Properties

Label 8021.2.a.d.1.3
Level $8021$
Weight $2$
Character 8021.1
Self dual yes
Analytic conductor $64.048$
Analytic rank $0$
Dimension $174$
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] = [8021,2,Mod(1,8021)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8021, 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("8021.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8021 = 13 \cdot 617 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8021.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: \(64.0480074613\)
Analytic rank: \(0\)
Dimension: \(174\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) \(=\) 8021.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.75277 q^{2} +3.20644 q^{3} +5.57777 q^{4} +3.17240 q^{5} -8.82662 q^{6} +4.29647 q^{7} -9.84879 q^{8} +7.28128 q^{9} +O(q^{10})\) \(q-2.75277 q^{2} +3.20644 q^{3} +5.57777 q^{4} +3.17240 q^{5} -8.82662 q^{6} +4.29647 q^{7} -9.84879 q^{8} +7.28128 q^{9} -8.73291 q^{10} -0.263402 q^{11} +17.8848 q^{12} +1.00000 q^{13} -11.8272 q^{14} +10.1721 q^{15} +15.9560 q^{16} -4.13601 q^{17} -20.0437 q^{18} +2.21339 q^{19} +17.6949 q^{20} +13.7764 q^{21} +0.725085 q^{22} -6.65465 q^{23} -31.5796 q^{24} +5.06413 q^{25} -2.75277 q^{26} +13.7277 q^{27} +23.9647 q^{28} +8.73232 q^{29} -28.0016 q^{30} -1.55098 q^{31} -24.2256 q^{32} -0.844582 q^{33} +11.3855 q^{34} +13.6301 q^{35} +40.6133 q^{36} +6.95318 q^{37} -6.09295 q^{38} +3.20644 q^{39} -31.2443 q^{40} +1.29749 q^{41} -37.9233 q^{42} -7.49702 q^{43} -1.46919 q^{44} +23.0991 q^{45} +18.3188 q^{46} -6.36833 q^{47} +51.1619 q^{48} +11.4596 q^{49} -13.9404 q^{50} -13.2619 q^{51} +5.57777 q^{52} +5.38401 q^{53} -37.7892 q^{54} -0.835616 q^{55} -42.3150 q^{56} +7.09710 q^{57} -24.0381 q^{58} -1.91259 q^{59} +56.7378 q^{60} +9.65500 q^{61} +4.26949 q^{62} +31.2838 q^{63} +34.7756 q^{64} +3.17240 q^{65} +2.32495 q^{66} +3.85262 q^{67} -23.0697 q^{68} -21.3378 q^{69} -37.5206 q^{70} -0.984946 q^{71} -71.7118 q^{72} -14.2783 q^{73} -19.1405 q^{74} +16.2379 q^{75} +12.3458 q^{76} -1.13170 q^{77} -8.82662 q^{78} +10.5549 q^{79} +50.6187 q^{80} +22.1732 q^{81} -3.57170 q^{82} -14.5510 q^{83} +76.8414 q^{84} -13.1211 q^{85} +20.6376 q^{86} +27.9997 q^{87} +2.59419 q^{88} +9.36383 q^{89} -63.5867 q^{90} +4.29647 q^{91} -37.1181 q^{92} -4.97312 q^{93} +17.5306 q^{94} +7.02175 q^{95} -77.6779 q^{96} -9.26914 q^{97} -31.5458 q^{98} -1.91790 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 174 q + 6 q^{2} + 37 q^{3} + 214 q^{4} + 10 q^{5} + 12 q^{6} + 28 q^{7} + 15 q^{8} + 211 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 174 q + 6 q^{2} + 37 q^{3} + 214 q^{4} + 10 q^{5} + 12 q^{6} + 28 q^{7} + 15 q^{8} + 211 q^{9} + 47 q^{10} + 47 q^{11} + 81 q^{12} + 174 q^{13} + 22 q^{14} + 26 q^{15} + 286 q^{16} + 27 q^{17} + 22 q^{18} + 91 q^{19} + 18 q^{20} + 8 q^{21} + 58 q^{22} + 62 q^{23} + 24 q^{24} + 244 q^{25} + 6 q^{26} + 139 q^{27} + 43 q^{28} + 42 q^{29} + 31 q^{30} + 82 q^{31} + 11 q^{32} + 12 q^{33} + 50 q^{34} + 74 q^{35} + 310 q^{36} + 47 q^{37} + 10 q^{38} + 37 q^{39} + 118 q^{40} + 16 q^{41} + 26 q^{42} + 136 q^{43} + 74 q^{44} + 18 q^{45} + 53 q^{46} + 15 q^{47} + 132 q^{48} + 254 q^{49} - 5 q^{50} + 121 q^{51} + 214 q^{52} + 39 q^{53} + 30 q^{54} + 188 q^{55} + 55 q^{56} + 11 q^{57} + 32 q^{58} + 58 q^{59} + 16 q^{60} + 128 q^{61} + 27 q^{62} + 42 q^{63} + 423 q^{64} + 10 q^{65} + 4 q^{66} + 132 q^{67} + 52 q^{68} + 63 q^{69} - 8 q^{70} + 78 q^{71} + 2 q^{72} + 21 q^{73} - 16 q^{74} + 188 q^{75} + 160 q^{76} + 20 q^{77} + 12 q^{78} + 232 q^{79} + 2 q^{80} + 302 q^{81} + 115 q^{82} + 18 q^{83} - 26 q^{84} + 47 q^{85} + 27 q^{86} + 127 q^{87} + 163 q^{88} + 14 q^{90} + 28 q^{91} + 68 q^{92} + 15 q^{93} + 91 q^{94} + 75 q^{95} - 26 q^{96} + 34 q^{97} - 60 q^{98} + 181 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.75277 −1.94651 −0.973253 0.229737i \(-0.926213\pi\)
−0.973253 + 0.229737i \(0.926213\pi\)
\(3\) 3.20644 1.85124 0.925621 0.378453i \(-0.123544\pi\)
0.925621 + 0.378453i \(0.123544\pi\)
\(4\) 5.57777 2.78888
\(5\) 3.17240 1.41874 0.709371 0.704836i \(-0.248979\pi\)
0.709371 + 0.704836i \(0.248979\pi\)
\(6\) −8.82662 −3.60345
\(7\) 4.29647 1.62391 0.811956 0.583719i \(-0.198403\pi\)
0.811956 + 0.583719i \(0.198403\pi\)
\(8\) −9.84879 −3.48207
\(9\) 7.28128 2.42709
\(10\) −8.73291 −2.76159
\(11\) −0.263402 −0.0794186 −0.0397093 0.999211i \(-0.512643\pi\)
−0.0397093 + 0.999211i \(0.512643\pi\)
\(12\) 17.8848 5.16290
\(13\) 1.00000 0.277350
\(14\) −11.8272 −3.16095
\(15\) 10.1721 2.62643
\(16\) 15.9560 3.98899
\(17\) −4.13601 −1.00313 −0.501565 0.865120i \(-0.667242\pi\)
−0.501565 + 0.865120i \(0.667242\pi\)
\(18\) −20.0437 −4.72435
\(19\) 2.21339 0.507785 0.253893 0.967232i \(-0.418289\pi\)
0.253893 + 0.967232i \(0.418289\pi\)
\(20\) 17.6949 3.95670
\(21\) 13.7764 3.00625
\(22\) 0.725085 0.154589
\(23\) −6.65465 −1.38759 −0.693795 0.720172i \(-0.744063\pi\)
−0.693795 + 0.720172i \(0.744063\pi\)
\(24\) −31.5796 −6.44616
\(25\) 5.06413 1.01283
\(26\) −2.75277 −0.539864
\(27\) 13.7277 2.64189
\(28\) 23.9647 4.52890
\(29\) 8.73232 1.62155 0.810775 0.585357i \(-0.199046\pi\)
0.810775 + 0.585357i \(0.199046\pi\)
\(30\) −28.0016 −5.11236
