\) −24.1739 −1.59746 −0.798728 0.601692i \(-0.794493\pi\)
−0.798728 + 0.601692i \(0.794493\pi\)
\(230\) 5.23490 0.345179
\(231\) 2.78017 0.182922
\(232\) 10.3666 0.680600
\(233\) 24.5284 1.60691 0.803455 0.595366i \(-0.202993\pi\)
0.803455 + 0.595366i \(0.202993\pi\)
\(234\) 0 0
\(235\) 18.5157 1.20783
\(236\) 8.45473 0.550356
\(237\) 17.3086 1.12431
\(238\) 4.34481 0.281632
\(239\) 14.0325 0.907689 0.453844 0.891081i \(-0.350052\pi\)
0.453844 + 0.891081i \(0.350052\pi\)
\(240\) −4.32304 −0.279051
\(241\) −10.4843 −0.675351 −0.337676 0.941263i \(-0.609641\pi\)
−0.337676 + 0.941263i \(0.609641\pi\)
\(242\) −8.30021 −0.533558
\(243\) 1.42327 0.0913029
\(244\) 3.52111 0.225416
\(245\) −2.55496 −0.163230
\(246\) −8.05861 −0.513798
\(247\) 0 0
\(248\) −11.0489 −0.701607
\(249\) 23.6383 1.49802
\(250\) 8.87130 0.561070
\(251\) 25.1782 1.58923 0.794617 0.607111i \(-0.207672\pi\)
0.794617 + 0.607111i \(0.207672\pi\)
\(252\) 0.137063 0.00863418
\(253\) 3.36658 0.211655
\(254\) −16.7114 −1.04857
\(255\) 18.7828 1.17623
\(256\) 1.00000 0.0625000
\(257\) 13.9825 0.872207 0.436103 0.899897i \(-0.356358\pi\)
0.436103 + 0.899897i \(0.356358\pi\)
\(258\) −2.21313 −0.137783
\(259\) 1.26875 0.0788363
\(260\) 0 0
\(261\) −1.42088 −0.0879502
\(262\) −5.85086 −0.361467
\(263\) −22.6829 −1.39869 −0.699345 0.714785i \(-0.746525\pi\)
−0.699345 + 0.714785i \(0.746525\pi\)
\(264\) −2.78017 −0.171107
\(265\) −3.31096 −0.203391
\(266\) 1.08815 0.0667185
\(267\) −6.66487 −0.407884
\(268\) −6.57002 −0.401328
\(269\) −28.0301 −1.70903 −0.854514 0.519429i \(-0.826145\pi\)
−0.854514 + 0.519429i \(0.826145\pi\)
\(270\) 13.5617 0.825337
\(271\) −0.161227 −0.00979381 −0.00489691 0.999988i \(-0.501559\pi\)
−0.00489691 + 0.999988i \(0.501559\pi\)
\(272\) −4.34481 −0.263443
\(273\) 0 0
\(274\) 20.7995 1.25655
\(275\) −2.51035 −0.151380
\(276\) −3.46681 −0.208678
\(277\) 27.1564 1.63167 0.815836 0.578283i \(-0.196277\pi\)
0.815836 + 0.578283i \(0.196277\pi\)
\(278\) 4.79225 0.287420
\(279\) 1.51440 0.0906648
\(280\) 2.55496 0.152688
\(281\) 15.4776 0.923314 0.461657 0.887058i \(-0.347255\pi\)
0.461657 + 0.887058i \(0.347255\pi\)
\(282\) −12.2620 −0.730194
\(283\) −13.7289 −0.816096 −0.408048 0.912961i \(-0.633790\pi\)
−0.408048 + 0.912961i \(0.633790\pi\)
\(284\) −15.4547 −0.917070
\(285\) 4.70410 0.278647
\(286\) 0 0
\(287\) 4.76271 0.281134
\(288\) −0.137063 −0.00807653
\(289\) 1.87741 0.110436
\(290\) −26.4862 −1.55532
\(291\) 20.3870 1.19511
\(292\) −5.18598 −0.303487
\(293\) −21.5948 −1.26158 −0.630790 0.775953i \(-0.717269\pi\)
−0.630790 + 0.775953i \(0.717269\pi\)
\(294\) 1.69202 0.0986807
\(295\) −21.6015 −1.25769
\(296\) −1.26875 −0.0737446
\(297\) 8.72156 0.506076
\(298\) 15.3448 0.888901
\(299\) 0 0
\(300\) 2.58509 0.149250
\(301\) 1.30798 0.0753906
\(302\) 0.521106 0.0299863
\(303\) −2.61058 −0.149974
\(304\) −1.08815 −0.0624095
\(305\) −8.99628 −0.515125
\(306\) 0.595515 0.0340433
\(307\) −3.11231 −0.177629 −0.0888144 0.996048i \(-0.528308\pi\)
−0.0888144 + 0.996048i \(0.528308\pi\)
\(308\) 1.64310 0.0936245
\(309\) −4.46309 −0.253896
\(310\) 28.2295 1.60333
\(311\) −11.0000 −0.623753 −0.311876 0.950123i \(-0.600957\pi\)
−0.311876 + 0.950123i \(0.600957\pi\)
\(312\) 0 0
\(313\) −30.3394 −1.71489 −0.857443 0.514579i \(-0.827948\pi\)
−0.857443 + 0.514579i \(0.827948\pi\)
\(314\) −23.0911 −1.30311
\(315\) −0.350191 −0.0197310
\(316\) 10.2295 0.575456
\(317\) −7.08144 −0.397733 −0.198867 0.980027i \(-0.563726\pi\)
−0.198867 + 0.980027i \(0.563726\pi\)
\(318\) 2.19269 0.122960
\(319\) −17.0334 −0.953686
\(320\) −2.55496 −0.142827
