Properties

Label 8281.2.a.bj.1.2
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
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] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, 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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.554958 q^{2} -0.801938 q^{3} -1.69202 q^{4} -2.80194 q^{5} -0.445042 q^{6} -2.04892 q^{8} -2.35690 q^{9} +O(q^{10})\) \(q+0.554958 q^{2} -0.801938 q^{3} -1.69202 q^{4} -2.80194 q^{5} -0.445042 q^{6} -2.04892 q^{8} -2.35690 q^{9} -1.55496 q^{10} +1.19806 q^{11} +1.35690 q^{12} +2.24698 q^{15} +2.24698 q^{16} -1.13706 q^{17} -1.30798 q^{18} +1.93900 q^{19} +4.74094 q^{20} +0.664874 q^{22} -4.60388 q^{23} +1.64310 q^{24} +2.85086 q^{25} +4.29590 q^{27} -7.89977 q^{29} +1.24698 q^{30} +5.89977 q^{31} +5.34481 q^{32} -0.960771 q^{33} -0.631023 q^{34} +3.98792 q^{36} -0.951083 q^{37} +1.07606 q^{38} +5.74094 q^{40} +3.31767 q^{41} +7.15883 q^{43} -2.02715 q^{44} +6.60388 q^{45} -2.55496 q^{46} -7.69202 q^{47} -1.80194 q^{48} +1.58211 q^{50} +0.911854 q^{51} +5.87263 q^{53} +2.38404 q^{54} -3.35690 q^{55} -1.55496 q^{57} -4.38404 q^{58} -0.0120816 q^{59} -3.80194 q^{60} +8.03684 q^{61} +3.27413 q^{62} -1.52781 q^{64} -0.533188 q^{66} +9.25667 q^{67} +1.92394 q^{68} +3.69202 q^{69} +13.7409 q^{71} +4.82908 q^{72} +12.8170 q^{73} -0.527811 q^{74} -2.28621 q^{75} -3.28083 q^{76} +0.807315 q^{79} -6.29590 q^{80} +3.62565 q^{81} +1.84117 q^{82} -16.3327 q^{83} +3.18598 q^{85} +3.97285 q^{86} +6.33513 q^{87} -2.45473 q^{88} -14.7289 q^{89} +3.66487 q^{90} +7.78986 q^{92} -4.73125 q^{93} -4.26875 q^{94} -5.43296 q^{95} -4.28621 q^{96} +3.13169 q^{97} -2.82371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 2 q^{3} - 4 q^{5} - q^{6} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 2 q^{3} - 4 q^{5} - q^{6} + 3 q^{8} - 3 q^{9} - 5 q^{10} + 8 q^{11} + 2 q^{15} + 2 q^{16} + 2 q^{17} - 9 q^{18} - 4 q^{19} + 3 q^{22} - 5 q^{23} + 9 q^{24} - 5 q^{25} - q^{27} - q^{29} - q^{30} - 5 q^{31} - 7 q^{32} + 10 q^{33} + 13 q^{34} - 7 q^{36} - 12 q^{37} - 12 q^{38} + 3 q^{40} - 7 q^{41} + 13 q^{43} + 11 q^{45} - 8 q^{46} - 18 q^{47} - q^{48} - q^{50} - q^{51} + q^{53} - 3 q^{54} - 6 q^{55} - 5 q^{57} - 3 q^{58} - 19 q^{59} - 7 q^{60} - 4 q^{61} - q^{62} - 11 q^{64} - 5 q^{66} + q^{67} + 21 q^{68} + 6 q^{69} + 27 q^{71} + 4 q^{72} + 9 q^{73} - 8 q^{74} - 15 q^{75} - 21 q^{76} - 5 q^{79} - 5 q^{80} - q^{81} + 14 q^{82} - 7 q^{83} - 5 q^{85} + 18 q^{86} + 18 q^{87} + 15 q^{88} - 11 q^{89} + 12 q^{90} - 22 q^{93} - 5 q^{94} + 3 q^{95} - 21 q^{96} + 7 q^{97} - q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.554958 0.392415 0.196207 0.980562i \(-0.437137\pi\)
0.196207 + 0.980562i \(0.437137\pi\)
\(3\) −0.801938 −0.462999 −0.231499 0.972835i \(-0.574363\pi\)
−0.231499 + 0.972835i \(0.574363\pi\)
\(4\) −1.69202 −0.846011
\(5\) −2.80194 −1.25306 −0.626532 0.779395i \(-0.715526\pi\)
−0.626532 + 0.779395i \(0.715526\pi\)
\(6\) −0.445042 −0.181688
\(7\) 0 0
\(8\) −2.04892 −0.724402
\(9\) −2.35690 −0.785632
\(10\) −1.55496 −0.491721
\(11\) 1.19806 0.361229 0.180615 0.983554i \(-0.442191\pi\)
0.180615 + 0.983554i \(0.442191\pi\)
\(12\) 1.35690 0.391702
\(13\) 0 0
\(14\) 0 0
\(15\) 2.24698 0.580168
\(16\) 2.24698 0.561745
\(17\) −1.13706 −0.275778 −0.137889 0.990448i \(-0.544032\pi\)
−0.137889 + 0.990448i \(0.544032\pi\)
\(18\) −1.30798 −0.308293
\(19\) 1.93900 0.444837 0.222419 0.974951i \(-0.428605\pi\)
0.222419 + 0.974951i \(0.428605\pi\)
\(20\) 4.74094 1.06011
\(21\) 0 0
\(22\) 0.664874 0.141752
\(23\) −4.60388 −0.959974 −0.479987 0.877275i \(-0.659359\pi\)
−0.479987 + 0.877275i \(0.659359\pi\)
\(24\) 1.64310 0.335397
\(25\) 2.85086 0.570171
\(26\) 0 0
\(27\) 4.29590 0.826746
\(28\) 0 0
\(29\) −7.89977 −1.46695 −0.733475 0.679716i \(-0.762103\pi\)
−0.733475 + 0.679716i \(0.762103\pi\)
\(30\) 1.24698 0.227666
\(31\) 5.89977 1.05963 0.529815 0.848113i \(-0.322261\pi\)
0.529815 + 0.848113i \(0.322261\pi\)
\(32\) 5.34481 0.944839
\(33\) −0.960771 −0.167249
\(34\) −0.631023 −0.108219
\(35\) 0 0
\(36\) 3.98792 0.664653
\(37\) −0.951083 −0.156357 −0.0781785 0.996939i \(-0.524910\pi\)
−0.0781785 + 0.996939i \(0.524910\pi\)
\(38\) 1.07606 0.174561
\(39\) 0 0
\(40\) 5.74094 0.907722
\(41\) 3.31767 0.518133 0.259066 0.965860i \(-0.416585\pi\)
0.259066 + 0.965860i \(0.416585\pi\)
\(42\) 0 0
\(43\) 7.15883 1.09171 0.545856 0.837879i \(-0.316205\pi\)
0.545856 + 0.837879i \(0.316205\pi\)
\(44\) −2.02715 −0.305604
\(45\) 6.60388 0.984448
\(46\) −2.55496 −0.376708
\(47\) −7.69202 −1.12200 −0.560998 0.827817i \(-0.689583\pi\)
−0.560998 + 0.827817i \(0.689583\pi\)
\(48\) −1.80194 −0.260087
\(49\) 0 0
\(50\) 1.58211 0.223743
\(51\) 0.911854 0.127685
\(52\) 0 0
\(53\) 5.87263 0.806667 0.403334 0.915053i \(-0.367851\pi\)
0.403334 + 0.915053i \(0.367851\pi\)
\(54\) 2.38404 0.324427
\(55\) −3.35690 −0.452644
\(56\) 0 0
\(57\) −1.55496 −0.205959
\(58\) −4.38404 −0.575653
\(59\) −0.0120816 −0.00157289 −0.000786444 1.00000i \(-0.500250\pi\)