\(31\) −1.55098 −0.278564 −0.139282 0.990253i \(-0.544479\pi\)
−0.139282 + 0.990253i \(0.544479\pi\)
\(32\) −24.2256 −4.28252
\(33\) −0.844582 −0.147023
\(34\) 11.3855 1.95260
\(35\) 13.6301 2.30391
\(36\) 40.6133 6.76888
\(37\) 6.95318 1.14310 0.571548 0.820569i \(-0.306343\pi\)
0.571548 + 0.820569i \(0.306343\pi\)
\(38\) −6.09295 −0.988407
\(39\) 3.20644 0.513442
\(40\) −31.2443 −4.94016
\(41\) 1.29749 0.202634 0.101317 0.994854i \(-0.467694\pi\)
0.101317 + 0.994854i \(0.467694\pi\)
\(42\) −37.9233 −5.85169
\(43\) −7.49702 −1.14328 −0.571642 0.820503i \(-0.693693\pi\)
−0.571642 + 0.820503i \(0.693693\pi\)
\(44\) −1.46919 −0.221489
\(45\) 23.0991 3.44342
\(46\) 18.3188 2.70095
\(47\) −6.36833 −0.928917 −0.464458 0.885595i \(-0.653751\pi\)
−0.464458 + 0.885595i \(0.653751\pi\)
\(48\) 51.1619 7.38458
\(49\) 11.4596 1.63709
\(50\) −13.9404 −1.97147
\(51\) −13.2619 −1.85703
\(52\) 5.57777 0.773497
\(53\) 5.38401 0.739551 0.369775 0.929121i \(-0.379435\pi\)
0.369775 + 0.929121i \(0.379435\pi\)
\(54\) −37.7892 −5.14246
\(55\) −0.835616 −0.112674
\(56\) −42.3150 −5.65458
\(57\) 7.09710 0.940033
\(58\) −24.0381 −3.15636
\(59\) −1.91259 −0.248998 −0.124499 0.992220i \(-0.539732\pi\)
−0.124499 + 0.992220i \(0.539732\pi\)
\(60\) 56.7378 7.32481
\(61\) 9.65500 1.23620 0.618098 0.786101i \(-0.287904\pi\)
0.618098 + 0.786101i \(0.287904\pi\)
\(62\) 4.26949 0.542226
\(63\) 31.2838 3.94139
\(64\) 34.7756 4.34695
\(65\) 3.17240 0.393488
\(66\) 2.32495 0.286181
\(67\) 3.85262 0.470673 0.235337 0.971914i \(-0.424381\pi\)
0.235337 + 0.971914i \(0.424381\pi\)
\(68\) −23.0697 −2.79761
\(69\) −21.3378 −2.56876
\(70\) −37.5206 −4.48457
\(71\) −0.984946 −0.116892 −0.0584458 0.998291i \(-0.518614\pi\)
−0.0584458 + 0.998291i \(0.518614\pi\)
\(72\) −71.7118 −8.45132
\(73\) −14.2783 −1.67115 −0.835575 0.549377i \(-0.814865\pi\)
−0.835575 + 0.549377i \(0.814865\pi\)
\(74\) −19.1405 −2.22504
\(75\) 16.2379 1.87499
\(76\) 12.3458 1.41615
\(77\) −1.13170 −0.128969
\(78\) −8.82662 −0.999418
\(79\) 10.5549 1.18752 0.593762 0.804641i \(-0.297642\pi\)
0.593762 + 0.804641i \(0.297642\pi\)
\(80\) 50.6187 5.65934
\(81\) 22.1732 2.46369
\(82\) −3.57170 −0.394428
\(83\) −14.5510 −1.59718 −0.798591 0.601874i \(-0.794421\pi\)
−0.798591 + 0.601874i \(0.794421\pi\)
\(84\) 76.8414 8.38409
\(85\) −13.1211 −1.42318
\(86\) 20.6376 2.22541
\(87\) 27.9997 3.00188
\(88\) 2.59419 0.276541
\(89\) 9.36383 0.992564 0.496282 0.868161i \(-0.334698\pi\)
0.496282 + 0.868161i \(0.334698\pi\)
\(90\) −63.5867 −6.70263
\(91\) 4.29647 0.450392
\(92\) −37.1181 −3.86983
\(93\) −4.97312 −0.515689
\(94\) 17.5306 1.80814
\(95\) 7.02175 0.720416
\(96\) −77.6779 −7.92797
\(97\) −9.26914 −0.941139 −0.470569 0.882363i \(-0.655951\pi\)
−0.470569 + 0.882363i \(0.655951\pi\)
\(98\) −31.5458 −3.18660
\(99\) −1.91790 −0.192756
\(100\) 28.2465 2.82465
\(101\) −3.87237 −0.385315 −0.192658 0.981266i \(-0.561711\pi\)
−0.192658 + 0.981266i \(0.561711\pi\)
\(102\) 36.5070 3.61473
\(103\) 6.34530 0.625221 0.312611 0.949881i \(-0.398797\pi\)
0.312611 + 0.949881i \(0.398797\pi\)
\(104\) −9.84879 −0.965753
\(105\) 43.7042 4.26509
\(106\) −14.8210 −1.43954
\(107\) −17.5470 −1.69633 −0.848166 0.529731i \(-0.822293\pi\)
−0.848166 + 0.529731i \(0.822293\pi\)
\(108\) 76.5698 7.36794
\(109\) −4.66196 −0.446534 −0.223267 0.974757i \(-0.571672\pi\)
−0.223267 + 0.974757i \(0.571672\pi\)
\(110\) 2.30026 0.219321
\(111\) 22.2950 2.11615
\(112\) 68.5542 6.47777
\(113\) 2.03834 0.191751 0.0958754 0.995393i \(-0.469435\pi\)
0.0958754 + 0.995393i \(0.469435\pi\)
\(114\) −19.5367 −1.82978
\(115\) −21.1112 −1.96863
\(116\) 48.7068 4.52232
\(117\) 7.28128 0.673155
\(118\) 5.26493 0.484676
\(119\) −17.7702 −1.62899
\(120\) −100.183 −9.14543
\(121\) −10.9306 −0.993693
\(122\) −26.5780 −2.40626
\(123\) 4.16033 0.375125
\(124\) −8.65099 −0.776882
\(125\) 0.203451 0.0181972
\(126\) −86.1172 −7.67193
\(127\) −3.00347 −0.266515 −0.133257 0.991081i \(-0.542544\pi\)
−0.133257 + 0.991081i \(0.542544\pi\)
\(128\) −47.2783 −4.17885
\(129\) −24.0388 −2.11650
\(130\) −8.73291 −0.765927
\(131\) 12.5394 1.09557 0.547786 0.836618i \(-0.315471\pi\)
0.547786 + 0.836618i \(0.315471\pi\)
\(132\) −4.71088 −0.410030
\(133\) 9.50974 0.824599
\(134\) −10.6054 −0.916168
\(135\) 43.5497 3.74816
\(136\) 40.7347 3.49297
\(137\) −18.3770 −1.57005 −0.785025 0.619464i \(-0.787350\pi\)
−0.785025 + 0.619464i \(0.787350\pi\)
\(138\) 58.7380 5.00011
\(139\) −15.9716 −1.35469 −0.677346 0.735665i \(-0.736870\pi\)
−0.677346 + 0.735665i \(0.736870\pi\)
\(140\) 76.0256 6.42534
\(141\) −20.4197 −1.71965
\(142\) 2.71134 0.227530
\(143\) −0.263402 −0.0220268
\(144\) 116.180 9.68165
\(145\) 27.7024 2.30056
\(146\) 39.3050 3.25290
\(147\) 36.7446 3.03065
\(148\) 38.7832 3.18796
\(149\) −16.1325 −1.32163 −0.660813 0.750550i \(-0.729789\pi\)
−0.660813 + 0.750550i \(0.729789\pi\)
\(150\) −44.6991 −3.64967
\(151\) −3.08634 −0.251163 −0.125582 0.992083i \(-0.540080\pi\)
−0.125582 + 0.992083i \(0.540080\pi\)
\(152\) −21.7992 −1.76815
\(153\) −30.1154 −2.43469
\(154\) 3.11530 0.251038
\(155\) −4.92032 −0.395210
\(156\) 17.8848 1.43193
\(157\) −1.38500 −0.110535 −0.0552676 0.998472i \(-0.517601\pi\)
−0.0552676 + 0.998472i \(0.517601\pi\)
\(158\) −29.0554 −2.31152
\(159\) 17.2635 1.36909
