Properties

Label 201.3.b.a.133.8
Level $201$
Weight $3$
Character 201.133
Analytic conductor $5.477$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,3,Mod(133,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.133");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 201.b (of order \(2\), degree \(1\), 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: \(5.47685331364\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 133.8
Character \(\chi\) \(=\) 201.133
Dual form 201.3.b.a.133.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.30530i q^{2} +1.73205i q^{3} +2.29618 q^{4} +2.19844i q^{5} +2.26085 q^{6} +9.13473i q^{7} -8.21843i q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.30530i q^{2} +1.73205i q^{3} +2.29618 q^{4} +2.19844i q^{5} +2.26085 q^{6} +9.13473i q^{7} -8.21843i q^{8} -3.00000 q^{9} +2.86963 q^{10} +4.26648i q^{11} +3.97711i q^{12} +14.1682i q^{13} +11.9236 q^{14} -3.80781 q^{15} -1.54281 q^{16} -2.27802 q^{17} +3.91591i q^{18} -8.04803 q^{19} +5.04803i q^{20} -15.8218 q^{21} +5.56905 q^{22} +19.0132 q^{23} +14.2347 q^{24} +20.1669 q^{25} +18.4938 q^{26} -5.19615i q^{27} +20.9750i q^{28} +25.2117 q^{29} +4.97035i q^{30} -9.14609i q^{31} -30.8599i q^{32} -7.38975 q^{33} +2.97351i q^{34} -20.0822 q^{35} -6.88855 q^{36} +3.39792 q^{37} +10.5051i q^{38} -24.5401 q^{39} +18.0677 q^{40} -43.7208i q^{41} +20.6523i q^{42} +34.0526i q^{43} +9.79661i q^{44} -6.59533i q^{45} -24.8180i q^{46} -6.95616 q^{47} -2.67222i q^{48} -34.4434 q^{49} -26.3239i q^{50} -3.94565i q^{51} +32.5328i q^{52} -87.9202i q^{53} -6.78255 q^{54} -9.37960 q^{55} +75.0732 q^{56} -13.9396i q^{57} -32.9089i q^{58} -84.3301 q^{59} -8.74344 q^{60} -23.7647i q^{61} -11.9384 q^{62} -27.4042i q^{63} -46.4527 q^{64} -31.1480 q^{65} +9.64587i q^{66} +(43.1590 + 51.2475i) q^{67} -5.23075 q^{68} +32.9318i q^{69} +26.2133i q^{70} -36.5444 q^{71} +24.6553i q^{72} -121.058 q^{73} -4.43532i q^{74} +34.9300i q^{75} -18.4798 q^{76} -38.9731 q^{77} +32.0322i q^{78} -93.7490i q^{79} -3.39177i q^{80} +9.00000 q^{81} -57.0689 q^{82} +50.1032 q^{83} -36.3298 q^{84} -5.00810i q^{85} +44.4490 q^{86} +43.6679i q^{87} +35.0637 q^{88} +52.8616 q^{89} -8.60890 q^{90} -129.423 q^{91} +43.6578 q^{92} +15.8415 q^{93} +9.07990i q^{94} -17.6931i q^{95} +53.4509 q^{96} -36.3620i q^{97} +44.9590i q^{98} -12.7994i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 52 q^{4} - 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 22 q - 52 q^{4} - 66 q^{9} - 36 q^{10} + 72 q^{14} + 12 q^{15} + 116 q^{16} - 14 q^{17} - 26 q^{19} + 48 q^{21} + 32 q^{22} - 82 q^{23} + 36 q^{24} - 62 q^{25} - 100 q^{26} + 102 q^{29} + 36 q^{33} - 180 q^{35} + 156 q^{36} + 106 q^{37} - 72 q^{39} + 76 q^{40} + 154 q^{47} + 146 q^{49} - 224 q^{55} - 452 q^{56} + 370 q^{59} - 24 q^{60} - 300 q^{62} + 148 q^{64} - 284 q^{65} - 134 q^{67} - 116 q^{68} + 160 q^{71} + 218 q^{73} + 480 q^{76} - 396 q^{77} + 198 q^{81} - 68 q^{82} + 128 q^{83} + 20 q^{86} - 856 q^{88} - 118 q^{89} + 108 q^{90} - 400 q^{91} + 804 q^{92} - 72 q^{93} - 144 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.30530i 0.652652i −0.945257 0.326326i \(-0.894189\pi\)
0.945257 0.326326i \(-0.105811\pi\)
\(3\) 1.73205i 0.577350i
\(4\) 2.29618 0.574046
\(5\) 2.19844i 0.439689i 0.975535 + 0.219844i \(0.0705549\pi\)
−0.975535 + 0.219844i \(0.929445\pi\)
\(6\) 2.26085 0.376809
\(7\) 9.13473i 1.30496i 0.757805 + 0.652481i \(0.226272\pi\)
−0.757805 + 0.652481i \(0.773728\pi\)
\(8\) 8.21843i 1.02730i
\(9\) −3.00000 −0.333333
\(10\) 2.86963 0.286963
\(11\) 4.26648i 0.387862i 0.981015 + 0.193931i \(0.0621237\pi\)
−0.981015 + 0.193931i \(0.937876\pi\)
\(12\) 3.97711i 0.331426i
\(13\) 14.1682i 1.08986i 0.838480 + 0.544932i \(0.183444\pi\)
−0.838480 + 0.544932i \(0.816556\pi\)
\(14\) 11.9236 0.851685
\(15\) −3.80781 −0.253854
\(16\) −1.54281 −0.0964255
\(17\) −2.27802 −0.134001 −0.0670006 0.997753i \(-0.521343\pi\)
−0.0670006 + 0.997753i \(0.521343\pi\)
\(18\) 3.91591i 0.217551i
\(19\) −8.04803 −0.423581 −0.211790 0.977315i \(-0.567929\pi\)
−0.211790 + 0.977315i \(0.567929\pi\)
\(20\) 5.04803i 0.252401i
\(21\) −15.8218 −0.753420
\(22\) 5.56905 0.253138
\(23\) 19.0132 0.826660 0.413330 0.910581i \(-0.364366\pi\)
0.413330 + 0.910581i \(0.364366\pi\)
\(24\) 14.2347 0.593114
\(25\) 20.1669 0.806674
\(26\) 18.4938 0.711301
\(27\) 5.19615i 0.192450i
\(28\) 20.9750i 0.749108i
\(29\) 25.2117 0.869367 0.434684 0.900583i \(-0.356860\pi\)
0.434684 + 0.900583i \(0.356860\pi\)
\(30\) 4.97035i 0.165678i
\(31\) 9.14609i 0.295035i −0.989059 0.147518i \(-0.952872\pi\)
0.989059 0.147518i \(-0.0471283\pi\)
\(32\) 30.8599i 0.964371i
\(33\) −7.38975 −0.223932
\(34\) 2.97351i 0.0874561i
\(35\) −20.0822 −0.573777
\(36\) −6.88855 −0.191349
\(37\) 3.39792 0.0918357 0.0459178 0.998945i \(-0.485379\pi\)
0.0459178 + 0.998945i \(0.485379\pi\)
\(38\) 10.5051i 0.276451i
\(39\) −24.5401 −0.629233
\(40\) 18.0677 0.451694
\(41\) 43.7208i 1.06636i −0.846002 0.533180i \(-0.820997\pi\)
0.846002 0.533180i \(-0.179003\pi\)
\(42\) 20.6523i 0.491721i
\(43\) 34.0526i 0.791921i 0.918267 + 0.395961i \(0.129588\pi\)
−0.918267 + 0.395961i \(0.870412\pi\)
\(44\) 9.79661i 0.222650i
\(45\) 6.59533i 0.146563i
\(46\) 24.8180i 0.539521i
\(47\) −6.95616 −0.148004 −0.0740018 0.997258i \(-0.523577\pi\)
−0.0740018 + 0.997258i \(0.523577\pi\)
\(48\) 2.67222i 0.0556713i
