Properties

Label 729.2.a.a.1.5
Level $729$
Weight $2$
Character 729.1
Self dual yes
Analytic conductor $5.821$
Analytic rank $1$
Dimension $6$
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] = [729,2,Mod(1,729)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("729.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.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: \(5.82109430735\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1397493.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.40162\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.05432 q^{2} -0.888399 q^{4} -1.74579 q^{5} +2.45925 q^{7} -3.04531 q^{8} +O(q^{10})\) \(q+1.05432 q^{2} -0.888399 q^{4} -1.74579 q^{5} +2.45925 q^{7} -3.04531 q^{8} -1.84063 q^{10} -1.25421 q^{11} -4.54903 q^{13} +2.59284 q^{14} -1.43395 q^{16} -6.64717 q^{17} +0.249156 q^{19} +1.55096 q^{20} -1.32235 q^{22} +0.842001 q^{23} -1.95223 q^{25} -4.79615 q^{26} -2.18479 q^{28} -0.512383 q^{29} -0.820004 q^{31} +4.57877 q^{32} -7.00828 q^{34} -4.29332 q^{35} +2.60806 q^{37} +0.262692 q^{38} +5.31646 q^{40} -8.15281 q^{41} +4.32714 q^{43} +1.11424 q^{44} +0.887743 q^{46} -5.30233 q^{47} -0.952106 q^{49} -2.05828 q^{50} +4.04135 q^{52} -10.4841 q^{53} +2.18959 q^{55} -7.48917 q^{56} -0.540218 q^{58} -3.00620 q^{59} +2.88317 q^{61} -0.864550 q^{62} +7.69541 q^{64} +7.94164 q^{65} +10.0863 q^{67} +5.90534 q^{68} -4.52655 q^{70} +0.0894756 q^{71} -5.32114 q^{73} +2.74974 q^{74} -0.221350 q^{76} -3.08442 q^{77} +4.77692 q^{79} +2.50337 q^{80} -8.59571 q^{82} +8.04066 q^{83} +11.6045 q^{85} +4.56222 q^{86} +3.81947 q^{88} +6.70377 q^{89} -11.1872 q^{91} -0.748033 q^{92} -5.59038 q^{94} -0.434974 q^{95} +5.49058 q^{97} -1.00383 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8} + 3 q^{10} - 12 q^{11} - 6 q^{14} - 3 q^{16} - 9 q^{17} + 3 q^{19} - 6 q^{20} + 6 q^{22} - 15 q^{23} - 6 q^{25} - 15 q^{26} - 6 q^{28} - 12 q^{29} - 12 q^{35} + 3 q^{37} + 3 q^{38} + 6 q^{40} - 15 q^{41} - 3 q^{44} + 3 q^{46} - 21 q^{47} - 12 q^{49} - 3 q^{50} + 12 q^{52} - 9 q^{53} - 6 q^{55} + 6 q^{56} - 12 q^{58} - 24 q^{59} - 9 q^{61} + 12 q^{62} - 12 q^{64} + 6 q^{65} - 9 q^{67} + 9 q^{68} + 15 q^{70} - 27 q^{71} - 6 q^{73} + 12 q^{74} + 6 q^{76} + 12 q^{77} + 21 q^{80} - 6 q^{82} - 12 q^{83} + 21 q^{86} + 12 q^{88} - 9 q^{89} - 6 q^{91} - 6 q^{92} + 6 q^{94} - 12 q^{95} + 45 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.05432 0.745520 0.372760 0.927928i \(-0.378411\pi\)
0.372760 + 0.927928i \(0.378411\pi\)
\(3\) 0 0
\(4\) −0.888399 −0.444200
\(5\) −1.74579 −0.780740 −0.390370 0.920658i \(-0.627653\pi\)
−0.390370 + 0.920658i \(0.627653\pi\)
\(6\) 0 0
\(7\) 2.45925 0.929508 0.464754 0.885440i \(-0.346143\pi\)
0.464754 + 0.885440i \(0.346143\pi\)
\(8\) −3.04531 −1.07668
\(9\) 0 0
\(10\) −1.84063 −0.582057
\(11\) −1.25421 −0.378159 −0.189080 0.981962i \(-0.560550\pi\)
−0.189080 + 0.981962i \(0.560550\pi\)
\(12\) 0 0
\(13\) −4.54903 −1.26167 −0.630837 0.775915i \(-0.717288\pi\)
−0.630837 + 0.775915i \(0.717288\pi\)
\(14\) 2.59284 0.692967
\(15\) 0 0
\(16\) −1.43395 −0.358487
\(17\) −6.64717 −1.61218 −0.806088 0.591796i \(-0.798419\pi\)
−0.806088 + 0.591796i \(0.798419\pi\)
\(18\) 0 0
\(19\) 0.249156 0.0571604 0.0285802 0.999592i \(-0.490901\pi\)
0.0285802 + 0.999592i \(0.490901\pi\)
\(20\) 1.55096 0.346804
\(21\) 0 0
\(22\) −1.32235 −0.281926
\(23\) 0.842001 0.175569 0.0877847 0.996139i \(-0.472021\pi\)
0.0877847 + 0.996139i \(0.472021\pi\)
\(24\) 0 0
\(25\) −1.95223 −0.390446
\(26\) −4.79615 −0.940603
\(27\) 0 0
\(28\) −2.18479 −0.412887
\(29\) −0.512383 −0.0951471 −0.0475736 0.998868i \(-0.515149\pi\)
−0.0475736 + 0.998868i \(0.515149\pi\)
\(30\) 0 0
\(31\) −0.820004 −0.147277 −0.0736385 0.997285i \(-0.523461\pi\)
−0.0736385 + 0.997285i \(0.523461\pi\)
\(32\) 4.57877 0.809421
\(33\) 0 0
\(34\) −7.00828 −1.20191
\(35\) −4.29332 −0.725704
\(36\) 0 0
\(37\) 2.60806 0.428763 0.214381 0.976750i \(-0.431226\pi\)
0.214381 + 0.976750i \(0.431226\pi\)
\(38\) 0.262692 0.0426142
\(39\) 0 0
\(40\) 5.31646 0.840607
\(41\) −8.15281 −1.27326 −0.636628 0.771171i \(-0.719671\pi\)
−0.636628 + 0.771171i \(0.719671\pi\)
\(42\) 0 0
\(43\) 4.32714 0.659883 0.329942 0.944001i \(-0.392971\pi\)
0.329942 + 0.944001i \(0.392971\pi\)
\(44\) 1.11424 0.167978
\(45\) 0 0
\(46\) 0.887743 0.130890
\(47\) −5.30233 −0.773425 −0.386713 0.922200i \(-0.626389\pi\)
−0.386713 + 0.922200i \(0.626389\pi\)
\(48\) 0 0
\(49\) −0.952106 −0.136015
\(50\) −2.05828 −0.291085
\(51\) 0 0
\(52\) 4.04135 0.560435
\(53\) −10.4841 −1.44010 −0.720052 0.693920i \(-0.755882\pi\)
−0.720052 + 0.693920i \(0.755882\pi\)
\(54\) 0 0
\(55\) 2.18959 0.295244
\(56\) −7.48917 −1.00078
\(57\) 0 0
\(58\) −0.540218 −0.0709341
\(59\) −3.00620 −0.391374 −0.195687 0.980666i \(-0.562694\pi\)