\(160\) −76.8533 −6.07578
\(161\) −28.5915 −2.25332
\(162\) −61.0378 −4.79559
\(163\) 0.685117 0.0536625 0.0268312 0.999640i \(-0.491458\pi\)
0.0268312 + 0.999640i \(0.491458\pi\)
\(164\) 7.23711 0.565123
\(165\) −2.67935 −0.208587
\(166\) 40.0557 3.10892
\(167\) −12.1220 −0.938031 −0.469015 0.883190i \(-0.655391\pi\)
−0.469015 + 0.883190i \(0.655391\pi\)
\(168\) −135.681 −10.4680
\(169\) 1.00000 0.0769231
\(170\) 36.1194 2.77023
\(171\) 16.1163 1.23244
\(172\) −41.8166 −3.18849
\(173\) 14.5772 1.10828 0.554140 0.832423i \(-0.313047\pi\)
0.554140 + 0.832423i \(0.313047\pi\)
\(174\) −77.0768 −5.84318
\(175\) 21.7579 1.64474
\(176\) −4.20283 −0.316800
\(177\) −6.13261 −0.460956
\(178\) −25.7765 −1.93203
\(179\) −0.995754 −0.0744261 −0.0372131 0.999307i \(-0.511848\pi\)
−0.0372131 + 0.999307i \(0.511848\pi\)
\(180\) 128.842 9.60329
\(181\) −16.9891 −1.26279 −0.631393 0.775463i \(-0.717517\pi\)
−0.631393 + 0.775463i \(0.717517\pi\)
\(182\) −11.8272 −0.876691
\(183\) 30.9582 2.28850
\(184\) 65.5402 4.83169
\(185\) 22.0583 1.62176
\(186\) 13.6899 1.00379
\(187\) 1.08943 0.0796671
\(188\) −35.5211 −2.59064
\(189\) 58.9805 4.29020
\(190\) −19.3293 −1.40229
\(191\) 27.1499 1.96450 0.982248 0.187586i \(-0.0600664\pi\)
0.982248 + 0.187586i \(0.0600664\pi\)
\(192\) 111.506 8.04726
\(193\) −24.0330 −1.72993 −0.864967 0.501828i \(-0.832661\pi\)
−0.864967 + 0.501828i \(0.832661\pi\)
\(194\) 25.5159 1.83193
\(195\) 10.1721 0.728441
\(196\) 63.9191 4.56565
\(197\) −16.4922 −1.17502 −0.587510 0.809217i \(-0.699892\pi\)
−0.587510 + 0.809217i \(0.699892\pi\)
\(198\) 5.27955 0.375201
\(199\) 18.6551 1.32243 0.661214 0.750198i \(-0.270042\pi\)
0.661214 + 0.750198i \(0.270042\pi\)
\(200\) −49.8756 −3.52673
\(201\) 12.3532 0.871329
\(202\) 10.6598 0.750018
\(203\) 37.5181 2.63326
\(204\) −73.9717 −5.17905
\(205\) 4.11616 0.287485
\(206\) −17.4672 −1.21700
\(207\) −48.4544 −3.36781
\(208\) 15.9560 1.10635
\(209\) −0.583009 −0.0403276
\(210\) −120.308 −8.30203
\(211\) 11.4372 0.787369 0.393684 0.919246i \(-0.371200\pi\)
0.393684 + 0.919246i \(0.371200\pi\)
\(212\) 30.0308 2.06252
\(213\) −3.15818 −0.216395
\(214\) 48.3029 3.30192
\(215\) −23.7835 −1.62202
\(216\) −135.201 −9.19927
\(217\) −6.66372 −0.452363
\(218\) 12.8333 0.869182
\(219\) −45.7826 −3.09370
\(220\) −4.66087 −0.314236
\(221\) −4.13601 −0.278218
\(222\) −61.3731 −4.11909
\(223\) 3.80732 0.254957 0.127478 0.991841i \(-0.459312\pi\)
0.127478 + 0.991841i \(0.459312\pi\)
\(224\) −104.084 −6.95443
\(225\) 36.8734 2.45822
\(226\) −5.61109 −0.373244
\(227\) 2.60376 0.172818 0.0864089 0.996260i \(-0.472461\pi\)
0.0864089 + 0.996260i \(0.472461\pi\)
\(228\) 39.5860 2.62164
\(229\) 28.1232 1.85843 0.929215 0.369538i \(-0.120484\pi\)
0.929215 + 0.369538i \(0.120484\pi\)
\(230\) 58.1144 3.83195
\(231\) −3.62872 −0.238752
\(232\) −86.0028 −5.64636
\(233\) −6.75743 −0.442694 −0.221347 0.975195i \(-0.571045\pi\)
−0.221347 + 0.975195i \(0.571045\pi\)
\(234\) −20.0437 −1.31030
\(235\) −20.2029 −1.31789
\(236\) −10.6680 −0.694427
\(237\) 33.8438 2.19839
\(238\) 48.9174 3.17085
\(239\) 17.0715 1.10427 0.552133 0.833756i \(-0.313814\pi\)
0.552133 + 0.833756i \(0.313814\pi\)
\(240\) 162.306 10.4768
\(241\) −8.81531 −0.567844 −0.283922 0.958847i \(-0.591636\pi\)
−0.283922 + 0.958847i \(0.591636\pi\)
\(242\) 30.0895 1.93423
\(243\) 29.9141 1.91899
\(244\) 53.8533 3.44761
\(245\) 36.3545 2.32261
\(246\) −11.4525 −0.730182
\(247\) 2.21339 0.140834
\(248\) 15.2752 0.969979
\(249\) −46.6570 −2.95677
\(250\) −0.560055 −0.0354210
\(251\) 16.3569 1.03244 0.516219 0.856457i \(-0.327339\pi\)
0.516219 + 0.856457i \(0.327339\pi\)
\(252\) 174.494 10.9921
\(253\) 1.75285 0.110200
\(254\) 8.26787 0.518772
\(255\) −42.0720 −2.63465
\(256\) 60.5953 3.78721
\(257\) 17.4386 1.08779 0.543897 0.839152i \(-0.316948\pi\)
0.543897 + 0.839152i \(0.316948\pi\)
\(258\) 66.1733 4.11977
\(259\) 29.8741 1.85629
\(260\) 17.6949 1.09739
\(261\) 63.5825 3.93566
\(262\) −34.5181 −2.13254
\(263\) 8.55463 0.527501 0.263751 0.964591i \(-0.415040\pi\)
0.263751 + 0.964591i \(0.415040\pi\)
\(264\) 8.31811 0.511945
\(265\) 17.0802 1.04923
\(266\) −26.1782 −1.60509
\(267\) 30.0246 1.83748
\(268\) 21.4890 1.31265
\(269\) −21.1997 −1.29257 −0.646283 0.763098i \(-0.723677\pi\)
−0.646283 + 0.763098i \(0.723677\pi\)
\(270\) −119.883 −7.29582
\(271\) 6.62808 0.402627 0.201314 0.979527i \(-0.435479\pi\)
0.201314 + 0.979527i \(0.435479\pi\)
\(272\) −65.9940 −4.00147
\(273\) 13.7764 0.833784
\(274\) 50.5876 3.05611
\(275\) −1.33390 −0.0804372
\(276\) −119.017 −7.16399
\(277\) −6.89556 −0.414314 −0.207157 0.978308i \(-0.566421\pi\)
−0.207157 + 0.978308i \(0.566421\pi\)
\(278\) 43.9662 2.63692
\(279\) −11.2931 −0.676100
\(280\) −134.240 −8.02238
\(281\) −30.7068 −1.83182 −0.915908 0.401388i \(-0.868528\pi\)
−0.915908 + 0.401388i \(0.868528\pi\)
\(282\) 56.2108 3.34731
\(283\) 23.8752 1.41924 0.709618 0.704587i \(-0.248868\pi\)
0.709618 + 0.704587i \(0.248868\pi\)
\(284\) −5.49380 −0.325997
\(285\) 22.5148 1.33366
\(286\) 0.725085 0.0428752
\(287\) 5.57463 0.329060
\(288\) −176.393 −10.3941
\(289\) 0.106572 0.00626894
\(290\) −76.2585 −4.47805
\(291\) −29.7210 −1.74228
\(292\) −79.6411 −4.66064
\(293\) 24.2474 1.41655 0.708274 0.705938i \(-0.249474\pi\)