\(49\) −34.4434 −0.702926
\(50\) 26.3239i 0.526477i
\(51\) 3.94565i 0.0773656i
\(52\) 32.5328i 0.625631i
\(53\) 87.9202i 1.65887i −0.558602 0.829436i \(-0.688662\pi\)
0.558602 0.829436i \(-0.311338\pi\)
\(54\) −6.78255 −0.125603
\(55\) −9.37960 −0.170538
\(56\) 75.0732 1.34059
\(57\) 13.9396i 0.244554i
\(58\) 32.9089i 0.567394i
\(59\) −84.3301 −1.42932 −0.714662 0.699470i \(-0.753420\pi\)
−0.714662 + 0.699470i \(0.753420\pi\)
\(60\) −8.74344 −0.145724
\(61\) 23.7647i 0.389584i −0.980845 0.194792i \(-0.937597\pi\)
0.980845 0.194792i \(-0.0624033\pi\)
\(62\) −11.9384 −0.192555
\(63\) 27.4042i 0.434987i
\(64\) −46.4527 −0.725824
\(65\) −31.1480 −0.479200
\(66\) 9.64587i 0.146150i
\(67\) 43.1590 + 51.2475i 0.644164 + 0.764888i
\(68\) −5.23075 −0.0769228
\(69\) 32.9318i 0.477273i
\(70\) 26.2133i 0.374476i
\(71\) −36.5444 −0.514710 −0.257355 0.966317i \(-0.582851\pi\)
−0.257355 + 0.966317i \(0.582851\pi\)
\(72\) 24.6553i 0.342435i
\(73\) −121.058 −1.65832 −0.829162 0.559008i \(-0.811182\pi\)
−0.829162 + 0.559008i \(0.811182\pi\)
\(74\) 4.43532i 0.0599367i
\(75\) 34.9300i 0.465733i
\(76\) −18.4798 −0.243155
\(77\) −38.9731 −0.506145
\(78\) 32.0322i 0.410670i
\(79\) 93.7490i 1.18670i −0.804946 0.593348i \(-0.797806\pi\)
0.804946 0.593348i \(-0.202194\pi\)
\(80\) 3.39177i 0.0423972i
\(81\) 9.00000 0.111111
\(82\) −57.0689 −0.695962
\(83\) 50.1032 0.603653 0.301826 0.953363i \(-0.402404\pi\)
0.301826 + 0.953363i \(0.402404\pi\)
\(84\) −36.3298 −0.432498
\(85\) 5.00810i 0.0589188i
\(86\) 44.4490 0.516849
\(87\) 43.6679i 0.501930i
\(88\) 35.0637 0.398452
\(89\) 52.8616 0.593951 0.296975 0.954885i \(-0.404022\pi\)
0.296975 + 0.954885i \(0.404022\pi\)
\(90\) −8.60890 −0.0956545
\(91\) −129.423 −1.42223
\(92\) 43.6578 0.474541
\(93\) 15.8415 0.170339
\(94\) 9.07990i 0.0965947i
\(95\) 17.6931i 0.186244i
\(96\) 53.4509 0.556780
\(97\) 36.3620i 0.374866i −0.982277 0.187433i \(-0.939983\pi\)
0.982277 0.187433i \(-0.0600168\pi\)
\(98\) 44.9590i 0.458765i
\(99\) 12.7994i 0.129287i
\(100\) 46.3068 0.463068
\(101\) 4.47109i 0.0442683i −0.999755 0.0221341i \(-0.992954\pi\)
0.999755 0.0221341i \(-0.00704609\pi\)
\(102\) −5.15027 −0.0504928
\(103\) 48.0708 0.466707 0.233353 0.972392i \(-0.425030\pi\)
0.233353 + 0.972392i \(0.425030\pi\)
\(104\) 116.441 1.11962
\(105\) 34.7834i 0.331270i
\(106\) −114.763 −1.08267
\(107\) 127.742 1.19385 0.596924 0.802298i \(-0.296389\pi\)
0.596924 + 0.802298i \(0.296389\pi\)
\(108\) 11.9313i 0.110475i
\(109\) 115.564i 1.06022i −0.847928 0.530112i \(-0.822150\pi\)
0.847928 0.530112i \(-0.177850\pi\)
\(110\) 12.2432i 0.111302i
\(111\) 5.88537i 0.0530214i
\(112\) 14.0931i 0.125832i
\(113\) 29.4439i 0.260566i 0.991477 + 0.130283i \(0.0415885\pi\)
−0.991477 + 0.130283i \(0.958411\pi\)
\(114\) −18.1954 −0.159609
\(115\) 41.7994i 0.363473i
\(116\) 57.8906 0.499057
\(117\) 42.5047i 0.363288i
\(118\) 110.076i 0.932851i
\(119\) 20.8091i 0.174866i
\(120\) 31.2942i 0.260785i
\(121\) 102.797 0.849563
\(122\) −31.0201 −0.254263
\(123\) 75.7266 0.615663
\(124\) 21.0011i 0.169364i
\(125\) 99.2967i 0.794374i
\(126\) −35.7708 −0.283895
\(127\) 214.525 1.68917 0.844585 0.535422i \(-0.179847\pi\)
0.844585 + 0.535422i \(0.179847\pi\)
\(128\) 62.8046i 0.490661i
\(129\) −58.9809 −0.457216
\(130\) 40.6576i 0.312751i
\(131\) −126.306 −0.964167 −0.482084 0.876125i \(-0.660120\pi\)
−0.482084 + 0.876125i \(0.660120\pi\)
\(132\) −16.9682 −0.128547
\(133\) 73.5166i 0.552757i
\(134\) 66.8935 56.3356i 0.499205 0.420415i
\(135\) 11.4234 0.0846181
\(136\) 18.7217i 0.137660i
\(137\) 37.0683i 0.270572i −0.990807 0.135286i \(-0.956805\pi\)
0.990807 0.135286i \(-0.0431953\pi\)
\(138\) 42.9860 0.311493
\(139\) 17.3661i 0.124936i 0.998047 + 0.0624679i \(0.0198971\pi\)
−0.998047 + 0.0624679i \(0.980103\pi\)
\(140\) −46.1124 −0.329374
\(141\) 12.0484i 0.0854499i
\(142\) 47.7015i 0.335926i
\(143\) −60.4484 −0.422716
\(144\) 4.62842 0.0321418
\(145\) 55.4264i 0.382251i
\(146\) 158.017i 1.08231i
\(147\) 59.6576i 0.405834i
\(148\) 7.80225 0.0527179
\(149\) −95.2384 −0.639184 −0.319592 0.947555i \(-0.603546\pi\)
−0.319592 + 0.947555i \(0.603546\pi\)
\(150\) 45.5943 0.303962
\(151\) 72.3267 0.478985 0.239493 0.970898i \(-0.423019\pi\)
0.239493 + 0.970898i \(0.423019\pi\)
\(152\) 66.1422i 0.435146i
\(153\) 6.83406 0.0446671
\(154\) 50.8717i 0.330336i
\(155\) 20.1071 0.129723
\(156\) −56.3485 −0.361208
\(157\) 226.414 1.44213 0.721065 0.692868i \(-0.243653\pi\)
0.721065 + 0.692868i \(0.243653\pi\)
\(158\) −122.371 −0.774499
\(159\) 152.282 0.957750
\(160\) 67.8437 0.424023
\(161\) 173.680i 1.07876i
\(162\) 11.7477i 0.0725168i
\(163\) 225.801 1.38528 0.692642 0.721281i \(-0.256446\pi\)
0.692642 + 0.721281i \(0.256446\pi\)
\(164\) 100.391i 0.612140i
\(165\) 16.2460i 0.0984603i
\(166\) 65.3998i 0.393975i
\(167\) −70.9723 −0.424984 −0.212492 0.977163i \(-0.568158\pi\)
−0.212492 + 0.977163i \(0.568158\pi\)
\(168\) 130.031i 0.773991i
\(169\) −31.7385 −0.187802
\(170\) −6.53708 −0.0384534
\(171\) 24.1441 0.141194
\(172\) 78.1911i 0.454599i
\(173\) −194.180 −1.12243 −0.561213 0.827672i \(-0.689665\pi\)
−0.561213 + 0.827672i \(0.689665\pi\)
\(174\) 56.9998 0.327585
\(175\) 184.219i 1.05268i
\(176\) 6.58235i 0.0373997i
\(177\) 146.064i 0.825221i
\(178\) 69.0004i 0.387643i
\(179\) 167.435i 0.935392i 0.883889 + 0.467696i \(0.154916\pi\)
−0.883889 + 0.467696i \(0.845084\pi\)
