Properties

Label 169.2.b.b.168.4
Level $169$
Weight $2$
Character 169.168
Analytic conductor $1.349$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,2,Mod(168,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.168");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 168.4
Root \(1.80194i\) of defining polynomial
Character \(\chi\) \(=\) 169.168
Dual form 169.2.b.b.168.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.554958i q^{2} +0.801938 q^{3} +1.69202 q^{4} +2.80194i q^{5} +0.445042i q^{6} -2.69202i q^{7} +2.04892i q^{8} -2.35690 q^{9} +O(q^{10})\) \(q+0.554958i q^{2} +0.801938 q^{3} +1.69202 q^{4} +2.80194i q^{5} +0.445042i q^{6} -2.69202i q^{7} +2.04892i q^{8} -2.35690 q^{9} -1.55496 q^{10} -1.19806i q^{11} +1.35690 q^{12} +1.49396 q^{14} +2.24698i q^{15} +2.24698 q^{16} -1.13706 q^{17} -1.30798i q^{18} -1.93900i q^{19} +4.74094i q^{20} -2.15883i q^{21} +0.664874 q^{22} +4.60388 q^{23} +1.64310i q^{24} -2.85086 q^{25} -4.29590 q^{27} -4.55496i q^{28} -7.89977 q^{29} -1.24698 q^{30} -5.89977i q^{31} +5.34481i q^{32} -0.960771i q^{33} -0.631023i q^{34} +7.54288 q^{35} -3.98792 q^{36} +0.951083i q^{37} +1.07606 q^{38} -5.74094 q^{40} -3.31767i q^{41} +1.19806 q^{42} -7.15883 q^{43} -2.02715i q^{44} -6.60388i q^{45} +2.55496i q^{46} -7.69202i q^{47} +1.80194 q^{48} -0.246980 q^{49} -1.58211i q^{50} -0.911854 q^{51} +5.87263 q^{53} -2.38404i q^{54} +3.35690 q^{55} +5.51573 q^{56} -1.55496i q^{57} -4.38404i q^{58} -0.0120816i q^{59} +3.80194i q^{60} -8.03684 q^{61} +3.27413 q^{62} +6.34481i q^{63} +1.52781 q^{64} +0.533188 q^{66} +9.25667i q^{67} -1.92394 q^{68} +3.69202 q^{69} +4.18598i q^{70} +13.7409i q^{71} -4.82908i q^{72} +12.8170i q^{73} -0.527811 q^{74} -2.28621 q^{75} -3.28083i q^{76} -3.22521 q^{77} +0.807315 q^{79} +6.29590i q^{80} +3.62565 q^{81} +1.84117 q^{82} +16.3327i q^{83} -3.65279i q^{84} -3.18598i q^{85} -3.97285i q^{86} -6.33513 q^{87} +2.45473 q^{88} -14.7289i q^{89} +3.66487 q^{90} +7.78986 q^{92} -4.73125i q^{93} +4.26875 q^{94} +5.43296 q^{95} +4.28621i q^{96} -3.13169i q^{97} -0.137063i q^{98} +2.82371i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} - 6 q^{9} - 10 q^{10} - 10 q^{14} + 4 q^{16} + 4 q^{17} + 6 q^{22} + 10 q^{23} + 10 q^{25} + 2 q^{27} - 2 q^{29} + 2 q^{30} + 8 q^{35} + 14 q^{36} - 24 q^{38} - 6 q^{40} + 16 q^{42} - 26 q^{43} + 2 q^{48} + 8 q^{49} + 2 q^{51} + 2 q^{53} + 12 q^{55} + 8 q^{56} + 8 q^{61} - 2 q^{62} + 22 q^{64} + 10 q^{66} - 42 q^{68} + 12 q^{69} - 16 q^{74} - 30 q^{75} - 16 q^{77} - 10 q^{79} - 2 q^{81} + 28 q^{82} - 36 q^{87} - 30 q^{88} + 24 q^{90} + 10 q^{94} - 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.554958i 0.392415i 0.980562 + 0.196207i \(0.0628626\pi\)
−0.980562 + 0.196207i \(0.937137\pi\)
\(3\) 0.801938 0.462999 0.231499 0.972835i \(-0.425637\pi\)
0.231499 + 0.972835i \(0.425637\pi\)
\(4\) 1.69202 0.846011
\(5\) 2.80194i 1.25306i 0.779395 + 0.626532i \(0.215526\pi\)
−0.779395 + 0.626532i \(0.784474\pi\)
\(6\) 0.445042i 0.181688i
\(7\) − 2.69202i − 1.01749i −0.860918 0.508744i \(-0.830110\pi\)
0.860918 0.508744i \(-0.169890\pi\)
\(8\) 2.04892i 0.724402i
\(9\) −2.35690 −0.785632
\(10\) −1.55496 −0.491721
\(11\) − 1.19806i − 0.361229i −0.983554 0.180615i \(-0.942191\pi\)
0.983554 0.180615i \(-0.0578087\pi\)
\(12\) 1.35690 0.391702
\(13\) 0 0
\(14\) 1.49396 0.399277
\(15\) 2.24698i 0.580168i
\(16\) 2.24698 0.561745
\(17\) −1.13706 −0.275778 −0.137889 0.990448i \(-0.544032\pi\)
−0.137889 + 0.990448i \(0.544032\pi\)
\(18\) − 1.30798i − 0.308293i
\(19\) − 1.93900i − 0.444837i −0.974951 0.222419i \(-0.928605\pi\)
0.974951 0.222419i \(-0.0713952\pi\)
\(20\) 4.74094i 1.06011i
\(21\) − 2.15883i − 0.471096i
\(22\) 0.664874 0.141752
\(23\) 4.60388 0.959974 0.479987 0.877275i \(-0.340641\pi\)
0.479987 + 0.877275i \(0.340641\pi\)
\(24\) 1.64310i 0.335397i
\(25\) −2.85086 −0.570171
\(26\) 0 0
\(27\) −4.29590 −0.826746
\(28\) − 4.55496i − 0.860806i
\(29\) −7.89977 −1.46695 −0.733475 0.679716i \(-0.762103\pi\)
−0.733475 + 0.679716i \(0.762103\pi\)
\(30\) −1.24698 −0.227666
\(31\) − 5.89977i − 1.05963i −0.848113 0.529815i \(-0.822261\pi\)
0.848113 0.529815i \(-0.177739\pi\)
\(32\) 5.34481i 0.944839i
\(33\) − 0.960771i − 0.167249i
\(34\) − 0.631023i − 0.108219i
\(35\) 7.54288 1.27498
\(36\) −3.98792 −0.664653
\(37\) 0.951083i 0.156357i 0.996939 + 0.0781785i \(0.0249104\pi\)
−0.996939 + 0.0781785i \(0.975090\pi\)
\(38\) 1.07606 0.174561
\(39\) 0 0
\(40\) −5.74094 −0.907722
\(41\) − 3.31767i − 0.518133i −0.965860 0.259066i \(-0.916585\pi\)
0.965860 0.259066i \(-0.0834148\pi\)
\(42\) 1.19806 0.184865
\(43\) −7.15883 −1.09171 −0.545856 0.837879i \(-0.683795\pi\)
−0.545856 + 0.837879i \(0.683795\pi\)
\(44\) − 2.02715i − 0.305604i
\(45\) − 6.60388i − 0.984448i
\(46\) 2.55496i 0.376708i
\(47\) − 7.69202i − 1.12200i −0.827817 0.560998i \(-0.810417\pi\)
0.827817 0.560998i \(-0.189583\pi\)
\(48\) 1.80194 0.260087
\(49\) −0.246980 −0.0352828
\(50\) − 1.58211i − 0.223743i
\(51\) −0.911854 −0.127685
\(52\) 0 0
\(53\) 5.87263 0.806667 0.403334 0.915053i \(-0.367851\pi\)
0.403334 + 0.915053i \(0.367851\pi\)
\(54\) − 2.38404i − 0.324427i
\(55\) 3.35690 0.452644
\(56\) 5.51573 0.737070
\(57\) − 1.55496i − 0.205959i
\(58\) − 4.38404i − 0.575653i
\(59\) − 0.0120816i − 0.00157289i −1.00000 0.000786444i \(-0.999750\pi\)
1.00000 0.000786444i \(-0.000250333\pi\)