−0.195687 + 0.980666i \(0.562694\pi\)
\(60\) 0 0
\(61\) 2.88317 0.369152 0.184576 0.982818i \(-0.440909\pi\)
0.184576 + 0.982818i \(0.440909\pi\)
\(62\) −0.864550 −0.109798
\(63\) 0 0
\(64\) 7.69541 0.961927
\(65\) 7.94164 0.985039
\(66\) 0 0
\(67\) 10.0863 1.23223 0.616117 0.787655i \(-0.288705\pi\)
0.616117 + 0.787655i \(0.288705\pi\)
\(68\) 5.90534 0.716128
\(69\) 0 0
\(70\) −4.52655 −0.541027
\(71\) 0.0894756 0.0106188 0.00530940 0.999986i \(-0.498310\pi\)
0.00530940 + 0.999986i \(0.498310\pi\)
\(72\) 0 0
\(73\) −5.32114 −0.622792 −0.311396 0.950280i \(-0.600797\pi\)
−0.311396 + 0.950280i \(0.600797\pi\)
\(74\) 2.74974 0.319651
\(75\) 0 0
\(76\) −0.221350 −0.0253906
\(77\) −3.08442 −0.351502
\(78\) 0 0
\(79\) 4.77692 0.537445 0.268723 0.963218i \(-0.413398\pi\)
0.268723 + 0.963218i \(0.413398\pi\)
\(80\) 2.50337 0.279885
\(81\) 0 0
\(82\) −8.59571 −0.949238
\(83\) 8.04066 0.882577 0.441289 0.897365i \(-0.354522\pi\)
0.441289 + 0.897365i \(0.354522\pi\)
\(84\) 0 0
\(85\) 11.6045 1.25869
\(86\) 4.56222 0.491956
\(87\) 0 0
\(88\) 3.81947 0.407157
\(89\) 6.70377 0.710598 0.355299 0.934753i \(-0.384379\pi\)
0.355299 + 0.934753i \(0.384379\pi\)
\(90\) 0 0
\(91\) −11.1872 −1.17274
\(92\) −0.748033 −0.0779878
\(93\) 0 0
\(94\) −5.59038 −0.576604
\(95\) −0.434974 −0.0446274
\(96\) 0 0
\(97\) 5.49058 0.557484 0.278742 0.960366i \(-0.410083\pi\)
0.278742 + 0.960366i \(0.410083\pi\)
\(98\) −1.00383 −0.101402
\(99\) 0 0
\(100\) 1.73436 0.173436
\(101\) 5.00546 0.498062 0.249031 0.968495i \(-0.419888\pi\)
0.249031 + 0.968495i \(0.419888\pi\)
\(102\) 0 0
\(103\) 11.6192 1.14487 0.572435 0.819950i \(-0.305999\pi\)
0.572435 + 0.819950i \(0.305999\pi\)
\(104\) 13.8532 1.35842
\(105\) 0 0
\(106\) −11.0537 −1.07363
\(107\) 19.4581 1.88109 0.940544 0.339673i \(-0.110316\pi\)
0.940544 + 0.339673i \(0.110316\pi\)
\(108\) 0 0
\(109\) 6.31515 0.604881 0.302441 0.953168i \(-0.402199\pi\)
0.302441 + 0.953168i \(0.402199\pi\)
\(110\) 2.30854 0.220110
\(111\) 0 0
\(112\) −3.52643 −0.333217
\(113\) −6.91572 −0.650577 −0.325288 0.945615i \(-0.605461\pi\)
−0.325288 + 0.945615i \(0.605461\pi\)
\(114\) 0 0
\(115\) −1.46995 −0.137074
\(116\) 0.455201 0.0422643
\(117\) 0 0
\(118\) −3.16951 −0.291777
\(119\) −16.3470 −1.49853
\(120\) 0 0
\(121\) −9.42695 −0.856995
\(122\) 3.03980 0.275211
\(123\) 0 0
\(124\) 0.728491 0.0654204
\(125\) 12.1371 1.08558
\(126\) 0 0
\(127\) 12.0232 1.06689 0.533445 0.845835i \(-0.320897\pi\)
0.533445 + 0.845835i \(0.320897\pi\)
\(128\) −1.04408 −0.0922848
\(129\) 0 0
\(130\) 8.37306 0.734366
\(131\) −14.0848 −1.23059 −0.615297 0.788295i \(-0.710964\pi\)
−0.615297 + 0.788295i \(0.710964\pi\)
\(132\) 0 0
\(133\) 0.612737 0.0531310
\(134\) 10.6342 0.918655
\(135\) 0 0
\(136\) 20.2427 1.73580
\(137\) 2.25814 0.192926 0.0964630 0.995337i \(-0.469247\pi\)
0.0964630 + 0.995337i \(0.469247\pi\)
\(138\) 0 0
\(139\) −7.97509 −0.676439 −0.338219 0.941067i \(-0.609825\pi\)
−0.338219 + 0.941067i \(0.609825\pi\)
\(140\) 3.81418 0.322357
\(141\) 0 0
\(142\) 0.0943364 0.00791653
\(143\) 5.70545 0.477114
\(144\) 0 0
\(145\) 0.894512 0.0742851
\(146\) −5.61021 −0.464304
\(147\) 0 0
\(148\) −2.31700 −0.190456
\(149\) −0.106938 −0.00876074 −0.00438037 0.999990i \(-0.501394\pi\)
−0.00438037 + 0.999990i \(0.501394\pi\)
\(150\) 0 0
\(151\) −20.2594 −1.64869 −0.824344 0.566090i \(-0.808456\pi\)
−0.824344 + 0.566090i \(0.808456\pi\)
\(152\) −0.758758 −0.0615434
\(153\) 0 0
\(154\) −3.25198 −0.262052
\(155\) 1.43155 0.114985
\(156\) 0 0
\(157\) −20.7498 −1.65602 −0.828009 0.560715i \(-0.810526\pi\)
−0.828009 + 0.560715i \(0.810526\pi\)
\(158\) 5.03642 0.400676
\(159\) 0 0
\(160\) −7.99356 −0.631947
\(161\) 2.07069 0.163193
\(162\) 0 0
\(163\) −20.1346 −1.57706 −0.788531 0.614995i \(-0.789158\pi\)
−0.788531 + 0.614995i \(0.789158\pi\)
\(164\) 7.24295 0.565580
\(165\) 0 0
\(166\) 8.47747 0.657979
\(167\) −19.8626 −1.53701 −0.768507 0.639841i \(-0.779000\pi\)
−0.768507 + 0.639841i \(0.779000\pi\)
\(168\) 0 0
\(169\) 7.69367 0.591821
\(170\) 12.2350 0.938378
\(171\) 0 0
\(172\) −3.84423 −0.293120
\(173\) 18.9251 1.43885 0.719425 0.694570i \(-0.244405\pi\)
0.719425 + 0.694570i \(0.244405\pi\)
\(174\) 0 0
\(175\) −4.80101 −0.362922
\(176\) 1.79848 0.135565
\(177\) 0 0
\(178\) 7.06795 0.529765
\(179\) 10.9137 0.815725 0.407863 0.913043i \(-0.366274\pi\)
0.407863 + 0.913043i \(0.366274\pi\)
\(180\) 0 0
\(181\) −17.9479 −1.33405 −0.667027 0.745033i \(-0.732433\pi\)
−0.667027 + 0.745033i \(0.732433\pi\)
\(182\) −11.7949 −0.874298
\(183\) 0 0
\(184\) −2.56415 −0.189032
\(185\) −4.55312 −0.334752
\(186\) 0 0
\(187\) 8.33697 0.609659
\(188\) 4.71059 0.343555
\(189\) 0 0
\(190\) −0.458604 −0.0332706
\(191\) −26.9661 −1.95120 −0.975598 0.219566i \(-0.929536\pi\)