\(321\) 17.1849 0.959168
\(322\) 2.04892 0.114182
\(323\) 4.72779 0.263061
\(324\) −8.57002 −0.476112
\(325\) 0 0
\(326\) 22.9028 1.26847
\(327\) −22.4306 −1.24041
\(328\) −4.76271 −0.262977
\(329\) 7.24698 0.399539
\(330\) 7.10321 0.391019
\(331\) 11.4722 0.630569 0.315284 0.948997i \(-0.397900\pi\)
0.315284 + 0.948997i \(0.397900\pi\)
\(332\) 13.9705 0.766729
\(333\) 0.173899 0.00952961
\(334\) −8.05861 −0.440947
\(335\) 16.7861 0.917125
\(336\) −1.69202 −0.0923073
\(337\) 3.96077 0.215757 0.107878 0.994164i \(-0.465594\pi\)
0.107878 + 0.994164i \(0.465594\pi\)
\(338\) 0 0
\(339\) −16.6843 −0.906164
\(340\) 11.1008 0.602026
\(341\) 18.1545 0.983122
\(342\) 0.149145 0.00806483
\(343\) −1.00000 −0.0539949
\(344\) −1.30798 −0.0705215
\(345\) 8.85756 0.476875
\(346\) 9.82669 0.528286
\(347\) 8.24698 0.442721 0.221361 0.975192i \(-0.428950\pi\)
0.221361 + 0.975192i \(0.428950\pi\)
\(348\) 17.5405 0.940269
\(349\) −14.5670 −0.779756 −0.389878 0.920867i \(-0.627483\pi\)
−0.389878 + 0.920867i \(0.627483\pi\)
\(350\) −1.52781 −0.0816649
\(351\) 0 0
\(352\) −1.64310 −0.0875777
\(353\) 3.04593 0.162119 0.0810593 0.996709i \(-0.474170\pi\)
0.0810593 + 0.996709i \(0.474170\pi\)
\(354\) 14.3056 0.760333
\(355\) 39.4862 2.09571
\(356\) −3.93900 −0.208767
\(357\) 7.35152 0.389084
\(358\) −19.2392 −1.01682
\(359\) 16.5851 0.875328 0.437664 0.899139i \(-0.355806\pi\)
0.437664 + 0.899139i \(0.355806\pi\)
\(360\) 0.350191 0.0184567
\(361\) −17.8159 −0.937681
\(362\) 9.57971 0.503499
\(363\) −14.0441 −0.737126
\(364\) 0 0
\(365\) 13.2500 0.693535
\(366\) 5.95779 0.311419
\(367\) 0.966148 0.0504325 0.0252163 0.999682i \(-0.491973\pi\)
0.0252163 + 0.999682i \(0.491973\pi\)
\(368\) −2.04892 −0.106807
\(369\) 0.652793 0.0339830
\(370\) 3.24160 0.168523
\(371\) −1.29590 −0.0672796
\(372\) −18.6950 −0.969291
\(373\) 32.0901 1.66156 0.830780 0.556600i \(-0.187895\pi\)
0.830780 + 0.556600i \(0.187895\pi\)
\(374\) 7.13898 0.369148
\(375\) 15.0104 0.775135
\(376\) −7.24698 −0.373734
\(377\) 0 0
\(378\) 5.30798 0.273013
\(379\) −0.972853 −0.0499721 −0.0249860 0.999688i \(-0.507954\pi\)
−0.0249860 + 0.999688i \(0.507954\pi\)
\(380\) 2.78017 0.142620
\(381\) −28.2760 −1.44863
\(382\) 1.51142 0.0773309
\(383\) −5.47650 −0.279836 −0.139918 0.990163i \(-0.544684\pi\)
−0.139918 + 0.990163i \(0.544684\pi\)
\(384\) 1.69202 0.0863456
\(385\) −4.19806 −0.213953
\(386\) −14.6310 −0.744699
\(387\) 0.179276 0.00911311
\(388\) 12.0489 0.611691
\(389\) −32.5066 −1.64815 −0.824076 0.566480i \(-0.808305\pi\)
−0.824076 + 0.566480i \(0.808305\pi\)
\(390\) 0 0
\(391\) 8.90217 0.450202
\(392\) 1.00000 0.0505076
\(393\) −9.89977 −0.499377
\(394\) 22.4480 1.13092
\(395\) −26.1360 −1.31504
\(396\) 0.225209 0.0113172
\(397\) −34.4674 −1.72987 −0.864935 0.501884i \(-0.832640\pi\)
−0.864935 + 0.501884i \(0.832640\pi\)
\(398\) −8.32975 −0.417533
\(399\) 1.84117 0.0921736
\(400\) 1.52781 0.0763906
\(401\) 12.9903 0.648705 0.324353 0.945936i \(-0.394854\pi\)
0.324353 + 0.945936i \(0.394854\pi\)
\(402\) −11.1166 −0.554447
\(403\) 0 0
\(404\) −1.54288 −0.0767610
\(405\) 21.8961 1.08802
\(406\) −10.3666 −0.514485
\(407\) 2.08469 0.103334
\(408\) −7.35152 −0.363954
\(409\) 31.4771 1.55644 0.778221 0.627990i \(-0.216122\pi\)
0.778221 + 0.627990i \(0.216122\pi\)
\(410\) 12.1685 0.600960
\(411\) 35.1933 1.73596
\(412\) −2.63773 −0.129951
\(413\) −8.45473 −0.416030
\(414\) 0.280831 0.0138021
\(415\) −35.6939 −1.75215
\(416\) 0 0
\(417\) 8.10859 0.397079
\(418\) 1.78794 0.0874509
\(419\) −22.4101 −1.09481 −0.547403 0.836869i \(-0.684384\pi\)
