Properties

Label 3762.2.a.bd.1.2
Level $3762$
Weight $2$
Character 3762.1
Self dual yes
Analytic conductor $30.040$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3762,2,Mod(1,3762)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3762, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3762.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3762.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: \(30.0397212404\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.772866\) of defining polynomial
Character \(\chi\) \(=\) 3762.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.22713 q^{5} +3.40268 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.22713 q^{5} +3.40268 q^{7} -1.00000 q^{8} +1.22713 q^{10} +1.00000 q^{11} -1.40268 q^{13} -3.40268 q^{14} +1.00000 q^{16} +4.80536 q^{17} -1.00000 q^{19} -1.22713 q^{20} -1.00000 q^{22} -6.80536 q^{23} -3.49414 q^{25} +1.40268 q^{26} +3.40268 q^{28} -8.03249 q^{29} -4.94841 q^{31} -1.00000 q^{32} -4.80536 q^{34} -4.17554 q^{35} +2.35109 q^{37} +1.00000 q^{38} +1.22713 q^{40} -2.94841 q^{41} -9.12395 q^{43} +1.00000 q^{44} +6.80536 q^{46} -5.25963 q^{47} +4.57822 q^{49} +3.49414 q^{50} -1.40268 q^{52} +4.45427 q^{53} -1.22713 q^{55} -3.40268 q^{56} +8.03249 q^{58} +14.5193 q^{59} +2.00000 q^{61} +4.94841 q^{62} +1.00000 q^{64} +1.72128 q^{65} +3.40268 q^{67} +4.80536 q^{68} +4.17554 q^{70} +7.57822 q^{71} -10.0650 q^{73} -2.35109 q^{74} -1.00000 q^{76} +3.40268 q^{77} -3.71390 q^{79} -1.22713 q^{80} +2.94841 q^{82} -10.6697 q^{83} -5.89682 q^{85} +9.12395 q^{86} -1.00000 q^{88} -14.9883 q^{89} -4.77287 q^{91} -6.80536 q^{92} +5.25963 q^{94} +1.22713 q^{95} +2.35109 q^{97} -4.57822 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} + q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} + q^{7} - 3 q^{8} + 5 q^{10} + 3 q^{11} + 5 q^{13} - q^{14} + 3 q^{16} - 4 q^{17} - 3 q^{19} - 5 q^{20} - 3 q^{22} - 2 q^{23} + 4 q^{25} - 5 q^{26} + q^{28} - 7 q^{29} - 3 q^{31} - 3 q^{32} + 4 q^{34} - 2 q^{35} - 14 q^{37} + 3 q^{38} + 5 q^{40} + 3 q^{41} - 5 q^{43} + 3 q^{44} + 2 q^{46} - 6 q^{49} - 4 q^{50} + 5 q^{52} + 16 q^{53} - 5 q^{55} - q^{56} + 7 q^{58} + 12 q^{59} + 6 q^{61} + 3 q^{62} + 3 q^{64} - 8 q^{65} + q^{67} - 4 q^{68} + 2 q^{70} + 3 q^{71} + 4 q^{73} + 14 q^{74} - 3 q^{76} + q^{77} + 2 q^{79} - 5 q^{80} - 3 q^{82} - 7 q^{83} + 6 q^{85} + 5 q^{86} - 3 q^{88} - 16 q^{89} - 13 q^{91} - 2 q^{92} + 5 q^{95} - 14 q^{97} + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.22713 −0.548791 −0.274396 0.961617i \(-0.588478\pi\)
−0.274396 + 0.961617i \(0.588478\pi\)
\(6\) 0 0
\(7\) 3.40268 1.28609 0.643046 0.765828i \(-0.277670\pi\)
0.643046 + 0.765828i \(0.277670\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.22713 0.388054
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.40268 −0.389033 −0.194517 0.980899i \(-0.562314\pi\)