0.708274 + 0.705938i \(0.249474\pi\)
\(294\) −101.150 −5.89917
\(295\) −6.06751 −0.353264
\(296\) −68.4804 −3.98034
\(297\) −3.61589 −0.209816
\(298\) 44.4092 2.57255
\(299\) −6.65465 −0.384848
\(300\) 90.5710 5.22912
\(301\) −32.2107 −1.85659
\(302\) 8.49601 0.488890
\(303\) −12.4165 −0.713311
\(304\) 35.3167 2.02555
\(305\) 30.6295 1.75384
\(306\) 82.9010 4.73914
\(307\) 25.4891 1.45474 0.727369 0.686247i \(-0.240743\pi\)
0.727369 + 0.686247i \(0.240743\pi\)
\(308\) −6.31234 −0.359679
\(309\) 20.3459 1.15744
\(310\) 13.5445 0.769278
\(311\) 23.0889 1.30925 0.654627 0.755952i \(-0.272826\pi\)
0.654627 + 0.755952i \(0.272826\pi\)
\(312\) −31.5796 −1.78784
\(313\) 3.94940 0.223233 0.111617 0.993751i \(-0.464397\pi\)
0.111617 + 0.993751i \(0.464397\pi\)
\(314\) 3.81260 0.215157
\(315\) 99.2447 5.59181
\(316\) 58.8730 3.31187
\(317\) −19.4039 −1.08983 −0.544917 0.838490i \(-0.683439\pi\)
−0.544917 + 0.838490i \(0.683439\pi\)
\(318\) −47.5226 −2.66494
\(319\) −2.30011 −0.128781
\(320\) 110.322 6.16720
\(321\) −56.2634 −3.14032
\(322\) 78.7059 4.38611
\(323\) −9.15458 −0.509375
\(324\) 123.677 6.87094
\(325\) 5.06413 0.280907
\(326\) −1.88597 −0.104454
\(327\) −14.9483 −0.826643
\(328\) −12.7787 −0.705587
\(329\) −27.3613 −1.50848
\(330\) 7.37566 0.406017
\(331\) 1.76385 0.0969502 0.0484751 0.998824i \(-0.484564\pi\)
0.0484751 + 0.998824i \(0.484564\pi\)
\(332\) −81.1622 −4.45435
\(333\) 50.6281 2.77440
\(334\) 33.3692 1.82588
\(335\) 12.2221 0.667763
\(336\) 219.815 11.9919
\(337\) −33.0416 −1.79989 −0.899946 0.436002i \(-0.856394\pi\)
−0.899946 + 0.436002i \(0.856394\pi\)
\(338\) −2.75277 −0.149731
\(339\) 6.53582 0.354977
\(340\) −73.1863 −3.96909
\(341\) 0.408530 0.0221231
\(342\) −44.3645 −2.39896
\(343\) 19.1606 1.03458
\(344\) 73.8365 3.98100
\(345\) −67.6919 −3.64441
\(346\) −40.1276 −2.15727
\(347\) −21.3470 −1.14597 −0.572984 0.819567i \(-0.694214\pi\)
−0.572984 + 0.819567i \(0.694214\pi\)
\(348\) 156.176 8.37190
\(349\) −31.1499 −1.66742 −0.833709 0.552204i \(-0.813787\pi\)
−0.833709 + 0.552204i \(0.813787\pi\)
\(350\) −59.8945 −3.20150
\(351\) 13.7277 0.732730
\(352\) 6.38106 0.340111
\(353\) 23.0819 1.22853 0.614263 0.789102i \(-0.289454\pi\)
0.614263 + 0.789102i \(0.289454\pi\)
\(354\) 16.8817 0.897252
\(355\) −3.12465 −0.165839
\(356\) 52.2293 2.76815
\(357\) −56.9792 −3.01566
\(358\) 2.74109 0.144871
\(359\) 0.0369949 0.00195252 0.000976259 1.00000i \(-0.499689\pi\)
0.000976259 1.00000i \(0.499689\pi\)
\(360\) −227.499 −11.9902
\(361\) −14.1009 −0.742154
\(362\) 46.7670 2.45802
\(363\) −35.0484 −1.83956
\(364\) 23.9647 1.25609
\(365\) −45.2965 −2.37093
\(366\) −85.2209 −4.45457
\(367\) −25.7029 −1.34168 −0.670841 0.741601i \(-0.734067\pi\)
−0.670841 + 0.741601i \(0.734067\pi\)
\(368\) −106.181 −5.53508
\(369\) 9.44740 0.491812
\(370\) −60.7215 −3.15676
\(371\) 23.1322 1.20097
\(372\) −27.7389 −1.43820
\(373\) 19.4748 1.00837 0.504183 0.863597i \(-0.331794\pi\)
0.504183 + 0.863597i \(0.331794\pi\)
\(374\) −2.99896 −0.155073
\(375\) 0.652355 0.0336875
\(376\) 62.7203 3.23455
\(377\) 8.73232 0.449737
\(378\) −162.360 −8.35090
\(379\) 25.7206 1.32118 0.660588 0.750748i \(-0.270307\pi\)
0.660588 + 0.750748i \(0.270307\pi\)
\(380\) 39.1657 2.00916
\(381\) −9.63045 −0.493383
\(382\) −74.7375 −3.82390
\(383\) 15.7957 0.807124 0.403562 0.914952i \(-0.367772\pi\)
0.403562 + 0.914952i \(0.367772\pi\)
\(384\) −151.595 −7.73607
\(385\) −3.59020 −0.182973
\(386\) 66.1575 3.36733
\(387\) −54.5879 −2.77486
\(388\) −51.7011 −2.62473
\(389\) 36.3914 1.84512 0.922559 0.385857i \(-0.126094\pi\)
0.922559 + 0.385857i \(0.126094\pi\)
\(390\) −28.0016 −1.41791
\(391\) 27.5237 1.39193
\(392\) −112.863 −5.70046
\(393\) 40.2069 2.02817
\(394\) 45.3993 2.28718
\(395\) 33.4845 1.68479
\(396\) −10.6976 −0.537575
\(397\) 10.5672 0.530354 0.265177 0.964200i \(-0.414570\pi\)
0.265177 + 0.964200i \(0.414570\pi\)
\(398\) −51.3534 −2.57411
\(399\) 30.4924 1.52653
\(400\) 80.8031 4.04015
\(401\) −27.0427 −1.35045 −0.675225 0.737612i \(-0.735954\pi\)
−0.675225 + 0.737612i \(0.735954\pi\)
\(402\) −34.0056 −1.69605
\(403\) −1.55098 −0.0772597
\(404\) −21.5992 −1.07460
\(405\) 70.3423 3.49534
\(406\) −103.279 −5.12565
\(407\) −1.83148 −0.0907831
\(408\) 130.613 6.46633
\(409\) 6.05597 0.299449 0.149724 0.988728i \(-0.452161\pi\)
0.149724 + 0.988728i \(0.452161\pi\)
\(410\) −11.3309 −0.559592
\(411\) −58.9247 −2.90654
\(412\) 35.3926 1.74367
\(413\) −8.21738 −0.404351
\(414\) 133.384 6.55546
\(415\) −46.1617 −2.26599
\(416\) −24.2256 −1.18776
\(417\) −51.2120 −2.50786
\(418\) 1.60489 0.0784979
\(419\) 8.89765 0.434679 0.217339 0.976096i \(-0.430262\pi\)
0.217339 + 0.976096i \(0.430262\pi\)
\(420\) 243.772 11.8949
\(421\) 22.5612 1.09957 0.549783 0.835308i \(-0.314711\pi\)
0.549783 + 0.835308i \(0.314711\pi\)
\(422\) −31.4840 −1.53262
\(423\) −46.3696 −2.25457
\(424\) −53.0260 −2.57517
\(425\) −20.9453 −1.01600
\(426\) 8.69374 0.421213
\(427\) 41.4824 2.00747
\(428\) −97.8730 −4.73087
\(429\) −0.844582 −0.0407768
\(430\) 65.4707 3.15728
\(431\) −11.4635 −0.552176 −0.276088 0.961132i \(-0.589038\pi\)
−0.276088 + 0.961132i \(0.589038\pi\)
\(432\) 219.038 10.5385
\(433\) 3.19450 0.153518 0.0767590 0.997050i \(-0.475543\pi\)