−0.547403 + 0.836869i \(0.684384\pi\)
\(420\) 4.32304 0.210943
\(421\) −17.2373 −0.840094 −0.420047 0.907502i \(-0.637986\pi\)
−0.420047 + 0.907502i \(0.637986\pi\)
\(422\) 10.4426 0.508340
\(423\) 0.993295 0.0482957
\(424\) 1.29590 0.0629343
\(425\) −6.63806 −0.321993
\(426\) −26.1497 −1.26696
\(427\) −3.52111 −0.170398
\(428\) 10.1564 0.490930
\(429\) 0 0
\(430\) 3.34183 0.161157
\(431\) 30.4349 1.46600 0.732998 0.680230i \(-0.238120\pi\)
0.732998 + 0.680230i \(0.238120\pi\)
\(432\) −5.30798 −0.255380
\(433\) 29.1812 1.40236 0.701179 0.712985i \(-0.252657\pi\)
0.701179 + 0.712985i \(0.252657\pi\)
\(434\) 11.0489 0.530365
\(435\) −44.8152 −2.14872
\(436\) −13.2567 −0.634879
\(437\) 2.22952 0.106652
\(438\) −8.77479 −0.419276
\(439\) −23.2349 −1.10894 −0.554471 0.832203i \(-0.687079\pi\)
−0.554471 + 0.832203i \(0.687079\pi\)
\(440\) 4.19806 0.200135
\(441\) −0.137063 −0.00652683
\(442\) 0 0
\(443\) −26.1371 −1.24181 −0.620905 0.783886i \(-0.713235\pi\)
−0.620905 + 0.783886i \(0.713235\pi\)
\(444\) −2.14675 −0.101880
\(445\) 10.0640 0.477079
\(446\) 15.9312 0.754366
\(447\) 25.9638 1.22804
\(448\) −1.00000 −0.0472456
\(449\) −4.29350 −0.202623 −0.101311 0.994855i \(-0.532304\pi\)
−0.101311 + 0.994855i \(0.532304\pi\)
\(450\) −0.209407 −0.00987154
\(451\) 7.82563 0.368494
\(452\) −9.86054 −0.463801
\(453\) 0.881723 0.0414270
\(454\) −5.53079 −0.259573
\(455\) 0 0
\(456\) −1.84117 −0.0862205
\(457\) −0.781495 −0.0365568 −0.0182784 0.999833i \(-0.505819\pi\)
−0.0182784 + 0.999833i \(0.505819\pi\)
\(458\) −24.1739 −1.12957
\(459\) 23.0622 1.07645
\(460\) 5.23490 0.244078
\(461\) −17.8103 −0.829508 −0.414754 0.909934i \(-0.636132\pi\)
−0.414754 + 0.909934i \(0.636132\pi\)
\(462\) 2.78017 0.129345
\(463\) −36.1226 −1.67876 −0.839380 0.543546i \(-0.817081\pi\)
−0.839380 + 0.543546i \(0.817081\pi\)
\(464\) 10.3666 0.481257
\(465\) 47.7650 2.21505
\(466\) 24.5284 1.13626
\(467\) −26.1142 −1.20842 −0.604211 0.796824i \(-0.706512\pi\)
−0.604211 + 0.796824i \(0.706512\pi\)
\(468\) 0 0
\(469\) 6.57002 0.303376
\(470\) 18.5157 0.854067
\(471\) −39.0707 −1.80028
\(472\) 8.45473 0.389161
\(473\) 2.14914 0.0988178
\(474\) 17.3086 0.795009
\(475\) −1.66248 −0.0762799
\(476\) 4.34481 0.199144
\(477\) −0.177620 −0.00813266
\(478\) 14.0325 0.641833
\(479\) 7.21983 0.329883 0.164941 0.986303i \(-0.447257\pi\)
0.164941 + 0.986303i \(0.447257\pi\)
\(480\) −4.32304 −0.197319
\(481\) 0 0
\(482\) −10.4843 −0.477545
\(483\) 3.46681 0.157745
\(484\) −8.30021 −0.377282
\(485\) −30.7845 −1.39785
\(486\) 1.42327 0.0645609
\(487\) −26.0858 −1.18206 −0.591029 0.806650i \(-0.701278\pi\)
−0.591029 + 0.806650i \(0.701278\pi\)
\(488\) 3.52111 0.159393
\(489\) 38.7520 1.75242
\(490\) −2.55496 −0.115421
\(491\) −10.9879 −0.495878 −0.247939 0.968776i \(-0.579753\pi\)
−0.247939 + 0.968776i \(0.579753\pi\)
\(492\) −8.05861 −0.363310
\(493\) −45.0409 −2.02854
\(494\) 0 0
\(495\) −0.575400 −0.0258623
\(496\) −11.0489 −0.496111
\(497\) 15.4547 0.693239
\(498\) 23.6383 1.05926
\(499\) −6.78150 −0.303581 −0.151791 0.988413i \(-0.548504\pi\)
−0.151791 + 0.988413i \(0.548504\pi\)
\(500\) 8.87130 0.396736
\(501\) −13.6353 −0.609182
\(502\) 25.1782 1.12376
\(503\) −22.0073 −0.981257 −0.490628 0.871369i \(-0.663233\pi\)
−0.490628 + 0.871369i \(0.663233\pi\)
\(504\) 0.137063 0.00610529
\(505\) 3.94198 0.175416
\(506\) 3.36658 0.149663
\(507\) 0 0
\(508\) −16.7114 −0.741448
\(509\) −5.02608 −0.222777 −0.111389 0.993777i \(-0.535530\pi\)
−0.111389 + 0.993777i \(0.535530\pi\)
\(510\) 18.7828 0.831717
\(511\) 5.18598 0.229414
\(512\) 1.00000 0.0441942
\(513\) 5.77586 0.255010