−0.975598 + 0.219566i \(0.929536\pi\)
\(192\) 0 0
\(193\) −17.1548 −1.23483 −0.617415 0.786638i \(-0.711820\pi\)
−0.617415 + 0.786638i \(0.711820\pi\)
\(194\) 5.78885 0.415615
\(195\) 0 0
\(196\) 0.845850 0.0604179
\(197\) 2.51225 0.178990 0.0894951 0.995987i \(-0.471475\pi\)
0.0894951 + 0.995987i \(0.471475\pi\)
\(198\) 0 0
\(199\) 18.5388 1.31418 0.657092 0.753810i \(-0.271786\pi\)
0.657092 + 0.753810i \(0.271786\pi\)
\(200\) 5.94514 0.420385
\(201\) 0 0
\(202\) 5.27739 0.371316
\(203\) −1.26008 −0.0884400
\(204\) 0 0
\(205\) 14.2331 0.994081
\(206\) 12.2504 0.853524
\(207\) 0 0
\(208\) 6.52307 0.452294
\(209\) −0.312495 −0.0216157
\(210\) 0 0
\(211\) −3.69118 −0.254111 −0.127056 0.991896i \(-0.540553\pi\)
−0.127056 + 0.991896i \(0.540553\pi\)
\(212\) 9.31408 0.639693
\(213\) 0 0
\(214\) 20.5152 1.40239
\(215\) −7.55427 −0.515197
\(216\) 0 0
\(217\) −2.01659 −0.136895
\(218\) 6.65822 0.450951
\(219\) 0 0
\(220\) −1.94523 −0.131147
\(221\) 30.2382 2.03404
\(222\) 0 0
\(223\) −21.2410 −1.42241 −0.711203 0.702987i \(-0.751849\pi\)
−0.711203 + 0.702987i \(0.751849\pi\)
\(224\) 11.2603 0.752363
\(225\) 0 0
\(226\) −7.29142 −0.485018
\(227\) −14.3400 −0.951783 −0.475891 0.879504i \(-0.657874\pi\)
−0.475891 + 0.879504i \(0.657874\pi\)
\(228\) 0 0
\(229\) 16.8858 1.11585 0.557923 0.829893i \(-0.311598\pi\)
0.557923 + 0.829893i \(0.311598\pi\)
\(230\) −1.54981 −0.102191
\(231\) 0 0
\(232\) 1.56037 0.102443
\(233\) 5.59945 0.366832 0.183416 0.983035i \(-0.441284\pi\)
0.183416 + 0.983035i \(0.441284\pi\)
\(234\) 0 0
\(235\) 9.25675 0.603844
\(236\) 2.67071 0.173848
\(237\) 0 0
\(238\) −17.2351 −1.11718
\(239\) −5.27427 −0.341164 −0.170582 0.985343i \(-0.554565\pi\)
−0.170582 + 0.985343i \(0.554565\pi\)
\(240\) 0 0
\(241\) −8.90248 −0.573459 −0.286730 0.958012i \(-0.592568\pi\)
−0.286730 + 0.958012i \(0.592568\pi\)
\(242\) −9.93907 −0.638907
\(243\) 0 0
\(244\) −2.56141 −0.163977
\(245\) 1.66217 0.106192
\(246\) 0 0
\(247\) −1.13342 −0.0721177
\(248\) 2.49717 0.158570
\(249\) 0 0
\(250\) 12.7965 0.809319
\(251\) −7.78021 −0.491082 −0.245541 0.969386i \(-0.578966\pi\)
−0.245541 + 0.969386i \(0.578966\pi\)
\(252\) 0 0
\(253\) −1.05605 −0.0663932
\(254\) 12.6764 0.795388
\(255\) 0 0
\(256\) −16.4916 −1.03073
\(257\) −20.4366 −1.27480 −0.637399 0.770534i \(-0.719989\pi\)
−0.637399 + 0.770534i \(0.719989\pi\)
\(258\) 0 0
\(259\) 6.41387 0.398538
\(260\) −7.05534 −0.437554
\(261\) 0 0
\(262\) −14.8500 −0.917433
\(263\) 11.2798 0.695543 0.347771 0.937579i \(-0.386939\pi\)
0.347771 + 0.937579i \(0.386939\pi\)
\(264\) 0 0
\(265\) 18.3030 1.12435
\(266\) 0.646023 0.0396102
\(267\) 0 0
\(268\) −8.96063 −0.547357
\(269\) −0.307761 −0.0187645 −0.00938226 0.999956i \(-0.502987\pi\)
−0.00938226 + 0.999956i \(0.502987\pi\)
\(270\) 0 0
\(271\) −2.22251 −0.135008 −0.0675040 0.997719i \(-0.521504\pi\)
−0.0675040 + 0.997719i \(0.521504\pi\)
\(272\) 9.53170 0.577944
\(273\) 0 0
\(274\) 2.38081 0.143830
\(275\) 2.44851 0.147651
\(276\) 0 0
\(277\) −23.3297 −1.40175 −0.700874 0.713285i \(-0.747206\pi\)
−0.700874 + 0.713285i \(0.747206\pi\)
\(278\) −8.40834 −0.504299
\(279\) 0 0
\(280\) 13.0745 0.781351
\(281\) 7.22546 0.431035 0.215517 0.976500i \(-0.430856\pi\)
0.215517 + 0.976500i \(0.430856\pi\)
\(282\) 0 0
\(283\) 7.12029 0.423257 0.211629 0.977350i \(-0.432123\pi\)
0.211629 + 0.977350i \(0.432123\pi\)
\(284\) −0.0794901 −0.00471687
\(285\) 0 0
\(286\) 6.01540 0.355698
\(287\) −20.0498 −1.18350
\(288\) 0 0
\(289\) 27.1849 1.59911
\(290\) 0.943106 0.0553811
\(291\) 0 0
\(292\) 4.72730 0.276644
\(293\) 0.552485 0.0322765 0.0161383 0.999870i \(-0.494863\pi\)
0.0161383 + 0.999870i \(0.494863\pi\)
\(294\) 0 0
\(295\) 5.24819 0.305561
\(296\) −7.94236 −0.461640
\(297\) 0 0
\(298\) −0.112748 −0.00653131
\(299\) −3.83029 −0.221511
\(300\) 0 0
\(301\) 10.6415 0.613367
\(302\) −21.3600 −1.22913
\(303\) 0 0
\(304\) −0.357277 −0.0204913
\(305\) −5.03340 −0.288212
\(306\) 0 0
\(307\) 6.72876 0.384031 0.192015 0.981392i \(-0.438498\pi\)
0.192015 + 0.981392i \(0.438498\pi\)
\(308\) 2.74020 0.156137
\(309\) 0 0
\(310\) 1.50932 0.0857236
\(311\) 15.4235 0.874584 0.437292 0.899320i \(-0.355938\pi\)
0.437292 + 0.899320i \(0.355938\pi\)
\(312\) 0 0
\(313\) −23.5609 −1.33174 −0.665870 0.746068i \(-0.731939\pi\)
−0.665870 + 0.746068i \(0.731939\pi\)
\(314\) −21.8771 −1.23459
\(315\) 0 0
\(316\) −4.24381 −0.238733
\(317\) 7.25204 0.407315 0.203658 0.979042i \(-0.434717\pi\)
0.203658 + 0.979042i \(0.434717\pi\)
\(318\) 0 0
\(319\) 0.642637 0.0359808
\(320\) −13.4346 −0.751014
\(321\) 0 0
\(322\) 2.18318 0.121664
\(323\) −1.65618 −0.0921525
\(324\) 0 0
\(325\) 8.88074 0.492615
\(326\) −21.2284 −1.17573
\(327\) 0 0
\(328\) 24.8279 1.37089
\(329\) −13.0397 −0.718905
\(330\) 0 0