\(180\) 15.1441i 0.0841338i
\(181\) −350.378 −1.93579 −0.967896 0.251350i \(-0.919126\pi\)
−0.967896 + 0.251350i \(0.919126\pi\)
\(182\) 168.936i 0.928221i
\(183\) 41.1616 0.224927
\(184\) 156.259i 0.849231i
\(185\) 7.47013i 0.0403791i
\(186\) 20.6779i 0.111172i
\(187\) 9.71912i 0.0519739i
\(188\) −15.9726 −0.0849608
\(189\) 47.4655 0.251140
\(190\) −23.0949 −0.121552
\(191\) 193.984i 1.01562i 0.861469 + 0.507810i \(0.169545\pi\)
−0.861469 + 0.507810i \(0.830455\pi\)
\(192\) 80.4585i 0.419055i
\(193\) −268.978 −1.39367 −0.696834 0.717233i \(-0.745408\pi\)
−0.696834 + 0.717233i \(0.745408\pi\)
\(194\) −47.4634 −0.244657
\(195\) 53.9500i 0.276666i
\(196\) −79.0883 −0.403511
\(197\) 190.688i 0.967957i 0.875080 + 0.483978i \(0.160809\pi\)
−0.875080 + 0.483978i \(0.839191\pi\)
\(198\) −16.7071 −0.0843795
\(199\) 50.1303 0.251911 0.125956 0.992036i \(-0.459800\pi\)
0.125956 + 0.992036i \(0.459800\pi\)
\(200\) 165.740i 0.828699i
\(201\) −88.7632 + 74.7535i −0.441608 + 0.371908i
\(202\) −5.83613 −0.0288918
\(203\) 230.302i 1.13449i
\(204\) 9.05993i 0.0444114i
\(205\) 96.1176 0.468866
\(206\) 62.7470i 0.304597i
\(207\) −57.0396 −0.275553
\(208\) 21.8588i 0.105091i
\(209\) 34.3367i 0.164291i
\(210\) −45.4028 −0.216204
\(211\) −45.4496 −0.215401 −0.107700 0.994183i \(-0.534349\pi\)
−0.107700 + 0.994183i \(0.534349\pi\)
\(212\) 201.881i 0.952268i
\(213\) 63.2968i 0.297168i
\(214\) 166.742i 0.779167i
\(215\) −74.8627 −0.348199
\(216\) −42.7042 −0.197705
\(217\) 83.5470 0.385009
\(218\) −150.847 −0.691957
\(219\) 209.678i 0.957434i
\(220\) −21.5373 −0.0978968
\(221\) 32.2755i 0.146043i
\(222\) 7.68219 0.0346045
\(223\) 192.524 0.863335 0.431668 0.902033i \(-0.357925\pi\)
0.431668 + 0.902033i \(0.357925\pi\)
\(224\) 281.897 1.25847
\(225\) −60.5006 −0.268891
\(226\) 38.4332 0.170059
\(227\) −126.627 −0.557827 −0.278913 0.960316i \(-0.589974\pi\)
−0.278913 + 0.960316i \(0.589974\pi\)
\(228\) 32.0079i 0.140385i
\(229\) 449.069i 1.96100i −0.196516 0.980501i \(-0.562963\pi\)
0.196516 0.980501i \(-0.437037\pi\)
\(230\) 54.5609 0.237221
\(231\) 67.5034i 0.292223i
\(232\) 207.200i 0.893104i
\(233\) 22.2708i 0.0955828i 0.998857 + 0.0477914i \(0.0152183\pi\)
−0.998857 + 0.0477914i \(0.984782\pi\)
\(234\) −55.4815 −0.237100
\(235\) 15.2927i 0.0650754i
\(236\) −193.637 −0.820498
\(237\) 162.378 0.685139
\(238\) −27.1622 −0.114127
\(239\) 351.927i 1.47250i −0.676710 0.736250i \(-0.736595\pi\)
0.676710 0.736250i \(-0.263405\pi\)
\(240\) 5.87473 0.0244780
\(241\) 67.0523 0.278225 0.139113 0.990277i \(-0.455575\pi\)
0.139113 + 0.990277i \(0.455575\pi\)
\(242\) 134.181i 0.554469i
\(243\) 15.5885i 0.0641500i
\(244\) 54.5680i 0.223639i
\(245\) 75.7217i 0.309068i
\(246\) 98.8462i 0.401814i
\(247\) 114.026i 0.461645i
\(248\) −75.1664 −0.303091
\(249\) 86.7812i 0.348519i
\(250\) 129.612 0.518449
\(251\) 286.257i 1.14047i 0.821482 + 0.570234i \(0.193147\pi\)
−0.821482 + 0.570234i \(0.806853\pi\)
\(252\) 62.9251i 0.249703i
\(253\) 81.1193i 0.320630i
\(254\) 280.020i 1.10244i
\(255\) 8.67428 0.0340168
\(256\) −267.790 −1.04605
\(257\) 220.859 0.859375 0.429688 0.902978i \(-0.358624\pi\)
0.429688 + 0.902978i \(0.358624\pi\)
\(258\) 76.9879i 0.298403i
\(259\) 31.0391i 0.119842i
\(260\) −71.5216 −0.275083
\(261\) −75.6350 −0.289789
\(262\) 164.868i 0.629265i
\(263\) −438.527 −1.66740 −0.833701 0.552217i \(-0.813782\pi\)
−0.833701 + 0.552217i \(0.813782\pi\)
\(264\) 60.7322i 0.230046i
\(265\) 193.287 0.729387
\(266\) −95.9615 −0.360757
\(267\) 91.5590i 0.342918i
\(268\) 99.1009 + 117.674i 0.369780 + 0.439081i
\(269\) −173.853 −0.646294 −0.323147 0.946349i \(-0.604741\pi\)
−0.323147 + 0.946349i \(0.604741\pi\)
\(270\) 14.9111i 0.0552261i
\(271\) 497.804i 1.83692i 0.395520 + 0.918458i \(0.370565\pi\)
−0.395520 + 0.918458i \(0.629435\pi\)
\(272\) 3.51455 0.0129211
\(273\) 224.167i 0.821125i
\(274\) −48.3854 −0.176589
\(275\) 86.0414i 0.312878i
\(276\) 75.6175i 0.273976i
\(277\) 99.6645 0.359800 0.179900 0.983685i \(-0.442423\pi\)
0.179900 + 0.983685i \(0.442423\pi\)
\(278\) 22.6680 0.0815396
\(279\) 27.4383i 0.0983450i
\(280\) 165.044i 0.589443i
\(281\) 86.7978i 0.308889i −0.988001 0.154445i \(-0.950641\pi\)
0.988001 0.154445i \(-0.0493588\pi\)
\(282\) −15.7269 −0.0557690
\(283\) 247.359 0.874061 0.437031 0.899447i \(-0.356030\pi\)
0.437031 + 0.899447i \(0.356030\pi\)
\(284\) −83.9127 −0.295467
\(285\) 30.6454 0.107528
\(286\) 78.9035i 0.275886i
\(287\) 399.378 1.39156
\(288\) 92.5796i 0.321457i
\(289\) −283.811 −0.982044
\(290\) 72.3482 0.249477
\(291\) 62.9808 0.216429
\(292\) −277.971 −0.951954
\(293\) 32.6690 0.111498 0.0557491 0.998445i \(-0.482245\pi\)
0.0557491 + 0.998445i \(0.482245\pi\)
\(294\) −77.8713 −0.264868
\(295\) 185.395i 0.628458i
\(296\) 27.9256i 0.0943431i
\(297\) 22.1693 0.0746440
\(298\) 124.315i 0.417165i
\(299\) 269.383i 0.900947i
\(300\) 80.2057i 0.267352i
\(301\) −311.062 −1.03343
\(302\) 94.4083i 0.312610i
\(303\) 7.74416 0.0255583
\(304\) 12.4166 0.0408440
\(305\) 52.2452 0.171296
\(306\) 8.92052i 0.0291520i
\(307\) 151.568 0.493707 0.246853 0.969053i \(-0.420603\pi\)
0.246853 + 0.969053i \(0.420603\pi\)
\(308\) −89.4894 −0.290550
\(309\) 83.2611i 0.269453i
\(310\) 26.2459i 0.0846643i
\(311\) 326.893i 1.05110i 0.850762 + 0.525552i \(0.176141\pi\)
−0.850762 + 0.525552i \(0.823859\pi\)
\(312\) 201.681i 0.646413i