\(514\) 13.9825 0.616743
\(515\) 6.73928 0.296968
\(516\) −2.21313 −0.0974275
\(517\) 11.9075 0.523693
\(518\) 1.26875 0.0557457
\(519\) 16.6270 0.729843
\(520\) 0 0
\(521\) −8.08144 −0.354054 −0.177027 0.984206i \(-0.556648\pi\)
−0.177027 + 0.984206i \(0.556648\pi\)
\(522\) −1.42088 −0.0621902
\(523\) −0.530204 −0.0231842 −0.0115921 0.999933i \(-0.503690\pi\)
−0.0115921 + 0.999933i \(0.503690\pi\)
\(524\) −5.85086 −0.255596
\(525\) −2.58509 −0.112823
\(526\) −22.6829 −0.989022
\(527\) 48.0055 2.09115
\(528\) −2.78017 −0.120991
\(529\) −18.8019 −0.817476
\(530\) −3.31096 −0.143819
\(531\) −1.15883 −0.0502891
\(532\) 1.08815 0.0471771
\(533\) 0 0
\(534\) −6.66487 −0.288417
\(535\) −25.9493 −1.12189
\(536\) −6.57002 −0.283782
\(537\) −32.5532 −1.40477
\(538\) −28.0301 −1.20846
\(539\) −1.64310 −0.0707735
\(540\) 13.5617 0.583601
\(541\) −14.5550 −0.625767 −0.312883 0.949792i \(-0.601295\pi\)
−0.312883 + 0.949792i \(0.601295\pi\)
\(542\) −0.161227 −0.00692527
\(543\) 16.2091 0.695598
\(544\) −4.34481 −0.186282
\(545\) 33.8702 1.45084
\(546\) 0 0
\(547\) −3.43727 −0.146967 −0.0734836 0.997296i \(-0.523412\pi\)
−0.0734836 + 0.997296i \(0.523412\pi\)
\(548\) 20.7995 0.888513
\(549\) −0.482615 −0.0205975
\(550\) −2.51035 −0.107042
\(551\) −11.2804 −0.480559
\(552\) −3.46681 −0.147557
\(553\) −10.2295 −0.435003
\(554\) 27.1564 1.15377
\(555\) 5.48486 0.232819
\(556\) 4.79225 0.203237
\(557\) 23.1304 0.980065 0.490032 0.871704i \(-0.336985\pi\)
0.490032 + 0.871704i \(0.336985\pi\)
\(558\) 1.51440 0.0641097
\(559\) 0 0
\(560\) 2.55496 0.107967
\(561\) 12.0793 0.509989
\(562\) 15.4776 0.652882
\(563\) 2.02656 0.0854092 0.0427046 0.999088i \(-0.486403\pi\)
0.0427046 + 0.999088i \(0.486403\pi\)
\(564\) −12.2620 −0.516325
\(565\) 25.1933 1.05989
\(566\) −13.7289 −0.577067
\(567\) 8.57002 0.359907
\(568\) −15.4547 −0.648466
\(569\) 23.5840 0.988694 0.494347 0.869265i \(-0.335407\pi\)
0.494347 + 0.869265i \(0.335407\pi\)
\(570\) 4.70410 0.197033
\(571\) −3.99462 −0.167170 −0.0835849 0.996501i \(-0.526637\pi\)
−0.0835849 + 0.996501i \(0.526637\pi\)
\(572\) 0 0
\(573\) 2.55735 0.106835
\(574\) 4.76271 0.198792
\(575\) −3.13036 −0.130545
\(576\) −0.137063 −0.00571097
\(577\) −37.0877 −1.54398 −0.771990 0.635635i \(-0.780738\pi\)
−0.771990 + 0.635635i \(0.780738\pi\)
\(578\) 1.87741 0.0780900
\(579\) −24.7560 −1.02882
\(580\) −26.4862 −1.09978
\(581\) −13.9705 −0.579592
\(582\) 20.3870 0.845069
\(583\) −2.12929 −0.0881863
\(584\) −5.18598 −0.214597
\(585\) 0 0
\(586\) −21.5948 −0.892072
\(587\) 32.5362 1.34291 0.671456 0.741045i \(-0.265669\pi\)
0.671456 + 0.741045i \(0.265669\pi\)
\(588\) 1.69202 0.0697778
\(589\) 12.0228 0.495392
\(590\) −21.6015 −0.889319
\(591\) 37.9825 1.56239
\(592\) −1.26875 −0.0521453
\(593\) 17.0871 0.701682 0.350841 0.936435i \(-0.385896\pi\)
0.350841 + 0.936435i \(0.385896\pi\)
\(594\) 8.72156 0.357850
\(595\) −11.1008 −0.455089
\(596\) 15.3448 0.628548
\(597\) −14.0941 −0.576834
\(598\) 0 0
\(599\) 21.3521 0.872424 0.436212 0.899844i \(-0.356320\pi\)
0.436212 + 0.899844i \(0.356320\pi\)
\(600\) 2.58509 0.105536
\(601\) −9.23251 −0.376602 −0.188301 0.982111i \(-0.560298\pi\)
−0.188301 + 0.982111i \(0.560298\pi\)
\(602\) 1.30798 0.0533092
\(603\) 0.900509 0.0366716
\(604\) 0.521106 0.0212035
\(605\) 21.2067 0.862174
\(606\) −2.61058 −0.106048
\(607\) 7.52303 0.305350 0.152675 0.988276i \(-0.451211\pi\)
0.152675 + 0.988276i \(0.451211\pi\)
\(608\) −1.08815 −0.0441301
\(609\) −17.5405 −0.710776
\(610\) −8.99628 −0.364249
\(611\) 0 0
\(612\) 0.595515 0.0240723