−0.194517 + 0.980899i \(0.562314\pi\)
\(14\) −3.40268 −0.909404
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.80536 1.16547 0.582735 0.812662i \(-0.301982\pi\)
0.582735 + 0.812662i \(0.301982\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −1.22713 −0.274396
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −6.80536 −1.41902 −0.709508 0.704698i \(-0.751083\pi\)
−0.709508 + 0.704698i \(0.751083\pi\)
\(24\) 0 0
\(25\) −3.49414 −0.698828
\(26\) 1.40268 0.275088
\(27\) 0 0
\(28\) 3.40268 0.643046
\(29\) −8.03249 −1.49160 −0.745798 0.666172i \(-0.767932\pi\)
−0.745798 + 0.666172i \(0.767932\pi\)
\(30\) 0 0
\(31\) −4.94841 −0.888761 −0.444380 0.895838i \(-0.646576\pi\)
−0.444380 + 0.895838i \(0.646576\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.80536 −0.824112
\(35\) −4.17554 −0.705796
\(36\) 0 0
\(37\) 2.35109 0.386517 0.193258 0.981148i \(-0.438094\pi\)
0.193258 + 0.981148i \(0.438094\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 1.22713 0.194027
\(41\) −2.94841 −0.460464 −0.230232 0.973136i \(-0.573949\pi\)
−0.230232 + 0.973136i \(0.573949\pi\)
\(42\) 0 0
\(43\) −9.12395 −1.39139 −0.695695 0.718337i \(-0.744903\pi\)
−0.695695 + 0.718337i \(0.744903\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 6.80536 1.00340
\(47\) −5.25963 −0.767195 −0.383598 0.923500i \(-0.625315\pi\)
−0.383598 + 0.923500i \(0.625315\pi\)
\(48\) 0 0
\(49\) 4.57822 0.654032
\(50\) 3.49414 0.494146
\(51\) 0 0
\(52\) −1.40268 −0.194517
\(53\) 4.45427 0.611841 0.305920 0.952057i \(-0.401036\pi\)
0.305920 + 0.952057i \(0.401036\pi\)
\(54\) 0 0
\(55\) −1.22713 −0.165467
\(56\) −3.40268 −0.454702
\(57\) 0 0
\(58\) 8.03249 1.05472
\(59\) 14.5193 1.89025 0.945123 0.326715i \(-0.105942\pi\)
0.945123 + 0.326715i \(0.105942\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 4.94841 0.628449
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.72128 0.213498
\(66\) 0 0
\(67\) 3.40268 0.415703 0.207852 0.978160i \(-0.433353\pi\)
0.207852 + 0.978160i \(0.433353\pi\)
\(68\) 4.80536 0.582735
\(69\) 0 0
\(70\) 4.17554 0.499073
\(71\) 7.57822 0.899370 0.449685 0.893187i \(-0.351536\pi\)
0.449685 + 0.893187i \(0.351536\pi\)
\(72\) 0 0
\(73\) −10.0650 −1.17802 −0.589009 0.808127i \(-0.700482\pi\)
−0.589009 + 0.808127i \(0.700482\pi\)
\(74\) −2.35109 −0.273309
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 3.40268 0.387771
\(78\) 0 0
\(79\) −3.71390 −0.417846 −0.208923 0.977932i \(-0.566996\pi\)
−0.208923 + 0.977932i \(0.566996\pi\)
\(80\) −1.22713 −0.137198
\(81\) 0 0
\(82\) 2.94841 0.325597
\(83\) −10.6697 −1.17115 −0.585575 0.810618i \(-0.699131\pi\)
−0.585575 + 0.810618i \(0.699131\pi\)
\(84\) 0 0
\(85\) −5.89682 −0.639600
\(86\) 9.12395 0.983861
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −14.9883 −1.58875 −0.794377 0.607425i \(-0.792203\pi\)
−0.794377 + 0.607425i \(0.792203\pi\)
\(90\) 0 0