−0.000786444 1.00000i \(0.500250\pi\)
\(60\) −3.80194 −0.490828
\(61\) 8.03684 1.02901 0.514506 0.857487i \(-0.327975\pi\)
0.514506 + 0.857487i \(0.327975\pi\)
\(62\) 3.27413 0.415815
\(63\) 0 0
\(64\) −1.52781 −0.190976
\(65\) 0 0
\(66\) −0.533188 −0.0656309
\(67\) 9.25667 1.13088 0.565441 0.824789i \(-0.308706\pi\)
0.565441 + 0.824789i \(0.308706\pi\)
\(68\) 1.92394 0.233311
\(69\) 3.69202 0.444467
\(70\) 0 0
\(71\) 13.7409 1.63075 0.815375 0.578934i \(-0.196531\pi\)
0.815375 + 0.578934i \(0.196531\pi\)
\(72\) 4.82908 0.569113
\(73\) 12.8170 1.50012 0.750058 0.661372i \(-0.230025\pi\)
0.750058 + 0.661372i \(0.230025\pi\)
\(74\) −0.527811 −0.0613568
\(75\) −2.28621 −0.263989
\(76\) −3.28083 −0.376337
\(77\) 0 0
\(78\) 0 0
\(79\) 0.807315 0.0908300 0.0454150 0.998968i \(-0.485539\pi\)
0.0454150 + 0.998968i \(0.485539\pi\)
\(80\) −6.29590 −0.703903
\(81\) 3.62565 0.402850
\(82\) 1.84117 0.203323
\(83\) −16.3327 −1.79275 −0.896375 0.443296i \(-0.853809\pi\)
−0.896375 + 0.443296i \(0.853809\pi\)
\(84\) 0 0
\(85\) 3.18598 0.345568
\(86\) 3.97285 0.428404
\(87\) 6.33513 0.679197
\(88\) −2.45473 −0.261675
\(89\) −14.7289 −1.56126 −0.780628 0.624996i \(-0.785101\pi\)
−0.780628 + 0.624996i \(0.785101\pi\)
\(90\) 3.66487 0.386312
\(91\) 0 0
\(92\) 7.78986 0.812149
\(93\) −4.73125 −0.490608
\(94\) −4.26875 −0.440288
\(95\) −5.43296 −0.557410
\(96\) −4.28621 −0.437459
\(97\) 3.13169 0.317975 0.158987 0.987281i \(-0.449177\pi\)
0.158987 + 0.987281i \(0.449177\pi\)
\(98\) 0 0
\(99\) −2.82371 −0.283793
\(100\) −4.82371 −0.482371
\(101\) −5.29052 −0.526426 −0.263213 0.964738i \(-0.584782\pi\)
−0.263213 + 0.964738i \(0.584782\pi\)
\(102\) 0.506041 0.0501055
\(103\) 13.5308 1.33323 0.666614 0.745403i \(-0.267743\pi\)
0.666614 + 0.745403i \(0.267743\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 3.25906 0.316548
\(107\) 5.63102 0.544371 0.272186 0.962245i \(-0.412253\pi\)
0.272186 + 0.962245i \(0.412253\pi\)
\(108\) −7.26875 −0.699436
\(109\) −4.17629 −0.400016 −0.200008 0.979794i \(-0.564097\pi\)
−0.200008 + 0.979794i \(0.564097\pi\)
\(110\) −1.86294 −0.177624
\(111\) 0.762709 0.0723931
\(112\) 0 0
\(113\) 7.64310 0.719003 0.359501 0.933145i \(-0.382947\pi\)
0.359501 + 0.933145i \(0.382947\pi\)
\(114\) −0.862937 −0.0808214
\(115\) 12.8998 1.20291
\(116\) 13.3666 1.24106
\(117\) 0 0
\(118\) −0.00670477 −0.000617224 0
\(119\) 0 0
\(120\) −4.60388 −0.420274
\(121\) −9.56465 −0.869513
\(122\) 4.46011 0.403799
\(123\) −2.66056 −0.239895
\(124\) −9.98254 −0.896459
\(125\) 6.02177 0.538604
\(126\) 0 0
\(127\) 6.77777 0.601430 0.300715 0.953714i \(-0.402775\pi\)
0.300715 + 0.953714i \(0.402775\pi\)
\(128\) −11.5375 −1.01978
\(129\) −5.74094 −0.505461
\(130\) 0 0
\(131\) 13.6799 1.19522 0.597611 0.801786i \(-0.296117\pi\)
0.597611 + 0.801786i \(0.296117\pi\)
\(132\) 1.62565 0.141494
\(133\) 0 0
\(134\) 5.13706 0.443775
\(135\) −12.0368 −1.03597
\(136\) 2.32975 0.199774
\(137\) −12.9879 −1.10963 −0.554816 0.831973i \(-0.687211\pi\)
−0.554816 + 0.831973i \(0.687211\pi\)
\(138\) 2.04892 0.174415
\(139\) −12.0465 −1.02177 −0.510886 0.859648i \(-0.670683\pi\)
−0.510886 + 0.859648i \(0.670683\pi\)
\(140\) 0 0
\(141\) 6.16852 0.519483
\(142\) 7.62565 0.639930
\(143\) 0 0
\(144\) −5.29590 −0.441325
\(145\) 22.1347 1.83818
\(146\) 7.11290 0.588668
\(147\) 0 0
\(148\) 1.60925 0.132280
\(149\) 0.740939 0.0607001 0.0303500 0.999539i \(-0.490338\pi\)
0.0303500 + 0.999539i \(0.490338\pi\)
\(150\) −1.26875 −0.103593
\(151\) 19.0737 1.55219 0.776097 0.630614i \(-0.217197\pi\)
0.776097 + 0.630614i \(0.217197\pi\)
\(152\) −3.97285 −0.322241
\(153\) 2.67994 0.216660
\(154\) 0 0
\(155\) −16.5308 −1.32779
\(156\) 0 0
\(157\) 4.02177 0.320972 0.160486 0.987038i \(-0.448694\pi\)
0.160486 + 0.987038i \(0.448694\pi\)
\(158\) 0.448026 0.0356430
\(159\) −4.70948 −0.373486
\(160\) −14.9758 −1.18394
\(161\) 0 0
\(162\) 2.01208 0.158084
\(163\) −15.1371 −1.18563 −0.592813 0.805340i \(-0.701983\pi\)
−0.592813 + 0.805340i \(0.701983\pi\)
\(164\) −5.61356 −0.438346
\(165\) 2.69202 0.209574
\(166\) −9.06398 −0.703502
\(167\) −6.26337 −0.484674 −0.242337 0.970192i \(-0.577914\pi\)
−0.242337 + 0.970192i \(0.577914\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1.76809 0.135606
\(171\) −4.57002 −0.349478
\(172\) −12.1129 −0.923600
\(173\) −16.3913 −1.24621 −0.623105 0.782138i \(-0.714129\pi\)
−0.623105 + 0.782138i \(0.714129\pi\)
\(174\) 3.51573 0.266527
\(175\) 0 0
\(176\) 2.69202 0.202919
\(177\) 0.00968868 0.000728246 0
\(178\) −8.17390 −0.612660
\(179\) −2.45473 −0.183475 −0.0917376 0.995783i \(-0.529242\pi\)
−0.0917376 + 0.995783i \(0.529242\pi\)
\(180\) −11.1739 −0.832853
\(181\) −11.8073 −0.877631 −0.438815 0.898577i \(-0.644602\pi\)
−0.438815 + 0.898577i \(0.644602\pi\)
\(182\) 0 0
\(183\) −6.44504 −0.476431
\(184\) 9.43296 0.695407
\(185\) 2.66487 0.195925
\(186\) −2.62565 −0.192522
\(187\) −1.36227 −0.0996192
\(188\) 13.0151 0.949221
\(189\) 0 0
\(190\) −3.01507 −0.218736