\(313\) 420.587i 1.34373i −0.740675 0.671864i \(-0.765494\pi\)
0.740675 0.671864i \(-0.234506\pi\)
\(314\) 295.539i 0.941208i
\(315\) 60.2466 0.191259
\(316\) 215.265i 0.681218i
\(317\) 387.365 1.22197 0.610985 0.791642i \(-0.290773\pi\)
0.610985 + 0.791642i \(0.290773\pi\)
\(318\) 198.774i 0.625077i
\(319\) 107.565i 0.337194i
\(320\) 102.124i 0.319136i
\(321\) 221.255i 0.689269i
\(322\) 226.706 0.704055
\(323\) 18.3336 0.0567603
\(324\) 20.6657 0.0637829
\(325\) 285.728i 0.879164i
\(326\) 294.739i 0.904108i
\(327\) 200.163 0.612120
\(328\) −359.316 −1.09548
\(329\) 63.5427i 0.193139i
\(330\) −21.2059 −0.0642603
\(331\) 402.032i 1.21460i −0.794474 0.607299i \(-0.792253\pi\)
0.794474 0.607299i \(-0.207747\pi\)
\(332\) 115.046 0.346524
\(333\) −10.1938 −0.0306119
\(334\) 92.6403i 0.277366i
\(335\) −112.665 + 94.8825i −0.336312 + 0.283231i
\(336\) 24.4100 0.0726489
\(337\) 110.892i 0.329057i 0.986372 + 0.164529i \(0.0526103\pi\)
−0.986372 + 0.164529i \(0.947390\pi\)
\(338\) 41.4284i 0.122569i
\(339\) −50.9984 −0.150438
\(340\) 11.4995i 0.0338221i
\(341\) 39.0216 0.114433
\(342\) 31.5154i 0.0921502i
\(343\) 132.971i 0.387671i
\(344\) 279.859 0.813544
\(345\) −72.3987 −0.209851
\(346\) 253.463i 0.732553i
\(347\) 529.552i 1.52609i 0.646347 + 0.763044i \(0.276296\pi\)
−0.646347 + 0.763044i \(0.723704\pi\)
\(348\) 100.269i 0.288131i
\(349\) −7.71435 −0.0221042 −0.0110521 0.999939i \(-0.503518\pi\)
−0.0110521 + 0.999939i \(0.503518\pi\)
\(350\) 240.461 0.687033
\(351\) 73.6202 0.209744
\(352\) 131.663 0.374043
\(353\) 184.044i 0.521370i −0.965424 0.260685i \(-0.916052\pi\)
0.965424 0.260685i \(-0.0839484\pi\)
\(354\) −190.658 −0.538582
\(355\) 80.3408i 0.226312i
\(356\) 121.380 0.340955
\(357\) 36.0424 0.100959
\(358\) 218.554 0.610485
\(359\) −177.535 −0.494527 −0.247264 0.968948i \(-0.579531\pi\)
−0.247264 + 0.968948i \(0.579531\pi\)
\(360\) −54.2032 −0.150565
\(361\) −296.229 −0.820579
\(362\) 457.350i 1.26340i
\(363\) 178.050i 0.490496i
\(364\) −297.179 −0.816425
\(365\) 266.138i 0.729146i
\(366\) 53.7284i 0.146799i
\(367\) 431.616i 1.17607i −0.808837 0.588033i \(-0.799903\pi\)
0.808837 0.588033i \(-0.200097\pi\)
\(368\) −29.3337 −0.0797111
\(369\) 131.162i 0.355453i
\(370\) 9.75079 0.0263535
\(371\) 803.127 2.16476
\(372\) 36.3750 0.0977821
\(373\) 518.092i 1.38899i 0.719499 + 0.694493i \(0.244371\pi\)
−0.719499 + 0.694493i \(0.755629\pi\)
\(374\) −12.6864 −0.0339209
\(375\) −171.987 −0.458632
\(376\) 57.1687i 0.152045i
\(377\) 357.204i 0.947492i
\(378\) 61.9568i 0.163907i
\(379\) 308.941i 0.815147i 0.913172 + 0.407573i \(0.133625\pi\)
−0.913172 + 0.407573i \(0.866375\pi\)
\(380\) 40.6267i 0.106912i
\(381\) 371.567i 0.975243i
\(382\) 253.207 0.662847
\(383\) 18.3045i 0.0477924i −0.999714 0.0238962i \(-0.992393\pi\)
0.999714 0.0238962i \(-0.00760712\pi\)
\(384\) 108.781 0.283283
\(385\) 85.6802i 0.222546i
\(386\) 351.098i 0.909579i
\(387\) 102.158i 0.263974i
\(388\) 83.4938i 0.215190i
\(389\) −406.794 −1.04574 −0.522872 0.852411i \(-0.675139\pi\)
−0.522872 + 0.852411i \(0.675139\pi\)
\(390\) −70.4211 −0.180567
\(391\) −43.3124 −0.110773
\(392\) 283.070i 0.722118i
\(393\) 218.768i 0.556662i
\(394\) 248.905 0.631739
\(395\) 206.102 0.521777
\(396\) 29.3898i 0.0742168i
\(397\) −512.886 −1.29190 −0.645952 0.763378i \(-0.723539\pi\)
−0.645952 + 0.763378i \(0.723539\pi\)
\(398\) 65.4353i 0.164410i
\(399\) 127.335 0.319134
\(400\) −31.1136 −0.0777839
\(401\) 531.314i 1.32497i −0.749074 0.662487i \(-0.769501\pi\)
0.749074 0.662487i \(-0.230499\pi\)
\(402\) 97.5760 + 115.863i 0.242726 + 0.288216i
\(403\) 129.584 0.321548
\(404\) 10.2665i 0.0254120i
\(405\) 19.7860i 0.0488543i
\(406\) 300.614 0.740428
\(407\) 14.4971i 0.0356195i
\(408\) −32.4270 −0.0794780
\(409\) 405.478i 0.991390i 0.868497 + 0.495695i \(0.165087\pi\)
−0.868497 + 0.495695i \(0.834913\pi\)
\(410\) 125.463i 0.306006i
\(411\) 64.2042 0.156215
\(412\) 110.379 0.267911
\(413\) 770.333i 1.86521i
\(414\) 74.4539i 0.179840i
\(415\) 110.149i 0.265419i
\(416\) 437.230 1.05103
\(417\) −30.0789 −0.0721317
\(418\) −44.8199 −0.107225
\(419\) 752.468 1.79587 0.897933 0.440131i \(-0.145068\pi\)
0.897933 + 0.440131i \(0.145068\pi\)
\(420\) 79.8690i 0.190164i
\(421\) 477.671 1.13461 0.567305 0.823508i \(-0.307986\pi\)
0.567305 + 0.823508i \(0.307986\pi\)
\(422\) 59.3255i 0.140582i
\(423\) 20.8685 0.0493345
\(424\) −722.566 −1.70416
\(425\) −45.9405 −0.108095
\(426\) −82.6215 −0.193947
\(427\) 217.084 0.508393
\(428\) 293.318 0.685324
\(429\) 104.700i 0.244055i
\(430\) 97.7186i 0.227252i
\(431\) −151.453 −0.351398 −0.175699 0.984444i \(-0.556219\pi\)
−0.175699 + 0.984444i \(0.556219\pi\)
\(432\) 8.01667i 0.0185571i
\(433\) 622.368i 1.43734i −0.695352 0.718669i \(-0.744752\pi\)
0.695352 0.718669i \(-0.255248\pi\)
\(434\) 109.054i 0.251277i
\(435\) −96.0013 −0.220693
\(436\) 265.357i 0.608617i
\(437\) −153.019 −0.350157
\(438\) −273.693 −0.624871
\(439\) −235.751 −0.537018 −0.268509 0.963277i \(-0.586531\pi\)
−0.268509 + 0.963277i \(0.586531\pi\)
\(440\) 77.0856i 0.175195i
\(441\) 103.330 0.234309
\(442\) −42.1293 −0.0953152
\(443\) 519.899i 1.17359i −0.809736 0.586794i \(-0.800390\pi\)
0.809736 0.586794i \(-0.199610\pi\)
\(444\) 13.5139i 0.0304367i
\(445\) 116.213i 0.261153i
\(446\) 251.302i 0.563457i
\(447\) 164.958i 0.369033i
\(448\) 424.333i 0.947173i