\(613\) 19.8465 0.801594 0.400797 0.916167i \(-0.368733\pi\)
0.400797 + 0.916167i \(0.368733\pi\)
\(614\) −3.11231 −0.125603
\(615\) 20.5894 0.830245
\(616\) 1.64310 0.0662026
\(617\) −0.214456 −0.00863366 −0.00431683 0.999991i \(-0.501374\pi\)
−0.00431683 + 0.999991i \(0.501374\pi\)
\(618\) −4.46309 −0.179532
\(619\) 45.6577 1.83514 0.917569 0.397577i \(-0.130149\pi\)
0.917569 + 0.397577i \(0.130149\pi\)
\(620\) 28.2295 1.13372
\(621\) 10.8756 0.436423
\(622\) −11.0000 −0.441060
\(623\) 3.93900 0.157813
\(624\) 0 0
\(625\) −30.3048 −1.21219
\(626\) −30.3394 −1.21261
\(627\) 3.02523 0.120816
\(628\) −23.0911 −0.921436
\(629\) 5.51248 0.219797
\(630\) −0.350191 −0.0139519
\(631\) −20.3448 −0.809914 −0.404957 0.914336i \(-0.632714\pi\)
−0.404957 + 0.914336i \(0.632714\pi\)
\(632\) 10.2295 0.406909
\(633\) 17.6692 0.702287
\(634\) −7.08144 −0.281240
\(635\) 42.6969 1.69438
\(636\) 2.19269 0.0869456
\(637\) 0 0
\(638\) −17.0334 −0.674358
\(639\) 2.11828 0.0837977
\(640\) −2.55496 −0.100994
\(641\) −21.3666 −0.843929 −0.421965 0.906612i \(-0.638659\pi\)
−0.421965 + 0.906612i \(0.638659\pi\)
\(642\) 17.1849 0.678235
\(643\) −13.0573 −0.514929 −0.257464 0.966288i \(-0.582887\pi\)
−0.257464 + 0.966288i \(0.582887\pi\)
\(644\) 2.04892 0.0807387
\(645\) 5.65445 0.222644
\(646\) 4.72779 0.186012
\(647\) 18.4838 0.726673 0.363337 0.931658i \(-0.381638\pi\)
0.363337 + 0.931658i \(0.381638\pi\)
\(648\) −8.57002 −0.336662
\(649\) −13.8920 −0.545309
\(650\) 0 0
\(651\) 18.6950 0.732715
\(652\) 22.9028 0.896941
\(653\) −33.1129 −1.29581 −0.647904 0.761722i \(-0.724354\pi\)
−0.647904 + 0.761722i \(0.724354\pi\)
\(654\) −22.4306 −0.877105
\(655\) 14.9487 0.584094
\(656\) −4.76271 −0.185953
\(657\) 0.710808 0.0277313
\(658\) 7.24698 0.282517
\(659\) −26.3545 −1.02663 −0.513313 0.858202i \(-0.671582\pi\)
−0.513313 + 0.858202i \(0.671582\pi\)
\(660\) 7.10321 0.276492
\(661\) 29.9745 1.16587 0.582937 0.812518i \(-0.301904\pi\)
0.582937 + 0.812518i \(0.301904\pi\)
\(662\) 11.4722 0.445879
\(663\) 0 0
\(664\) 13.9705 0.542159
\(665\) −2.78017 −0.107810
\(666\) 0.173899 0.00673845
\(667\) −21.2403 −0.822427
\(668\) −8.05861 −0.311797
\(669\) 26.9560 1.04218
\(670\) 16.7861 0.648505
\(671\) −5.78554 −0.223348
\(672\) −1.69202 −0.0652711
\(673\) 8.85756 0.341434 0.170717 0.985320i \(-0.445392\pi\)
0.170717 + 0.985320i \(0.445392\pi\)
\(674\) 3.96077 0.152563
\(675\) −8.10959 −0.312138
\(676\) 0 0
\(677\) −18.0575 −0.694008 −0.347004 0.937864i \(-0.612801\pi\)
−0.347004 + 0.937864i \(0.612801\pi\)
\(678\) −16.6843 −0.640755
\(679\) −12.0489 −0.462395
\(680\) 11.1008 0.425697
\(681\) −9.35822 −0.358608
\(682\) 18.1545 0.695173
\(683\) 35.2137 1.34742 0.673708 0.738998i \(-0.264701\pi\)
0.673708 + 0.738998i \(0.264701\pi\)
\(684\) 0.149145 0.00570270
\(685\) −53.1420 −2.03045
\(686\) −1.00000 −0.0381802
\(687\) −40.9028 −1.56054
\(688\) −1.30798 −0.0498662
\(689\) 0 0
\(690\) 8.85756 0.337202
\(691\) −10.3120 −0.392288 −0.196144 0.980575i \(-0.562842\pi\)
−0.196144 + 0.980575i \(0.562842\pi\)
\(692\) 9.82669 0.373555
\(693\) −0.225209 −0.00855500
\(694\) 8.24698 0.313051
\(695\) −12.2440 −0.464441
\(696\) 17.5405 0.664870
\(697\) 20.6931 0.783807
\(698\) −14.5670 −0.551371
\(699\) 41.5026 1.56977
\(700\) −1.52781 −0.0577458
\(701\) 27.6025 1.04253 0.521267 0.853394i \(-0.325460\pi\)
0.521267 + 0.853394i \(0.325460\pi\)
\(702\) 0 0
\(703\) 1.38059 0.0520698
\(704\) −1.64310 −0.0619268
\(705\) 31.3290 1.17992
\(706\) 3.04593 0.114635
\(707\) 1.54288 0.0580258
\(708\) 14.3056 0.537637
\(709\) −25.0218 −0.939712 −0.469856 0.882743i \(-0.655694\pi\)