\(60\) 3.80194i 0.490828i
\(61\) −8.03684 −1.02901 −0.514506 0.857487i \(-0.672025\pi\)
−0.514506 + 0.857487i \(0.672025\pi\)
\(62\) 3.27413 0.415815
\(63\) 6.34481i 0.799371i
\(64\) 1.52781 0.190976
\(65\) 0 0
\(66\) 0.533188 0.0656309
\(67\) 9.25667i 1.13088i 0.824789 + 0.565441i \(0.191294\pi\)
−0.824789 + 0.565441i \(0.808706\pi\)
\(68\) −1.92394 −0.233311
\(69\) 3.69202 0.444467
\(70\) 4.18598i 0.500320i
\(71\) 13.7409i 1.63075i 0.578934 + 0.815375i \(0.303469\pi\)
−0.578934 + 0.815375i \(0.696531\pi\)
\(72\) − 4.82908i − 0.569113i
\(73\) 12.8170i 1.50012i 0.661372 + 0.750058i \(0.269975\pi\)
−0.661372 + 0.750058i \(0.730025\pi\)
\(74\) −0.527811 −0.0613568
\(75\) −2.28621 −0.263989
\(76\) − 3.28083i − 0.376337i
\(77\) −3.22521 −0.367547
\(78\) 0 0
\(79\) 0.807315 0.0908300 0.0454150 0.998968i \(-0.485539\pi\)
0.0454150 + 0.998968i \(0.485539\pi\)
\(80\) 6.29590i 0.703903i
\(81\) 3.62565 0.402850
\(82\) 1.84117 0.203323
\(83\) 16.3327i 1.79275i 0.443296 + 0.896375i \(0.353809\pi\)
−0.443296 + 0.896375i \(0.646191\pi\)
\(84\) − 3.65279i − 0.398552i
\(85\) − 3.18598i − 0.345568i
\(86\) − 3.97285i − 0.428404i
\(87\) −6.33513 −0.679197
\(88\) 2.45473 0.261675
\(89\) − 14.7289i − 1.56126i −0.624996 0.780628i \(-0.714899\pi\)
0.624996 0.780628i \(-0.285101\pi\)
\(90\) 3.66487 0.386312
\(91\) 0 0
\(92\) 7.78986 0.812149
\(93\) − 4.73125i − 0.490608i
\(94\) 4.26875 0.440288
\(95\) 5.43296 0.557410
\(96\) 4.28621i 0.437459i
\(97\) − 3.13169i − 0.317975i −0.987281 0.158987i \(-0.949177\pi\)
0.987281 0.158987i \(-0.0508229\pi\)
\(98\) − 0.137063i − 0.0138455i
\(99\) 2.82371i 0.283793i
\(100\) −4.82371 −0.482371
\(101\) −5.29052 −0.526426 −0.263213 0.964738i \(-0.584782\pi\)
−0.263213 + 0.964738i \(0.584782\pi\)
\(102\) − 0.506041i − 0.0501055i
\(103\) 13.5308 1.33323 0.666614 0.745403i \(-0.267743\pi\)
0.666614 + 0.745403i \(0.267743\pi\)
\(104\) 0 0
\(105\) 6.04892 0.590314
\(106\) 3.25906i 0.316548i
\(107\) 5.63102 0.544371 0.272186 0.962245i \(-0.412253\pi\)
0.272186 + 0.962245i \(0.412253\pi\)
\(108\) −7.26875 −0.699436
\(109\) − 4.17629i − 0.400016i −0.979794 0.200008i \(-0.935903\pi\)
0.979794 0.200008i \(-0.0640969\pi\)
\(110\) 1.86294i 0.177624i
\(111\) 0.762709i 0.0723931i
\(112\) − 6.04892i − 0.571569i
\(113\) 7.64310 0.719003 0.359501 0.933145i \(-0.382947\pi\)
0.359501 + 0.933145i \(0.382947\pi\)
\(114\) 0.862937 0.0808214
\(115\) 12.8998i 1.20291i
\(116\) −13.3666 −1.24106
\(117\) 0 0
\(118\) 0.00670477 0.000617224 0
\(119\) 3.06100i 0.280601i
\(120\) −4.60388 −0.420274
\(121\) 9.56465 0.869513
\(122\) − 4.46011i − 0.403799i
\(123\) − 2.66056i − 0.239895i
\(124\) − 9.98254i − 0.896459i
\(125\) 6.02177i 0.538604i
\(126\) −3.52111 −0.313685
\(127\) −6.77777 −0.601430 −0.300715 0.953714i \(-0.597225\pi\)
−0.300715 + 0.953714i \(0.597225\pi\)
\(128\) 11.5375i 1.01978i
\(129\) −5.74094 −0.505461
\(130\) 0 0
\(131\) −13.6799 −1.19522 −0.597611 0.801786i \(-0.703883\pi\)
−0.597611 + 0.801786i \(0.703883\pi\)
\(132\) − 1.62565i − 0.141494i
\(133\) −5.21983 −0.452617
\(134\) −5.13706 −0.443775
\(135\) − 12.0368i − 1.03597i
\(136\) − 2.32975i − 0.199774i
\(137\) 12.9879i 1.10963i 0.831973 + 0.554816i \(0.187211\pi\)
−0.831973 + 0.554816i \(0.812789\pi\)
\(138\) 2.04892i 0.174415i
\(139\) 12.0465 1.02177 0.510886 0.859648i \(-0.329317\pi\)
0.510886 + 0.859648i \(0.329317\pi\)
\(140\) 12.7627 1.07865
\(141\) − 6.16852i − 0.519483i
\(142\) −7.62565 −0.639930
\(143\) 0 0
\(144\) −5.29590 −0.441325
\(145\) − 22.1347i − 1.83818i
\(146\) −7.11290 −0.588668
\(147\) −0.198062 −0.0163359
\(148\) 1.60925i 0.132280i
\(149\) 0.740939i 0.0607001i 0.999539 + 0.0303500i \(0.00966220\pi\)
−0.999539 + 0.0303500i \(0.990338\pi\)
\(150\) − 1.26875i − 0.103593i
\(151\) − 19.0737i − 1.55219i −0.630614 0.776097i \(-0.717197\pi\)
0.630614 0.776097i \(-0.282803\pi\)
\(152\) 3.97285 0.322241
\(153\) 2.67994 0.216660
\(154\) − 1.78986i − 0.144231i
\(155\) 16.5308 1.32779
\(156\) 0 0
\(157\) −4.02177 −0.320972 −0.160486 0.987038i \(-0.551306\pi\)
−0.160486 + 0.987038i \(0.551306\pi\)
\(158\) 0.448026i 0.0356430i
\(159\) 4.70948 0.373486
\(160\) −14.9758 −1.18394
\(161\) − 12.3937i − 0.976763i
\(162\) 2.01208i 0.158084i
\(163\) 15.1371i 1.18563i 0.805340 + 0.592813i \(0.201983\pi\)
−0.805340 + 0.592813i \(0.798017\pi\)
\(164\) − 5.61356i − 0.438346i
\(165\) 2.69202 0.209574
\(166\) −9.06398 −0.703502
\(167\) − 6.26337i − 0.484674i −0.970192 0.242337i \(-0.922086\pi\)
0.970192 0.242337i \(-0.0779140\pi\)
\(168\) 4.42327 0.341263
\(169\) 0 0
\(170\) 1.76809 0.135606
\(171\) 4.57002i 0.349478i
\(172\) −12.1129 −0.923600
\(173\) −16.3913 −1.24621 −0.623105 0.782138i \(-0.714129\pi\)
−0.623105 + 0.782138i \(0.714129\pi\)
\(174\) − 3.51573i − 0.266527i
\(175\) 7.67456i 0.580142i
\(176\) − 2.69202i − 0.202919i
\(177\) − 0.00968868i 0 0.000728246i
\(178\) 8.17390 0.612660
\(179\) 2.45473 0.183475 0.0917376 0.995783i \(-0.470758\pi\)
0.0917376 + 0.995783i \(0.470758\pi\)
\(180\) − 11.1739i − 0.832853i
\(181\) −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
\(184\) 9.43296i 0.695407i
\(185\) −2.66487 −0.195925
\(186\) 2.62565 0.192522
\(187\) 1.36227i 0.0996192i
\(188\) − 13.0151i − 0.949221i
\(189\) 11.5646i 0.841204i
\(190\) 3.01507i 0.218736i
\(191\) −8.99330 −0.650732 −0.325366 0.945588i \(-0.605488\pi\)
−0.325366 + 0.945588i \(0.605488\pi\)