\(449\) −381.092 −0.848756 −0.424378 0.905485i \(-0.639507\pi\)
−0.424378 + 0.905485i \(0.639507\pi\)
\(450\) 78.9716i 0.175492i
\(451\) 186.534 0.413600
\(452\) 67.6086i 0.149577i
\(453\) 125.274i 0.276542i
\(454\) 165.286i 0.364066i
\(455\) 284.529i 0.625338i
\(456\) −114.562 −0.251232
\(457\) −202.700 −0.443544 −0.221772 0.975099i \(-0.571184\pi\)
−0.221772 + 0.975099i \(0.571184\pi\)
\(458\) −586.172 −1.27985
\(459\) 11.8369i 0.0257885i
\(460\) 95.9791i 0.208650i
\(461\) 361.109 0.783316 0.391658 0.920111i \(-0.371902\pi\)
0.391658 + 0.920111i \(0.371902\pi\)
\(462\) −88.1125 −0.190720
\(463\) 488.310i 1.05466i −0.849659 0.527332i \(-0.823192\pi\)
0.849659 0.527332i \(-0.176808\pi\)
\(464\) −38.8967 −0.0838292
\(465\) 34.8266i 0.0748959i
\(466\) 29.0702 0.0623823
\(467\) 697.376 1.49331 0.746655 0.665211i \(-0.231658\pi\)
0.746655 + 0.665211i \(0.231658\pi\)
\(468\) 97.5985i 0.208544i
\(469\) −468.132 + 394.246i −0.998149 + 0.840609i
\(470\) −19.9616 −0.0424716
\(471\) 392.161i 0.832614i
\(472\) 693.061i 1.46835i
\(473\) −145.285 −0.307156
\(474\) 211.953i 0.447157i
\(475\) −162.303 −0.341691
\(476\) 47.7815i 0.100381i
\(477\) 263.761i 0.552957i
\(478\) −459.372 −0.961029
\(479\) −612.073 −1.27781 −0.638907 0.769284i \(-0.720613\pi\)
−0.638907 + 0.769284i \(0.720613\pi\)
\(480\) 117.509i 0.244810i
\(481\) 48.1425i 0.100088i
\(482\) 87.5236i 0.181584i
\(483\) −300.823 −0.622823
\(484\) 236.041 0.487688
\(485\) 79.9398 0.164824
\(486\) 20.3477 0.0418676
\(487\) 347.453i 0.713456i 0.934208 + 0.356728i \(0.116108\pi\)
−0.934208 + 0.356728i \(0.883892\pi\)
\(488\) −195.308 −0.400222
\(489\) 391.100i 0.799795i
\(490\) −98.8398 −0.201714
\(491\) −437.982 −0.892020 −0.446010 0.895028i \(-0.647155\pi\)
−0.446010 + 0.895028i \(0.647155\pi\)
\(492\) 173.882 0.353419
\(493\) −57.4327 −0.116496
\(494\) −148.839 −0.301293
\(495\) 28.1388 0.0568461
\(496\) 14.1107i 0.0284489i
\(497\) 333.823i 0.671677i
\(498\) 113.276 0.227461
\(499\) 56.5316i 0.113290i 0.998394 + 0.0566449i \(0.0180403\pi\)
−0.998394 + 0.0566449i \(0.981960\pi\)
\(500\) 228.003i 0.456007i
\(501\) 122.928i 0.245364i
\(502\) 373.653 0.744328
\(503\) 593.739i 1.18040i −0.807259 0.590198i \(-0.799050\pi\)
0.807259 0.590198i \(-0.200950\pi\)
\(504\) −225.219 −0.446864
\(505\) 9.82944 0.0194642
\(506\) 105.885 0.209260
\(507\) 54.9727i 0.108427i
\(508\) 492.588 0.969661
\(509\) −964.817 −1.89551 −0.947757 0.318992i \(-0.896656\pi\)
−0.947757 + 0.318992i \(0.896656\pi\)
\(510\) 11.3226i 0.0222011i
\(511\) 1105.83i 2.16405i
\(512\) 98.3287i 0.192048i
\(513\) 41.8188i 0.0815181i
\(514\) 288.288i 0.560873i
\(515\) 105.681i 0.205206i
\(516\) −135.431 −0.262463
\(517\) 29.6783i 0.0574049i
\(518\) 40.5154 0.0782151
\(519\) 336.329i 0.648033i
\(520\) 255.988i 0.492284i
\(521\) 860.145i 1.65095i −0.564439 0.825475i \(-0.690907\pi\)
0.564439 0.825475i \(-0.309093\pi\)
\(522\) 98.7266i 0.189131i
\(523\) −534.293 −1.02159 −0.510796 0.859702i \(-0.670649\pi\)
−0.510796 + 0.859702i \(0.670649\pi\)
\(524\) −290.022 −0.553476
\(525\) −319.076 −0.607764
\(526\) 572.410i 1.08823i
\(527\) 20.8350i 0.0395350i
\(528\) 11.4010 0.0215928
\(529\) −167.499 −0.316633
\(530\) 252.299i 0.476035i
\(531\) 252.990 0.476441
\(532\) 168.808i 0.317308i
\(533\) 619.446 1.16219
\(534\) 119.512 0.223806
\(535\) 280.833i 0.524921i
\(536\) 421.174 354.699i 0.785772 0.661752i
\(537\) −290.006 −0.540049
\(538\) 226.931i 0.421805i
\(539\) 146.952i 0.272638i
\(540\) 26.2303 0.0485747
\(541\) 18.7475i 0.0346534i 0.999850 + 0.0173267i \(0.00551553\pi\)
−0.999850 + 0.0173267i \(0.994484\pi\)
\(542\) 649.785 1.19887
\(543\) 606.873i 1.11763i
\(544\) 70.2994i 0.129227i
\(545\) 254.062 0.466168
\(546\) −292.606 −0.535908
\(547\) 244.601i 0.447168i −0.974685 0.223584i \(-0.928224\pi\)
0.974685 0.223584i \(-0.0717758\pi\)
\(548\) 85.1156i 0.155321i
\(549\) 71.2940i 0.129861i
\(550\) 112.310 0.204200
\(551\) −202.904 −0.368247
\(552\) 270.648 0.490304
\(553\) 856.372 1.54859
\(554\) 130.092i 0.234824i
\(555\) −12.9386 −0.0233129
\(556\) 39.8757i 0.0717189i
\(557\) −798.593 −1.43374 −0.716870 0.697207i \(-0.754426\pi\)
−0.716870 + 0.697207i \(0.754426\pi\)
\(558\) 35.8152 0.0641850
\(559\) −482.465 −0.863086
\(560\) 30.9830 0.0553267
\(561\) 16.8340 0.0300071
\(562\) −113.297 −0.201597
\(563\) 910.796i 1.61775i 0.587978 + 0.808877i \(0.299924\pi\)
−0.587978 + 0.808877i \(0.700076\pi\)
\(564\) 27.6654i 0.0490521i
\(565\) −64.7307 −0.114568
\(566\) 322.879i 0.570457i
\(567\) 82.2126i 0.144996i
\(568\) 300.338i 0.528763i
\(569\) −88.1892 −0.154990 −0.0774949 0.996993i \(-0.524692\pi\)
−0.0774949 + 0.996993i \(0.524692\pi\)
\(570\) 40.0016i 0.0701782i
\(571\) −520.918 −0.912291 −0.456146 0.889905i \(-0.650770\pi\)
−0.456146 + 0.889905i \(0.650770\pi\)
\(572\) −138.801 −0.242658
\(573\) −335.989 −0.586369
\(574\) 521.309i 0.908204i
\(575\) 383.436 0.666846
\(576\) 139.358 0.241941
\(577\) 317.606i 0.550443i 0.961381 + 0.275222i \(0.0887513\pi\)
−0.961381 + 0.275222i \(0.911249\pi\)
\(578\) 370.459i 0.640932i
\(579\) 465.883i 0.804634i
\(580\) 127.269i 0.219430i
\(581\) 457.679i 0.787744i
\(582\) 82.2091i 0.141253i
\(583\) 375.109 0.643412
\(584\) 994.904i 1.70360i
\(585\) 93.4441 0.159733
\(586\) 42.6429i 0.0727694i
\(587\) 1082.75i 1.84455i 0.386532 + 0.922276i \(0.373673\pi\)
−0.386532 + 0.922276i \(0.626327\pi\)
\(588\) 136.985i 0.232967i