\(191\) −8.99330 −0.650732 −0.325366 0.945588i \(-0.605488\pi\)
−0.325366 + 0.945588i \(0.605488\pi\)
\(192\) 1.22521 0.0884219
\(193\) −13.5254 −0.973581 −0.486790 0.873519i \(-0.661832\pi\)
−0.486790 + 0.873519i \(0.661832\pi\)
\(194\) 1.73795 0.124778
\(195\) 0 0
\(196\) 0 0
\(197\) −12.9758 −0.924490 −0.462245 0.886752i \(-0.652956\pi\)
−0.462245 + 0.886752i \(0.652956\pi\)
\(198\) −1.56704 −0.111365
\(199\) 13.5864 0.963116 0.481558 0.876414i \(-0.340071\pi\)
0.481558 + 0.876414i \(0.340071\pi\)
\(200\) −5.84117 −0.413033
\(201\) −7.42327 −0.523597
\(202\) −2.93602 −0.206577
\(203\) 0 0
\(204\) −1.54288 −0.108023
\(205\) −9.29590 −0.649254
\(206\) 7.50902 0.523179
\(207\) 10.8509 0.754187
\(208\) 0 0
\(209\) 2.32304 0.160688
\(210\) 0 0
\(211\) 10.4601 0.720103 0.360052 0.932932i \(-0.382759\pi\)
0.360052 + 0.932932i \(0.382759\pi\)
\(212\) −9.93661 −0.682449
\(213\) −11.0194 −0.755035
\(214\) 3.12498 0.213619
\(215\) −20.0586 −1.36799
\(216\) −8.80194 −0.598896
\(217\) 0 0
\(218\) −2.31767 −0.156972
\(219\) −10.2784 −0.694553
\(220\) 5.67994 0.382941
\(221\) 0 0
\(222\) 0.423272 0.0284081
\(223\) −11.4058 −0.763790 −0.381895 0.924206i \(-0.624728\pi\)
−0.381895 + 0.924206i \(0.624728\pi\)
\(224\) 0 0
\(225\) −6.71917 −0.447945
\(226\) 4.24160 0.282147
\(227\) 10.6407 0.706249 0.353124 0.935576i \(-0.385119\pi\)
0.353124 + 0.935576i \(0.385119\pi\)
\(228\) 2.63102 0.174244
\(229\) −1.13946 −0.0752974 −0.0376487 0.999291i \(-0.511987\pi\)
−0.0376487 + 0.999291i \(0.511987\pi\)
\(230\) 7.15883 0.472040
\(231\) 0 0
\(232\) 16.1860 1.06266
\(233\) −10.8509 −0.710863 −0.355432 0.934702i \(-0.615666\pi\)
−0.355432 + 0.934702i \(0.615666\pi\)
\(234\) 0 0
\(235\) 21.5526 1.40593
\(236\) 0.0204423 0.00133068
\(237\) −0.647416 −0.0420542
\(238\) 0 0
\(239\) 11.9293 0.771643 0.385822 0.922573i \(-0.373918\pi\)
0.385822 + 0.922573i \(0.373918\pi\)
\(240\) 5.04892 0.325906
\(241\) −3.64848 −0.235019 −0.117510 0.993072i \(-0.537491\pi\)
−0.117510 + 0.993072i \(0.537491\pi\)
\(242\) −5.30798 −0.341210
\(243\) −15.7952 −1.01326
\(244\) −13.5985 −0.870555
\(245\) 0 0
\(246\) −1.47650 −0.0941383
\(247\) 0 0
\(248\) −12.0881 −0.767598
\(249\) 13.0978 0.830042
\(250\) 3.34183 0.211356
\(251\) −1.37329 −0.0866813 −0.0433406 0.999060i \(-0.513800\pi\)
−0.0433406 + 0.999060i \(0.513800\pi\)
\(252\) 0 0
\(253\) −5.51573 −0.346771
\(254\) 3.76138 0.236010
\(255\) −2.55496 −0.159998
\(256\) −3.34721 −0.209200
\(257\) −29.4359 −1.83616 −0.918082 0.396391i \(-0.870263\pi\)
−0.918082 + 0.396391i \(0.870263\pi\)
\(258\) −3.18598 −0.198350
\(259\) 0 0
\(260\) 0 0
\(261\) 18.6189 1.15248
\(262\) 7.59179 0.469023
\(263\) 10.6963 0.659564 0.329782 0.944057i \(-0.393025\pi\)
0.329782 + 0.944057i \(0.393025\pi\)
\(264\) 1.96854 0.121155
\(265\) −16.4547 −1.01081
\(266\) 0 0
\(267\) 11.8116 0.722860
\(268\) −15.6625 −0.956738
\(269\) 10.1860 0.621050 0.310525 0.950565i \(-0.399495\pi\)
0.310525 + 0.950565i \(0.399495\pi\)
\(270\) −6.67994 −0.406528
\(271\) −29.4523 −1.78910 −0.894551 0.446966i \(-0.852505\pi\)
−0.894551 + 0.446966i \(0.852505\pi\)
\(272\) −2.55496 −0.154917
\(273\) 0 0
\(274\) −7.20775 −0.435436
\(275\) 3.41550 0.205963
\(276\) −6.24698 −0.376024
\(277\) −10.2446 −0.615538 −0.307769 0.951461i \(-0.599582\pi\)
−0.307769 + 0.951461i \(0.599582\pi\)
\(278\) −6.68532 −0.400959
\(279\) −13.9051 −0.832480
\(280\) 0 0
\(281\) 11.5646 0.689889 0.344944 0.938623i \(-0.387898\pi\)
0.344944 + 0.938623i \(0.387898\pi\)
\(282\) 3.42327 0.203853
\(283\) 30.7090 1.82546 0.912730 0.408562i \(-0.133970\pi\)
0.912730 + 0.408562i \(0.133970\pi\)
\(284\) −23.2500 −1.37963
\(285\) 4.35690 0.258080
\(286\) 0 0
\(287\) 0 0
\(288\) −12.5972 −0.742295
\(289\) −15.7071 −0.923946
\(290\) 12.2838 0.721330
\(291\) −2.51142 −0.147222
\(292\) −21.6866 −1.26911
\(293\) −18.6082 −1.08710 −0.543551 0.839376i \(-0.682921\pi\)
−0.543551 + 0.839376i \(0.682921\pi\)
\(294\) 0 0
\(295\) 0.0338518 0.00197093
\(296\) 1.94869 0.113265
\(297\) 5.14675 0.298645
\(298\) 0.411190 0.0238196
\(299\) 0 0
\(300\) 3.86831 0.223337
\(301\) 0 0
\(302\) 10.5851 0.609103
\(303\) 4.24267 0.243735
\(304\) 4.35690 0.249885
\(305\) −22.5187 −1.28942
\(306\) 1.48725 0.0850207
\(307\) 8.94438 0.510483 0.255241 0.966877i \(-0.417845\pi\)
0.255241 + 0.966877i \(0.417845\pi\)
\(308\) 0 0
\(309\) −10.8509 −0.617284
\(310\) −9.17390 −0.521042
\(311\) −21.0398 −1.19306 −0.596529 0.802591i \(-0.703454\pi\)
−0.596529 + 0.802591i \(0.703454\pi\)
\(312\) 0 0
\(313\) 7.12737 0.402863 0.201432 0.979503i \(-0.435441\pi\)
0.201432 + 0.979503i \(0.435441\pi\)
\(314\) 2.23191 0.125954
\(315\) 0 0
\(316\) −1.36599 −0.0768431
\(317\) −23.9651 −1.34601 −0.673007 0.739636i \(-0.734997\pi\)
−0.673007 + 0.739636i \(0.734997\pi\)
\(318\) −2.61356 −0.146561
\(319\) −9.46442 −0.529906
\(320\) 4.28083 0.239306
\(321\) −4.51573 −0.252043
\(322\) 0 0
\(323\) −2.20477 −0.122677
\(324\) −6.13467 −0.340815
\(325\) 0 0
\(326\) −8.40044 −0.465257
\(327\) 3.34913 0.185207