\(192\) 1.22521 0.0884219
\(193\) 13.5254i 0.973581i 0.873519 + 0.486790i \(0.161832\pi\)
−0.873519 + 0.486790i \(0.838168\pi\)
\(194\) 1.73795 0.124778
\(195\) 0 0
\(196\) −0.417895 −0.0298496
\(197\) − 12.9758i − 0.924490i −0.886752 0.462245i \(-0.847044\pi\)
0.886752 0.462245i \(-0.152956\pi\)
\(198\) −1.56704 −0.111365
\(199\) 13.5864 0.963116 0.481558 0.876414i \(-0.340071\pi\)
0.481558 + 0.876414i \(0.340071\pi\)
\(200\) − 5.84117i − 0.413033i
\(201\) 7.42327i 0.523597i
\(202\) − 2.93602i − 0.206577i
\(203\) 21.2664i 1.49261i
\(204\) −1.54288 −0.108023
\(205\) 9.29590 0.649254
\(206\) 7.50902i 0.523179i
\(207\) −10.8509 −0.754187
\(208\) 0 0
\(209\) −2.32304 −0.160688
\(210\) 3.35690i 0.231648i
\(211\) 10.4601 0.720103 0.360052 0.932932i \(-0.382759\pi\)
0.360052 + 0.932932i \(0.382759\pi\)
\(212\) 9.93661 0.682449
\(213\) 11.0194i 0.755035i
\(214\) 3.12498i 0.213619i
\(215\) − 20.0586i − 1.36799i
\(216\) − 8.80194i − 0.598896i
\(217\) −15.8823 −1.07816
\(218\) 2.31767 0.156972
\(219\) 10.2784i 0.694553i
\(220\) 5.67994 0.382941
\(221\) 0 0
\(222\) −0.423272 −0.0284081
\(223\) 11.4058i 0.763790i 0.924206 + 0.381895i \(0.124728\pi\)
−0.924206 + 0.381895i \(0.875272\pi\)
\(224\) 14.3884 0.961362
\(225\) 6.71917 0.447945
\(226\) 4.24160i 0.282147i
\(227\) − 10.6407i − 0.706249i −0.935576 0.353124i \(-0.885119\pi\)
0.935576 0.353124i \(-0.114881\pi\)
\(228\) − 2.63102i − 0.174244i
\(229\) − 1.13946i − 0.0752974i −0.999291 0.0376487i \(-0.988013\pi\)
0.999291 0.0376487i \(-0.0119868\pi\)
\(230\) −7.15883 −0.472040
\(231\) −2.58642 −0.170174
\(232\) − 16.1860i − 1.06266i
\(233\) 10.8509 0.710863 0.355432 0.934702i \(-0.384334\pi\)
0.355432 + 0.934702i \(0.384334\pi\)
\(234\) 0 0
\(235\) 21.5526 1.40593
\(236\) − 0.0204423i − 0.00133068i
\(237\) 0.647416 0.0420542
\(238\) −1.69873 −0.110112
\(239\) 11.9293i 0.771643i 0.922573 + 0.385822i \(0.126082\pi\)
−0.922573 + 0.385822i \(0.873918\pi\)
\(240\) 5.04892i 0.325906i
\(241\) − 3.64848i − 0.235019i −0.993072 0.117510i \(-0.962509\pi\)
0.993072 0.117510i \(-0.0374911\pi\)
\(242\) 5.30798i 0.341210i
\(243\) 15.7952 1.01326
\(244\) −13.5985 −0.870555
\(245\) − 0.692021i − 0.0442116i
\(246\) 1.47650 0.0941383
\(247\) 0 0
\(248\) 12.0881 0.767598
\(249\) 13.0978i 0.830042i
\(250\) −3.34183 −0.211356
\(251\) −1.37329 −0.0866813 −0.0433406 0.999060i \(-0.513800\pi\)
−0.0433406 + 0.999060i \(0.513800\pi\)
\(252\) 10.7356i 0.676277i
\(253\) − 5.51573i − 0.346771i
\(254\) − 3.76138i − 0.236010i
\(255\) − 2.55496i − 0.159998i
\(256\) −3.34721 −0.209200
\(257\) −29.4359 −1.83616 −0.918082 0.396391i \(-0.870263\pi\)
−0.918082 + 0.396391i \(0.870263\pi\)
\(258\) − 3.18598i − 0.198350i
\(259\) 2.56033 0.159091
\(260\) 0 0
\(261\) 18.6189 1.15248
\(262\) − 7.59179i − 0.469023i
\(263\) 10.6963 0.659564 0.329782 0.944057i \(-0.393025\pi\)
0.329782 + 0.944057i \(0.393025\pi\)
\(264\) 1.96854 0.121155
\(265\) 16.4547i 1.01081i
\(266\) − 2.89679i − 0.177613i
\(267\) − 11.8116i − 0.722860i
\(268\) 15.6625i 0.956738i
\(269\) −10.1860 −0.621050 −0.310525 0.950565i \(-0.600505\pi\)
−0.310525 + 0.950565i \(0.600505\pi\)
\(270\) 6.67994 0.406528
\(271\) − 29.4523i − 1.78910i −0.446966 0.894551i \(-0.647495\pi\)
0.446966 0.894551i \(-0.352505\pi\)
\(272\) −2.55496 −0.154917
\(273\) 0 0
\(274\) −7.20775 −0.435436
\(275\) 3.41550i 0.205963i
\(276\) 6.24698 0.376024
\(277\) 10.2446 0.615538 0.307769 0.951461i \(-0.400418\pi\)
0.307769 + 0.951461i \(0.400418\pi\)
\(278\) 6.68532i 0.400959i
\(279\) 13.9051i 0.832480i
\(280\) 15.4547i 0.923597i
\(281\) − 11.5646i − 0.689889i −0.938623 0.344944i \(-0.887898\pi\)
0.938623 0.344944i \(-0.112102\pi\)
\(282\) 3.42327 0.203853
\(283\) 30.7090 1.82546 0.912730 0.408562i \(-0.133970\pi\)
0.912730 + 0.408562i \(0.133970\pi\)
\(284\) 23.2500i 1.37963i
\(285\) 4.35690 0.258080
\(286\) 0 0
\(287\) −8.93123 −0.527194
\(288\) − 12.5972i − 0.742295i
\(289\) −15.7071 −0.923946
\(290\) 12.2838 0.721330
\(291\) − 2.51142i − 0.147222i
\(292\) 21.6866i 1.26911i
\(293\) − 18.6082i − 1.08710i −0.839376 0.543551i \(-0.817079\pi\)
0.839376 0.543551i \(-0.182921\pi\)
\(294\) − 0.109916i − 0.00641045i
\(295\) 0.0338518 0.00197093
\(296\) −1.94869 −0.113265
\(297\) 5.14675i 0.298645i
\(298\) −0.411190 −0.0238196
\(299\) 0 0
\(300\) −3.86831 −0.223337
\(301\) 19.2717i 1.11080i
\(302\) 10.5851 0.609103
\(303\) −4.24267 −0.243735
\(304\) − 4.35690i − 0.249885i
\(305\) − 22.5187i − 1.28942i
\(306\) 1.48725i 0.0850207i
\(307\) 8.94438i 0.510483i 0.966877 + 0.255241i \(0.0821549\pi\)
−0.966877 + 0.255241i \(0.917845\pi\)
\(308\) −5.45712 −0.310948
\(309\) 10.8509 0.617284
\(310\) 9.17390i 0.521042i
\(311\) −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 \(-0.564559\pi\)
−0.201432 + 0.979503i \(0.564559\pi\)
\(314\) − 2.23191i − 0.125954i
\(315\) −17.7778 −1.00166
\(316\) 1.36599 0.0768431
\(317\) − 23.9651i − 1.34601i −0.739636 0.673007i \(-0.765003\pi\)
0.739636 0.673007i \(-0.234997\pi\)
\(318\) 2.61356i 0.146561i
\(319\) 9.46442i 0.529906i
\(320\) 4.28083i 0.239306i
\(321\) 4.51573 0.252043
\(322\) 6.87800 0.383296
\(323\) 2.20477i 0.122677i
\(324\) 6.13467 0.340815
\(325\) 0 0
\(326\) −8.40044 −0.465257
\(327\) − 3.34913i − 0.185207i
\(328\) 6.79763 0.375336
\(329\) −20.7071 −1.14162
\(330\) 1.49396i 0.0822397i
\(331\) − 2.89546i − 0.159149i −0.996829 0.0795745i \(-0.974644\pi\)