\(589\) 73.6080i 0.124971i
\(590\) −241.997 −0.410164
\(591\) −330.280 −0.558850
\(592\) −5.24234 −0.00885530
\(593\) 17.7957i 0.0300096i −0.999887 0.0150048i \(-0.995224\pi\)
0.999887 0.0150048i \(-0.00477636\pi\)
\(594\) 28.9376i 0.0487165i
\(595\) 45.7476 0.0768868
\(596\) −218.685 −0.366921
\(597\) 86.8283i 0.145441i
\(598\) 351.627 0.588004
\(599\) 945.801i 1.57897i −0.613773 0.789483i \(-0.710349\pi\)
0.613773 0.789483i \(-0.289651\pi\)
\(600\) 287.070 0.478450
\(601\) 674.073 1.12158 0.560792 0.827956i \(-0.310497\pi\)
0.560792 + 0.827956i \(0.310497\pi\)
\(602\) 406.030i 0.674468i
\(603\) −129.477 153.742i −0.214721 0.254963i
\(604\) 166.075 0.274959
\(605\) 225.994i 0.373543i
\(606\) 10.1085i 0.0166807i
\(607\) −500.176 −0.824013 −0.412007 0.911181i \(-0.635172\pi\)
−0.412007 + 0.911181i \(0.635172\pi\)
\(608\) 248.361i 0.408489i
\(609\) −398.894 −0.654999
\(610\) 68.1959i 0.111796i
\(611\) 98.5565i 0.161304i
\(612\) 15.6923 0.0256409
\(613\) −1068.83 −1.74360 −0.871800 0.489861i \(-0.837047\pi\)
−0.871800 + 0.489861i \(0.837047\pi\)
\(614\) 197.842i 0.322218i
\(615\) 166.481i 0.270700i
\(616\) 320.298i 0.519964i
\(617\) 525.779 0.852153 0.426077 0.904687i \(-0.359895\pi\)
0.426077 + 0.904687i \(0.359895\pi\)
\(618\) 108.681 0.175859
\(619\) 494.969 0.799626 0.399813 0.916597i \(-0.369075\pi\)
0.399813 + 0.916597i \(0.369075\pi\)
\(620\) 46.1697 0.0744672
\(621\) 98.7954i 0.159091i
\(622\) 426.695 0.686004
\(623\) 482.877i 0.775083i
\(624\) 37.8606 0.0606741
\(625\) 285.873 0.457397
\(626\) −548.993 −0.876986
\(627\) 59.4730 0.0948532
\(628\) 519.889 0.827848
\(629\) −7.74053 −0.0123061
\(630\) 78.6400i 0.124825i
\(631\) 196.595i 0.311561i 0.987792 + 0.155781i \(0.0497893\pi\)
−0.987792 + 0.155781i \(0.950211\pi\)
\(632\) −770.469 −1.21910
\(633\) 78.7210i 0.124362i
\(634\) 505.628i 0.797521i
\(635\) 471.620i 0.742708i
\(636\) 349.668 0.549792
\(637\) 488.001i 0.766093i
\(638\) 140.405 0.220070
\(639\) 109.633 0.171570
\(640\) 138.072 0.215738
\(641\) 1180.92i 1.84231i −0.389193 0.921156i \(-0.627246\pi\)
0.389193 0.921156i \(-0.372754\pi\)
\(642\) 288.805 0.449852
\(643\) −246.496 −0.383353 −0.191677 0.981458i \(-0.561392\pi\)
−0.191677 + 0.981458i \(0.561392\pi\)
\(644\) 398.802i 0.619258i
\(645\) 129.666i 0.201033i
\(646\) 23.9309i 0.0370447i
\(647\) 1138.40i 1.75951i 0.475429 + 0.879754i \(0.342293\pi\)
−0.475429 + 0.879754i \(0.657707\pi\)
\(648\) 73.9659i 0.114145i
\(649\) 359.793i 0.554380i
\(650\) 372.962 0.573788
\(651\) 144.708i 0.222285i
\(652\) 518.481 0.795217
\(653\) 316.692i 0.484980i −0.970154 0.242490i \(-0.922036\pi\)
0.970154 0.242490i \(-0.0779642\pi\)
\(654\) 261.274i 0.399501i
\(655\) 277.676i 0.423933i
\(656\) 67.4528i 0.102824i
\(657\) 363.173 0.552775
\(658\) −82.9425 −0.126052
\(659\) −373.268 −0.566416 −0.283208 0.959059i \(-0.591399\pi\)
−0.283208 + 0.959059i \(0.591399\pi\)
\(660\) 37.3037i 0.0565207i
\(661\) 180.368i 0.272872i −0.990649 0.136436i \(-0.956435\pi\)
0.990649 0.136436i \(-0.0435648\pi\)
\(662\) −524.773 −0.792709
\(663\) 55.9028 0.0843179
\(664\) 411.769i 0.620134i
\(665\) 161.622 0.243041
\(666\) 13.3059i 0.0199789i
\(667\) 479.354 0.718672
\(668\) −162.965 −0.243960
\(669\) 333.461i 0.498447i
\(670\) 123.850 + 147.061i 0.184851 + 0.219495i
\(671\) 101.391 0.151105
\(672\) 488.260i 0.726577i
\(673\) 882.215i 1.31087i 0.755252 + 0.655434i \(0.227514\pi\)
−0.755252 + 0.655434i \(0.772486\pi\)
\(674\) 144.748 0.214760
\(675\) 104.790i 0.155244i
\(676\) −72.8774 −0.107807
\(677\) 552.316i 0.815829i 0.913020 + 0.407914i \(0.133744\pi\)
−0.913020 + 0.407914i \(0.866256\pi\)
\(678\) 66.5683i 0.0981834i
\(679\) 332.157 0.489186
\(680\) −41.1587 −0.0605275
\(681\) 219.324i 0.322061i
\(682\) 50.9350i 0.0746847i
\(683\) 251.021i 0.367526i 0.982971 + 0.183763i \(0.0588280\pi\)
−0.982971 + 0.183763i \(0.941172\pi\)
\(684\) 55.4393 0.0810516
\(685\) 81.4925 0.118967
\(686\) 173.568 0.253014
\(687\) 777.811 1.13218
\(688\) 52.5367i 0.0763614i
\(689\) 1245.67 1.80794
\(690\) 94.5023i 0.136960i
\(691\) −608.855 −0.881122 −0.440561 0.897723i \(-0.645220\pi\)
−0.440561 + 0.897723i \(0.645220\pi\)
\(692\) −445.872 −0.644324
\(693\) 116.919 0.168715
\(694\) 691.226 0.996003
\(695\) −38.1783 −0.0549328
\(696\) 358.881 0.515634
\(697\) 99.5968i 0.142894i
\(698\) 10.0696i 0.0144263i
\(699\) −38.5742 −0.0551848
\(700\) 423.000i 0.604286i
\(701\) 762.979i 1.08841i −0.838951 0.544207i \(-0.816830\pi\)
0.838951 0.544207i \(-0.183170\pi\)
\(702\) 96.0967i 0.136890i
\(703\) −27.3466 −0.0388998
\(704\) 198.190i 0.281519i
\(705\) 26.4878 0.0375713
\(706\) −240.233 −0.340273
\(707\) 40.8423 0.0577684
\(708\) 335.390i 0.473715i
\(709\) 618.375 0.872179 0.436090 0.899903i \(-0.356363\pi\)
0.436090 + 0.899903i \(0.356363\pi\)
\(710\) −104.869 −0.147703
\(711\) 281.247i 0.395565i
\(712\) 434.439i 0.610168i
\(713\) 173.896i 0.243894i
\(714\) 47.0463i 0.0658912i
\(715\) 132.892i 0.185863i
\(716\) 384.462i 0.536958i
\(717\) 609.556 0.850148
\(718\) 231.737i 0.322754i
\(719\) −949.789 −1.32099 −0.660493 0.750832i \(-0.729653\pi\)
−0.660493 + 0.750832i \(0.729653\pi\)
\(720\) 10.1753i 0.0141324i
\(721\) 439.114i 0.609035i
\(722\) 386.669i 0.535553i
\(723\) 116.138i 0.160633i
\(724\) −804.533 −1.11123
\(725\) 508.440 0.701296
\(726\) 232.409 0.320123
\(727\) 259.467i 0.356901i −0.983949 0.178451i \(-0.942892\pi\)