0.0767590 + 0.997050i \(0.475543\pi\)
\(434\) 18.3437 0.880527
\(435\) 88.8263 4.25889
\(436\) −26.0033 −1.24533
\(437\) −14.7293 −0.704598
\(438\) 126.029 6.02190
\(439\) −21.0644 −1.00535 −0.502676 0.864475i \(-0.667651\pi\)
−0.502676 + 0.864475i \(0.667651\pi\)
\(440\) 8.22980 0.392340
\(441\) 83.4408 3.97337
\(442\) 11.3855 0.541553
\(443\) 8.00724 0.380435 0.190218 0.981742i \(-0.439081\pi\)
0.190218 + 0.981742i \(0.439081\pi\)
\(444\) 124.356 5.90169
\(445\) 29.7058 1.40819
\(446\) −10.4807 −0.496275
\(447\) −51.7280 −2.44665
\(448\) 149.412 7.05907
\(449\) −37.6623 −1.77739 −0.888697 0.458496i \(-0.848388\pi\)
−0.888697 + 0.458496i \(0.848388\pi\)
\(450\) −101.504 −4.78495
\(451\) −0.341761 −0.0160929
\(452\) 11.3694 0.534771
\(453\) −9.89619 −0.464964
\(454\) −7.16757 −0.336391
\(455\) 13.6301 0.638990
\(456\) −69.8978 −3.27326
\(457\) 30.8556 1.44337 0.721683 0.692224i \(-0.243369\pi\)
0.721683 + 0.692224i \(0.243369\pi\)
\(458\) −77.4167 −3.61745
\(459\) −56.7778 −2.65016
\(460\) −117.753 −5.49029
\(461\) 19.4842 0.907470 0.453735 0.891137i \(-0.350091\pi\)
0.453735 + 0.891137i \(0.350091\pi\)
\(462\) 9.98905 0.464733
\(463\) 10.8825 0.505750 0.252875 0.967499i \(-0.418624\pi\)
0.252875 + 0.967499i \(0.418624\pi\)
\(464\) 139.333 6.46835
\(465\) −15.7767 −0.731629
\(466\) 18.6017 0.861707
\(467\) 25.1089 1.16190 0.580951 0.813939i \(-0.302681\pi\)
0.580951 + 0.813939i \(0.302681\pi\)
\(468\) 40.6133 1.87735
\(469\) 16.5527 0.764331
\(470\) 55.6140 2.56528
\(471\) −4.44093 −0.204627
\(472\) 18.8367 0.867029
\(473\) 1.97473 0.0907980
\(474\) −93.1645 −4.27919
\(475\) 11.2089 0.514298
\(476\) −99.1182 −4.54307
\(477\) 39.2025 1.79496
\(478\) −46.9941 −2.14946
\(479\) −17.0339 −0.778300 −0.389150 0.921174i \(-0.627231\pi\)
−0.389150 + 0.921174i \(0.627231\pi\)
\(480\) −246.426 −11.2477
\(481\) 6.95318 0.317038
\(482\) 24.2666 1.10531
\(483\) −91.6770 −4.17145
\(484\) −60.9685 −2.77129
\(485\) −29.4054 −1.33523
\(486\) −82.3467 −3.73532
\(487\) −34.9925 −1.58566 −0.792830 0.609443i \(-0.791393\pi\)
−0.792830 + 0.609443i \(0.791393\pi\)
\(488\) −95.0900 −4.30452
\(489\) 2.19679 0.0993422
\(490\) −100.076 −4.52097
\(491\) −22.8146 −1.02961 −0.514804 0.857308i \(-0.672135\pi\)
−0.514804 + 0.857308i \(0.672135\pi\)
\(492\) 23.2054 1.04618
\(493\) −36.1170 −1.62663
\(494\) −6.09295 −0.274135
\(495\) −6.08435 −0.273471
\(496\) −24.7473 −1.11119
\(497\) −4.23179 −0.189822
\(498\) 128.436 5.75537
\(499\) −6.12172 −0.274046 −0.137023 0.990568i \(-0.543753\pi\)
−0.137023 + 0.990568i \(0.543753\pi\)
\(500\) 1.13480 0.0507500
\(501\) −38.8686 −1.73652
\(502\) −45.0268 −2.00965
\(503\) −6.01625 −0.268251 −0.134126 0.990964i \(-0.542823\pi\)
−0.134126 + 0.990964i \(0.542823\pi\)
\(504\) −308.107 −13.7242
\(505\) −12.2847 −0.546662
\(506\) −4.82519 −0.214506
\(507\) 3.20644 0.142403
\(508\) −16.7526 −0.743278
\(509\) −1.37514 −0.0609519 −0.0304759 0.999536i \(-0.509702\pi\)
−0.0304759 + 0.999536i \(0.509702\pi\)
\(510\) 115.815 5.12836
\(511\) −61.3463 −2.71380
\(512\) −72.2486 −3.19297
\(513\) 30.3847 1.34152
\(514\) −48.0047 −2.11740
\(515\) 20.1298 0.887027
\(516\) −134.083 −5.90266
\(517\) 1.67743 0.0737732
\(518\) −82.2367 −3.61327
\(519\) 46.7408 2.05169
\(520\) −31.2443 −1.37015
\(521\) 5.75248 0.252021 0.126010 0.992029i \(-0.459783\pi\)
0.126010 + 0.992029i \(0.459783\pi\)
\(522\) −175.028 −7.66078
\(523\) −25.3691 −1.10931 −0.554657 0.832079i \(-0.687150\pi\)
−0.554657 + 0.832079i \(0.687150\pi\)
\(524\) 69.9419 3.05542
\(525\) 69.7654 3.04481
\(526\) −23.5490 −1.02678
\(527\) 6.41486 0.279436
\(528\) −13.4761 −0.586473
\(529\) 21.2844 0.925407
\(530\) −47.0181 −2.04233
\(531\) −13.9261 −0.604342
\(532\) 53.0431 2.29971
\(533\) 1.29749 0.0562006
\(534\) −82.6510 −3.57666
\(535\) −55.6661 −2.40666
\(536\) −37.9437 −1.63892
\(537\) −3.19283 −0.137781
\(538\) 58.3579 2.51599
\(539\) −3.01848 −0.130015
\(540\) 242.910 10.4532
\(541\) −2.94516 −0.126622 −0.0633112 0.997994i \(-0.520166\pi\)
−0.0633112 + 0.997994i \(0.520166\pi\)
\(542\) −18.2456 −0.783716
\(543\) −54.4744 −2.33772
\(544\) 100.197 4.29592
\(545\) −14.7896 −0.633517
\(546\) −37.9233 −1.62297
\(547\) −39.4876 −1.68837 −0.844185 0.536053i \(-0.819915\pi\)
−0.844185 + 0.536053i \(0.819915\pi\)
\(548\) −102.502 −4.37869
\(549\) 70.3007 3.00036
\(550\) 3.67193 0.156572
\(551\) 19.3280 0.823400
\(552\) 210.151 8.94462
\(553\) 45.3490 1.92843
\(554\) 18.9819 0.806464
\(555\) 70.7286 3.00226
\(556\) −89.0858 −3.77808
\(557\) 5.88433 0.249327 0.124664 0.992199i \(-0.460215\pi\)
0.124664 + 0.992199i \(0.460215\pi\)
\(558\) 31.0874 1.31603
\(559\) −7.49702 −0.317090
\(560\) 217.482 9.19027
\(561\) 3.49320 0.147483
\(562\) 84.5290 3.56564
\(563\) 18.8957 0.796358 0.398179 0.917308i \(-0.369642\pi\)
0.398179 + 0.917308i \(0.369642\pi\)
\(564\) −113.896 −4.79590
\(565\) 6.46643 0.272045
\(566\) −65.7231 −2.76255
\(567\) 95.2665 4.00081
\(568\) 9.70053 0.407025
\(569\) 7.76020 0.325324 0.162662 0.986682i \(-0.447992\pi\)
0.162662 + 0.986682i \(0.447992\pi\)
\(570\) −61.9783 −2.59598
\(571\) −10.0504 −0.420597 −0.210299 0.977637i \(-0.567444\pi\)
−0.210299 + 0.977637i \(0.567444\pi\)
\(572\) −1.46919 −0.0614301
\(573\) 87.0546 3.63676
\(574\) −15.3457 −0.640517