\(328\) −6.79763 −0.375336
\(329\) 0 0
\(330\) 1.49396 0.0822397
\(331\) −2.89546 −0.159149 −0.0795745 0.996829i \(-0.525356\pi\)
−0.0795745 + 0.996829i \(0.525356\pi\)
\(332\) 27.6353 1.51669
\(333\) 2.24160 0.122839
\(334\) −3.47591 −0.190193
\(335\) −25.9366 −1.41707
\(336\) 0 0
\(337\) −3.10560 −0.169173 −0.0845865 0.996416i \(-0.526957\pi\)
−0.0845865 + 0.996416i \(0.526957\pi\)
\(338\) 0 0
\(339\) −6.12929 −0.332898
\(340\) −5.39075 −0.292354
\(341\) 7.06829 0.382770
\(342\) −2.53617 −0.137140
\(343\) 0 0
\(344\) −14.6679 −0.790838
\(345\) −10.3448 −0.556946
\(346\) −9.09651 −0.489031
\(347\) −11.3787 −0.610839 −0.305419 0.952218i \(-0.598797\pi\)
−0.305419 + 0.952218i \(0.598797\pi\)
\(348\) −10.7192 −0.574608
\(349\) −3.34721 −0.179172 −0.0895859 0.995979i \(-0.528554\pi\)
−0.0895859 + 0.995979i \(0.528554\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6.40342 0.341303
\(353\) 0.637727 0.0339428 0.0169714 0.999856i \(-0.494598\pi\)
0.0169714 + 0.999856i \(0.494598\pi\)
\(354\) 0.00537681 0.000285774 0
\(355\) −38.5013 −2.04343
\(356\) 24.9215 1.32084
\(357\) 0 0
\(358\) −1.36227 −0.0719983
\(359\) 21.4590 1.13256 0.566282 0.824211i \(-0.308381\pi\)
0.566282 + 0.824211i \(0.308381\pi\)
\(360\) −13.5308 −0.713136
\(361\) −15.2403 −0.802120
\(362\) −6.55257 −0.344395
\(363\) 7.67025 0.402584
\(364\) 0 0
\(365\) −35.9124 −1.87974
\(366\) −3.57673 −0.186959
\(367\) 9.38703 0.489999 0.244999 0.969523i \(-0.421212\pi\)
0.244999 + 0.969523i \(0.421212\pi\)
\(368\) −10.3448 −0.539261
\(369\) −7.81940 −0.407062
\(370\) 1.47889 0.0768840
\(371\) 0 0
\(372\) 8.00538 0.415059
\(373\) 27.7265 1.43562 0.717811 0.696238i \(-0.245144\pi\)
0.717811 + 0.696238i \(0.245144\pi\)
\(374\) −0.756004 −0.0390921
\(375\) −4.82908 −0.249373
\(376\) 15.7603 0.812776
\(377\) 0 0
\(378\) 0 0
\(379\) −35.8702 −1.84253 −0.921265 0.388935i \(-0.872843\pi\)
−0.921265 + 0.388935i \(0.872843\pi\)
\(380\) 9.19269 0.471575
\(381\) −5.43535 −0.278462
\(382\) −4.99090 −0.255357
\(383\) 4.85517 0.248087 0.124044 0.992277i \(-0.460414\pi\)
0.124044 + 0.992277i \(0.460414\pi\)
\(384\) 9.25236 0.472157
\(385\) 0 0
\(386\) −7.50604 −0.382047
\(387\) −16.8726 −0.857684
\(388\) −5.29888 −0.269010
\(389\) 2.38537 0.120943 0.0604716 0.998170i \(-0.480740\pi\)
0.0604716 + 0.998170i \(0.480740\pi\)
\(390\) 0 0
\(391\) 5.23490 0.264740
\(392\) 0 0
\(393\) −10.9705 −0.553387
\(394\) −7.20105 −0.362783
\(395\) −2.26205 −0.113816
\(396\) 4.77777 0.240092
\(397\) −15.2664 −0.766196 −0.383098 0.923708i \(-0.625143\pi\)
−0.383098 + 0.923708i \(0.625143\pi\)
\(398\) 7.53989 0.377941
\(399\) 0 0
\(400\) 6.40581 0.320291
\(401\) −12.7584 −0.637124 −0.318562 0.947902i \(-0.603200\pi\)
−0.318562 + 0.947902i \(0.603200\pi\)
\(402\) −4.11960 −0.205467
\(403\) 0 0
\(404\) 8.95167 0.445362
\(405\) −10.1588 −0.504797
\(406\) 0 0
\(407\) −1.13946 −0.0564807
\(408\) −1.86831 −0.0924953
\(409\) −25.3588 −1.25391 −0.626956 0.779054i \(-0.715700\pi\)
−0.626956 + 0.779054i \(0.715700\pi\)
\(410\) −5.15883 −0.254777
\(411\) 10.4155 0.513759
\(412\) −22.8944 −1.12793
\(413\) 0 0
\(414\) 6.02177 0.295954
\(415\) 45.7633 2.24643
\(416\) 0 0
\(417\) 9.66056 0.473080
\(418\) 1.28919 0.0630565
\(419\) 11.6673 0.569983 0.284992 0.958530i \(-0.408009\pi\)
0.284992 + 0.958530i \(0.408009\pi\)
\(420\) 0 0
\(421\) 8.29291 0.404172 0.202086 0.979368i \(-0.435228\pi\)
0.202086 + 0.979368i \(0.435228\pi\)
\(422\) 5.80492 0.282579
\(423\) 18.1293 0.881476
\(424\) −12.0325 −0.584351
\(425\) −3.24160 −0.157241
\(426\) −6.11529 −0.296287
\(427\) 0 0
\(428\) −9.52781 −0.460544
\(429\) 0 0
\(430\) −11.1317 −0.536818
\(431\) −0.932296 −0.0449071 −0.0224536 0.999748i \(-0.507148\pi\)
−0.0224536 + 0.999748i \(0.507148\pi\)
\(432\) 9.65279 0.464420
\(433\) 13.3502 0.641569 0.320785 0.947152i \(-0.396053\pi\)
0.320785 + 0.947152i \(0.396053\pi\)
\(434\) 0 0
\(435\) −17.7506 −0.851077
\(436\) 7.06638 0.338418
\(437\) −8.92692 −0.427032
\(438\) −5.70410 −0.272553
\(439\) −13.9922 −0.667813 −0.333906 0.942606i \(-0.608367\pi\)
−0.333906 + 0.942606i \(0.608367\pi\)
\(440\) 6.87800 0.327896
\(441\) 0 0
\(442\) 0 0
\(443\) 23.7017 1.12610 0.563051 0.826422i \(-0.309627\pi\)
0.563051 + 0.826422i \(0.309627\pi\)
\(444\) −1.29052 −0.0612454
\(445\) 41.2693 1.95635
\(446\) −6.32975 −0.299722
\(447\) −0.594187 −0.0281041
\(448\) 0 0
\(449\) 12.5864 0.593990 0.296995 0.954879i \(-0.404016\pi\)
0.296995 + 0.954879i \(0.404016\pi\)
\(450\) −3.72886 −0.175780
\(451\) 3.97477 0.187165
\(452\) −12.9323 −0.608284
\(453\) −15.2959 −0.718664
\(454\) 5.90515 0.277142
\(455\) 0 0
\(456\) 3.18598 0.149197
\(457\) 33.6383 1.57353 0.786767 0.617250i \(-0.211753\pi\)
0.786767 + 0.617250i \(0.211753\pi\)
\(458\) −0.632351 −0.0295478
\(459\) −4.88471 −0.227999
\(460\) −21.8267 −1.01767
\(461\) −1.40283 −0.0653363 −0.0326681 0.999466i \(-0.510400\pi\)
−0.0326681 + 0.999466i \(0.510400\pi\)
\(462\) 0 0
\(463\) 15.2010 0.706453 0.353226 0.935538i \(-0.385085\pi\)