−0.469856 + 0.882743i \(0.655694\pi\)
\(710\) 39.4862 1.48189
\(711\) −1.40209 −0.0525826
\(712\) −3.93900 −0.147620
\(713\) 22.6383 0.847812
\(714\) 7.35152 0.275124
\(715\) 0 0
\(716\) −19.2392 −0.719003
\(717\) 23.7433 0.886711
\(718\) 16.5851 0.618950
\(719\) −42.3347 −1.57882 −0.789408 0.613869i \(-0.789612\pi\)
−0.789408 + 0.613869i \(0.789612\pi\)
\(720\) 0.350191 0.0130509
\(721\) 2.63773 0.0982341
\(722\) −17.8159 −0.663041
\(723\) −17.7396 −0.659743
\(724\) 9.57971 0.356027
\(725\) 15.8382 0.588215
\(726\) −14.0441 −0.521227
\(727\) −6.22090 −0.230720 −0.115360 0.993324i \(-0.536802\pi\)
−0.115360 + 0.993324i \(0.536802\pi\)
\(728\) 0 0
\(729\) 28.1183 1.04142
\(730\) 13.2500 0.490403
\(731\) 5.68292 0.210191
\(732\) 5.95779 0.220206
\(733\) 6.42652 0.237369 0.118684 0.992932i \(-0.462132\pi\)
0.118684 + 0.992932i \(0.462132\pi\)
\(734\) 0.966148 0.0356612
\(735\) −4.32304 −0.159458
\(736\) −2.04892 −0.0755241
\(737\) 10.7952 0.397648
\(738\) 0.652793 0.0240296
\(739\) 0.244587 0.00899727 0.00449863 0.999990i \(-0.498568\pi\)
0.00449863 + 0.999990i \(0.498568\pi\)
\(740\) 3.24160 0.119164
\(741\) 0 0
\(742\) −1.29590 −0.0475739
\(743\) −27.1414 −0.995720 −0.497860 0.867257i \(-0.665881\pi\)
−0.497860 + 0.867257i \(0.665881\pi\)
\(744\) −18.6950 −0.685392
\(745\) −39.2054 −1.43637
\(746\) 32.0901 1.17490
\(747\) −1.91484 −0.0700603
\(748\) 7.13898 0.261027
\(749\) −10.1564 −0.371108
\(750\) 15.0104 0.548103
\(751\) 17.1903 0.627283 0.313641 0.949542i \(-0.398451\pi\)
0.313641 + 0.949542i \(0.398451\pi\)
\(752\) −7.24698 −0.264270
\(753\) 42.6021 1.55251
\(754\) 0 0
\(755\) −1.33140 −0.0484548
\(756\) 5.30798 0.193049
\(757\) 34.7778 1.26402 0.632010 0.774960i \(-0.282230\pi\)
0.632010 + 0.774960i \(0.282230\pi\)
\(758\) −0.972853 −0.0353356
\(759\) 5.69633 0.206764
\(760\) 2.78017 0.100847
\(761\) 15.0441 0.545349 0.272675 0.962106i \(-0.412092\pi\)
0.272675 + 0.962106i \(0.412092\pi\)
\(762\) −28.2760 −1.02433
\(763\) 13.2567 0.479924
\(764\) 1.51142 0.0546812
\(765\) −1.52152 −0.0550105
\(766\) −5.47650 −0.197874
\(767\) 0 0
\(768\) 1.69202 0.0610556
\(769\) 28.8170 1.03917 0.519584 0.854419i \(-0.326087\pi\)
0.519584 + 0.854419i \(0.326087\pi\)
\(770\) −4.19806 −0.151288
\(771\) 23.6588 0.852049
\(772\) −14.6310 −0.526582
\(773\) 30.0428 1.08056 0.540282 0.841484i \(-0.318317\pi\)
0.540282 + 0.841484i \(0.318317\pi\)
\(774\) 0.179276 0.00644394
\(775\) −16.8807 −0.606371
\(776\) 12.0489 0.432531
\(777\) 2.14675 0.0770143
\(778\) −32.5066 −1.16542
\(779\) 5.18252 0.185683
\(780\) 0 0
\(781\) 25.3937 0.908659
\(782\) 8.90217 0.318341
\(783\) −55.0256 −1.96645
\(784\) 1.00000 0.0357143
\(785\) 58.9969 2.10569
\(786\) −9.89977 −0.353113
\(787\) −11.1946 −0.399045 −0.199522 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(788\) 22.4480 0.799678
\(789\) −38.3800 −1.36636
\(790\) −26.1360 −0.929877
\(791\) 9.86054 0.350601
\(792\) 0.225209 0.00800247
\(793\) 0 0
\(794\) −34.4674 −1.22320
\(795\) −5.60222 −0.198690
\(796\) −8.32975 −0.295240
\(797\) −3.46788 −0.122839 −0.0614193 0.998112i \(-0.519563\pi\)
−0.0614193 + 0.998112i \(0.519563\pi\)
\(798\) 1.84117 0.0651766
\(799\) 31.4868 1.11392
\(800\) 1.52781 0.0540163
\(801\) 0.539893 0.0190762
\(802\) 12.9903 0.458704
\(803\) 8.52111 0.300703
\(804\) −11.1166 −0.392053
\(805\) −5.23490 −0.184506
\(806\) 0 0
\(807\) −47.4276 −1.66953
\(808\) −1.54288 −0.0542782
\(809\) 42.5303 1.49529 0.747643 0.664101i \(-0.231185\pi\)
0.747643 + 0.664101i \(0.231185\pi\)
\(810\) 21.8961 0.769349
\(811\) −21.2519 −0.746255 −0.373127 0.927780i \(-0.621715\pi\)