\(575\) −33.7000 −1.40539
\(576\) 253.211 10.5505
\(577\) −21.6345 −0.900656 −0.450328 0.892863i \(-0.648693\pi\)
−0.450328 + 0.892863i \(0.648693\pi\)
\(578\) −0.293368 −0.0122025
\(579\) −77.0605 −3.20253
\(580\) 154.518 6.41600
\(581\) −62.5180 −2.59368
\(582\) 81.8152 3.39135
\(583\) −1.41816 −0.0587341
\(584\) 140.624 5.81906
\(585\) 23.0991 0.955032
\(586\) −66.7476 −2.75732
\(587\) −22.8695 −0.943927 −0.471963 0.881618i \(-0.656455\pi\)
−0.471963 + 0.881618i \(0.656455\pi\)
\(588\) 204.953 8.45212
\(589\) −3.43291 −0.141451
\(590\) 16.7025 0.687630
\(591\) −52.8813 −2.17525
\(592\) 110.945 4.55980
\(593\) 42.5958 1.74920 0.874600 0.484845i \(-0.161124\pi\)
0.874600 + 0.484845i \(0.161124\pi\)
\(594\) 9.95374 0.408407
\(595\) −56.3743 −2.31112
\(596\) −89.9834 −3.68586
\(597\) 59.8166 2.44813
\(598\) 18.3188 0.749109
\(599\) −28.6069 −1.16885 −0.584423 0.811449i \(-0.698679\pi\)
−0.584423 + 0.811449i \(0.698679\pi\)
\(600\) −159.923 −6.52884
\(601\) −0.863777 −0.0352342 −0.0176171 0.999845i \(-0.505608\pi\)
−0.0176171 + 0.999845i \(0.505608\pi\)
\(602\) 88.6688 3.61387
\(603\) 28.0520 1.14237
\(604\) −17.2149 −0.700465
\(605\) −34.6763 −1.40979
\(606\) 34.1799 1.38846
\(607\) 34.2041 1.38830 0.694152 0.719829i \(-0.255780\pi\)
0.694152 + 0.719829i \(0.255780\pi\)
\(608\) −53.6205 −2.17460
\(609\) 120.300 4.87479
\(610\) −84.3162 −3.41386
\(611\) −6.36833 −0.257635
\(612\) −167.977 −6.79007
\(613\) 2.10108 0.0848619 0.0424309 0.999099i \(-0.486490\pi\)
0.0424309 + 0.999099i \(0.486490\pi\)
\(614\) −70.1656 −2.83166
\(615\) 13.1982 0.532205
\(616\) 11.1458 0.449079
\(617\) −1.00000 −0.0402585
\(618\) −56.0076 −2.25295
\(619\) 41.1047 1.65214 0.826069 0.563569i \(-0.190572\pi\)
0.826069 + 0.563569i \(0.190572\pi\)
\(620\) −27.4444 −1.10219
\(621\) −91.3529 −3.66587
\(622\) −63.5587 −2.54847
\(623\) 40.2314 1.61184
\(624\) 51.1619 2.04811
\(625\) −24.6752 −0.987009
\(626\) −10.8718 −0.434525
\(627\) −1.86939 −0.0746561
\(628\) −7.72522 −0.308270
\(629\) −28.7584 −1.14667
\(630\) −273.198 −10.8845
\(631\) −23.4955 −0.935341 −0.467671 0.883903i \(-0.654907\pi\)
−0.467671 + 0.883903i \(0.654907\pi\)
\(632\) −103.953 −4.13504
\(633\) 36.6727 1.45761
\(634\) 53.4146 2.12137
\(635\) −9.52821 −0.378115
\(636\) 96.2920 3.81822
\(637\) 11.4596 0.454047
\(638\) 6.33168 0.250673
\(639\) −7.17167 −0.283707
\(640\) −149.986 −5.92871
\(641\) 30.7230 1.21349 0.606743 0.794898i \(-0.292476\pi\)
0.606743 + 0.794898i \(0.292476\pi\)
\(642\) 154.881 6.11265
\(643\) −30.3677 −1.19759 −0.598793 0.800904i \(-0.704353\pi\)
−0.598793 + 0.800904i \(0.704353\pi\)
\(644\) −159.477 −6.28426
\(645\) −76.2606 −3.00276
\(646\) 25.2005 0.991501
\(647\) 20.5713 0.808741 0.404370 0.914595i \(-0.367491\pi\)
0.404370 + 0.914595i \(0.367491\pi\)
\(648\) −218.379 −8.57875
\(649\) 0.503780 0.0197751
\(650\) −13.9404 −0.546788
\(651\) −21.3669 −0.837433
\(652\) 3.82142 0.149658
\(653\) −39.5346 −1.54711 −0.773554 0.633731i \(-0.781523\pi\)
−0.773554 + 0.633731i \(0.781523\pi\)
\(654\) 41.1493 1.60907
\(655\) 39.7800 1.55433
\(656\) 20.7027 0.808305
\(657\) −103.964 −4.05604
\(658\) 75.3196 2.93626
\(659\) −14.3977 −0.560853 −0.280427 0.959875i \(-0.590476\pi\)
−0.280427 + 0.959875i \(0.590476\pi\)
\(660\) −14.9448 −0.581726
\(661\) −34.7869 −1.35305 −0.676526 0.736418i \(-0.736516\pi\)
−0.676526 + 0.736418i \(0.736516\pi\)
\(662\) −4.85549 −0.188714
\(663\) −13.2619 −0.515049
\(664\) 143.310 5.56150
\(665\) 30.1687 1.16989
\(666\) −139.368 −5.40039
\(667\) −58.1105 −2.25005
\(668\) −67.6139 −2.61606
\(669\) 12.2080 0.471987
\(670\) −33.6446 −1.29980
\(671\) −2.54314 −0.0981769
\(672\) −333.741 −12.8743
\(673\) −28.7714 −1.10906 −0.554528 0.832165i \(-0.687101\pi\)
−0.554528 + 0.832165i \(0.687101\pi\)
\(674\) 90.9561 3.50350
\(675\) 69.5188 2.67578
\(676\) 5.57777 0.214530
\(677\) 10.2175 0.392690 0.196345 0.980535i \(-0.437093\pi\)
0.196345 + 0.980535i \(0.437093\pi\)
\(678\) −17.9916 −0.690964
\(679\) −39.8246 −1.52833
\(680\) 129.227 4.95562
\(681\) 8.34882 0.319927
\(682\) −1.12459 −0.0430628
\(683\) −31.1954 −1.19366 −0.596830 0.802368i \(-0.703573\pi\)
−0.596830 + 0.802368i \(0.703573\pi\)
\(684\) 89.8929 3.43714
\(685\) −58.2991 −2.22749
\(686\) −52.7449 −2.01381
\(687\) 90.1753 3.44040
\(688\) −119.622 −4.56055
\(689\) 5.38401 0.205114
\(690\) 186.341 7.09387
\(691\) −10.7091 −0.407392 −0.203696 0.979034i \(-0.565295\pi\)
−0.203696 + 0.979034i \(0.565295\pi\)
\(692\) 81.3080 3.09087
\(693\) −8.24020 −0.313019
\(694\) 58.7635 2.23063
\(695\) −50.6683 −1.92196
\(696\) −275.763 −10.4528
\(697\) −5.36644 −0.203268
\(698\) 85.7487 3.24564
\(699\) −21.6673 −0.819534
\(700\) 121.360 4.58699
\(701\) 0.411144 0.0155287 0.00776435 0.999970i \(-0.497529\pi\)
0.00776435 + 0.999970i \(0.497529\pi\)
\(702\) −37.7892 −1.42626
\(703\) 15.3901 0.580447
\(704\) −9.15996 −0.345229
\(705\) −64.7795 −2.43974
\(706\) −63.5393 −2.39133
\(707\) −16.6375 −0.625718
\(708\) −34.2063 −1.28555
\(709\) 13.3940 0.503021 0.251510 0.967855i \(-0.419073\pi\)
0.251510 + 0.967855i \(0.419073\pi\)
\(710\) 8.60144 0.322806
\(711\) 76.8535 2.88223
\(712\) −92.2224 −3.45618
\(713\) 10.3212 0.386532
\(714\) 156.851 5.87000
\(715\) −0.835616 −0.0312503
\(716\) −5.55408 −0.207566