\(331\) 29.0345 1.59588 0.797939 0.602738i \(-0.205924\pi\)
0.797939 + 0.602738i \(0.205924\pi\)
\(332\) −7.14332 −0.392040
\(333\) 0 0
\(334\) −20.9416 −1.14588
\(335\) −17.6085 −0.962053
\(336\) 0 0
\(337\) −1.15910 −0.0631400 −0.0315700 0.999502i \(-0.510051\pi\)
−0.0315700 + 0.999502i \(0.510051\pi\)
\(338\) 8.11163 0.441214
\(339\) 0 0
\(340\) −10.3095 −0.559109
\(341\) 1.02846 0.0556942
\(342\) 0 0
\(343\) −19.5562 −1.05594
\(344\) −13.1775 −0.710483
\(345\) 0 0
\(346\) 19.9532 1.07269
\(347\) 5.88971 0.316176 0.158088 0.987425i \(-0.449467\pi\)
0.158088 + 0.987425i \(0.449467\pi\)
\(348\) 0 0
\(349\) −30.5927 −1.63759 −0.818795 0.574087i \(-0.805357\pi\)
−0.818795 + 0.574087i \(0.805357\pi\)
\(350\) −5.06182 −0.270566
\(351\) 0 0
\(352\) −5.74276 −0.306090
\(353\) 36.9613 1.96725 0.983625 0.180227i \(-0.0576833\pi\)
0.983625 + 0.180227i \(0.0576833\pi\)
\(354\) 0 0
\(355\) −0.156205 −0.00829052
\(356\) −5.95562 −0.315647
\(357\) 0 0
\(358\) 11.5065 0.608140
\(359\) −26.3761 −1.39207 −0.696037 0.718006i \(-0.745055\pi\)
−0.696037 + 0.718006i \(0.745055\pi\)
\(360\) 0 0
\(361\) −18.9379 −0.996733
\(362\) −18.9229 −0.994564
\(363\) 0 0
\(364\) 9.93869 0.520929
\(365\) 9.28958 0.486239
\(366\) 0 0
\(367\) −11.3131 −0.590541 −0.295270 0.955414i \(-0.595410\pi\)
−0.295270 + 0.955414i \(0.595410\pi\)
\(368\) −1.20739 −0.0629394
\(369\) 0 0
\(370\) −4.80047 −0.249564
\(371\) −25.7830 −1.33859
\(372\) 0 0
\(373\) 5.84408 0.302595 0.151297 0.988488i \(-0.451655\pi\)
0.151297 + 0.988488i \(0.451655\pi\)
\(374\) 8.78987 0.454513
\(375\) 0 0
\(376\) 16.1473 0.832731
\(377\) 2.33085 0.120045
\(378\) 0 0
\(379\) 24.3265 1.24957 0.624783 0.780798i \(-0.285187\pi\)
0.624783 + 0.780798i \(0.285187\pi\)
\(380\) 0.386430 0.0198235
\(381\) 0 0
\(382\) −28.4310 −1.45466
\(383\) −3.81605 −0.194991 −0.0974955 0.995236i \(-0.531083\pi\)
−0.0974955 + 0.995236i \(0.531083\pi\)
\(384\) 0 0
\(385\) 5.38474 0.274432
\(386\) −18.0867 −0.920591
\(387\) 0 0
\(388\) −4.87782 −0.247634
\(389\) 10.8418 0.549704 0.274852 0.961487i \(-0.411371\pi\)
0.274852 + 0.961487i \(0.411371\pi\)
\(390\) 0 0
\(391\) −5.59692 −0.283049
\(392\) 2.89946 0.146445
\(393\) 0 0
\(394\) 2.64872 0.133441
\(395\) −8.33948 −0.419605
\(396\) 0 0
\(397\) −10.5092 −0.527442 −0.263721 0.964599i \(-0.584950\pi\)
−0.263721 + 0.964599i \(0.584950\pi\)
\(398\) 19.5460 0.979751
\(399\) 0 0
\(400\) 2.79939 0.139970
\(401\) 14.3656 0.717383 0.358691 0.933456i \(-0.383223\pi\)
0.358691 + 0.933456i \(0.383223\pi\)
\(402\) 0 0
\(403\) 3.73022 0.185816
\(404\) −4.44685 −0.221239
\(405\) 0 0
\(406\) −1.32853 −0.0659338
\(407\) −3.27106 −0.162141
\(408\) 0 0
\(409\) 17.6823 0.874332 0.437166 0.899381i \(-0.355982\pi\)
0.437166 + 0.899381i \(0.355982\pi\)
\(410\) 15.0063 0.741108
\(411\) 0 0
\(412\) −10.3225 −0.508551
\(413\) −7.39299 −0.363785
\(414\) 0 0
\(415\) −14.0373 −0.689063
\(416\) −20.8290 −1.02122
\(417\) 0 0
\(418\) −0.329471 −0.0161150
\(419\) 9.13376 0.446214 0.223107 0.974794i \(-0.428380\pi\)
0.223107 + 0.974794i \(0.428380\pi\)
\(420\) 0 0
\(421\) 24.1949 1.17919 0.589594 0.807700i \(-0.299288\pi\)
0.589594 + 0.807700i \(0.299288\pi\)
\(422\) −3.89170 −0.189445
\(423\) 0 0
\(424\) 31.9274 1.55053
\(425\) 12.9768 0.629467
\(426\) 0 0
\(427\) 7.09043 0.343130
\(428\) −17.2866 −0.835578
\(429\) 0 0
\(430\) −7.96466 −0.384090
\(431\) −29.5332 −1.42256 −0.711282 0.702907i \(-0.751885\pi\)
−0.711282 + 0.702907i \(0.751885\pi\)
\(432\) 0 0
\(433\) 0.669754 0.0321863 0.0160932 0.999870i \(-0.494877\pi\)
0.0160932 + 0.999870i \(0.494877\pi\)
\(434\) −2.12614 −0.102058
\(435\) 0 0
\(436\) −5.61037 −0.268688
\(437\) 0.209790 0.0100356
\(438\) 0 0
\(439\) −6.34887 −0.303015 −0.151507 0.988456i \(-0.548413\pi\)
−0.151507 + 0.988456i \(0.548413\pi\)
\(440\) −6.66798 −0.317883
\(441\) 0 0
\(442\) 31.8809 1.51642
\(443\) 15.3539 0.729487 0.364743 0.931108i \(-0.381157\pi\)
0.364743 + 0.931108i \(0.381157\pi\)
\(444\) 0 0
\(445\) −11.7034 −0.554792
\(446\) −22.3950 −1.06043
\(447\) 0 0
\(448\) 18.9249 0.894118
\(449\) −32.0398 −1.51205 −0.756027 0.654541i \(-0.772862\pi\)
−0.756027 + 0.654541i \(0.772862\pi\)
\(450\) 0 0
\(451\) 10.2254 0.481494
\(452\) 6.14392 0.288986
\(453\) 0 0
\(454\) −15.1191 −0.709573
\(455\) 19.5304 0.915601
\(456\) 0 0
\(457\) 19.1680 0.896643 0.448321 0.893872i \(-0.352022\pi\)
0.448321 + 0.893872i \(0.352022\pi\)
\(458\) 17.8031 0.831886
\(459\) 0 0
\(460\) 1.30591 0.0608882
\(461\) −5.22029 −0.243133 −0.121567 0.992583i \(-0.538792\pi\)
−0.121567 + 0.992583i \(0.538792\pi\)
\(462\) 0 0
\(463\) 1.69739 0.0788844 0.0394422 0.999222i \(-0.487442\pi\)
0.0394422 + 0.999222i \(0.487442\pi\)
\(464\) 0.734731 0.0341090
\(465\) 0 0
\(466\) 5.90364 0.273481