\(91\) −4.77287 −0.500332
\(92\) −6.80536 −0.709508
\(93\) 0 0
\(94\) 5.25963 0.542489
\(95\) 1.22713 0.125901
\(96\) 0 0
\(97\) 2.35109 0.238717 0.119358 0.992851i \(-0.461916\pi\)
0.119358 + 0.992851i \(0.461916\pi\)
\(98\) −4.57822 −0.462470
\(99\) 0 0
\(100\) −3.49414 −0.349414
\(101\) −14.0650 −1.39952 −0.699759 0.714379i \(-0.746709\pi\)
−0.699759 + 0.714379i \(0.746709\pi\)
\(102\) 0 0
\(103\) −15.5782 −1.53497 −0.767484 0.641068i \(-0.778492\pi\)
−0.767484 + 0.641068i \(0.778492\pi\)
\(104\) 1.40268 0.137544
\(105\) 0 0
\(106\) −4.45427 −0.432637
\(107\) 4.35109 0.420636 0.210318 0.977633i \(-0.432550\pi\)
0.210318 + 0.977633i \(0.432550\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 1.22713 0.117003
\(111\) 0 0
\(112\) 3.40268 0.321523
\(113\) 19.9618 1.87785 0.938924 0.344124i \(-0.111824\pi\)
0.938924 + 0.344124i \(0.111824\pi\)
\(114\) 0 0
\(115\) 8.35109 0.778743
\(116\) −8.03249 −0.745798
\(117\) 0 0
\(118\) −14.5193 −1.33661
\(119\) 16.3511 1.49890
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) −4.94841 −0.444380
\(125\) 10.4235 0.932302
\(126\) 0 0
\(127\) 11.7139 1.03944 0.519720 0.854337i \(-0.326036\pi\)
0.519720 + 0.854337i \(0.326036\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −1.72128 −0.150966
\(131\) −1.85695 −0.162242 −0.0811211 0.996704i \(-0.525850\pi\)
−0.0811211 + 0.996704i \(0.525850\pi\)
\(132\) 0 0
\(133\) −3.40268 −0.295050
\(134\) −3.40268 −0.293947
\(135\) 0 0
\(136\) −4.80536 −0.412056
\(137\) −10.5973 −0.905390 −0.452695 0.891665i \(-0.649537\pi\)
−0.452695 + 0.891665i \(0.649537\pi\)
\(138\) 0 0
\(139\) 14.7347 1.24978 0.624889 0.780713i \(-0.285144\pi\)
0.624889 + 0.780713i \(0.285144\pi\)
\(140\) −4.17554 −0.352898
\(141\) 0 0
\(142\) −7.57822 −0.635950
\(143\) −1.40268 −0.117298
\(144\) 0 0
\(145\) 9.85695 0.818575
\(146\) 10.0650 0.832984
\(147\) 0 0
\(148\) 2.35109 0.193258
\(149\) −18.3511 −1.50338 −0.751690 0.659517i \(-0.770761\pi\)
−0.751690 + 0.659517i \(0.770761\pi\)
\(150\) 0 0
\(151\) −7.64891 −0.622460 −0.311230 0.950335i \(-0.600741\pi\)
−0.311230 + 0.950335i \(0.600741\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −3.40268 −0.274196
\(155\) 6.07236 0.487744
\(156\) 0 0
\(157\) 8.38358 0.669083 0.334541 0.942381i \(-0.391419\pi\)
0.334541 + 0.942381i \(0.391419\pi\)
\(158\) 3.71390 0.295462
\(159\) 0 0
\(160\) 1.22713 0.0970135
\(161\) −23.1564 −1.82498
\(162\) 0 0
\(163\) −14.2479 −1.11598 −0.557991 0.829847i \(-0.688428\pi\)
−0.557991 + 0.829847i \(0.688428\pi\)
\(164\) −2.94841 −0.230232
\(165\) 0 0
\(166\) 10.6697 0.828128
\(167\) 10.8703 0.841172 0.420586 0.907253i \(-0.361824\pi\)
0.420586 + 0.907253i \(0.361824\pi\)
\(168\) 0 0
\(169\) −11.0325 −0.848653
\(170\) 5.89682 0.452265
\(171\) 0 0
\(172\) −9.12395 −0.695695
\(173\) −3.39530 −0.258140 −0.129070 0.991635i \(-0.541199\pi\)
−0.129070 + 0.991635i \(0.541199\pi\)
\(174\) 0 0