\(717\) 54.7389 2.04426
\(718\) −0.101839 −0.00380059
\(719\) −11.2494 −0.419532 −0.209766 0.977752i \(-0.567270\pi\)
−0.209766 + 0.977752i \(0.567270\pi\)
\(720\) 368.569 13.7358
\(721\) 27.2624 1.01530
\(722\) 38.8167 1.44461
\(723\) −28.2658 −1.05122
\(724\) −94.7610 −3.52176
\(725\) 44.2216 1.64235
\(726\) 96.4804 3.58072
\(727\) 40.5949 1.50558 0.752790 0.658260i \(-0.228707\pi\)
0.752790 + 0.658260i \(0.228707\pi\)
\(728\) −42.3150 −1.56830
\(729\) 29.3982 1.08882
\(730\) 124.691 4.61502
\(731\) 31.0077 1.14686
\(732\) 172.678 6.38235
\(733\) 26.3672 0.973893 0.486947 0.873432i \(-0.338111\pi\)
0.486947 + 0.873432i \(0.338111\pi\)
\(734\) 70.7543 2.61159
\(735\) 116.569 4.29970
\(736\) 161.213 5.94238
\(737\) −1.01479 −0.0373802
\(738\) −26.0066 −0.957315
\(739\) −12.2731 −0.451474 −0.225737 0.974188i \(-0.572479\pi\)
−0.225737 + 0.974188i \(0.572479\pi\)
\(740\) 123.036 4.52289
\(741\) 7.09710 0.260718
\(742\) −63.6778 −2.33769
\(743\) 21.9240 0.804312 0.402156 0.915571i \(-0.368261\pi\)
0.402156 + 0.915571i \(0.368261\pi\)
\(744\) 48.9792 1.79567
\(745\) −51.1788 −1.87505
\(746\) −53.6097 −1.96279
\(747\) −105.950 −3.87651
\(748\) 6.07660 0.222182
\(749\) −75.3901 −2.75469
\(750\) −1.79579 −0.0655728
\(751\) 35.5762 1.29819 0.649096 0.760706i \(-0.275147\pi\)
0.649096 + 0.760706i \(0.275147\pi\)
\(752\) −101.613 −3.70544
\(753\) 52.4475 1.91129
\(754\) −24.0381 −0.875416
\(755\) −9.79112 −0.356335
\(756\) 328.980 11.9649
\(757\) −7.87619 −0.286265 −0.143132 0.989704i \(-0.545717\pi\)
−0.143132 + 0.989704i \(0.545717\pi\)
\(758\) −70.8029 −2.57168
\(759\) 5.62040 0.204008
\(760\) −69.1557 −2.50854
\(761\) 8.79464 0.318806 0.159403 0.987214i \(-0.449043\pi\)
0.159403 + 0.987214i \(0.449043\pi\)
\(762\) 26.5105 0.960373
\(763\) −20.0299 −0.725133
\(764\) 151.436 5.47875
\(765\) −95.5383 −3.45419
\(766\) −43.4821 −1.57107
\(767\) −1.91259 −0.0690596
\(768\) 194.295 7.01103
\(769\) 28.4647 1.02646 0.513232 0.858250i \(-0.328448\pi\)
0.513232 + 0.858250i \(0.328448\pi\)
\(770\) 9.88300 0.356159
\(771\) 55.9160 2.01377
\(772\) −134.051 −4.82459
\(773\) −29.9567 −1.07747 −0.538733 0.842476i \(-0.681097\pi\)
−0.538733 + 0.842476i \(0.681097\pi\)
\(774\) 150.268 5.40128
\(775\) −7.85435 −0.282137
\(776\) 91.2898 3.27711
\(777\) 95.7896 3.43643
\(778\) −100.177 −3.59153
\(779\) 2.87185 0.102895
\(780\) 56.7378 2.03154
\(781\) 0.259436 0.00928337
\(782\) −75.7665 −2.70941
\(783\) 119.875 4.28397
\(784\) 182.849 6.53033
\(785\) −4.39378 −0.156821
\(786\) −110.680 −3.94784
\(787\) 45.6797 1.62831 0.814153 0.580651i \(-0.197202\pi\)
0.814153 + 0.580651i \(0.197202\pi\)
\(788\) −91.9897 −3.27700
\(789\) 27.4299 0.976532
\(790\) −92.1753 −3.27945
\(791\) 8.75765 0.311386
\(792\) 18.8890 0.671191
\(793\) 9.65500 0.342859
\(794\) −29.0892 −1.03234
\(795\) 54.7668 1.94238
\(796\) 104.054 3.68810
\(797\) −8.20775 −0.290733 −0.145367 0.989378i \(-0.546436\pi\)
−0.145367 + 0.989378i \(0.546436\pi\)
\(798\) −83.9388 −2.97140
\(799\) 26.3395 0.931824
\(800\) −122.682 −4.33745
\(801\) 68.1807 2.40905
\(802\) 74.4426 2.62866
\(803\) 3.76093 0.132720
\(804\) 68.9034 2.43004
\(805\) −90.7037 −3.19688
\(806\) 4.26949 0.150386
\(807\) −67.9755 −2.39285
\(808\) 38.1381 1.34169
\(809\) −45.9737 −1.61635 −0.808175 0.588943i \(-0.799544\pi\)
−0.808175 + 0.588943i \(0.799544\pi\)
\(810\) −193.637 −6.80369
\(811\) 3.55060 0.124678 0.0623392 0.998055i \(-0.480144\pi\)
0.0623392 + 0.998055i \(0.480144\pi\)
\(812\) 209.267 7.34384
\(813\) 21.2526 0.745360
\(814\) 5.04165 0.176710
\(815\) 2.17347 0.0761332
\(816\) −211.606 −7.40769
\(817\) −16.5938 −0.580543
\(818\) −16.6707 −0.582878
\(819\) 31.2838 1.09314
\(820\) 22.9590 0.801763
\(821\) 22.2205 0.775502 0.387751 0.921764i \(-0.373252\pi\)
0.387751 + 0.921764i \(0.373252\pi\)
\(822\) 162.206 5.65760
\(823\) 27.4312 0.956192 0.478096 0.878308i \(-0.341327\pi\)
0.478096 + 0.878308i \(0.341327\pi\)
\(824\) −62.4935 −2.17707
\(825\) −4.27708 −0.148909
\(826\) 22.6206 0.787071
\(827\) −26.4634 −0.920222 −0.460111 0.887861i \(-0.652190\pi\)
−0.460111 + 0.887861i \(0.652190\pi\)
\(828\) −270.267 −9.39244
\(829\) −38.3939 −1.33347 −0.666737 0.745293i \(-0.732310\pi\)
−0.666737 + 0.745293i \(0.732310\pi\)
\(830\) 127.073 4.41076
\(831\) −22.1102 −0.766995
\(832\) 34.7756 1.20563
\(833\) −47.3971 −1.64221
\(834\) 140.975 4.88157
\(835\) −38.4560 −1.33082
\(836\) −3.25189 −0.112469
\(837\) −21.2913 −0.735936
\(838\) −24.4932 −0.846105
\(839\) −28.9443 −0.999270 −0.499635 0.866236i \(-0.666532\pi\)
−0.499635 + 0.866236i \(0.666532\pi\)
\(840\) −430.433 −14.8514
\(841\) 47.2534 1.62943
\(842\) −62.1059 −2.14031
\(843\) −98.4597 −3.39113
\(844\) 63.7940 2.19588
\(845\) 3.17240 0.109134
\(846\) 127.645 4.38853
\(847\) −46.9630 −1.61367
\(848\) 85.9071 2.95006
\(849\) 76.5546 2.62735
\(850\) 57.6577 1.97764
\(851\) −46.2710 −1.58615
\(852\) −17.6156 −0.603499
\(853\) 42.5657 1.45742 0.728710 0.684822i \(-0.240120\pi\)
0.728710 + 0.684822i \(0.240120\pi\)
\(854\) −114.192 −3.90756
\(855\) 51.1273 1.74852
\(856\) 172.817 5.90675
\(857\) 54.2756 1.85402 0.927009 0.375039i \(-0.122371\pi\)
0.927009 + 0.375039i \(0.122371\pi\)
\(858\) 2.32495 0.0793723
\(859\) 24.4559 0.834424 0.417212 0.908809i \(-0.363007\pi\)