0.996829 0.0795745i \(-0.0253561\pi\)
\(332\) 27.6353i 1.51669i
\(333\) − 2.24160i − 0.122839i
\(334\) 3.47591 0.190193
\(335\) −25.9366 −1.41707
\(336\) − 4.85086i − 0.264636i
\(337\) 3.10560 0.169173 0.0845865 0.996416i \(-0.473043\pi\)
0.0845865 + 0.996416i \(0.473043\pi\)
\(338\) 0 0
\(339\) 6.12929 0.332898
\(340\) − 5.39075i − 0.292354i
\(341\) −7.06829 −0.382770
\(342\) −2.53617 −0.137140
\(343\) − 18.1793i − 0.981589i
\(344\) − 14.6679i − 0.790838i
\(345\) 10.3448i 0.556946i
\(346\) − 9.09651i − 0.489031i
\(347\) −11.3787 −0.610839 −0.305419 0.952218i \(-0.598797\pi\)
−0.305419 + 0.952218i \(0.598797\pi\)
\(348\) −10.7192 −0.574608
\(349\) − 3.34721i − 0.179172i −0.995979 0.0895859i \(-0.971446\pi\)
0.995979 0.0895859i \(-0.0285544\pi\)
\(350\) −4.25906 −0.227656
\(351\) 0 0
\(352\) 6.40342 0.341303
\(353\) − 0.637727i − 0.0339428i −0.999856 0.0169714i \(-0.994598\pi\)
0.999856 0.0169714i \(-0.00540242\pi\)
\(354\) 0.00537681 0.000285774 0
\(355\) −38.5013 −2.04343
\(356\) − 24.9215i − 1.32084i
\(357\) 2.45473i 0.129918i
\(358\) 1.36227i 0.0719983i
\(359\) − 21.4590i − 1.13256i −0.824211 0.566282i \(-0.808381\pi\)
0.824211 0.566282i \(-0.191619\pi\)
\(360\) 13.5308 0.713136
\(361\) 15.2403 0.802120
\(362\) − 6.55257i − 0.344395i
\(363\) 7.67025 0.402584
\(364\) 0 0
\(365\) −35.9124 −1.87974
\(366\) − 3.57673i − 0.186959i
\(367\) −9.38703 −0.489999 −0.244999 0.969523i \(-0.578788\pi\)
−0.244999 + 0.969523i \(0.578788\pi\)
\(368\) 10.3448 0.539261
\(369\) 7.81940i 0.407062i
\(370\) − 1.47889i − 0.0768840i
\(371\) − 15.8092i − 0.820775i
\(372\) − 8.00538i − 0.415059i
\(373\) 27.7265 1.43562 0.717811 0.696238i \(-0.245144\pi\)
0.717811 + 0.696238i \(0.245144\pi\)
\(374\) −0.756004 −0.0390921
\(375\) 4.82908i 0.249373i
\(376\) 15.7603 0.812776
\(377\) 0 0
\(378\) −6.41789 −0.330101
\(379\) − 35.8702i − 1.84253i −0.388935 0.921265i \(-0.627157\pi\)
0.388935 0.921265i \(-0.372843\pi\)
\(380\) 9.19269 0.471575
\(381\) −5.43535 −0.278462
\(382\) − 4.99090i − 0.255357i
\(383\) − 4.85517i − 0.248087i −0.992277 0.124044i \(-0.960414\pi\)
0.992277 0.124044i \(-0.0395863\pi\)
\(384\) 9.25236i 0.472157i
\(385\) − 9.03684i − 0.460560i
\(386\) −7.50604 −0.382047
\(387\) 16.8726 0.857684
\(388\) − 5.29888i − 0.269010i
\(389\) −2.38537 −0.120943 −0.0604716 0.998170i \(-0.519260\pi\)
−0.0604716 + 0.998170i \(0.519260\pi\)
\(390\) 0 0
\(391\) −5.23490 −0.264740
\(392\) − 0.506041i − 0.0255589i
\(393\) −10.9705 −0.553387
\(394\) 7.20105 0.362783
\(395\) 2.26205i 0.113816i
\(396\) 4.77777i 0.240092i
\(397\) − 15.2664i − 0.766196i −0.923708 0.383098i \(-0.874857\pi\)
0.923708 0.383098i \(-0.125143\pi\)
\(398\) 7.53989i 0.377941i
\(399\) −4.18598 −0.209561
\(400\) −6.40581 −0.320291
\(401\) 12.7584i 0.637124i 0.947902 + 0.318562i \(0.103200\pi\)
−0.947902 + 0.318562i \(0.896800\pi\)
\(402\) −4.11960 −0.205467
\(403\) 0 0
\(404\) −8.95167 −0.445362
\(405\) 10.1588i 0.504797i
\(406\) −11.8019 −0.585720
\(407\) 1.13946 0.0564807
\(408\) − 1.86831i − 0.0924953i
\(409\) 25.3588i 1.25391i 0.779054 + 0.626956i \(0.215700\pi\)
−0.779054 + 0.626956i \(0.784300\pi\)
\(410\) 5.15883i 0.254777i
\(411\) 10.4155i 0.513759i
\(412\) 22.8944 1.12793
\(413\) −0.0325239 −0.00160040
\(414\) − 6.02177i − 0.295954i
\(415\) −45.7633 −2.24643
\(416\) 0 0
\(417\) 9.66056 0.473080
\(418\) − 1.28919i − 0.0630565i
\(419\) −11.6673 −0.569983 −0.284992 0.958530i \(-0.591991\pi\)
−0.284992 + 0.958530i \(0.591991\pi\)
\(420\) 10.2349 0.499412
\(421\) 8.29291i 0.404172i 0.979368 + 0.202086i \(0.0647720\pi\)
−0.979368 + 0.202086i \(0.935228\pi\)
\(422\) 5.80492i 0.282579i
\(423\) 18.1293i 0.881476i
\(424\) 12.0325i 0.584351i
\(425\) 3.24160 0.157241
\(426\) −6.11529 −0.296287
\(427\) 21.6353i 1.04701i
\(428\) 9.52781 0.460544
\(429\) 0 0
\(430\) 11.1317 0.536818
\(431\) − 0.932296i − 0.0449071i −0.999748 0.0224536i \(-0.992852\pi\)
0.999748 0.0224536i \(-0.00714779\pi\)
\(432\) −9.65279 −0.464420
\(433\) 13.3502 0.641569 0.320785 0.947152i \(-0.396053\pi\)
0.320785 + 0.947152i \(0.396053\pi\)
\(434\) − 8.81402i − 0.423086i
\(435\) − 17.7506i − 0.851077i
\(436\) − 7.06638i − 0.338418i
\(437\) − 8.92692i − 0.427032i
\(438\) −5.70410 −0.272553
\(439\) −13.9922 −0.667813 −0.333906 0.942606i \(-0.608367\pi\)
−0.333906 + 0.942606i \(0.608367\pi\)
\(440\) 6.87800i 0.327896i
\(441\) 0.582105 0.0277193
\(442\) 0 0
\(443\) 23.7017 1.12610 0.563051 0.826422i \(-0.309627\pi\)
0.563051 + 0.826422i \(0.309627\pi\)
\(444\) 1.29052i 0.0612454i
\(445\) 41.2693 1.95635
\(446\) −6.32975 −0.299722
\(447\) 0.594187i 0.0281041i
\(448\) − 4.11290i − 0.194316i
\(449\) − 12.5864i − 0.593990i −0.954879 0.296995i \(-0.904016\pi\)
0.954879 0.296995i \(-0.0959844\pi\)
\(450\) 3.72886i 0.175780i
\(451\) −3.97477 −0.187165
\(452\) 12.9323 0.608284
\(453\) − 15.2959i − 0.718664i
\(454\) 5.90515 0.277142
\(455\) 0 0
\(456\) 3.18598 0.149197
\(457\) 33.6383i 1.57353i 0.617250 + 0.786767i \(0.288247\pi\)
−0.617250 + 0.786767i \(0.711753\pi\)
\(458\) 0.632351 0.0295478
\(459\) 4.88471 0.227999
\(460\) 21.8267i 1.01767i
\(461\) 1.40283i 0.0653363i 0.999466 + 0.0326681i \(0.0104004\pi\)
−0.999466 + 0.0326681i \(0.989600\pi\)
\(462\) − 1.43535i − 0.0667787i
\(463\) − 15.2010i − 0.706453i −0.935538 0.353226i \(-0.885085\pi\)
0.935538 0.353226i \(-0.114915\pi\)
\(464\) −17.7506 −0.824052
\(465\) 13.2567 0.614763