\(175\) −11.8894 −0.898757
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 14.9883 1.12342
\(179\) 0.311217 0.0232614 0.0116307 0.999932i \(-0.496298\pi\)
0.0116307 + 0.999932i \(0.496298\pi\)
\(180\) 0 0
\(181\) −15.3246 −1.13907 −0.569535 0.821967i \(-0.692877\pi\)
−0.569535 + 0.821967i \(0.692877\pi\)
\(182\) 4.77287 0.353788
\(183\) 0 0
\(184\) 6.80536 0.501698
\(185\) −2.88510 −0.212117
\(186\) 0 0
\(187\) 4.80536 0.351403
\(188\) −5.25963 −0.383598
\(189\) 0 0
\(190\) −1.22713 −0.0890257
\(191\) −26.8703 −1.94427 −0.972135 0.234422i \(-0.924680\pi\)
−0.972135 + 0.234422i \(0.924680\pi\)
\(192\) 0 0
\(193\) 18.5665 1.33645 0.668223 0.743961i \(-0.267055\pi\)
0.668223 + 0.743961i \(0.267055\pi\)
\(194\) −2.35109 −0.168798
\(195\) 0 0
\(196\) 4.57822 0.327016
\(197\) −14.9085 −1.06219 −0.531095 0.847312i \(-0.678219\pi\)
−0.531095 + 0.847312i \(0.678219\pi\)
\(198\) 0 0
\(199\) 15.1564 1.07441 0.537206 0.843451i \(-0.319480\pi\)
0.537206 + 0.843451i \(0.319480\pi\)
\(200\) 3.49414 0.247073
\(201\) 0 0
\(202\) 14.0650 0.989609
\(203\) −27.3320 −1.91833
\(204\) 0 0
\(205\) 3.61810 0.252699
\(206\) 15.5782 1.08539
\(207\) 0 0
\(208\) −1.40268 −0.0972583
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 11.1564 0.768041 0.384021 0.923324i \(-0.374539\pi\)
0.384021 + 0.923324i \(0.374539\pi\)
\(212\) 4.45427 0.305920
\(213\) 0 0
\(214\) −4.35109 −0.297434
\(215\) 11.1963 0.763583
\(216\) 0 0
\(217\) −16.8378 −1.14303
\(218\) −6.00000 −0.406371
\(219\) 0 0
\(220\) −1.22713 −0.0827334
\(221\) −6.74037 −0.453407
\(222\) 0 0
\(223\) 1.61072 0.107861 0.0539307 0.998545i \(-0.482825\pi\)
0.0539307 + 0.998545i \(0.482825\pi\)
\(224\) −3.40268 −0.227351
\(225\) 0 0
\(226\) −19.9618 −1.32784
\(227\) 16.7672 1.11288 0.556438 0.830889i \(-0.312168\pi\)
0.556438 + 0.830889i \(0.312168\pi\)
\(228\) 0 0
\(229\) −8.55913 −0.565603 −0.282801 0.959178i \(-0.591264\pi\)
−0.282801 + 0.959178i \(0.591264\pi\)
\(230\) −8.35109 −0.550654
\(231\) 0 0
\(232\) 8.03249 0.527359
\(233\) −5.44255 −0.356553 −0.178277 0.983980i \(-0.557052\pi\)
−0.178277 + 0.983980i \(0.557052\pi\)
\(234\) 0 0
\(235\) 6.45427 0.421030
\(236\) 14.5193 0.945123
\(237\) 0 0
\(238\) −16.3511 −1.05988
\(239\) −11.2921 −0.730426 −0.365213 0.930924i \(-0.619004\pi\)
−0.365213 + 0.930924i \(0.619004\pi\)
\(240\) 0 0
\(241\) 11.4101 0.734987 0.367493 0.930026i \(-0.380216\pi\)
0.367493 + 0.930026i \(0.380216\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −5.61810 −0.358927
\(246\) 0 0
\(247\) 1.40268 0.0892503
\(248\) 4.94841 0.314224
\(249\) 0 0
\(250\) −10.4235 −0.659237
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −6.80536 −0.427849
\(254\) −11.7139 −0.734995
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −24.2331 −1.51162 −0.755811 0.654790i \(-0.772757\pi\)
−0.755811 + 0.654790i \(0.772757\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 1.72128 0.106749