0.983949 0.178451i \(-0.0571085\pi\)
\(728\) 1063.65i 1.46106i
\(729\) −27.0000 −0.0370370
\(730\) −347.391 −0.475878
\(731\) 77.5726i 0.106118i
\(732\) 94.5145 0.129118
\(733\) 212.263i 0.289581i −0.989462 0.144790i \(-0.953749\pi\)
0.989462 0.144790i \(-0.0462508\pi\)
\(734\) −563.390 −0.767561
\(735\) 131.154 0.178441
\(736\) 586.745i 0.797208i
\(737\) −218.646 + 184.137i −0.296670 + 0.249846i
\(738\) 171.207 0.231987
\(739\) 322.025i 0.435758i −0.975976 0.217879i \(-0.930086\pi\)
0.975976 0.217879i \(-0.0699137\pi\)
\(740\) 17.1528i 0.0231794i
\(741\) 197.499 0.266531
\(742\) 1048.32i 1.41284i
\(743\) −779.375 −1.04896 −0.524478 0.851424i \(-0.675740\pi\)
−0.524478 + 0.851424i \(0.675740\pi\)
\(744\) 130.192i 0.174989i
\(745\) 209.376i 0.281042i
\(746\) 676.267 0.906525
\(747\) −150.309 −0.201218
\(748\) 22.3169i 0.0298354i
\(749\) 1166.89i 1.55793i
\(750\) 224.495i 0.299327i
\(751\) −39.6346 −0.0527758 −0.0263879 0.999652i \(-0.508401\pi\)
−0.0263879 + 0.999652i \(0.508401\pi\)
\(752\) 10.7320 0.0142713
\(753\) −495.812 −0.658449
\(754\) 466.260 0.618382
\(755\) 159.006i 0.210604i
\(756\) 108.989 0.144166
\(757\) 1401.87i 1.85188i 0.377668 + 0.925941i \(0.376726\pi\)
−0.377668 + 0.925941i \(0.623274\pi\)
\(758\) 403.261 0.532007
\(759\) −140.503 −0.185116
\(760\) −145.410 −0.191329
\(761\) 151.310 0.198831 0.0994153 0.995046i \(-0.468303\pi\)
0.0994153 + 0.995046i \(0.468303\pi\)
\(762\) 485.008 0.636494
\(763\) 1055.65 1.38355
\(764\) 445.422i 0.583013i
\(765\) 15.0243i 0.0196396i
\(766\) −23.8929 −0.0311918
\(767\) 1194.81i 1.55777i
\(768\) 463.826i 0.603940i
\(769\) 461.300i 0.599870i −0.953960 0.299935i \(-0.903035\pi\)
0.953960 0.299935i \(-0.0969650\pi\)
\(770\) −111.839 −0.145245
\(771\) 382.540i 0.496160i
\(772\) −617.622 −0.800029
\(773\) 679.969 0.879649 0.439824 0.898084i \(-0.355041\pi\)
0.439824 + 0.898084i \(0.355041\pi\)
\(774\) −133.347 −0.172283
\(775\) 184.448i 0.237997i
\(776\) −298.839 −0.385101
\(777\) −53.7613 −0.0691908
\(778\) 530.990i 0.682507i
\(779\) 351.866i 0.451690i
\(780\) 123.879i 0.158819i
\(781\) 155.916i 0.199636i
\(782\) 56.5359i 0.0722965i
\(783\) 131.004i 0.167310i
\(784\) 53.1395 0.0677799
\(785\) 497.759i 0.634088i
\(786\) −285.559 −0.363307
\(787\) 790.065i 1.00389i −0.864898 0.501947i \(-0.832617\pi\)
0.864898 0.501947i \(-0.167383\pi\)
\(788\) 437.854i 0.555652i
\(789\) 759.550i 0.962675i
\(790\) 269.025i 0.340538i
\(791\) −268.962 −0.340028
\(792\) −105.191 −0.132817
\(793\) 336.703 0.424594
\(794\) 669.472i 0.843163i
\(795\) 334.784i 0.421112i
\(796\) 115.108 0.144609
\(797\) 888.931 1.11535 0.557673 0.830061i \(-0.311694\pi\)
0.557673 + 0.830061i \(0.311694\pi\)
\(798\) 166.210i 0.208283i
\(799\) 15.8463 0.0198326
\(800\) 622.347i 0.777933i
\(801\) −158.585 −0.197984
\(802\) −693.526 −0.864746
\(803\) 516.490i 0.643200i
\(804\) −203.817 + 171.648i −0.253503 + 0.213492i
\(805\) −381.826 −0.474319
\(806\) 169.146i 0.209859i
\(807\) 301.122i 0.373138i
\(808\) −36.7454 −0.0454769
\(809\) 752.068i 0.929626i −0.885409 0.464813i \(-0.846121\pi\)
0.885409 0.464813i \(-0.153879\pi\)
\(810\) 25.8267 0.0318848
\(811\) 1099.10i 1.35524i 0.735411 + 0.677621i \(0.236989\pi\)
−0.735411 + 0.677621i \(0.763011\pi\)
\(812\) 528.815i 0.651250i
\(813\) −862.222 −1.06054
\(814\) 18.9232 0.0232471
\(815\) 496.411i 0.609094i
\(816\) 6.08738i 0.00746002i
\(817\) 274.057i 0.335443i
\(818\) 529.272 0.647032
\(819\) 388.269 0.474077
\(820\) 220.704 0.269151
\(821\) 570.245 0.694573 0.347287 0.937759i \(-0.387103\pi\)
0.347287 + 0.937759i \(0.387103\pi\)
\(822\) 83.8059i 0.101954i
\(823\) −967.829 −1.17598 −0.587988 0.808870i \(-0.700080\pi\)
−0.587988 + 0.808870i \(0.700080\pi\)
\(824\) 395.066i 0.479450i
\(825\) −149.028 −0.180640
\(826\) −1005.52 −1.21733
\(827\) −1005.58 −1.21594 −0.607969 0.793961i \(-0.708015\pi\)
−0.607969 + 0.793961i \(0.708015\pi\)
\(828\) −130.973 −0.158180
\(829\) −63.3910 −0.0764668 −0.0382334 0.999269i \(-0.512173\pi\)
−0.0382334 + 0.999269i \(0.512173\pi\)
\(830\) 143.778 0.173226
\(831\) 172.624i 0.207730i
\(832\) 658.153i 0.791049i
\(833\) 78.4626 0.0941929
\(834\) 39.2621i 0.0470769i
\(835\) 156.028i 0.186860i
\(836\) 78.8435i 0.0943104i
\(837\) −47.5245 −0.0567795
\(838\) 982.199i 1.17208i
\(839\) −1014.89 −1.20964 −0.604818 0.796363i \(-0.706754\pi\)
−0.604818 + 0.796363i \(0.706754\pi\)
\(840\) −285.865 −0.340315
\(841\) −205.372 −0.244200
\(842\) 623.505i 0.740505i
\(843\) 150.338 0.178337
\(844\) −104.361 −0.123650
\(845\) 69.7753i 0.0825743i
\(846\) 27.2397i 0.0321982i
\(847\) 939.025i 1.10865i
\(848\) 135.644i 0.159958i
\(849\) 428.439i 0.504639i
\(850\) 59.9663i 0.0705486i
\(851\) 64.6053 0.0759169
\(852\) 145.341i 0.170588i
\(853\) 30.4715 0.0357228 0.0178614 0.999840i \(-0.494314\pi\)
0.0178614 + 0.999840i \(0.494314\pi\)
\(854\) 283.360i 0.331803i
\(855\) 53.0794i 0.0620812i
\(856\) 1049.84i 1.22644i
\(857\) 116.993i 0.136515i −0.997668 0.0682574i \(-0.978256\pi\)
0.997668 0.0682574i \(-0.0217439\pi\)
\(858\) −136.665 −0.159283
\(859\) 571.567 0.665387 0.332694 0.943035i \(-0.392043\pi\)
0.332694 + 0.943035i \(0.392043\pi\)
\(860\) −171.899 −0.199882
\(861\) 691.742i 0.803417i
\(862\) 197.691i 0.229340i
\(863\) −1438.95 −1.66739 −0.833693 0.552229i \(-0.813778\pi\)
−0.833693 + 0.552229i \(0.813778\pi\)
\(864\) −160.353 −0.185593