\(467\) 19.6827 0.910808 0.455404 0.890285i \(-0.349495\pi\)
0.455404 + 0.890285i \(0.349495\pi\)
\(468\) 0 0
\(469\) 24.8046 1.14537
\(470\) 9.75962 0.450178
\(471\) 0 0
\(472\) 9.15482 0.421385
\(473\) −5.42716 −0.249541
\(474\) 0 0
\(475\) −0.486410 −0.0223180
\(476\) 14.5227 0.665646
\(477\) 0 0
\(478\) −5.56080 −0.254345
\(479\) −29.2534 −1.33662 −0.668311 0.743882i \(-0.732983\pi\)
−0.668311 + 0.743882i \(0.732983\pi\)
\(480\) 0 0
\(481\) −11.8641 −0.540959
\(482\) −9.38610 −0.427525
\(483\) 0 0
\(484\) 8.37489 0.380677
\(485\) −9.58538 −0.435250
\(486\) 0 0
\(487\) −20.5056 −0.929199 −0.464600 0.885521i \(-0.653802\pi\)
−0.464600 + 0.885521i \(0.653802\pi\)
\(488\) −8.78016 −0.397459
\(489\) 0 0
\(490\) 1.75247 0.0791686
\(491\) 17.5270 0.790982 0.395491 0.918470i \(-0.370574\pi\)
0.395491 + 0.918470i \(0.370574\pi\)
\(492\) 0 0
\(493\) 3.40590 0.153394
\(494\) −1.19499 −0.0537652
\(495\) 0 0
\(496\) 1.17584 0.0527969
\(497\) 0.220043 0.00987026
\(498\) 0 0
\(499\) −19.1060 −0.855301 −0.427651 0.903944i \(-0.640659\pi\)
−0.427651 + 0.903944i \(0.640659\pi\)
\(500\) −10.7826 −0.482212
\(501\) 0 0
\(502\) −8.20287 −0.366112
\(503\) −10.9676 −0.489022 −0.244511 0.969646i \(-0.578627\pi\)
−0.244511 + 0.969646i \(0.578627\pi\)
\(504\) 0 0
\(505\) −8.73847 −0.388857
\(506\) −1.11342 −0.0494975
\(507\) 0 0
\(508\) −10.6814 −0.473912
\(509\) −19.8994 −0.882023 −0.441012 0.897501i \(-0.645380\pi\)
−0.441012 + 0.897501i \(0.645380\pi\)
\(510\) 0 0
\(511\) −13.0860 −0.578890
\(512\) −15.2994 −0.676143
\(513\) 0 0
\(514\) −21.5468 −0.950387
\(515\) −20.2846 −0.893845
\(516\) 0 0
\(517\) 6.65026 0.292478
\(518\) 6.76230 0.297118
\(519\) 0 0
\(520\) −24.1848 −1.06057
\(521\) 35.1167 1.53849 0.769244 0.638955i \(-0.220633\pi\)
0.769244 + 0.638955i \(0.220633\pi\)
\(522\) 0 0
\(523\) −14.2454 −0.622907 −0.311453 0.950261i \(-0.600816\pi\)
−0.311453 + 0.950261i \(0.600816\pi\)
\(524\) 12.5129 0.546629
\(525\) 0 0
\(526\) 11.8926 0.518541
\(527\) 5.45070 0.237436
\(528\) 0 0
\(529\) −22.2910 −0.969175
\(530\) 19.2973 0.838223
\(531\) 0 0
\(532\) −0.544355 −0.0236008
\(533\) 37.0874 1.60643
\(534\) 0 0
\(535\) −33.9697 −1.46864
\(536\) −30.7158 −1.32672
\(537\) 0 0
\(538\) −0.324480 −0.0139893
\(539\) 1.19414 0.0514354
\(540\) 0 0
\(541\) −13.2368 −0.569094 −0.284547 0.958662i \(-0.591843\pi\)
−0.284547 + 0.958662i \(0.591843\pi\)
\(542\) −2.34325 −0.100651
\(543\) 0 0
\(544\) −30.4359 −1.30493
\(545\) −11.0249 −0.472255
\(546\) 0 0
\(547\) −16.3049 −0.697148 −0.348574 0.937281i \(-0.613334\pi\)
−0.348574 + 0.937281i \(0.613334\pi\)
\(548\) −2.00613 −0.0856976
\(549\) 0 0
\(550\) 2.58152 0.110077
\(551\) −0.127663 −0.00543864
\(552\) 0 0
\(553\) 11.7476 0.499560
\(554\) −24.5971 −1.04503
\(555\) 0 0
\(556\) 7.08507 0.300474
\(557\) 30.8972 1.30915 0.654577 0.755995i \(-0.272847\pi\)
0.654577 + 0.755995i \(0.272847\pi\)
\(558\) 0 0
\(559\) −19.6843 −0.832557
\(560\) 6.15640 0.260155
\(561\) 0 0
\(562\) 7.61798 0.321345
\(563\) 26.7759 1.12847 0.564236 0.825614i \(-0.309171\pi\)
0.564236 + 0.825614i \(0.309171\pi\)
\(564\) 0 0
\(565\) 12.0734 0.507931
\(566\) 7.50710 0.315547
\(567\) 0 0
\(568\) −0.272481 −0.0114331
\(569\) 19.0606 0.799064 0.399532 0.916719i \(-0.369173\pi\)
0.399532 + 0.916719i \(0.369173\pi\)
\(570\) 0 0
\(571\) −18.7742 −0.785676 −0.392838 0.919608i \(-0.628507\pi\)
−0.392838 + 0.919608i \(0.628507\pi\)
\(572\) −5.06872 −0.211934
\(573\) 0 0
\(574\) −21.1390 −0.882324
\(575\) −1.64378 −0.0685503
\(576\) 0 0
\(577\) −4.85962 −0.202309 −0.101154 0.994871i \(-0.532254\pi\)
−0.101154 + 0.994871i \(0.532254\pi\)
\(578\) 28.6617 1.19217
\(579\) 0 0
\(580\) −0.794683 −0.0329974
\(581\) 19.7740 0.820362
\(582\) 0 0
\(583\) 13.1493 0.544589
\(584\) 16.2045 0.670548
\(585\) 0 0
\(586\) 0.582499 0.0240628
\(587\) −32.7973 −1.35369 −0.676846 0.736125i \(-0.736654\pi\)
−0.676846 + 0.736125i \(0.736654\pi\)
\(588\) 0 0
\(589\) −0.204309 −0.00841841
\(590\) 5.53330 0.227802
\(591\) 0 0
\(592\) −3.73983 −0.153706
\(593\) 17.3446 0.712258 0.356129 0.934437i \(-0.384096\pi\)
0.356129 + 0.934437i \(0.384096\pi\)
\(594\) 0 0
\(595\) 28.5384 1.16996
\(596\) 0.0950040 0.00389151
\(597\) 0 0
\(598\) −4.03837 −0.165141
\(599\) 24.4079 0.997280 0.498640 0.866809i \(-0.333833\pi\)
0.498640 + 0.866809i \(0.333833\pi\)
\(600\) 0 0
\(601\) 7.87027 0.321035 0.160517 0.987033i \(-0.448684\pi\)
0.160517 + 0.987033i \(0.448684\pi\)
\(602\) 11.2196 0.457277
\(603\) 0 0
\(604\) 17.9984 0.732346
\(605\) 16.4574 0.669090
\(606\) 0 0
\(607\) −10.2375 −0.415526 −0.207763 0.978179i \(-0.566618\pi\)
−0.207763 + 0.978179i \(0.566618\pi\)
\(608\) 1.14083 0.0462668
\(609\) 0 0
\(610\) −5.30684 −0.214868
\(611\) 24.1205 0.975810
\(612\) 0 0
\(613\) 2.23507 0.0902736 0.0451368 0.998981i \(-0.485628\pi\)