0.417212 + 0.908809i \(0.363007\pi\)
\(860\) −132.659 −4.52364
\(861\) 17.8747 0.609169
\(862\) 31.5563 1.07481
\(863\) 54.7805 1.86475 0.932375 0.361492i \(-0.117733\pi\)
0.932375 + 0.361492i \(0.117733\pi\)
\(864\) −332.561 −11.3140
\(865\) 46.2446 1.57236
\(866\) −8.79374 −0.298823
\(867\) 0.341717 0.0116053
\(868\) −37.1687 −1.26159
\(869\) −2.78019 −0.0943115
\(870\) −244.519 −8.28996
\(871\) 3.85262 0.130541
\(872\) 45.9146 1.55487
\(873\) −67.4912 −2.28423
\(874\) 40.5465 1.37150
\(875\) 0.874122 0.0295507
\(876\) −255.365 −8.62797
\(877\) −29.0422 −0.980686 −0.490343 0.871529i \(-0.663129\pi\)
−0.490343 + 0.871529i \(0.663129\pi\)
\(878\) 57.9857 1.95692
\(879\) 77.7479 2.62237
\(880\) −13.3330 −0.449457
\(881\) −15.5702 −0.524574 −0.262287 0.964990i \(-0.584477\pi\)
−0.262287 + 0.964990i \(0.584477\pi\)
\(882\) −229.694 −7.73419
\(883\) −1.48410 −0.0499440 −0.0249720 0.999688i \(-0.507950\pi\)
−0.0249720 + 0.999688i \(0.507950\pi\)
\(884\) −23.0697 −0.775918
\(885\) −19.4551 −0.653977
\(886\) −22.0421 −0.740520
\(887\) 42.4750 1.42617 0.713085 0.701078i \(-0.247297\pi\)
0.713085 + 0.701078i \(0.247297\pi\)
\(888\) −219.579 −7.36857
\(889\) −12.9043 −0.432796
\(890\) −81.7735 −2.74105
\(891\) −5.84046 −0.195663
\(892\) 21.2363 0.711045
\(893\) −14.0956 −0.471690
\(894\) 142.395 4.76242
\(895\) −3.15893 −0.105591
\(896\) −203.130 −6.78609
\(897\) −21.3378 −0.712447
\(898\) 103.676 3.45971
\(899\) −13.5436 −0.451705
\(900\) 205.671 6.85570
\(901\) −22.2683 −0.741865
\(902\) 0.940792 0.0313250
\(903\) −103.282 −3.43700
\(904\) −20.0752 −0.667690
\(905\) −53.8961 −1.79157
\(906\) 27.2420 0.905054
\(907\) 27.1168 0.900398 0.450199 0.892928i \(-0.351353\pi\)
0.450199 + 0.892928i \(0.351353\pi\)
\(908\) 14.5232 0.481969
\(909\) −28.1958 −0.935196
\(910\) −37.5206 −1.24380
\(911\) −22.1250 −0.733033 −0.366516 0.930412i \(-0.619450\pi\)
−0.366516 + 0.930412i \(0.619450\pi\)
\(912\) 113.241 3.74978
\(913\) 3.83276 0.126846
\(914\) −84.9386 −2.80952
\(915\) 98.2118 3.24678
\(916\) 156.864 5.18295
\(917\) 53.8751 1.77911
\(918\) 156.297 5.15856
\(919\) 16.3792 0.540300 0.270150 0.962818i \(-0.412927\pi\)
0.270150 + 0.962818i \(0.412927\pi\)
\(920\) 207.920 6.85492
\(921\) 81.7292 2.69307
\(922\) −53.6356 −1.76640
\(923\) −0.984946 −0.0324199
\(924\) −20.2402 −0.665852
\(925\) 35.2118 1.15776
\(926\) −29.9569 −0.984446
\(927\) 46.2019 1.51747
\(928\) −211.545 −6.94432
\(929\) 23.3057 0.764635 0.382318 0.924031i \(-0.375126\pi\)
0.382318 + 0.924031i \(0.375126\pi\)
\(930\) 43.4298 1.42412
\(931\) 25.3646 0.831290
\(932\) −37.6914 −1.23462
\(933\) 74.0334 2.42374
\(934\) −69.1192 −2.26165
\(935\) 3.45611 0.113027
\(936\) −71.7118 −2.34397
\(937\) 31.6897 1.03526 0.517629 0.855605i \(-0.326815\pi\)
0.517629 + 0.855605i \(0.326815\pi\)
\(938\) −45.5658 −1.48778
\(939\) 12.6635 0.413259
\(940\) −112.687 −3.67545
\(941\) −36.4720 −1.18895 −0.594477 0.804113i \(-0.702641\pi\)
−0.594477 + 0.804113i \(0.702641\pi\)
\(942\) 12.2249 0.398308
\(943\) −8.63435 −0.281173
\(944\) −30.5172 −0.993251
\(945\) 187.110 6.08669
\(946\) −5.43598 −0.176739
\(947\) 27.1913 0.883598 0.441799 0.897114i \(-0.354340\pi\)
0.441799 + 0.897114i \(0.354340\pi\)
\(948\) 188.773 6.13106
\(949\) −14.2783 −0.463493
\(950\) −30.8555 −1.00108
\(951\) −62.2176 −2.01754
\(952\) 175.015 5.67227
\(953\) −43.1808 −1.39876 −0.699381 0.714749i \(-0.746541\pi\)
−0.699381 + 0.714749i \(0.746541\pi\)
\(954\) −107.916 −3.49390
\(955\) 86.1303 2.78711
\(956\) 95.2210 3.07967
\(957\) −7.37516 −0.238405
\(958\) 46.8906 1.51497
\(959\) −78.9560 −2.54962
\(960\) 353.742 11.4170
\(961\) −28.5945 −0.922402
\(962\) −19.1405 −0.617116
\(963\) −127.765 −4.11716
\(964\) −49.1697 −1.58365
\(965\) −76.2424 −2.45433
\(966\) 252.366 8.11974
\(967\) −13.3760 −0.430142 −0.215071 0.976598i \(-0.568998\pi\)
−0.215071 + 0.976598i \(0.568998\pi\)
\(968\) 107.653 3.46011
\(969\) −29.3537 −0.942975
\(970\) 80.9466 2.59904
\(971\) 8.24212 0.264502 0.132251 0.991216i \(-0.457779\pi\)
0.132251 + 0.991216i \(0.457779\pi\)
\(972\) 166.854 5.35184
\(973\) −68.6214 −2.19990
\(974\) 96.3264 3.08650
\(975\) 16.2379 0.520027
\(976\) 154.055 4.93117
\(977\) −2.11960 −0.0678120 −0.0339060 0.999425i \(-0.510795\pi\)
−0.0339060 + 0.999425i \(0.510795\pi\)
\(978\) −6.04726 −0.193370
\(979\) −2.46645 −0.0788281
\(980\) 202.777 6.47748
\(981\) −33.9450 −1.08378
\(982\) 62.8034 2.00414
\(983\) 25.0743 0.799747 0.399874 0.916570i \(-0.369054\pi\)
0.399874 + 0.916570i \(0.369054\pi\)
\(984\) −40.9742 −1.30621
\(985\) −52.3199 −1.66705
\(986\) 99.4218 3.16624
\(987\) −87.7325 −2.79256
\(988\) 12.3458 0.392771
\(989\) 49.8900 1.58641
\(990\) 16.7489 0.532313
\(991\) −26.7446 −0.849572 −0.424786 0.905294i \(-0.639651\pi\)
−0.424786 + 0.905294i \(0.639651\pi\)
\(992\) 37.5733 1.19295
\(993\) 5.65570 0.179478
\(994\) 11.6492 0.369489
\(995\) 59.1816 1.87618
\(996\) −260.242 −8.24608
\(997\) 7.40486 0.234514 0.117257 0.993102i \(-0.462590\pi\)
0.117257 + 0.993102i \(0.462590\pi\)
\(998\) 16.8517 0.533432
\(999\) 95.4511 3.01994
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 8021.2.a.d.1.3 174
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8021.2.a.d.1.3 174 1.1 even 1 trivial