\(261\) 0 0
\(262\) 1.85695 0.114723
\(263\) −20.5665 −1.26819 −0.634093 0.773257i \(-0.718626\pi\)
−0.634093 + 0.773257i \(0.718626\pi\)
\(264\) 0 0
\(265\) −5.46599 −0.335773
\(266\) 3.40268 0.208632
\(267\) 0 0
\(268\) 3.40268 0.207852
\(269\) 1.36281 0.0830918 0.0415459 0.999137i \(-0.486772\pi\)
0.0415459 + 0.999137i \(0.486772\pi\)
\(270\) 0 0
\(271\) −23.1963 −1.40908 −0.704538 0.709666i \(-0.748846\pi\)
−0.704538 + 0.709666i \(0.748846\pi\)
\(272\) 4.80536 0.291368
\(273\) 0 0
\(274\) 10.5973 0.640208
\(275\) −3.49414 −0.210705
\(276\) 0 0
\(277\) 31.0385 1.86492 0.932462 0.361269i \(-0.117656\pi\)
0.932462 + 0.361269i \(0.117656\pi\)
\(278\) −14.7347 −0.883727
\(279\) 0 0
\(280\) 4.17554 0.249537
\(281\) 2.77287 0.165415 0.0827076 0.996574i \(-0.473643\pi\)
0.0827076 + 0.996574i \(0.473643\pi\)
\(282\) 0 0
\(283\) −9.23451 −0.548935 −0.274467 0.961596i \(-0.588502\pi\)
−0.274467 + 0.961596i \(0.588502\pi\)
\(284\) 7.57822 0.449685
\(285\) 0 0
\(286\) 1.40268 0.0829421
\(287\) −10.0325 −0.592199
\(288\) 0 0
\(289\) 6.09146 0.358321
\(290\) −9.85695 −0.578820
\(291\) 0 0
\(292\) −10.0650 −0.589009
\(293\) −17.7390 −1.03632 −0.518162 0.855283i \(-0.673384\pi\)
−0.518162 + 0.855283i \(0.673384\pi\)
\(294\) 0 0
\(295\) −17.8171 −1.03735
\(296\) −2.35109 −0.136654
\(297\) 0 0
\(298\) 18.3511 1.06305
\(299\) 9.54573 0.552044
\(300\) 0 0
\(301\) −31.0459 −1.78946
\(302\) 7.64891 0.440145
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) −2.45427 −0.140531
\(306\) 0 0
\(307\) 20.9735 1.19702 0.598511 0.801115i \(-0.295759\pi\)
0.598511 + 0.801115i \(0.295759\pi\)
\(308\) 3.40268 0.193886
\(309\) 0 0
\(310\) −6.07236 −0.344887
\(311\) 3.36281 0.190687 0.0953436 0.995444i \(-0.469605\pi\)
0.0953436 + 0.995444i \(0.469605\pi\)
\(312\) 0 0
\(313\) −4.94103 −0.279284 −0.139642 0.990202i \(-0.544595\pi\)
−0.139642 + 0.990202i \(0.544595\pi\)
\(314\) −8.38358 −0.473113
\(315\) 0 0
\(316\) −3.71390 −0.208923
\(317\) −14.2861 −0.802388 −0.401194 0.915993i \(-0.631405\pi\)
−0.401194 + 0.915993i \(0.631405\pi\)
\(318\) 0 0
\(319\) −8.03249 −0.449733
\(320\) −1.22713 −0.0685989
\(321\) 0 0
\(322\) 23.1564 1.29046
\(323\) −4.80536 −0.267377
\(324\) 0 0
\(325\) 4.90116 0.271867
\(326\) 14.2479 0.789119
\(327\) 0 0
\(328\) 2.94841 0.162799
\(329\) −17.8968 −0.986684
\(330\) 0 0
\(331\) −31.9293 −1.75499 −0.877497 0.479582i \(-0.840788\pi\)
−0.877497 + 0.479582i \(0.840788\pi\)
\(332\) −10.6697 −0.585575
\(333\) 0 0
\(334\) −10.8703 −0.594799
\(335\) −4.17554 −0.228134
\(336\) 0 0
\(337\) −8.84523 −0.481830 −0.240915 0.970546i \(-0.577448\pi\)
−0.240915 + 0.970546i \(0.577448\pi\)
\(338\) 11.0325 0.600088
\(339\) 0 0
\(340\) −5.89682 −0.319800
\(341\) −4.94841 −0.267971
\(342\) 0 0
\(343\) −8.24053 −0.444947
\(344\) 9.12395 0.491931
\(345\) 0 0
\(346\) 3.39530 0.182532