0.0451368 + 0.998981i \(0.485628\pi\)
\(614\) 7.09430 0.286303
\(615\) 0 0
\(616\) 9.39301 0.378455
\(617\) 33.9757 1.36781 0.683905 0.729571i \(-0.260280\pi\)
0.683905 + 0.729571i \(0.260280\pi\)
\(618\) 0 0
\(619\) −28.8560 −1.15982 −0.579910 0.814681i \(-0.696912\pi\)
−0.579910 + 0.814681i \(0.696912\pi\)
\(620\) −1.27179 −0.0510763
\(621\) 0 0
\(622\) 16.2613 0.652020
\(623\) 16.4862 0.660507
\(624\) 0 0
\(625\) −11.4277 −0.457107
\(626\) −24.8408 −0.992838
\(627\) 0 0
\(628\) 18.4341 0.735602
\(629\) −17.3362 −0.691241
\(630\) 0 0
\(631\) −3.14078 −0.125032 −0.0625162 0.998044i \(-0.519913\pi\)
−0.0625162 + 0.998044i \(0.519913\pi\)
\(632\) −14.5472 −0.578657
\(633\) 0 0
\(634\) 7.64601 0.303662
\(635\) −20.9900 −0.832964
\(636\) 0 0
\(637\) 4.33116 0.171607
\(638\) 0.677549 0.0268244
\(639\) 0 0
\(640\) 1.82275 0.0720504
\(641\) −31.8225 −1.25691 −0.628457 0.777844i \(-0.716313\pi\)
−0.628457 + 0.777844i \(0.716313\pi\)
\(642\) 0 0
\(643\) −12.8471 −0.506639 −0.253319 0.967383i \(-0.581522\pi\)
−0.253319 + 0.967383i \(0.581522\pi\)
\(644\) −1.83960 −0.0724903
\(645\) 0 0
\(646\) −1.74616 −0.0687016
\(647\) 28.2444 1.11040 0.555200 0.831717i \(-0.312642\pi\)
0.555200 + 0.831717i \(0.312642\pi\)
\(648\) 0 0
\(649\) 3.77042 0.148002
\(650\) 9.36319 0.367254
\(651\) 0 0
\(652\) 17.8875 0.700530
\(653\) 25.9905 1.01709 0.508543 0.861036i \(-0.330184\pi\)
0.508543 + 0.861036i \(0.330184\pi\)
\(654\) 0 0
\(655\) 24.5891 0.960774
\(656\) 11.6907 0.456446
\(657\) 0 0
\(658\) −13.7481 −0.535958
\(659\) −47.6178 −1.85493 −0.927463 0.373914i \(-0.878015\pi\)
−0.927463 + 0.373914i \(0.878015\pi\)
\(660\) 0 0
\(661\) −0.876508 −0.0340922 −0.0170461 0.999855i \(-0.505426\pi\)
−0.0170461 + 0.999855i \(0.505426\pi\)
\(662\) 30.6118 1.18976
\(663\) 0 0
\(664\) −24.4863 −0.950253
\(665\) −1.06971 −0.0414815
\(666\) 0 0
\(667\) −0.431427 −0.0167049
\(668\) 17.6459 0.682741
\(669\) 0 0
\(670\) −18.5650 −0.717230
\(671\) −3.61611 −0.139598
\(672\) 0 0
\(673\) 37.3394 1.43933 0.719665 0.694321i \(-0.244295\pi\)
0.719665 + 0.694321i \(0.244295\pi\)
\(674\) −1.22206 −0.0470721
\(675\) 0 0
\(676\) −6.83505 −0.262886
\(677\) −13.7192 −0.527270 −0.263635 0.964622i \(-0.584922\pi\)
−0.263635 + 0.964622i \(0.584922\pi\)
\(678\) 0 0
\(679\) 13.5027 0.518185
\(680\) −35.3394 −1.35521
\(681\) 0 0
\(682\) 1.08433 0.0415211
\(683\) 49.9887 1.91276 0.956381 0.292121i \(-0.0943611\pi\)
0.956381 + 0.292121i \(0.0943611\pi\)
\(684\) 0 0
\(685\) −3.94223 −0.150625
\(686\) −20.6186 −0.787221
\(687\) 0 0
\(688\) −6.20490 −0.236560
\(689\) 47.6925 1.81694
\(690\) 0 0
\(691\) 23.8151 0.905967 0.452984 0.891519i \(-0.350360\pi\)
0.452984 + 0.891519i \(0.350360\pi\)
\(692\) −16.8131 −0.639137
\(693\) 0 0
\(694\) 6.20967 0.235716
\(695\) 13.9228 0.528122
\(696\) 0 0
\(697\) 54.1931 2.05271
\(698\) −32.2546 −1.22086
\(699\) 0 0
\(700\) 4.26521 0.161210
\(701\) 34.4493 1.30113 0.650565 0.759450i \(-0.274532\pi\)
0.650565 + 0.759450i \(0.274532\pi\)
\(702\) 0 0
\(703\) 0.649815 0.0245082
\(704\) −9.65169 −0.363762
\(705\) 0 0
\(706\) 38.9692 1.46662
\(707\) 12.3097 0.462953
\(708\) 0 0
\(709\) 15.5233 0.582989 0.291494 0.956573i \(-0.405848\pi\)
0.291494 + 0.956573i \(0.405848\pi\)
\(710\) −0.164691 −0.00618075
\(711\) 0 0
\(712\) −20.4151 −0.765087
\(713\) −0.690444 −0.0258573
\(714\) 0 0
\(715\) −9.96050 −0.372502
\(716\) −9.69569 −0.362345
\(717\) 0 0
\(718\) −27.8089 −1.03782
\(719\) 12.0537 0.449528 0.224764 0.974413i \(-0.427839\pi\)
0.224764 + 0.974413i \(0.427839\pi\)
\(720\) 0 0
\(721\) 28.5744 1.06417
\(722\) −19.9667 −0.743084
\(723\) 0 0
\(724\) 15.9449 0.592586
\(725\) 1.00029 0.0371498
\(726\) 0 0
\(727\) −31.6302 −1.17310 −0.586550 0.809913i \(-0.699514\pi\)
−0.586550 + 0.809913i \(0.699514\pi\)
\(728\) 34.0685 1.26266
\(729\) 0 0
\(730\) 9.79423 0.362501
\(731\) −28.7633 −1.06385
\(732\) 0 0
\(733\) −19.0308 −0.702918 −0.351459 0.936203i \(-0.614314\pi\)
−0.351459 + 0.936203i \(0.614314\pi\)
\(734\) −11.9277 −0.440260
\(735\) 0 0
\(736\) 3.85533 0.142109
\(737\) −12.6503 −0.465981
\(738\) 0 0
\(739\) 16.6007 0.610668 0.305334 0.952245i \(-0.401232\pi\)
0.305334 + 0.952245i \(0.401232\pi\)
\(740\) 4.04499 0.148697
\(741\) 0 0
\(742\) −27.1837 −0.997944
\(743\) −33.3035 −1.22179 −0.610894 0.791712i \(-0.709190\pi\)
−0.610894 + 0.791712i \(0.709190\pi\)
\(744\) 0 0
\(745\) 0.186692 0.00683985
\(746\) 6.16156 0.225591
\(747\) 0 0
\(748\) −7.40655 −0.270810
\(749\) 47.8523 1.74849
\(750\) 0 0
\(751\) 27.7816 1.01376 0.506882 0.862015i \(-0.330798\pi\)
0.506882 + 0.862015i \(0.330798\pi\)
\(752\) 7.60328 0.277263
\(753\) 0 0
\(754\) 2.45747 0.0894957
\(755\) 35.3686 1.28720
\(756\) 0 0
\(757\) −3.12036 −0.113411 −0.0567057 0.998391i \(-0.518060\pi\)