0.353226 + 0.935538i \(0.385085\pi\)
\(464\) −17.7506 −0.824052
\(465\) 13.2567 0.614763
\(466\) −6.02177 −0.278953
\(467\) 39.3414 1.82050 0.910250 0.414058i \(-0.135889\pi\)
0.910250 + 0.414058i \(0.135889\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 11.9608 0.551709
\(471\) −3.22521 −0.148610
\(472\) 0.0247542 0.00113940
\(473\) 8.57673 0.394358
\(474\) −0.359289 −0.0165027
\(475\) 5.52781 0.253633
\(476\) 0 0
\(477\) −13.8412 −0.633743
\(478\) 6.62027 0.302804
\(479\) −22.3690 −1.02206 −0.511032 0.859561i \(-0.670737\pi\)
−0.511032 + 0.859561i \(0.670737\pi\)
\(480\) 12.0097 0.548165
\(481\) 0 0
\(482\) −2.02475 −0.0922250
\(483\) 0 0
\(484\) 16.1836 0.735618
\(485\) −8.77479 −0.398443
\(486\) −8.76569 −0.397620
\(487\) −22.9205 −1.03863 −0.519313 0.854584i \(-0.673812\pi\)
−0.519313 + 0.854584i \(0.673812\pi\)
\(488\) −16.4668 −0.745418
\(489\) 12.1390 0.548944
\(490\) 0 0
\(491\) 1.84356 0.0831987 0.0415993 0.999134i \(-0.486755\pi\)
0.0415993 + 0.999134i \(0.486755\pi\)
\(492\) 4.50173 0.202954
\(493\) 8.98254 0.404553
\(494\) 0 0
\(495\) 7.91185 0.355611
\(496\) 13.2567 0.595242
\(497\) 0 0
\(498\) 7.26875 0.325720
\(499\) 12.0344 0.538736 0.269368 0.963037i \(-0.413185\pi\)
0.269368 + 0.963037i \(0.413185\pi\)
\(500\) −10.1890 −0.455664
\(501\) 5.02284 0.224404
\(502\) −0.762118 −0.0340150
\(503\) 30.5056 1.36018 0.680088 0.733130i \(-0.261942\pi\)
0.680088 + 0.733130i \(0.261942\pi\)
\(504\) 0 0
\(505\) 14.8237 0.659646
\(506\) −3.06100 −0.136078
\(507\) 0 0
\(508\) −11.4681 −0.508816
\(509\) −1.51142 −0.0669924 −0.0334962 0.999439i \(-0.510664\pi\)
−0.0334962 + 0.999439i \(0.510664\pi\)
\(510\) −1.41789 −0.0627854
\(511\) 0 0
\(512\) 21.2174 0.937687
\(513\) 8.32975 0.367767
\(514\) −16.3357 −0.720538
\(515\) −37.9124 −1.67062
\(516\) 9.71379 0.427626
\(517\) −9.21552 −0.405298
\(518\) 0 0
\(519\) 13.1448 0.576994
\(520\) 0 0
\(521\) 5.64012 0.247098 0.123549 0.992338i \(-0.460572\pi\)
0.123549 + 0.992338i \(0.460572\pi\)
\(522\) 10.3327 0.452251
\(523\) 31.7506 1.38836 0.694179 0.719802i \(-0.255768\pi\)
0.694179 + 0.719802i \(0.255768\pi\)
\(524\) −23.1468 −1.01117
\(525\) 0 0
\(526\) 5.93602 0.258823
\(527\) −6.70841 −0.292223
\(528\) −2.15883 −0.0939512
\(529\) −1.80433 −0.0784492
\(530\) −9.13169 −0.396655
\(531\) 0.0284750 0.00123571
\(532\) 0 0
\(533\) 0 0
\(534\) 6.55496 0.283661
\(535\) −15.7778 −0.682133
\(536\) −18.9661 −0.819213
\(537\) 1.96854 0.0849488
\(538\) 5.65279 0.243709
\(539\) 0 0
\(540\) 20.3666 0.876438
\(541\) −24.3297 −1.04602 −0.523009 0.852327i \(-0.675191\pi\)
−0.523009 + 0.852327i \(0.675191\pi\)
\(542\) −16.3448 −0.702070
\(543\) 9.46873 0.406342
\(544\) −6.07739 −0.260566
\(545\) 11.7017 0.501246
\(546\) 0 0
\(547\) −8.18896 −0.350135 −0.175067 0.984556i \(-0.556014\pi\)
−0.175067 + 0.984556i \(0.556014\pi\)
\(548\) 21.9758 0.938761
\(549\) −18.9420 −0.808424
\(550\) 1.89546 0.0808227
\(551\) −15.3177 −0.652555
\(552\) −7.56465 −0.321973
\(553\) 0 0
\(554\) −5.68532 −0.241546
\(555\) −2.13706 −0.0907133
\(556\) 20.3830 0.864431
\(557\) −25.3327 −1.07338 −0.536691 0.843779i \(-0.680326\pi\)
−0.536691 + 0.843779i \(0.680326\pi\)
\(558\) −7.71678 −0.326677
\(559\) 0 0
\(560\) 0 0
\(561\) 1.09246 0.0461236
\(562\) 6.41789 0.270723
\(563\) 25.3937 1.07022 0.535109 0.844783i \(-0.320270\pi\)
0.535109 + 0.844783i \(0.320270\pi\)
\(564\) −10.4373 −0.439488
\(565\) −21.4155 −0.900957
\(566\) 17.0422 0.716338
\(567\) 0 0
\(568\) −28.1540 −1.18132
\(569\) −31.1347 −1.30523 −0.652617 0.757688i \(-0.726329\pi\)
−0.652617 + 0.757688i \(0.726329\pi\)
\(570\) 2.41789 0.101274
\(571\) −20.5090 −0.858276 −0.429138 0.903239i \(-0.641183\pi\)
−0.429138 + 0.903239i \(0.641183\pi\)
\(572\) 0 0
\(573\) 7.21206 0.301288
\(574\) 0 0
\(575\) −13.1250 −0.547350
\(576\) 3.60089 0.150037
\(577\) 15.6890 0.653143 0.326572 0.945172i \(-0.394107\pi\)
0.326572 + 0.945172i \(0.394107\pi\)
\(578\) −8.71678 −0.362570
\(579\) 10.8465 0.450767
\(580\) −37.4523 −1.55512
\(581\) 0 0
\(582\) −1.39373 −0.0577720
\(583\) 7.03577 0.291392
\(584\) −26.2610 −1.08669
\(585\) 0 0
\(586\) −10.3268 −0.426595
\(587\) 30.5687 1.26171 0.630853 0.775903i \(-0.282705\pi\)
0.630853 + 0.775903i \(0.282705\pi\)
\(588\) 0 0
\(589\) 11.4397 0.471363
\(590\) 0.0187864 0.000773422 0
\(591\) 10.4058 0.428038
\(592\) −2.13706 −0.0878328
\(593\) −29.6883 −1.21915 −0.609576 0.792727i \(-0.708660\pi\)
−0.609576 + 0.792727i \(0.708660\pi\)
\(594\) 2.85623 0.117193
\(595\) 0 0
\(596\) −1.25368 −0.0513529
\(597\) −10.8955 −0.445922
\(598\) 0 0
\(599\) 24.2325 0.990113 0.495057 0.868861i \(-0.335147\pi\)
0.495057 + 0.868861i \(0.335147\pi\)
\(600\) 4.68425 0.191234
\(601\) −16.4819 −0.672310 −0.336155 0.941807i \(-0.609127\pi\)
−0.336155 + 0.941807i \(0.609127\pi\)
\(602\) 0 0
\(603\) −21.8170 −0.888457
\(604\) −32.2731 −1.31317
\(605\) 26.7995 1.08956
\(606\) 2.35450 0.0956451
\(607\) −1.43190 −0.0581188 −0.0290594 0.999578i \(-0.509251\pi\)
−0.0290594 + 0.999578i \(0.509251\pi\)
\(608\) 10.3636 0.420300