\(466\) 6.02177i 0.278953i
\(467\) 39.3414 1.82050 0.910250 0.414058i \(-0.135889\pi\)
0.910250 + 0.414058i \(0.135889\pi\)
\(468\) 0 0
\(469\) 24.9191 1.15066
\(470\) 11.9608i 0.551709i
\(471\) −3.22521 −0.148610
\(472\) 0.0247542 0.00113940
\(473\) 8.57673i 0.394358i
\(474\) 0.359289i 0.0165027i
\(475\) 5.52781i 0.253633i
\(476\) 5.17928i 0.237392i
\(477\) −13.8412 −0.633743
\(478\) −6.62027 −0.302804
\(479\) − 22.3690i − 1.02206i −0.859561 0.511032i \(-0.829263\pi\)
0.859561 0.511032i \(-0.170737\pi\)
\(480\) −12.0097 −0.548165
\(481\) 0 0
\(482\) 2.02475 0.0922250
\(483\) − 9.93900i − 0.452240i
\(484\) 16.1836 0.735618
\(485\) 8.77479 0.398443
\(486\) 8.76569i 0.397620i
\(487\) − 22.9205i − 1.03863i −0.854584 0.519313i \(-0.826188\pi\)
0.854584 0.519313i \(-0.173812\pi\)
\(488\) − 16.4668i − 0.745418i
\(489\) 12.1390i 0.548944i
\(490\) 0.384043 0.0173493
\(491\) −1.84356 −0.0831987 −0.0415993 0.999134i \(-0.513245\pi\)
−0.0415993 + 0.999134i \(0.513245\pi\)
\(492\) − 4.50173i − 0.202954i
\(493\) 8.98254 0.404553
\(494\) 0 0
\(495\) −7.91185 −0.355611
\(496\) − 13.2567i − 0.595242i
\(497\) 36.9909 1.65927
\(498\) −7.26875 −0.325720
\(499\) 12.0344i 0.538736i 0.963037 + 0.269368i \(0.0868147\pi\)
−0.963037 + 0.269368i \(0.913185\pi\)
\(500\) 10.1890i 0.455664i
\(501\) − 5.02284i − 0.224404i
\(502\) − 0.762118i − 0.0340150i
\(503\) −30.5056 −1.36018 −0.680088 0.733130i \(-0.738058\pi\)
−0.680088 + 0.733130i \(0.738058\pi\)
\(504\) −13.0000 −0.579066
\(505\) − 14.8237i − 0.659646i
\(506\) 3.06100 0.136078
\(507\) 0 0
\(508\) −11.4681 −0.508816
\(509\) 1.51142i 0.0669924i 0.999439 + 0.0334962i \(0.0106642\pi\)
−0.999439 + 0.0334962i \(0.989336\pi\)
\(510\) 1.41789 0.0627854
\(511\) 34.5036 1.52635
\(512\) 21.2174i 0.937687i
\(513\) 8.32975i 0.367767i
\(514\) − 16.3357i − 0.720538i
\(515\) 37.9124i 1.67062i
\(516\) −9.71379 −0.427626
\(517\) −9.21552 −0.405298
\(518\) 1.42088i 0.0624298i
\(519\) −13.1448 −0.576994
\(520\) 0 0
\(521\) −5.64012 −0.247098 −0.123549 0.992338i \(-0.539428\pi\)
−0.123549 + 0.992338i \(0.539428\pi\)
\(522\) 10.3327i 0.452251i
\(523\) −31.7506 −1.38836 −0.694179 0.719802i \(-0.744232\pi\)
−0.694179 + 0.719802i \(0.744232\pi\)
\(524\) −23.1468 −1.01117
\(525\) 6.15452i 0.268605i
\(526\) 5.93602i 0.258823i
\(527\) 6.70841i 0.292223i
\(528\) − 2.15883i − 0.0939512i
\(529\) −1.80433 −0.0784492
\(530\) −9.13169 −0.396655
\(531\) 0.0284750i 0.00123571i
\(532\) −8.83207 −0.382919
\(533\) 0 0
\(534\) 6.55496 0.283661
\(535\) 15.7778i 0.682133i
\(536\) −18.9661 −0.819213
\(537\) 1.96854 0.0849488
\(538\) − 5.65279i − 0.243709i
\(539\) 0.295897i 0.0127452i
\(540\) − 20.3666i − 0.876438i
\(541\) 24.3297i 1.04602i 0.852327 + 0.523009i \(0.175191\pi\)
−0.852327 + 0.523009i \(0.824809\pi\)
\(542\) 16.3448 0.702070
\(543\) −9.46873 −0.406342
\(544\) − 6.07739i − 0.260566i
\(545\) 11.7017 0.501246
\(546\) 0 0
\(547\) −8.18896 −0.350135 −0.175067 0.984556i \(-0.556014\pi\)
−0.175067 + 0.984556i \(0.556014\pi\)
\(548\) 21.9758i 0.938761i
\(549\) 18.9420 0.808424
\(550\) −1.89546 −0.0808227
\(551\) 15.3177i 0.652555i
\(552\) 7.56465i 0.321973i
\(553\) − 2.17331i − 0.0924185i
\(554\) 5.68532i 0.241546i
\(555\) −2.13706 −0.0907133
\(556\) 20.3830 0.864431
\(557\) 25.3327i 1.07338i 0.843779 + 0.536691i \(0.180326\pi\)
−0.843779 + 0.536691i \(0.819674\pi\)
\(558\) −7.71678 −0.326677
\(559\) 0 0
\(560\) 16.9487 0.716213
\(561\) 1.09246i 0.0461236i
\(562\) 6.41789 0.270723
\(563\) 25.3937 1.07022 0.535109 0.844783i \(-0.320270\pi\)
0.535109 + 0.844783i \(0.320270\pi\)
\(564\) − 10.4373i − 0.439488i
\(565\) 21.4155i 0.900957i
\(566\) 17.0422i 0.716338i
\(567\) − 9.76032i − 0.409895i
\(568\) −28.1540 −1.18132
\(569\) 31.1347 1.30523 0.652617 0.757688i \(-0.273671\pi\)
0.652617 + 0.757688i \(0.273671\pi\)
\(570\) 2.41789i 0.101274i
\(571\) 20.5090 0.858276 0.429138 0.903239i \(-0.358817\pi\)
0.429138 + 0.903239i \(0.358817\pi\)
\(572\) 0 0
\(573\) −7.21206 −0.301288
\(574\) − 4.95646i − 0.206879i
\(575\) −13.1250 −0.547350
\(576\) −3.60089 −0.150037
\(577\) − 15.6890i − 0.653143i −0.945172 0.326572i \(-0.894107\pi\)
0.945172 0.326572i \(-0.105893\pi\)
\(578\) − 8.71678i − 0.362570i
\(579\) 10.8465i 0.450767i
\(580\) − 37.4523i − 1.55512i
\(581\) 43.9681 1.82410
\(582\) 1.39373 0.0577720
\(583\) − 7.03577i − 0.291392i
\(584\) −26.2610 −1.08669
\(585\) 0 0
\(586\) 10.3268 0.426595
\(587\) − 30.5687i − 1.26171i −0.775903 0.630853i \(-0.782705\pi\)
0.775903 0.630853i \(-0.217295\pi\)
\(588\) −0.335126 −0.0138203
\(589\) −11.4397 −0.471363
\(590\) 0.0187864i 0 0.000773422i
\(591\) − 10.4058i − 0.428038i
\(592\) 2.13706i 0.0878328i
\(593\) − 29.6883i − 1.21915i −0.792727 0.609576i \(-0.791340\pi\)
0.792727 0.609576i \(-0.208660\pi\)
\(594\) −2.85623 −0.117193
\(595\) −8.57673 −0.351612
\(596\) 1.25368i 0.0513529i
\(597\) 10.8955 0.445922
\(598\) 0 0
\(599\) 24.2325 0.990113 0.495057 0.868861i \(-0.335147\pi\)
0.495057 + 0.868861i \(0.335147\pi\)
\(600\) − 4.68425i − 0.191234i
\(601\) 16.4819 0.672310 0.336155 0.941807i \(-0.390873\pi\)
0.336155 + 0.941807i \(0.390873\pi\)
\(602\) −10.6950 −0.435896
\(603\) − 21.8170i − 0.888457i
\(604\) − 32.2731i − 1.31317i
\(605\) 26.7995i 1.08956i
\(606\) − 2.35450i − 0.0956451i
\(607\) 1.43190 0.0581188 0.0290594 0.999578i \(-0.490749\pi\)
0.0290594 + 0.999578i \(0.490749\pi\)
\(608\) 10.3636 0.420300