−0.0567057 + 0.998391i \(0.518060\pi\)
\(758\) 25.6480 0.931577
\(759\) 0 0
\(760\) 1.32463 0.0480494
\(761\) −43.9592 −1.59352 −0.796760 0.604296i \(-0.793454\pi\)
−0.796760 + 0.604296i \(0.793454\pi\)
\(762\) 0 0
\(763\) 15.5305 0.562242
\(764\) 23.9566 0.866720
\(765\) 0 0
\(766\) −4.02336 −0.145370
\(767\) 13.6753 0.493787
\(768\) 0 0
\(769\) 3.71286 0.133889 0.0669445 0.997757i \(-0.478675\pi\)
0.0669445 + 0.997757i \(0.478675\pi\)
\(770\) 5.67726 0.204594
\(771\) 0 0
\(772\) 15.2403 0.548511
\(773\) −8.96903 −0.322594 −0.161297 0.986906i \(-0.551568\pi\)
−0.161297 + 0.986906i \(0.551568\pi\)
\(774\) 0 0
\(775\) 1.60083 0.0575036
\(776\) −16.7205 −0.600232
\(777\) 0 0
\(778\) 11.4308 0.409815
\(779\) −2.03132 −0.0727797
\(780\) 0 0
\(781\) −0.112222 −0.00401560
\(782\) −5.90098 −0.211018
\(783\) 0 0
\(784\) 1.36527 0.0487597
\(785\) 36.2248 1.29292
\(786\) 0 0
\(787\) 43.8514 1.56313 0.781567 0.623821i \(-0.214421\pi\)
0.781567 + 0.623821i \(0.214421\pi\)
\(788\) −2.23188 −0.0795074
\(789\) 0 0
\(790\) −8.79253 −0.312824
\(791\) −17.0075 −0.604716
\(792\) 0 0
\(793\) −13.1156 −0.465750
\(794\) −11.0801 −0.393219
\(795\) 0 0
\(796\) −16.4699 −0.583760
\(797\) 12.0160 0.425629 0.212815 0.977093i \(-0.431737\pi\)
0.212815 + 0.977093i \(0.431737\pi\)
\(798\) 0 0
\(799\) 35.2455 1.24690
\(800\) −8.93881 −0.316035
\(801\) 0 0
\(802\) 15.1460 0.534823
\(803\) 6.67384 0.235515
\(804\) 0 0
\(805\) −3.61498 −0.127411
\(806\) 3.93286 0.138529
\(807\) 0 0
\(808\) −15.2432 −0.536254
\(809\) −8.02937 −0.282298 −0.141149 0.989988i \(-0.545080\pi\)
−0.141149 + 0.989988i \(0.545080\pi\)
\(810\) 0 0
\(811\) −12.8345 −0.450681 −0.225341 0.974280i \(-0.572349\pi\)
−0.225341 + 0.974280i \(0.572349\pi\)
\(812\) 1.11945 0.0392850
\(813\) 0 0
\(814\) −3.44876 −0.120879
\(815\) 35.1507 1.23127
\(816\) 0 0
\(817\) 1.07813 0.0377192
\(818\) 18.6428 0.651832
\(819\) 0 0
\(820\) −12.6447 −0.441570
\(821\) −29.6654 −1.03533 −0.517665 0.855584i \(-0.673198\pi\)
−0.517665 + 0.855584i \(0.673198\pi\)
\(822\) 0 0
\(823\) 49.5407 1.72688 0.863441 0.504451i \(-0.168305\pi\)
0.863441 + 0.504451i \(0.168305\pi\)
\(824\) −35.3840 −1.23266
\(825\) 0 0
\(826\) −7.79462 −0.271209
\(827\) 40.8431 1.42025 0.710126 0.704074i \(-0.248638\pi\)
0.710126 + 0.704074i \(0.248638\pi\)
\(828\) 0 0
\(829\) −9.45276 −0.328308 −0.164154 0.986435i \(-0.552489\pi\)
−0.164154 + 0.986435i \(0.552489\pi\)
\(830\) −14.7999 −0.513711
\(831\) 0 0
\(832\) −35.0067 −1.21364
\(833\) 6.32881 0.219280
\(834\) 0 0
\(835\) 34.6759 1.20001
\(836\) 0.277620 0.00960170
\(837\) 0 0
\(838\) 9.62995 0.332661
\(839\) −12.2869 −0.424191 −0.212095 0.977249i \(-0.568029\pi\)
−0.212095 + 0.977249i \(0.568029\pi\)
\(840\) 0 0
\(841\) −28.7375 −0.990947
\(842\) 25.5093 0.879108
\(843\) 0 0
\(844\) 3.27924 0.112876
\(845\) −13.4315 −0.462058
\(846\) 0 0
\(847\) −23.1832 −0.796584
\(848\) 15.0337 0.516259
\(849\) 0 0
\(850\) 13.6817 0.469280
\(851\) 2.19599 0.0752776
\(852\) 0 0
\(853\) −30.8659 −1.05683 −0.528413 0.848987i \(-0.677213\pi\)
−0.528413 + 0.848987i \(0.677213\pi\)
\(854\) 7.47562 0.255810
\(855\) 0 0
\(856\) −59.2560 −2.02533
\(857\) 11.1460 0.380741 0.190371 0.981712i \(-0.439031\pi\)
0.190371 + 0.981712i \(0.439031\pi\)
\(858\) 0 0
\(859\) 4.14868 0.141551 0.0707755 0.997492i \(-0.477453\pi\)
0.0707755 + 0.997492i \(0.477453\pi\)
\(860\) 6.71121 0.228850
\(861\) 0 0
\(862\) −31.1376 −1.06055
\(863\) −47.2534 −1.60852 −0.804262 0.594275i \(-0.797439\pi\)
−0.804262 + 0.594275i \(0.797439\pi\)
\(864\) 0 0
\(865\) −33.0392 −1.12337
\(866\) 0.706138 0.0239956
\(867\) 0 0
\(868\) 1.79154 0.0608088
\(869\) −5.99127 −0.203240
\(870\) 0 0
\(871\) −45.8827 −1.55468
\(872\) −19.2316 −0.651264
\(873\) 0 0
\(874\) 0.221187 0.00748175
\(875\) 29.8481 1.00905
\(876\) 0 0
\(877\) 35.1504 1.18695 0.593473 0.804854i \(-0.297756\pi\)
0.593473 + 0.804854i \(0.297756\pi\)
\(878\) −6.69377 −0.225904
\(879\) 0 0
\(880\) −3.13976 −0.105841
\(881\) −19.3596 −0.652242 −0.326121 0.945328i \(-0.605742\pi\)
−0.326121 + 0.945328i \(0.605742\pi\)
\(882\) 0 0
\(883\) −13.7860 −0.463937 −0.231969 0.972723i \(-0.574517\pi\)
−0.231969 + 0.972723i \(0.574517\pi\)
\(884\) −26.8636 −0.903519
\(885\) 0 0
\(886\) 16.1880 0.543847
\(887\) −29.8175 −1.00117 −0.500586 0.865687i \(-0.666882\pi\)
−0.500586 + 0.865687i \(0.666882\pi\)
\(888\) 0 0
\(889\) 29.5681 0.991683
\(890\) −12.3391 −0.413609
\(891\) 0 0
\(892\) 18.8705 0.631832
\(893\) −1.32111 −0.0442092
\(894\) 0 0
\(895\) −19.0529 −0.636869
\(896\) −2.56766 −0.0857795
\(897\) 0 0
\(898\) −33.7804 −1.12727
\(899\) 0.420156 0.0140130
\(900\) 0 0
\(901\) 69.6897 2.32170
\(902\) 10.7809 0.358963
\(903\) 0 0
\(904\) 21.0605 0.700463
\(905\) 31.3331 1.04155
\(906\) 0 0