\(609\) 0 0
\(610\) −12.4969 −0.505986
\(611\) 0 0
\(612\) −4.53452 −0.183297
\(613\) −3.84846 −0.155438 −0.0777190 0.996975i \(-0.524764\pi\)
−0.0777190 + 0.996975i \(0.524764\pi\)
\(614\) 4.96376 0.200321
\(615\) 7.45473 0.300604
\(616\) 0 0
\(617\) 15.0388 0.605437 0.302719 0.953080i \(-0.402106\pi\)
0.302719 + 0.953080i \(0.402106\pi\)
\(618\) −6.02177 −0.242231
\(619\) 12.8170 0.515159 0.257579 0.966257i \(-0.417075\pi\)
0.257579 + 0.966257i \(0.417075\pi\)
\(620\) 27.9705 1.12332
\(621\) −19.7778 −0.793655
\(622\) −11.6762 −0.468174
\(623\) 0 0
\(624\) 0 0
\(625\) −31.1269 −1.24508
\(626\) 3.95539 0.158089
\(627\) −1.86294 −0.0743985
\(628\) −6.80492 −0.271546
\(629\) 1.08144 0.0431199
\(630\) 0 0
\(631\) −25.7517 −1.02516 −0.512579 0.858640i \(-0.671310\pi\)
−0.512579 + 0.858640i \(0.671310\pi\)
\(632\) −1.65412 −0.0657974
\(633\) −8.38835 −0.333407
\(634\) −13.2996 −0.528195
\(635\) −18.9909 −0.753631
\(636\) 7.96854 0.315973
\(637\) 0 0
\(638\) −5.25236 −0.207943
\(639\) −32.3860 −1.28117
\(640\) 32.3274 1.27785
\(641\) 24.4571 0.965998 0.482999 0.875621i \(-0.339547\pi\)
0.482999 + 0.875621i \(0.339547\pi\)
\(642\) −2.50604 −0.0989055
\(643\) −9.97344 −0.393314 −0.196657 0.980472i \(-0.563009\pi\)
−0.196657 + 0.980472i \(0.563009\pi\)
\(644\) 0 0
\(645\) 16.0858 0.633376
\(646\) −1.22355 −0.0481401
\(647\) −11.8431 −0.465600 −0.232800 0.972525i \(-0.574789\pi\)
−0.232800 + 0.972525i \(0.574789\pi\)
\(648\) −7.42865 −0.291825
\(649\) −0.0144745 −0.000568173 0
\(650\) 0 0
\(651\) 0 0
\(652\) 25.6122 1.00305
\(653\) −7.47411 −0.292484 −0.146242 0.989249i \(-0.546718\pi\)
−0.146242 + 0.989249i \(0.546718\pi\)
\(654\) 1.85862 0.0726780
\(655\) −38.3303 −1.49769
\(656\) 7.45473 0.291058
\(657\) −30.2083 −1.17854
\(658\) 0 0
\(659\) 34.1739 1.33123 0.665613 0.746297i \(-0.268170\pi\)
0.665613 + 0.746297i \(0.268170\pi\)
\(660\) −4.55496 −0.177302
\(661\) 33.6088 1.30723 0.653615 0.756827i \(-0.273252\pi\)
0.653615 + 0.756827i \(0.273252\pi\)
\(662\) −1.60686 −0.0624524
\(663\) 0 0
\(664\) 33.4644 1.29867
\(665\) 0 0
\(666\) 1.24400 0.0482039
\(667\) 36.3696 1.40824
\(668\) 10.5978 0.410040
\(669\) 9.14675 0.353634
\(670\) −14.3937 −0.556078
\(671\) 9.62863 0.371709
\(672\) 0 0
\(673\) 48.0320 1.85150 0.925750 0.378137i \(-0.123435\pi\)
0.925750 + 0.378137i \(0.123435\pi\)
\(674\) −1.72348 −0.0663860
\(675\) 12.2470 0.471386
\(676\) 0 0
\(677\) 33.6582 1.29359 0.646794 0.762665i \(-0.276109\pi\)
0.646794 + 0.762665i \(0.276109\pi\)
\(678\) −3.40150 −0.130634
\(679\) 0 0
\(680\) −6.52781 −0.250330
\(681\) −8.53319 −0.326992
\(682\) 3.92261 0.150204
\(683\) −15.9041 −0.608553 −0.304276 0.952584i \(-0.598415\pi\)
−0.304276 + 0.952584i \(0.598415\pi\)
\(684\) 7.73258 0.295663
\(685\) 36.3913 1.39044
\(686\) 0 0
\(687\) 0.913773 0.0348626
\(688\) 16.0858 0.613264
\(689\) 0 0
\(690\) −5.74094 −0.218554
\(691\) 33.1903 1.26262 0.631309 0.775531i \(-0.282518\pi\)
0.631309 + 0.775531i \(0.282518\pi\)
\(692\) 27.7345 1.05431
\(693\) 0 0
\(694\) −6.31468 −0.239702
\(695\) 33.7536 1.28035
\(696\) −12.9801 −0.492011
\(697\) −3.77240 −0.142890
\(698\) −1.85756 −0.0703097
\(699\) 8.70171 0.329129
\(700\) 0 0
\(701\) −14.9129 −0.563253 −0.281627 0.959524i \(-0.590874\pi\)
−0.281627 + 0.959524i \(0.590874\pi\)
\(702\) 0 0
\(703\) −1.84415 −0.0695534
\(704\) −1.83041 −0.0689863
\(705\) −17.2838 −0.650946
\(706\) 0.353912 0.0133197
\(707\) 0 0
\(708\) −0.0163935 −0.000616104 0
\(709\) −38.4312 −1.44331 −0.721656 0.692252i \(-0.756619\pi\)
−0.721656 + 0.692252i \(0.756619\pi\)
\(710\) −21.3666 −0.801874
\(711\) −1.90276 −0.0713589
\(712\) 30.1782 1.13098
\(713\) −27.1618 −1.01722
\(714\) 0 0
\(715\) 0 0
\(716\) 4.15346 0.155222
\(717\) −9.56657 −0.357270
\(718\) 11.9089 0.444435
\(719\) 11.4373 0.426538 0.213269 0.976993i \(-0.431589\pi\)
0.213269 + 0.976993i \(0.431589\pi\)
\(720\) 14.8388 0.553008
\(721\) 0 0
\(722\) −8.45771 −0.314764
\(723\) 2.92585 0.108814
\(724\) 19.9782 0.742485
\(725\) −22.5211 −0.836413
\(726\) 4.25667 0.157980
\(727\) −3.63640 −0.134867 −0.0674333 0.997724i \(-0.521481\pi\)
−0.0674333 + 0.997724i \(0.521481\pi\)
\(728\) 0 0
\(729\) 1.78986 0.0662910
\(730\) −19.9299 −0.737639
\(731\) −8.14005 −0.301071
\(732\) 10.9051 0.403066
\(733\) −3.52217 −0.130094 −0.0650472 0.997882i \(-0.520720\pi\)
−0.0650472 + 0.997882i \(0.520720\pi\)
\(734\) 5.20941 0.192283
\(735\) 0 0
\(736\) −24.6069 −0.907021
\(737\) 11.0901 0.408508
\(738\) −4.33944 −0.159737
\(739\) −0.420288 −0.0154605 −0.00773027 0.999970i \(-0.502461\pi\)
−0.00773027 + 0.999970i \(0.502461\pi\)
\(740\) −4.50902 −0.165755
\(741\) 0 0
\(742\) 0 0
\(743\) 25.3623 0.930452 0.465226 0.885192i \(-0.345973\pi\)
0.465226 + 0.885192i \(0.345973\pi\)
\(744\) 9.69394 0.355397
\(745\) −2.07606 −0.0760611
\(746\) 15.3870 0.563359
\(747\) 38.4946 1.40844
\(748\) 2.30499 0.0842790
\(749\) 0 0
\(750\) −2.67994 −0.0978576
\(751\) 0.650874 0.0237507 0.0118754 0.999929i \(-0.496220\pi\)
0.0118754 + 0.999929i \(0.496220\pi\)