\(609\) 17.0543i 0.691075i
\(610\) 12.4969 0.505986
\(611\) 0 0
\(612\) 4.53452 0.183297
\(613\) − 3.84846i − 0.155438i −0.996975 0.0777190i \(-0.975236\pi\)
0.996975 0.0777190i \(-0.0247637\pi\)
\(614\) −4.96376 −0.200321
\(615\) 7.45473 0.300604
\(616\) − 6.60819i − 0.266251i
\(617\) 15.0388i 0.605437i 0.953080 + 0.302719i \(0.0978943\pi\)
−0.953080 + 0.302719i \(0.902106\pi\)
\(618\) 6.02177i 0.242231i
\(619\) 12.8170i 0.515159i 0.966257 + 0.257579i \(0.0829249\pi\)
−0.966257 + 0.257579i \(0.917075\pi\)
\(620\) 27.9705 1.12332
\(621\) −19.7778 −0.793655
\(622\) − 11.6762i − 0.468174i
\(623\) −39.6504 −1.58856
\(624\) 0 0
\(625\) −31.1269 −1.24508
\(626\) − 3.95539i − 0.158089i
\(627\) −1.86294 −0.0743985
\(628\) −6.80492 −0.271546
\(629\) − 1.08144i − 0.0431199i
\(630\) − 9.86592i − 0.393068i
\(631\) 25.7517i 1.02516i 0.858640 + 0.512579i \(0.171310\pi\)
−0.858640 + 0.512579i \(0.828690\pi\)
\(632\) 1.65412i 0.0657974i
\(633\) 8.38835 0.333407
\(634\) 13.2996 0.528195
\(635\) − 18.9909i − 0.753631i
\(636\) 7.96854 0.315973
\(637\) 0 0
\(638\) −5.25236 −0.207943
\(639\) − 32.3860i − 1.28117i
\(640\) −32.3274 −1.27785
\(641\) −24.4571 −0.965998 −0.482999 0.875621i \(-0.660453\pi\)
−0.482999 + 0.875621i \(0.660453\pi\)
\(642\) 2.50604i 0.0989055i
\(643\) 9.97344i 0.393314i 0.980472 + 0.196657i \(0.0630086\pi\)
−0.980472 + 0.196657i \(0.936991\pi\)
\(644\) − 20.9705i − 0.826352i
\(645\) − 16.0858i − 0.633376i
\(646\) −1.22355 −0.0481401
\(647\) −11.8431 −0.465600 −0.232800 0.972525i \(-0.574789\pi\)
−0.232800 + 0.972525i \(0.574789\pi\)
\(648\) 7.42865i 0.291825i
\(649\) −0.0144745 −0.000568173 0
\(650\) 0 0
\(651\) −12.7366 −0.499188
\(652\) 25.6122i 1.00305i
\(653\) −7.47411 −0.292484 −0.146242 0.989249i \(-0.546718\pi\)
−0.146242 + 0.989249i \(0.546718\pi\)
\(654\) 1.85862 0.0726780
\(655\) − 38.3303i − 1.49769i
\(656\) − 7.45473i − 0.291058i
\(657\) − 30.2083i − 1.17854i
\(658\) − 11.4916i − 0.447988i
\(659\) 34.1739 1.33123 0.665613 0.746297i \(-0.268170\pi\)
0.665613 + 0.746297i \(0.268170\pi\)
\(660\) 4.55496 0.177302
\(661\) 33.6088i 1.30723i 0.756827 + 0.653615i \(0.226748\pi\)
−0.756827 + 0.653615i \(0.773252\pi\)
\(662\) 1.60686 0.0624524
\(663\) 0 0
\(664\) −33.4644 −1.29867
\(665\) − 14.6256i − 0.567158i
\(666\) 1.24400 0.0482039
\(667\) −36.3696 −1.40824
\(668\) − 10.5978i − 0.410040i
\(669\) 9.14675i 0.353634i
\(670\) − 14.3937i − 0.556078i
\(671\) 9.62863i 0.371709i
\(672\) 11.5386 0.445110
\(673\) −48.0320 −1.85150 −0.925750 0.378137i \(-0.876565\pi\)
−0.925750 + 0.378137i \(0.876565\pi\)
\(674\) 1.72348i 0.0663860i
\(675\) 12.2470 0.471386
\(676\) 0 0
\(677\) −33.6582 −1.29359 −0.646794 0.762665i \(-0.723891\pi\)
−0.646794 + 0.762665i \(0.723891\pi\)
\(678\) 3.40150i 0.130634i
\(679\) −8.43057 −0.323535
\(680\) 6.52781 0.250330
\(681\) − 8.53319i − 0.326992i
\(682\) − 3.92261i − 0.150204i
\(683\) 15.9041i 0.608553i 0.952584 + 0.304276i \(0.0984147\pi\)
−0.952584 + 0.304276i \(0.901585\pi\)
\(684\) 7.73258i 0.295663i
\(685\) −36.3913 −1.39044
\(686\) 10.0887 0.385190
\(687\) − 0.913773i − 0.0348626i
\(688\) −16.0858 −0.613264
\(689\) 0 0
\(690\) −5.74094 −0.218554
\(691\) − 33.1903i − 1.26262i −0.775531 0.631309i \(-0.782518\pi\)
0.775531 0.631309i \(-0.217482\pi\)
\(692\) −27.7345 −1.05431
\(693\) 7.60148 0.288756
\(694\) − 6.31468i − 0.239702i
\(695\) 33.7536i 1.28035i
\(696\) − 12.9801i − 0.492011i
\(697\) 3.77240i 0.142890i
\(698\) 1.85756 0.0703097
\(699\) 8.70171 0.329129
\(700\) 12.9855i 0.490807i
\(701\) 14.9129 0.563253 0.281627 0.959524i \(-0.409126\pi\)
0.281627 + 0.959524i \(0.409126\pi\)
\(702\) 0 0
\(703\) 1.84415 0.0695534
\(704\) − 1.83041i − 0.0689863i
\(705\) 17.2838 0.650946
\(706\) 0.353912 0.0133197
\(707\) 14.2422i 0.535633i
\(708\) − 0.0163935i 0 0.000616104i
\(709\) 38.4312i 1.44331i 0.692252 + 0.721656i \(0.256619\pi\)
−0.692252 + 0.721656i \(0.743381\pi\)
\(710\) − 21.3666i − 0.801874i
\(711\) −1.90276 −0.0713589
\(712\) 30.1782 1.13098
\(713\) − 27.1618i − 1.01722i
\(714\) −1.36227 −0.0509818
\(715\) 0 0
\(716\) 4.15346 0.155222
\(717\) 9.56657i 0.357270i
\(718\) 11.9089 0.444435
\(719\) 11.4373 0.426538 0.213269 0.976993i \(-0.431589\pi\)
0.213269 + 0.976993i \(0.431589\pi\)
\(720\) − 14.8388i − 0.553008i
\(721\) − 36.4252i − 1.35654i
\(722\) 8.45771i 0.314764i
\(723\) − 2.92585i − 0.108814i
\(724\) −19.9782 −0.742485
\(725\) 22.5211 0.836413
\(726\) 4.25667i 0.157980i
\(727\) −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
\(730\) − 19.9299i − 0.737639i
\(731\) 8.14005 0.301071
\(732\) −10.9051 −0.403066
\(733\) 3.52217i 0.130094i 0.997882 + 0.0650472i \(0.0207198\pi\)
−0.997882 + 0.0650472i \(0.979280\pi\)
\(734\) − 5.20941i − 0.192283i
\(735\) − 0.554958i − 0.0204699i
\(736\) 24.6069i 0.907021i
\(737\) 11.0901 0.408508
\(738\) −4.33944 −0.159737
\(739\) 0.420288i 0.0154605i 0.999970 + 0.00773027i \(0.00246064\pi\)
−0.999970 + 0.00773027i \(0.997539\pi\)
\(740\) −4.50902 −0.165755
\(741\) 0 0
\(742\) 8.77346 0.322084
\(743\) 25.3623i 0.930452i 0.885192 + 0.465226i \(0.154027\pi\)
−0.885192 + 0.465226i \(0.845973\pi\)
\(744\) 9.69394 0.355397
\(745\) −2.07606 −0.0760611
\(746\) 15.3870i 0.563359i
\(747\) − 38.4946i − 1.40844i
\(748\) 2.30499i 0.0842790i
\(749\) − 15.1588i − 0.553892i
\(750\) −2.67994 −0.0978576
\(751\) −0.650874 −0.0237507 −0.0118754 0.999929i \(-0.503780\pi\)