\(907\) −6.23157 −0.206916 −0.103458 0.994634i \(-0.532991\pi\)
−0.103458 + 0.994634i \(0.532991\pi\)
\(908\) 12.7397 0.422781
\(909\) 0 0
\(910\) 20.5914 0.682599
\(911\) −31.6395 −1.04826 −0.524131 0.851638i \(-0.675610\pi\)
−0.524131 + 0.851638i \(0.675610\pi\)
\(912\) 0 0
\(913\) −10.0847 −0.333755
\(914\) 20.2093 0.668465
\(915\) 0 0
\(916\) −15.0013 −0.495658
\(917\) −34.6380 −1.14385
\(918\) 0 0
\(919\) 36.0031 1.18763 0.593816 0.804601i \(-0.297621\pi\)
0.593816 + 0.804601i \(0.297621\pi\)
\(920\) 4.47647 0.147585
\(921\) 0 0
\(922\) −5.50388 −0.181261
\(923\) −0.407027 −0.0133975
\(924\) 0 0
\(925\) −5.09153 −0.167408
\(926\) 1.78960 0.0588099
\(927\) 0 0
\(928\) −2.34609 −0.0770141
\(929\) 25.4477 0.834913 0.417456 0.908697i \(-0.362922\pi\)
0.417456 + 0.908697i \(0.362922\pi\)
\(930\) 0 0
\(931\) −0.237223 −0.00777468
\(932\) −4.97454 −0.162947
\(933\) 0 0
\(934\) 20.7520 0.679026
\(935\) −14.5546 −0.475985
\(936\) 0 0
\(937\) 28.3048 0.924677 0.462338 0.886704i \(-0.347011\pi\)
0.462338 + 0.886704i \(0.347011\pi\)
\(938\) 26.1521 0.853897
\(939\) 0 0
\(940\) −8.22369 −0.268227
\(941\) 8.29444 0.270391 0.135196 0.990819i \(-0.456834\pi\)
0.135196 + 0.990819i \(0.456834\pi\)
\(942\) 0 0
\(943\) −6.86468 −0.223545
\(944\) 4.31074 0.140303
\(945\) 0 0
\(946\) −5.72199 −0.186038
\(947\) −44.4506 −1.44445 −0.722225 0.691658i \(-0.756881\pi\)
−0.722225 + 0.691658i \(0.756881\pi\)
\(948\) 0 0
\(949\) 24.2060 0.785761
\(950\) −0.512834 −0.0166385
\(951\) 0 0
\(952\) 49.7818 1.61344
\(953\) −9.67149 −0.313290 −0.156645 0.987655i \(-0.550068\pi\)
−0.156645 + 0.987655i \(0.550068\pi\)
\(954\) 0 0
\(955\) 47.0770 1.52338
\(956\) 4.68566 0.151545
\(957\) 0 0
\(958\) −30.8426 −0.996479
\(959\) 5.55332 0.179326
\(960\) 0 0
\(961\) −30.3276 −0.978309
\(962\) −12.5087 −0.403296
\(963\) 0 0
\(964\) 7.90895 0.254730
\(965\) 29.9486 0.964081
\(966\) 0 0
\(967\) −33.1151 −1.06491 −0.532455 0.846459i \(-0.678730\pi\)
−0.532455 + 0.846459i \(0.678730\pi\)
\(968\) 28.7080 0.922710
\(969\) 0 0
\(970\) −10.1061 −0.324487
\(971\) −27.4309 −0.880298 −0.440149 0.897925i \(-0.645074\pi\)
−0.440149 + 0.897925i \(0.645074\pi\)
\(972\) 0 0
\(973\) −19.6127 −0.628755
\(974\) −21.6196 −0.692737
\(975\) 0 0
\(976\) −4.13432 −0.132336
\(977\) 36.3776 1.16382 0.581911 0.813253i \(-0.302305\pi\)
0.581911 + 0.813253i \(0.302305\pi\)
\(978\) 0 0
\(979\) −8.40796 −0.268719
\(980\) −1.47667 −0.0471706
\(981\) 0 0
\(982\) 18.4791 0.589693
\(983\) 42.1372 1.34397 0.671984 0.740566i \(-0.265442\pi\)
0.671984 + 0.740566i \(0.265442\pi\)
\(984\) 0 0
\(985\) −4.38585 −0.139745
\(986\) 3.59092 0.114358
\(987\) 0 0
\(988\) 1.00693 0.0320347
\(989\) 3.64346 0.115855
\(990\) 0 0
\(991\) 25.5409 0.811333 0.405667 0.914021i \(-0.367039\pi\)
0.405667 + 0.914021i \(0.367039\pi\)
\(992\) −3.75461 −0.119209
\(993\) 0 0
\(994\) 0.231996 0.00735848
\(995\) −32.3649 −1.02604
\(996\) 0 0
\(997\) 23.5329 0.745294 0.372647 0.927973i \(-0.378450\pi\)
0.372647 + 0.927973i \(0.378450\pi\)
\(998\) −20.1439 −0.637644
\(999\) 0 0
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 729.2.a.a.1.5 6
3.2 odd 2 729.2.a.d.1.2 6
9.2 odd 6 729.2.c.b.244.5 12
9.4 even 3 729.2.c.e.487.2 12
9.5 odd 6 729.2.c.b.487.5 12
9.7 even 3 729.2.c.e.244.2 12
27.2 odd 18 243.2.e.a.190.2 12
27.4 even 9 243.2.e.c.136.2 12
27.5 odd 18 81.2.e.a.73.1 12
27.7 even 9 243.2.e.c.109.2 12
27.11 odd 18 81.2.e.a.10.1 12
27.13 even 9 243.2.e.d.55.1 12
27.14 odd 18 243.2.e.a.55.2 12
27.16 even 9 27.2.e.a.13.2 12
27.20 odd 18 243.2.e.b.109.1 12
27.22 even 9 27.2.e.a.25.2 yes 12
27.23 odd 18 243.2.e.b.136.1 12
27.25 even 9 243.2.e.d.190.1 12
108.43 odd 18 432.2.u.c.337.2 12
108.103 odd 18 432.2.u.c.241.2 12
135.22 odd 36 675.2.u.b.349.3 24
135.43 odd 36 675.2.u.b.499.3 24
135.49 even 18 675.2.l.c.376.1 12
135.97 odd 36 675.2.u.b.499.2 24
135.103 odd 36 675.2.u.b.349.2 24
135.124 even 18 675.2.l.c.526.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.13.2 12 27.16 even 9
27.2.e.a.25.2 yes 12 27.22 even 9
81.2.e.a.10.1 12 27.11 odd 18
81.2.e.a.73.1 12 27.5 odd 18
243.2.e.a.55.2 12 27.14 odd 18
243.2.e.a.190.2 12 27.2 odd 18
243.2.e.b.109.1 12 27.20 odd 18
243.2.e.b.136.1 12 27.23 odd 18
243.2.e.c.109.2 12 27.7 even 9
243.2.e.c.136.2 12 27.4 even 9
243.2.e.d.55.1 12 27.13 even 9
243.2.e.d.190.1 12 27.25 even 9
432.2.u.c.241.2 12 108.103 odd 18
432.2.u.c.337.2 12 108.43 odd 18
675.2.l.c.376.1 12 135.49 even 18
675.2.l.c.526.1 12 135.124 even 18
675.2.u.b.349.2 24 135.103 odd 36
675.2.u.b.349.3 24 135.22 odd 36
675.2.u.b.499.2 24 135.97 odd 36
675.2.u.b.499.3 24 135.43 odd 36
729.2.a.a.1.5 6 1.1 even 1 trivial
729.2.a.d.1.2 6 3.2 odd 2
729.2.c.b.244.5 12 9.2 odd 6
729.2.c.b.487.5 12 9.5 odd 6
729.2.c.e.244.2 12 9.7 even 3
729.2.c.e.487.2 12 9.4 even 3