\(865\) 426.893i 0.493518i
\(866\) −812.378 −0.938081
\(867\) 491.574i 0.566983i
\(868\) 191.839 0.221013
\(869\) 399.978 0.460274
\(870\) 125.311i 0.144035i
\(871\) −726.085 + 611.486i −0.833623 + 0.702050i
\(872\) −949.758 −1.08917
\(873\) 109.086i 0.124955i
\(874\) 199.736i 0.228531i
\(875\) −907.049 −1.03663
\(876\) 481.459i 0.549611i
\(877\) −473.773 −0.540220 −0.270110 0.962829i \(-0.587060\pi\)
−0.270110 + 0.962829i \(0.587060\pi\)
\(878\) 307.727i 0.350486i
\(879\) 56.5843i 0.0643735i
\(880\) 14.4709 0.0164442
\(881\) 775.606 0.880370 0.440185 0.897907i \(-0.354913\pi\)
0.440185 + 0.897907i \(0.354913\pi\)
\(882\) 134.877i 0.152922i
\(883\) 926.204i 1.04893i 0.851432 + 0.524464i \(0.175734\pi\)
−0.851432 + 0.524464i \(0.824266\pi\)
\(884\) 74.1105i 0.0838354i
\(885\) 321.113 0.362840
\(886\) −678.626 −0.765944
\(887\) −469.230 −0.529008 −0.264504 0.964385i \(-0.585208\pi\)
−0.264504 + 0.964385i \(0.585208\pi\)
\(888\) 48.3685 0.0544690
\(889\) 1959.62i 2.20430i
\(890\) 151.693 0.170442
\(891\) 38.3983i 0.0430957i
\(892\) 442.070 0.495594
\(893\) 55.9834 0.0626914
\(894\) −215.320 −0.240850
\(895\) −368.097 −0.411281
\(896\) 573.703 0.640294
\(897\) −466.585 −0.520162
\(898\) 497.440i 0.553942i
\(899\) 230.588i 0.256494i
\(900\) −138.920 −0.154356
\(901\) 200.284i 0.222291i
\(902\) 243.483i 0.269937i
\(903\) 538.775i 0.596650i
\(904\) 241.983 0.267680
\(905\) 770.287i 0.851146i
\(906\) 163.520 0.180486
\(907\) 254.129 0.280186 0.140093 0.990138i \(-0.455260\pi\)
0.140093 + 0.990138i \(0.455260\pi\)
\(908\) −290.758 −0.320218
\(909\) 13.4133i 0.0147561i
\(910\) −371.396 −0.408128
\(911\) 288.361 0.316533 0.158266 0.987396i \(-0.449410\pi\)
0.158266 + 0.987396i \(0.449410\pi\)
\(912\) 21.5061i 0.0235813i
\(913\) 213.764i 0.234134i
\(914\) 264.584i 0.289480i
\(915\) 90.4914i 0.0988977i
\(916\) 1031.15i 1.12570i
\(917\) 1153.77i 1.25820i
\(918\) 15.4508 0.0168309
\(919\) 517.961i 0.563614i −0.959471 0.281807i \(-0.909066\pi\)
0.959471 0.281807i \(-0.0909337\pi\)
\(920\) 343.525 0.373397
\(921\) 262.523i 0.285042i
\(922\) 471.356i 0.511232i
\(923\) 517.769i 0.560963i
\(924\) 155.000i 0.167749i
\(925\) 68.5253 0.0740815
\(926\) −637.392 −0.688329
\(927\) −144.212 −0.155569
\(928\) 778.029i 0.838393i
\(929\) 648.006i 0.697531i −0.937210 0.348766i \(-0.886601\pi\)
0.937210 0.348766i \(-0.113399\pi\)
\(930\) 45.4593 0.0488809
\(931\) 277.201 0.297746
\(932\) 51.1378i 0.0548689i
\(933\) −566.196 −0.606855
\(934\) 910.287i 0.974612i
\(935\) 21.3669 0.0228523
\(936\) −349.322 −0.373207
\(937\) 1311.04i 1.39918i 0.714542 + 0.699592i \(0.246635\pi\)
−0.714542 + 0.699592i \(0.753365\pi\)
\(938\) 514.610 + 611.054i 0.548625 + 0.651444i
\(939\) 728.478 0.775802
\(940\) 35.1149i 0.0373563i
\(941\) 311.971i 0.331531i 0.986165 + 0.165765i \(0.0530095\pi\)
−0.986165 + 0.165765i \(0.946991\pi\)
\(942\) 511.889 0.543407
\(943\) 831.271i 0.881518i
\(944\) 130.105 0.137823
\(945\) 104.350i 0.110423i
\(946\) 189.641i 0.200466i
\(947\) 393.116 0.415117 0.207559 0.978223i \(-0.433448\pi\)
0.207559 + 0.978223i \(0.433448\pi\)
\(948\) 372.850 0.393301
\(949\) 1715.17i 1.80735i
\(950\) 211.855i 0.223006i
\(951\) 670.935i 0.705505i
\(952\) −171.018 −0.179641
\(953\) 534.730 0.561102 0.280551 0.959839i \(-0.409483\pi\)
0.280551 + 0.959839i \(0.409483\pi\)
\(954\) 344.288 0.360888
\(955\) −426.462 −0.446557
\(956\) 808.090i 0.845282i
\(957\) −186.308 −0.194679
\(958\) 798.941i 0.833967i
\(959\) 338.609 0.353086
\(960\) 176.883 0.184254
\(961\) 877.349 0.912954
\(962\) 62.8405 0.0653228
\(963\) −383.225 −0.397949
\(964\) 153.964 0.159714
\(965\) 591.332i 0.612780i
\(966\) 392.666i 0.406486i
\(967\) 1743.86 1.80338 0.901688 0.432387i \(-0.142329\pi\)
0.901688 + 0.432387i \(0.142329\pi\)
\(968\) 844.831i 0.872760i
\(969\) 31.7547i 0.0327706i
\(970\) 104.346i 0.107573i
\(971\) 933.434 0.961312 0.480656 0.876909i \(-0.340399\pi\)
0.480656 + 0.876909i \(0.340399\pi\)
\(972\) 35.7940i 0.0368251i
\(973\) −158.635 −0.163036
\(974\) 453.531 0.465638
\(975\) −494.896 −0.507586
\(976\) 36.6643i 0.0375659i
\(977\) 1076.60 1.10195 0.550974 0.834522i \(-0.314256\pi\)
0.550974 + 0.834522i \(0.314256\pi\)
\(978\) 510.503 0.521987
\(979\) 225.533i 0.230371i
\(980\) 173.871i 0.177419i
\(981\) 346.693i 0.353408i
\(982\) 571.699i 0.582178i
\(983\) 1507.23i 1.53330i −0.642068 0.766648i \(-0.721923\pi\)
0.642068 0.766648i \(-0.278077\pi\)
\(984\) 622.354i 0.632473i
\(985\) −419.216 −0.425600
\(986\) 74.9670i 0.0760315i
\(987\) 110.059 0.111509
\(988\) 261.825i 0.265005i
\(989\) 647.449i 0.654650i
\(990\) 36.7297i 0.0371007i
\(991\) 42.8503i 0.0432394i −0.999766 0.0216197i \(-0.993118\pi\)
0.999766 0.0216197i \(-0.00688230\pi\)
\(992\) −282.247 −0.284523
\(993\) 696.339 0.701248
\(994\) −435.741 −0.438371
\(995\) 110.209i 0.110762i
\(996\) 199.266i 0.200066i
\(997\) 618.244 0.620104 0.310052 0.950720i \(-0.399654\pi\)
0.310052 + 0.950720i \(0.399654\pi\)
\(998\) 73.7909 0.0739388
\(999\) 17.6561i 0.0176738i
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 201.3.b.a.133.8 22
3.2 odd 2 603.3.b.e.334.15 22
67.66 odd 2 inner 201.3.b.a.133.15 yes 22
201.200 even 2 603.3.b.e.334.8 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
201.3.b.a.133.8 22 1.1 even 1 trivial
201.3.b.a.133.15 yes 22 67.66 odd 2 inner
603.3.b.e.334.8 22 201.200 even 2
603.3.b.e.334.15 22 3.2 odd 2