−0.0118754 + 0.999929i \(0.503780\pi\)
\(752\) − 17.2838i − 0.630276i
\(753\) −1.10129 −0.0401333
\(754\) 0 0
\(755\) 53.4432 1.94500
\(756\) 19.5676i 0.711668i
\(757\) −16.7909 −0.610276 −0.305138 0.952308i \(-0.598703\pi\)
−0.305138 + 0.952308i \(0.598703\pi\)
\(758\) 19.9065 0.723036
\(759\) − 4.42327i − 0.160555i
\(760\) 11.1317i 0.403789i
\(761\) 30.9221i 1.12093i 0.828179 + 0.560463i \(0.189377\pi\)
−0.828179 + 0.560463i \(0.810623\pi\)
\(762\) − 3.01639i − 0.109272i
\(763\) −11.2427 −0.407012
\(764\) −15.2168 −0.550526
\(765\) 7.50902i 0.271489i
\(766\) 2.69441 0.0973531
\(767\) 0 0
\(768\) −2.68425 −0.0968596
\(769\) 43.7689i 1.57835i 0.614169 + 0.789174i \(0.289491\pi\)
−0.614169 + 0.789174i \(0.710509\pi\)
\(770\) 5.01507 0.180730
\(771\) −23.6058 −0.850142
\(772\) 22.8853i 0.823660i
\(773\) 42.4209i 1.52577i 0.646532 + 0.762886i \(0.276218\pi\)
−0.646532 + 0.762886i \(0.723782\pi\)
\(774\) 9.36360i 0.336568i
\(775\) 16.8194i 0.604171i
\(776\) 6.41657 0.230341
\(777\) 2.05323 0.0736592
\(778\) − 1.32378i − 0.0474598i
\(779\) −6.43296 −0.230485
\(780\) 0 0
\(781\) 16.4625 0.589075
\(782\) − 2.90515i − 0.103888i
\(783\) 33.9366 1.21280
\(784\) −0.554958 −0.0198199
\(785\) − 11.2687i − 0.402199i
\(786\) − 6.08815i − 0.217157i
\(787\) − 36.0116i − 1.28368i −0.766841 0.641838i \(-0.778172\pi\)
0.766841 0.641838i \(-0.221828\pi\)
\(788\) − 21.9554i − 0.782129i
\(789\) 8.57779 0.305378
\(790\) −1.25534 −0.0446630
\(791\) − 20.5754i − 0.731577i
\(792\) −5.78554 −0.205580
\(793\) 0 0
\(794\) 8.47219 0.300667
\(795\) 13.1957i 0.468002i
\(796\) 22.9885 0.814806
\(797\) 31.7101 1.12323 0.561614 0.827399i \(-0.310181\pi\)
0.561614 + 0.827399i \(0.310181\pi\)
\(798\) − 2.32304i − 0.0822349i
\(799\) 8.74632i 0.309422i
\(800\) − 15.2373i − 0.538720i
\(801\) 34.7144i 1.22657i
\(802\) −7.08038 −0.250017
\(803\) 15.3556 0.541886
\(804\) 12.5603i 0.442969i
\(805\) 34.7265 1.22395
\(806\) 0 0
\(807\) −8.16852 −0.287546
\(808\) − 10.8398i − 0.381344i
\(809\) −45.2814 −1.59201 −0.796005 0.605290i \(-0.793057\pi\)
−0.796005 + 0.605290i \(0.793057\pi\)
\(810\) −5.63773 −0.198090
\(811\) 42.8635i 1.50514i 0.658511 + 0.752571i \(0.271187\pi\)
−0.658511 + 0.752571i \(0.728813\pi\)
\(812\) 35.9831i 1.26276i
\(813\) − 23.6189i − 0.828352i
\(814\) 0.632351i 0.0221639i
\(815\) −42.4131 −1.48567
\(816\) −2.04892 −0.0717265
\(817\) 13.8810i 0.485634i
\(818\) −14.0731 −0.492054
\(819\) 0 0
\(820\) 15.7289 0.549276
\(821\) 7.82776i 0.273191i 0.990627 + 0.136595i \(0.0436160\pi\)
−0.990627 + 0.136595i \(0.956384\pi\)
\(822\) −5.78017 −0.201606
\(823\) 36.7754 1.28191 0.640955 0.767579i \(-0.278539\pi\)
0.640955 + 0.767579i \(0.278539\pi\)
\(824\) 27.7235i 0.965793i
\(825\) 2.73902i 0.0953604i
\(826\) − 0.0180494i 0 0.000628019i
\(827\) 47.3293i 1.64580i 0.568186 + 0.822900i \(0.307645\pi\)
−0.568186 + 0.822900i \(0.692355\pi\)
\(828\) −18.3599 −0.638050
\(829\) −25.2687 −0.877620 −0.438810 0.898580i \(-0.644600\pi\)
−0.438810 + 0.898580i \(0.644600\pi\)
\(830\) − 25.3967i − 0.881533i
\(831\) 8.21552 0.284993
\(832\) 0 0
\(833\) 0.280831 0.00973023
\(834\) 5.36121i 0.185643i
\(835\) 17.5496 0.607328
\(836\) −3.93064 −0.135944
\(837\) 25.3448i 0.876045i
\(838\) − 6.47484i − 0.223670i
\(839\) − 37.6883i − 1.30114i −0.759444 0.650572i \(-0.774529\pi\)
0.759444 0.650572i \(-0.225471\pi\)
\(840\) 12.3937i 0.427624i
\(841\) 33.4064 1.15194
\(842\) −4.60222 −0.158603
\(843\) − 9.27413i − 0.319418i
\(844\) 17.6987 0.609215
\(845\) 0 0
\(846\) −10.0610 −0.345904
\(847\) − 25.7482i − 0.884720i
\(848\) 13.1957 0.453141
\(849\) 24.6267 0.845187
\(850\) 1.79895i 0.0617036i
\(851\) 4.37867i 0.150099i
\(852\) 18.6450i 0.638768i
\(853\) − 31.0121i − 1.06183i −0.847424 0.530917i \(-0.821848\pi\)
0.847424 0.530917i \(-0.178152\pi\)
\(854\) −12.0067 −0.410861
\(855\) −12.8049 −0.437919
\(856\) 11.5375i 0.394344i
\(857\) −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 \(-0.595457\pi\)
−0.295413 + 0.955370i \(0.595457\pi\)
\(860\) − 33.9396i − 1.15733i
\(861\) −7.16229 −0.244090
\(862\) 0.517385 0.0176222
\(863\) 3.46383i 0.117910i 0.998261 + 0.0589550i \(0.0187769\pi\)
−0.998261 + 0.0589550i \(0.981223\pi\)
\(864\) − 22.9608i − 0.781141i
\(865\) − 45.9275i − 1.56158i
\(866\) 7.40880i 0.251761i
\(867\) −12.5961 −0.427786
\(868\) −26.8732 −0.912136
\(869\) − 0.967213i − 0.0328105i
\(870\) 9.85086 0.333975
\(871\) 0 0
\(872\) 8.55688 0.289772
\(873\) 7.38106i 0.249811i
\(874\) 4.95407 0.167574
\(875\) 16.2107 0.548023
\(876\) 17.3913i 0.587599i
\(877\) − 57.2549i − 1.93336i −0.255989 0.966680i \(-0.582401\pi\)
0.255989 0.966680i \(-0.417599\pi\)
\(878\) − 7.76510i − 0.262059i
\(879\) − 14.9226i − 0.503327i
\(880\) 7.54288 0.254270
\(881\) 43.1782 1.45471 0.727355 0.686261i \(-0.240749\pi\)
0.727355 + 0.686261i \(0.240749\pi\)
\(882\) 0.323044i 0.0108775i
\(883\) −49.9560 −1.68115 −0.840576 0.541693i \(-0.817784\pi\)
−0.840576 + 0.541693i \(0.817784\pi\)
\(884\) 0 0
\(885\) 0.0271471 0.000912539 0
\(886\) 13.1535i 0.441899i
\(887\) 17.6746 0.593454 0.296727 0.954962i \(-0.404105\pi\)
0.296727 + 0.954962i \(0.404105\pi\)
\(888\) −1.56273 −0.0524417
\(889\) 18.2459i 0.611948i
\(890\) 22.9028i 0.767702i
\(891\) − 4.34375i − 0.145521i
\(892\) 19.2989i 0.646174i
\(893\) −14.9148 −0.499106
\(894\) −0.329749 −0.0110284
\(895\) 6.87800i 0.229906i
\(896\) 31.0592 1.03761