−0.373127 + 0.927780i \(0.621715\pi\)
\(812\) −10.3666 −0.363796
\(813\) −0.272799 −0.00956747
\(814\) 2.08469 0.0730683
\(815\) −58.5156 −2.04971
\(816\) −7.35152 −0.257355
\(817\) 1.42327 0.0497940
\(818\) 31.4771 1.10057
\(819\) 0 0
\(820\) 12.1685 0.424943
\(821\) 2.81295 0.0981728 0.0490864 0.998795i \(-0.484369\pi\)
0.0490864 + 0.998795i \(0.484369\pi\)
\(822\) 35.1933 1.22751
\(823\) 34.5244 1.20344 0.601722 0.798706i \(-0.294482\pi\)
0.601722 + 0.798706i \(0.294482\pi\)
\(824\) −2.63773 −0.0918896
\(825\) −4.24757 −0.147881
\(826\) −8.45473 −0.294178
\(827\) −14.8877 −0.517696 −0.258848 0.965918i \(-0.583343\pi\)
−0.258848 + 0.965918i \(0.583343\pi\)
\(828\) 0.280831 0.00975957
\(829\) −27.6606 −0.960691 −0.480346 0.877079i \(-0.659489\pi\)
−0.480346 + 0.877079i \(0.659489\pi\)
\(830\) −35.6939 −1.23895
\(831\) 45.9493 1.59396
\(832\) 0 0
\(833\) −4.34481 −0.150539
\(834\) 8.10859 0.280777
\(835\) 20.5894 0.712526
\(836\) 1.78794 0.0618371
\(837\) 58.6474 2.02715
\(838\) −22.4101 −0.774145
\(839\) 9.57673 0.330625 0.165313 0.986241i \(-0.447137\pi\)
0.165313 + 0.986241i \(0.447137\pi\)
\(840\) 4.32304 0.149159
\(841\) 78.4661 2.70573
\(842\) −17.2373 −0.594036
\(843\) 26.1884 0.901975
\(844\) 10.4426 0.359451
\(845\) 0 0
\(846\) 0.993295 0.0341502
\(847\) 8.30021 0.285199
\(848\) 1.29590 0.0445013
\(849\) −23.2295 −0.797235
\(850\) −6.63806 −0.227683
\(851\) 2.59956 0.0891119
\(852\) −26.1497 −0.895875
\(853\) −39.5816 −1.35525 −0.677625 0.735408i \(-0.736991\pi\)
−0.677625 + 0.735408i \(0.736991\pi\)
\(854\) −3.52111 −0.120490
\(855\) −0.381059 −0.0130319
\(856\) 10.1564 0.347140
\(857\) −17.6547 −0.603073 −0.301537 0.953455i \(-0.597500\pi\)
−0.301537 + 0.953455i \(0.597500\pi\)
\(858\) 0 0
\(859\) 37.6862 1.28584 0.642918 0.765935i \(-0.277724\pi\)
0.642918 + 0.765935i \(0.277724\pi\)
\(860\) 3.34183 0.113955
\(861\) 8.05861 0.274637
\(862\) 30.4349 1.03662
\(863\) −25.9758 −0.884228 −0.442114 0.896959i \(-0.645771\pi\)
−0.442114 + 0.896959i \(0.645771\pi\)
\(864\) −5.30798 −0.180581
\(865\) −25.1068 −0.853656
\(866\) 29.1812 0.991617
\(867\) 3.17662 0.107884
\(868\) 11.0489 0.375025
\(869\) −16.8082 −0.570178
\(870\) −44.8152 −1.51938
\(871\) 0 0
\(872\) −13.2567 −0.448927
\(873\) −1.65146 −0.0558936
\(874\) 2.22952 0.0754147
\(875\) −8.87130 −0.299905
\(876\) −8.77479 −0.296473
\(877\) −29.8133 −1.00672 −0.503362 0.864076i \(-0.667904\pi\)
−0.503362 + 0.864076i \(0.667904\pi\)
\(878\) −23.2349 −0.784140
\(879\) −36.5388 −1.23242
\(880\) 4.19806 0.141517
\(881\) 56.3769 1.89938 0.949692 0.313184i \(-0.101396\pi\)
0.949692 + 0.313184i \(0.101396\pi\)
\(882\) −0.137063 −0.00461516
\(883\) −33.0726 −1.11298 −0.556491 0.830854i \(-0.687853\pi\)
−0.556491 + 0.830854i \(0.687853\pi\)
\(884\) 0 0
\(885\) −36.5502 −1.22862
\(886\) −26.1371 −0.878092
\(887\) −42.7469 −1.43530 −0.717650 0.696404i \(-0.754782\pi\)
−0.717650 + 0.696404i \(0.754782\pi\)
\(888\) −2.14675 −0.0720403
\(889\) 16.7114 0.560482
\(890\) 10.0640 0.337345
\(891\) 14.0814 0.471746
\(892\) 15.9312 0.533417
\(893\) 7.88577 0.263887
\(894\) 25.9638 0.868358
\(895\) 49.1554 1.64308
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −4.29350 −0.143276
\(899\) −114.540 −3.82011
\(900\) −0.209407 −0.00698023
\(901\) −5.63043 −0.187577
\(902\) 7.82563 0.260565
\(903\) 2.21313 0.0736483
\(904\) −9.86054 −0.327957
\(905\) −24.4758 −0.813602
\(906\) 0.881723 0.0292933
\(907\) −30.7881 −1.02230 −0.511150 0.859491i \(-0.670780\pi\)
−0.511150 + 0.859491i \(0.670780\pi\)
\(908\) −5.53079 −0.183546
\(909\) 0.211472 0.00701408