\(752\) −17.2838 −0.630276
\(753\) 1.10129 0.0401333
\(754\) 0 0
\(755\) −53.4432 −1.94500
\(756\) 0 0
\(757\) −16.7909 −0.610276 −0.305138 0.952308i \(-0.598703\pi\)
−0.305138 + 0.952308i \(0.598703\pi\)
\(758\) −19.9065 −0.723036
\(759\) 4.42327 0.160555
\(760\) 11.1317 0.403789
\(761\) 30.9221 1.12093 0.560463 0.828179i \(-0.310623\pi\)
0.560463 + 0.828179i \(0.310623\pi\)
\(762\) −3.01639 −0.109272
\(763\) 0 0
\(764\) 15.2168 0.550526
\(765\) −7.50902 −0.271489
\(766\) 2.69441 0.0973531
\(767\) 0 0
\(768\) 2.68425 0.0968596
\(769\) −43.7689 −1.57835 −0.789174 0.614169i \(-0.789491\pi\)
−0.789174 + 0.614169i \(0.789491\pi\)
\(770\) 0 0
\(771\) 23.6058 0.850142
\(772\) 22.8853 0.823660
\(773\) −42.4209 −1.52577 −0.762886 0.646532i \(-0.776218\pi\)
−0.762886 + 0.646532i \(0.776218\pi\)
\(774\) −9.36360 −0.336568
\(775\) 16.8194 0.604171
\(776\) −6.41657 −0.230341
\(777\) 0 0
\(778\) 1.32378 0.0474598
\(779\) 6.43296 0.230485
\(780\) 0 0
\(781\) 16.4625 0.589075
\(782\) 2.90515 0.103888
\(783\) −33.9366 −1.21280
\(784\) 0 0
\(785\) −11.2687 −0.402199
\(786\) −6.08815 −0.217157
\(787\) −36.0116 −1.28368 −0.641838 0.766841i \(-0.721828\pi\)
−0.641838 + 0.766841i \(0.721828\pi\)
\(788\) 21.9554 0.782129
\(789\) −8.57779 −0.305378
\(790\) −1.25534 −0.0446630
\(791\) 0 0
\(792\) 5.78554 0.205580
\(793\) 0 0
\(794\) −8.47219 −0.300667
\(795\) 13.1957 0.468002
\(796\) −22.9885 −0.814806
\(797\) 31.7101 1.12323 0.561614 0.827399i \(-0.310181\pi\)
0.561614 + 0.827399i \(0.310181\pi\)
\(798\) 0 0
\(799\) 8.74632 0.309422
\(800\) 15.2373 0.538720
\(801\) 34.7144 1.22657
\(802\) −7.08038 −0.250017
\(803\) 15.3556 0.541886
\(804\) 12.5603 0.442969
\(805\) 0 0
\(806\) 0 0
\(807\) −8.16852 −0.287546
\(808\) 10.8398 0.381344
\(809\) −45.2814 −1.59201 −0.796005 0.605290i \(-0.793057\pi\)
−0.796005 + 0.605290i \(0.793057\pi\)
\(810\) −5.63773 −0.198090
\(811\) −42.8635 −1.50514 −0.752571 0.658511i \(-0.771187\pi\)
−0.752571 + 0.658511i \(0.771187\pi\)
\(812\) 0 0
\(813\) 23.6189 0.828352
\(814\) −0.632351 −0.0221639
\(815\) 42.4131 1.48567
\(816\) 2.04892 0.0717265
\(817\) 13.8810 0.485634
\(818\) −14.0731 −0.492054
\(819\) 0 0
\(820\) 15.7289 0.549276
\(821\) 7.82776 0.273191 0.136595 0.990627i \(-0.456384\pi\)
0.136595 + 0.990627i \(0.456384\pi\)
\(822\) 5.78017 0.201606
\(823\) −36.7754 −1.28191 −0.640955 0.767579i \(-0.721461\pi\)
−0.640955 + 0.767579i \(0.721461\pi\)
\(824\) −27.7235 −0.965793
\(825\) −2.73902 −0.0953604
\(826\) 0 0
\(827\) −47.3293 −1.64580 −0.822900 0.568186i \(-0.807645\pi\)
−0.822900 + 0.568186i \(0.807645\pi\)
\(828\) −18.3599 −0.638050
\(829\) −25.2687 −0.877620 −0.438810 0.898580i \(-0.644600\pi\)
−0.438810 + 0.898580i \(0.644600\pi\)
\(830\) 25.3967 0.881533
\(831\) 8.21552 0.284993
\(832\) 0 0
\(833\) 0 0
\(834\) 5.36121 0.185643
\(835\) 17.5496 0.607328
\(836\) −3.93064 −0.135944
\(837\) 25.3448 0.876045
\(838\) 6.47484 0.223670
\(839\) −37.6883 −1.30114 −0.650572 0.759444i \(-0.725471\pi\)
−0.650572 + 0.759444i \(0.725471\pi\)
\(840\) 0 0
\(841\) 33.4064 1.15194
\(842\) 4.60222 0.158603
\(843\) −9.27413 −0.319418
\(844\) −17.6987 −0.609215
\(845\) 0 0
\(846\) 10.0610 0.345904
\(847\) 0 0
\(848\) 13.1957 0.453141
\(849\) −24.6267 −0.845187
\(850\) −1.79895 −0.0617036
\(851\) 4.37867 0.150099
\(852\) 18.6450 0.638768
\(853\) −31.0121 −1.06183 −0.530917 0.847424i \(-0.678152\pi\)
−0.530917 + 0.847424i \(0.678152\pi\)
\(854\) 0 0
\(855\) 12.8049 0.437919
\(856\) −11.5375 −0.394344
\(857\) −12.4692 −0.425940 −0.212970 0.977059i \(-0.568314\pi\)
−0.212970 + 0.977059i \(0.568314\pi\)
\(858\) 0 0
\(859\) 17.3163 0.590826 0.295413 0.955370i \(-0.404543\pi\)
0.295413 + 0.955370i \(0.404543\pi\)
\(860\) 33.9396 1.15733
\(861\) 0 0
\(862\) −0.517385 −0.0176222
\(863\) 3.46383 0.117910 0.0589550 0.998261i \(-0.481223\pi\)
0.0589550 + 0.998261i \(0.481223\pi\)
\(864\) 22.9608 0.781141
\(865\) 45.9275 1.56158
\(866\) 7.40880 0.251761
\(867\) 12.5961 0.427786
\(868\) 0 0
\(869\) 0.967213 0.0328105
\(870\) −9.85086 −0.333975
\(871\) 0 0
\(872\) 8.55688 0.289772
\(873\) −7.38106 −0.249811
\(874\) −4.95407 −0.167574
\(875\) 0 0
\(876\) 17.3913 0.587599
\(877\) −57.2549 −1.93336 −0.966680 0.255989i \(-0.917599\pi\)
−0.966680 + 0.255989i \(0.917599\pi\)
\(878\) −7.76510 −0.262059
\(879\) 14.9226 0.503327
\(880\) −7.54288 −0.254270
\(881\) 43.1782 1.45471 0.727355 0.686261i \(-0.240749\pi\)
0.727355 + 0.686261i \(0.240749\pi\)
\(882\) 0 0
\(883\) 49.9560 1.68115 0.840576 0.541693i \(-0.182216\pi\)
0.840576 + 0.541693i \(0.182216\pi\)
\(884\) 0 0
\(885\) −0.0271471 −0.000912539 0
\(886\) 13.1535 0.441899
\(887\) −17.6746 −0.593454 −0.296727 0.954962i \(-0.595895\pi\)
−0.296727 + 0.954962i \(0.595895\pi\)
\(888\) −1.56273 −0.0524417
\(889\) 0 0
\(890\) 22.9028 0.767702
\(891\) 4.34375 0.145521
\(892\) 19.2989 0.646174
\(893\) −14.9148 −0.499106
\(894\) −0.329749 −0.0110284
\(895\) 6.87800 0.229906
\(896\) 0 0
\(897\) 0 0
\(898\) 6.98493 0.233090