\(897\) 0 0
\(898\) 6.98493 0.233090
\(899\) 46.6069i 1.55443i
\(900\) 11.3690 0.378966
\(901\) −6.67755 −0.222461
\(902\) − 2.20583i − 0.0734462i
\(903\) 15.4547i 0.514301i
\(904\) 15.6601i 0.520847i
\(905\) − 33.0834i − 1.09973i
\(906\) 8.48858 0.282014
\(907\) −7.73423 −0.256811 −0.128406 0.991722i \(-0.540986\pi\)
−0.128406 + 0.991722i \(0.540986\pi\)
\(908\) − 18.0043i − 0.597494i
\(909\) 12.4692 0.413577
\(910\) 0 0
\(911\) 39.6179 1.31260 0.656299 0.754501i \(-0.272121\pi\)
0.656299 + 0.754501i \(0.272121\pi\)
\(912\) − 3.49396i − 0.115697i
\(913\) 19.5676 0.647594
\(914\) −18.6679 −0.617478
\(915\) − 18.0586i − 0.596999i
\(916\) − 1.92798i − 0.0637024i
\(917\) 36.8267i 1.21612i
\(918\) 2.71081i 0.0894700i
\(919\) 14.6213 0.482313 0.241157 0.970486i \(-0.422473\pi\)
0.241157 + 0.970486i \(0.422473\pi\)
\(920\) −26.4306 −0.871390
\(921\) 7.17283i 0.236353i
\(922\) −0.778512 −0.0256389
\(923\) 0 0
\(924\) −4.37627 −0.143969
\(925\) − 2.71140i − 0.0891502i
\(926\) 8.43594 0.277222
\(927\) −31.8907 −1.04743
\(928\) − 42.2228i − 1.38603i
\(929\) − 3.55735i − 0.116713i −0.998296 0.0583565i \(-0.981414\pi\)
0.998296 0.0583565i \(-0.0185860\pi\)
\(930\) 7.35690i 0.241242i
\(931\) 0.478894i 0.0156951i
\(932\) 18.3599 0.601398
\(933\) −16.8726 −0.552385
\(934\) 21.8328i 0.714391i
\(935\) −3.81700 −0.124829
\(936\) 0 0
\(937\) 34.5526 1.12878 0.564392 0.825507i \(-0.309111\pi\)
0.564392 + 0.825507i \(0.309111\pi\)
\(938\) 13.8291i 0.451536i
\(939\) −5.71571 −0.186525
\(940\) 36.4674 1.18944
\(941\) − 20.6233i − 0.672299i −0.941809 0.336149i \(-0.890875\pi\)
0.941809 0.336149i \(-0.109125\pi\)
\(942\) − 1.78986i − 0.0583167i
\(943\) − 15.2741i − 0.497394i
\(944\) − 0.0271471i 0 0.000883562i
\(945\) −32.4034 −1.05408
\(946\) −4.75973 −0.154752
\(947\) 29.4999i 0.958619i 0.877646 + 0.479309i \(0.159113\pi\)
−0.877646 + 0.479309i \(0.840887\pi\)
\(948\) 1.09544 0.0355783
\(949\) 0 0
\(950\) −3.06770 −0.0995295
\(951\) − 19.2185i − 0.623203i
\(952\) −6.27173 −0.203268
\(953\) −26.2389 −0.849963 −0.424981 0.905202i \(-0.639719\pi\)
−0.424981 + 0.905202i \(0.639719\pi\)
\(954\) − 7.68127i − 0.248690i
\(955\) − 25.1987i − 0.815409i
\(956\) 20.1847i 0.652818i
\(957\) 7.58987i 0.245346i
\(958\) 12.4138 0.401073
\(959\) 34.9638 1.12904
\(960\) 3.43296i 0.110798i
\(961\) −3.80731 −0.122817
\(962\) 0 0
\(963\) −13.2717 −0.427676
\(964\) − 6.17331i − 0.198829i
\(965\) −37.8974 −1.21996
\(966\) 5.51573 0.177466
\(967\) − 17.5176i − 0.563330i −0.959513 0.281665i \(-0.909113\pi\)
0.959513 0.281665i \(-0.0908866\pi\)
\(968\) 19.5972i 0.629877i
\(969\) 1.76809i 0.0567991i
\(970\) 4.86964i 0.156355i
\(971\) 20.5120 0.658262 0.329131 0.944284i \(-0.393244\pi\)
0.329131 + 0.944284i \(0.393244\pi\)
\(972\) 26.7259 0.857233
\(973\) − 32.4295i − 1.03964i
\(974\) 12.7199 0.407572
\(975\) 0 0
\(976\) −18.0586 −0.578042
\(977\) − 25.4450i − 0.814059i −0.913415 0.407030i \(-0.866565\pi\)
0.913415 0.407030i \(-0.133435\pi\)
\(978\) −6.73663 −0.215414
\(979\) −17.6461 −0.563971
\(980\) − 1.17092i − 0.0374035i
\(981\) 9.84309i 0.314266i
\(982\) − 1.02310i − 0.0326484i
\(983\) − 39.5244i − 1.26063i −0.776339 0.630316i \(-0.782926\pi\)
0.776339 0.630316i \(-0.217074\pi\)
\(984\) 5.45127 0.173780
\(985\) 36.3575 1.15845
\(986\) 4.98493i 0.158753i
\(987\) −16.6058 −0.528568
\(988\) 0 0
\(989\) −32.9584 −1.04802
\(990\) − 4.39075i − 0.139547i
\(991\) −29.8377 −0.947826 −0.473913 0.880572i \(-0.657159\pi\)
−0.473913 + 0.880572i \(0.657159\pi\)
\(992\) 31.5332 1.00118
\(993\) − 2.32198i − 0.0736858i
\(994\) 20.5284i 0.651121i
\(995\) 38.0683i 1.20685i
\(996\) 22.1618i 0.702224i
\(997\) −4.93123 −0.156174 −0.0780868 0.996947i \(-0.524881\pi\)
−0.0780868 + 0.996947i \(0.524881\pi\)
\(998\) −6.67861 −0.211408
\(999\) − 4.08575i − 0.129268i
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 169.2.b.b.168.4 6
3.2 odd 2 1521.2.b.l.1351.3 6
4.3 odd 2 2704.2.f.o.337.2 6
13.2 odd 12 169.2.c.b.22.2 6
13.3 even 3 169.2.e.b.147.3 12
13.4 even 6 169.2.e.b.23.3 12
13.5 odd 4 169.2.a.c.1.2 yes 3
13.6 odd 12 169.2.c.b.146.2 6
13.7 odd 12 169.2.c.c.146.2 6
13.8 odd 4 169.2.a.b.1.2 3
13.9 even 3 169.2.e.b.23.4 12
13.10 even 6 169.2.e.b.147.4 12
13.11 odd 12 169.2.c.c.22.2 6
13.12 even 2 inner 169.2.b.b.168.3 6
39.5 even 4 1521.2.a.o.1.2 3
39.8 even 4 1521.2.a.r.1.2 3
39.38 odd 2 1521.2.b.l.1351.4 6
52.31 even 4 2704.2.a.ba.1.1 3
52.47 even 4 2704.2.a.z.1.1 3
52.51 odd 2 2704.2.f.o.337.1 6
65.34 odd 4 4225.2.a.bg.1.2 3
65.44 odd 4 4225.2.a.bb.1.2 3
91.34 even 4 8281.2.a.bf.1.2 3
91.83 even 4 8281.2.a.bj.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.2 3 13.8 odd 4
169.2.a.c.1.2 yes 3 13.5 odd 4
169.2.b.b.168.3 6 13.12 even 2 inner
169.2.b.b.168.4 6 1.1 even 1 trivial
169.2.c.b.22.2 6 13.2 odd 12
169.2.c.b.146.2 6 13.6 odd 12
169.2.c.c.22.2 6 13.11 odd 12
169.2.c.c.146.2 6 13.7 odd 12
169.2.e.b.23.3 12 13.4 even 6
169.2.e.b.23.4 12 13.9 even 3
169.2.e.b.147.3 12 13.3 even 3
169.2.e.b.147.4 12 13.10 even 6
1521.2.a.o.1.2 3 39.5 even 4
1521.2.a.r.1.2 3 39.8 even 4
1521.2.b.l.1351.3 6 3.2 odd 2
1521.2.b.l.1351.4 6 39.38 odd 2
2704.2.a.z.1.1 3 52.47 even 4
2704.2.a.ba.1.1 3 52.31 even 4
2704.2.f.o.337.1 6 52.51 odd 2
2704.2.f.o.337.2 6 4.3 odd 2
4225.2.a.bb.1.2 3 65.44 odd 4
4225.2.a.bg.1.2 3 65.34 odd 4
8281.2.a.bf.1.2 3 91.34 even 4
8281.2.a.bj.1.2 3 91.83 even 4