\(910\) 0 0
\(911\) −11.8901 −0.393936 −0.196968 0.980410i \(-0.563110\pi\)
−0.196968 + 0.980410i \(0.563110\pi\)
\(912\) −1.84117 −0.0609671
\(913\) −22.9549 −0.759697
\(914\) −0.781495 −0.0258496
\(915\) −15.2219 −0.503220
\(916\) −24.1739 −0.798728
\(917\) 5.85086 0.193212
\(918\) 23.0622 0.761165
\(919\) −42.9705 −1.41746 −0.708732 0.705477i \(-0.750733\pi\)
−0.708732 + 0.705477i \(0.750733\pi\)
\(920\) 5.23490 0.172589
\(921\) −5.26609 −0.173524
\(922\) −17.8103 −0.586551
\(923\) 0 0
\(924\) 2.78017 0.0914608
\(925\) −1.93841 −0.0637345
\(926\) −36.1226 −1.18706
\(927\) 0.361536 0.0118744
\(928\) 10.3666 0.340300
\(929\) −15.0519 −0.493837 −0.246918 0.969036i \(-0.579418\pi\)
−0.246918 + 0.969036i \(0.579418\pi\)
\(930\) 47.7650 1.56627
\(931\) −1.08815 −0.0356625
\(932\) 24.5284 0.803455
\(933\) −18.6122 −0.609337
\(934\) −26.1142 −0.854484
\(935\) −18.2398 −0.596505
\(936\) 0 0
\(937\) 7.25965 0.237162 0.118581 0.992944i \(-0.462165\pi\)
0.118581 + 0.992944i \(0.462165\pi\)
\(938\) 6.57002 0.214519
\(939\) −51.3350 −1.67525
\(940\) 18.5157 0.603917
\(941\) −25.4588 −0.829932 −0.414966 0.909837i \(-0.636207\pi\)
−0.414966 + 0.909837i \(0.636207\pi\)
\(942\) −39.0707 −1.27299
\(943\) 9.75840 0.317777
\(944\) 8.45473 0.275178
\(945\) −13.5617 −0.441161
\(946\) 2.14914 0.0698747
\(947\) −30.9831 −1.00682 −0.503408 0.864049i \(-0.667921\pi\)
−0.503408 + 0.864049i \(0.667921\pi\)
\(948\) 17.3086 0.562156
\(949\) 0 0
\(950\) −1.66248 −0.0539380
\(951\) −11.9820 −0.388541
\(952\) 4.34481 0.140816
\(953\) 12.4179 0.402255 0.201128 0.979565i \(-0.435539\pi\)
0.201128 + 0.979565i \(0.435539\pi\)
\(954\) −0.177620 −0.00575066
\(955\) −3.86161 −0.124959
\(956\) 14.0325 0.453844
\(957\) −28.8208 −0.931645
\(958\) 7.21983 0.233262
\(959\) −20.7995 −0.671652
\(960\) −4.32304 −0.139526
\(961\) 91.0786 2.93802
\(962\) 0 0
\(963\) −1.39208 −0.0448590
\(964\) −10.4843 −0.337676
\(965\) 37.3817 1.20336
\(966\) 3.46681 0.111543
\(967\) 54.8426 1.76362 0.881810 0.471605i \(-0.156325\pi\)
0.881810 + 0.471605i \(0.156325\pi\)
\(968\) −8.30021 −0.266779
\(969\) 7.99953 0.256982
\(970\) −30.7845 −0.988430
\(971\) 14.5338 0.466411 0.233206 0.972427i \(-0.425078\pi\)
0.233206 + 0.972427i \(0.425078\pi\)
\(972\) 1.42327 0.0456515
\(973\) −4.79225 −0.153632
\(974\) −26.0858 −0.835841
\(975\) 0 0
\(976\) 3.52111 0.112708
\(977\) 42.8866 1.37206 0.686032 0.727571i \(-0.259351\pi\)
0.686032 + 0.727571i \(0.259351\pi\)
\(978\) 38.7520 1.23915
\(979\) 6.47219 0.206852
\(980\) −2.55496 −0.0816151
\(981\) 1.81700 0.0580124
\(982\) −10.9879 −0.350639
\(983\) 10.7918 0.344204 0.172102 0.985079i \(-0.444944\pi\)
0.172102 + 0.985079i \(0.444944\pi\)
\(984\) −8.05861 −0.256899
\(985\) −57.3538 −1.82744
\(986\) −45.0409 −1.43439
\(987\) 12.2620 0.390305
\(988\) 0 0
\(989\) 2.67994 0.0852171
\(990\) −0.575400 −0.0182874
\(991\) −44.6045 −1.41691 −0.708454 0.705757i \(-0.750607\pi\)
−0.708454 + 0.705757i \(0.750607\pi\)
\(992\) −11.0489 −0.350803
\(993\) 19.4112 0.615995
\(994\) 15.4547 0.490194
\(995\) 21.2822 0.674690
\(996\) 23.6383 0.749009
\(997\) 41.8751 1.32620 0.663099 0.748531i \(-0.269241\pi\)
0.663099 + 0.748531i \(0.269241\pi\)
\(998\) −6.78150 −0.214664
\(999\) 6.73450 0.213070
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 2366.2.a.bb.1.3 yes 3
13.5 odd 4 2366.2.d.o.337.3 6
13.8 odd 4 2366.2.d.o.337.6 6
13.12 even 2 2366.2.a.w.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2366.2.a.w.1.3 3 13.12 even 2
2366.2.a.bb.1.3 yes 3 1.1 even 1 trivial
2366.2.d.o.337.3 6 13.5 odd 4
2366.2.d.o.337.6 6 13.8 odd 4