\(899\) −46.6069 −1.55443
\(900\) 11.3690 0.378966
\(901\) −6.67755 −0.222461
\(902\) 2.20583 0.0734462
\(903\) 0 0
\(904\) −15.6601 −0.520847
\(905\) 33.0834 1.09973
\(906\) −8.48858 −0.282014
\(907\) 7.73423 0.256811 0.128406 0.991722i \(-0.459014\pi\)
0.128406 + 0.991722i \(0.459014\pi\)
\(908\) −18.0043 −0.597494
\(909\) 12.4692 0.413577
\(910\) 0 0
\(911\) 39.6179 1.31260 0.656299 0.754501i \(-0.272121\pi\)
0.656299 + 0.754501i \(0.272121\pi\)
\(912\) −3.49396 −0.115697
\(913\) −19.5676 −0.647594
\(914\) 18.6679 0.617478
\(915\) 18.0586 0.596999
\(916\) 1.92798 0.0637024
\(917\) 0 0
\(918\) −2.71081 −0.0894700
\(919\) 14.6213 0.482313 0.241157 0.970486i \(-0.422473\pi\)
0.241157 + 0.970486i \(0.422473\pi\)
\(920\) −26.4306 −0.871390
\(921\) −7.17283 −0.236353
\(922\) −0.778512 −0.0256389
\(923\) 0 0
\(924\) 0 0
\(925\) −2.71140 −0.0891502
\(926\) 8.43594 0.277222
\(927\) −31.8907 −1.04743
\(928\) −42.2228 −1.38603
\(929\) 3.55735 0.116713 0.0583565 0.998296i \(-0.481414\pi\)
0.0583565 + 0.998296i \(0.481414\pi\)
\(930\) 7.35690 0.241242
\(931\) 0 0
\(932\) 18.3599 0.601398
\(933\) 16.8726 0.552385
\(934\) 21.8328 0.714391
\(935\) 3.81700 0.124829
\(936\) 0 0
\(937\) −34.5526 −1.12878 −0.564392 0.825507i \(-0.690889\pi\)
−0.564392 + 0.825507i \(0.690889\pi\)
\(938\) 0 0
\(939\) −5.71571 −0.186525
\(940\) −36.4674 −1.18944
\(941\) 20.6233 0.672299 0.336149 0.941809i \(-0.390875\pi\)
0.336149 + 0.941809i \(0.390875\pi\)
\(942\) −1.78986 −0.0583167
\(943\) −15.2741 −0.497394
\(944\) −0.0271471 −0.000883562 0
\(945\) 0 0
\(946\) 4.75973 0.154752
\(947\) −29.4999 −0.958619 −0.479309 0.877646i \(-0.659113\pi\)
−0.479309 + 0.877646i \(0.659113\pi\)
\(948\) 1.09544 0.0355783
\(949\) 0 0
\(950\) 3.06770 0.0995295
\(951\) 19.2185 0.623203
\(952\) 0 0
\(953\) 26.2389 0.849963 0.424981 0.905202i \(-0.360281\pi\)
0.424981 + 0.905202i \(0.360281\pi\)
\(954\) −7.68127 −0.248690
\(955\) 25.1987 0.815409
\(956\) −20.1847 −0.652818
\(957\) 7.58987 0.245346
\(958\) −12.4138 −0.401073
\(959\) 0 0
\(960\) −3.43296 −0.110798
\(961\) 3.80731 0.122817
\(962\) 0 0
\(963\) −13.2717 −0.427676
\(964\) 6.17331 0.198829
\(965\) 37.8974 1.21996
\(966\) 0 0
\(967\) −17.5176 −0.563330 −0.281665 0.959513i \(-0.590887\pi\)
−0.281665 + 0.959513i \(0.590887\pi\)
\(968\) 19.5972 0.629877
\(969\) 1.76809 0.0567991
\(970\) −4.86964 −0.156355
\(971\) −20.5120 −0.658262 −0.329131 0.944284i \(-0.606756\pi\)
−0.329131 + 0.944284i \(0.606756\pi\)
\(972\) 26.7259 0.857233
\(973\) 0 0
\(974\) −12.7199 −0.407572
\(975\) 0 0
\(976\) 18.0586 0.578042
\(977\) −25.4450 −0.814059 −0.407030 0.913415i \(-0.633435\pi\)
−0.407030 + 0.913415i \(0.633435\pi\)
\(978\) 6.73663 0.215414
\(979\) −17.6461 −0.563971
\(980\) 0 0
\(981\) 9.84309 0.314266
\(982\) 1.02310 0.0326484
\(983\) −39.5244 −1.26063 −0.630316 0.776339i \(-0.717074\pi\)
−0.630316 + 0.776339i \(0.717074\pi\)
\(984\) 5.45127 0.173780
\(985\) 36.3575 1.15845
\(986\) 4.98493 0.158753
\(987\) 0 0
\(988\) 0 0
\(989\) −32.9584 −1.04802
\(990\) 4.39075 0.139547
\(991\) −29.8377 −0.947826 −0.473913 0.880572i \(-0.657159\pi\)
−0.473913 + 0.880572i \(0.657159\pi\)
\(992\) 31.5332 1.00118
\(993\) 2.32198 0.0736858
\(994\) 0 0
\(995\) −38.0683 −1.20685
\(996\) −22.1618 −0.702224
\(997\) 4.93123 0.156174 0.0780868 0.996947i \(-0.475119\pi\)
0.0780868 + 0.996947i \(0.475119\pi\)
\(998\) 6.67861 0.211408
\(999\) −4.08575 −0.129268
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 8281.2.a.bj.1.2 3
7.6 odd 2 169.2.a.c.1.2 yes 3
13.12 even 2 8281.2.a.bf.1.2 3
21.20 even 2 1521.2.a.o.1.2 3
28.27 even 2 2704.2.a.ba.1.1 3
35.34 odd 2 4225.2.a.bb.1.2 3
91.6 even 12 169.2.e.b.23.3 12
91.20 even 12 169.2.e.b.23.4 12
91.34 even 4 169.2.b.b.168.4 6
91.41 even 12 169.2.e.b.147.4 12
91.48 odd 6 169.2.c.b.146.2 6
91.55 odd 6 169.2.c.b.22.2 6
91.62 odd 6 169.2.c.c.22.2 6
91.69 odd 6 169.2.c.c.146.2 6
91.76 even 12 169.2.e.b.147.3 12
91.83 even 4 169.2.b.b.168.3 6
91.90 odd 2 169.2.a.b.1.2 3
273.83 odd 4 1521.2.b.l.1351.4 6
273.125 odd 4 1521.2.b.l.1351.3 6
273.272 even 2 1521.2.a.r.1.2 3
364.83 odd 4 2704.2.f.o.337.1 6
364.307 odd 4 2704.2.f.o.337.2 6
364.363 even 2 2704.2.a.z.1.1 3
455.454 odd 2 4225.2.a.bg.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.2 3 91.90 odd 2
169.2.a.c.1.2 yes 3 7.6 odd 2
169.2.b.b.168.3 6 91.83 even 4
169.2.b.b.168.4 6 91.34 even 4
169.2.c.b.22.2 6 91.55 odd 6
169.2.c.b.146.2 6 91.48 odd 6
169.2.c.c.22.2 6 91.62 odd 6
169.2.c.c.146.2 6 91.69 odd 6
169.2.e.b.23.3 12 91.6 even 12
169.2.e.b.23.4 12 91.20 even 12
169.2.e.b.147.3 12 91.76 even 12
169.2.e.b.147.4 12 91.41 even 12
1521.2.a.o.1.2 3 21.20 even 2
1521.2.a.r.1.2 3 273.272 even 2
1521.2.b.l.1351.3 6 273.125 odd 4
1521.2.b.l.1351.4 6 273.83 odd 4
2704.2.a.z.1.1 3 364.363 even 2
2704.2.a.ba.1.1 3 28.27 even 2
2704.2.f.o.337.1 6 364.83 odd 4
2704.2.f.o.337.2 6 364.307 odd 4
4225.2.a.bb.1.2 3 35.34 odd 2
4225.2.a.bg.1.2 3 455.454 odd 2
8281.2.a.bf.1.2 3 13.12 even 2
8281.2.a.bj.1.2 3 1.1 even 1 trivial