\(347\) −5.27439 −0.283144 −0.141572 0.989928i \(-0.545216\pi\)
−0.141572 + 0.989928i \(0.545216\pi\)
\(348\) 0 0
\(349\) −10.9085 −0.583921 −0.291960 0.956430i \(-0.594308\pi\)
−0.291960 + 0.956430i \(0.594308\pi\)
\(350\) 11.8894 0.635517
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 22.1300 1.17786 0.588930 0.808184i \(-0.299549\pi\)
0.588930 + 0.808184i \(0.299549\pi\)
\(354\) 0 0
\(355\) −9.29950 −0.493566
\(356\) −14.9883 −0.794377
\(357\) 0 0
\(358\) −0.311217 −0.0164483
\(359\) 16.3910 0.865082 0.432541 0.901614i \(-0.357617\pi\)
0.432541 + 0.901614i \(0.357617\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 15.3246 0.805444
\(363\) 0 0
\(364\) −4.77287 −0.250166
\(365\) 12.3511 0.646486
\(366\) 0 0
\(367\) −32.3511 −1.68871 −0.844357 0.535782i \(-0.820017\pi\)
−0.844357 + 0.535782i \(0.820017\pi\)
\(368\) −6.80536 −0.354754
\(369\) 0 0
\(370\) 2.88510 0.149989
\(371\) 15.1564 0.786883
\(372\) 0 0
\(373\) 17.5782 0.910166 0.455083 0.890449i \(-0.349610\pi\)
0.455083 + 0.890449i \(0.349610\pi\)
\(374\) −4.80536 −0.248479
\(375\) 0 0
\(376\) 5.25963 0.271245
\(377\) 11.2670 0.580280
\(378\) 0 0
\(379\) 8.93365 0.458891 0.229445 0.973322i \(-0.426309\pi\)
0.229445 + 0.973322i \(0.426309\pi\)
\(380\) 1.22713 0.0629507
\(381\) 0 0
\(382\) 26.8703 1.37481
\(383\) 16.0399 0.819599 0.409800 0.912176i \(-0.365599\pi\)
0.409800 + 0.912176i \(0.365599\pi\)
\(384\) 0 0
\(385\) −4.17554 −0.212805
\(386\) −18.5665 −0.945010
\(387\) 0 0
\(388\) 2.35109 0.119358
\(389\) −17.2271 −0.873450 −0.436725 0.899595i \(-0.643862\pi\)
−0.436725 + 0.899595i \(0.643862\pi\)
\(390\) 0 0
\(391\) −32.7022 −1.65382
\(392\) −4.57822 −0.231235
\(393\) 0 0
\(394\) 14.9085 0.751081
\(395\) 4.55745 0.229310
\(396\) 0 0
\(397\) 15.0784 0.756762 0.378381 0.925650i \(-0.376481\pi\)
0.378381 + 0.925650i \(0.376481\pi\)
\(398\) −15.1564 −0.759724
\(399\) 0 0
\(400\) −3.49414 −0.174707
\(401\) 13.1564 0.657002 0.328501 0.944504i \(-0.393457\pi\)
0.328501 + 0.944504i \(0.393457\pi\)
\(402\) 0 0
\(403\) 6.94103 0.345757
\(404\) −14.0650 −0.699759
\(405\) 0 0
\(406\) 27.3320 1.35646
\(407\) 2.35109 0.116539
\(408\) 0 0
\(409\) −16.3836 −0.810116 −0.405058 0.914291i \(-0.632749\pi\)
−0.405058 + 0.914291i \(0.632749\pi\)
\(410\) −3.61810 −0.178685
\(411\) 0 0
\(412\) −15.5782 −0.767484
\(413\) 49.4044 2.43103
\(414\) 0 0
\(415\) 13.0931 0.642717
\(416\) 1.40268 0.0687720
\(417\) 0 0
\(418\) 1.00000 0.0489116
\(419\) −1.12966 −0.0551874 −0.0275937 0.999619i \(-0.508784\pi\)
−0.0275937 + 0.999619i \(0.508784\pi\)
\(420\) 0 0
\(421\) −2.63719 −0.128529 −0.0642645 0.997933i \(-0.520470\pi\)
−0.0642645 + 0.997933i \(0.520470\pi\)
\(422\) −11.1564 −0.543087
\(423\) 0 0
\(424\) −4.45427 −0.216318
\(425\) −16.7906 −0.814464
\(426\) 0 0
\(427\) 6.80536 0.329334
\(428\) 4.35109 0.210318
\(429\) 0 0
\(430\) −11.1963 −0.539934
\(431\) 28.4958 1.37260 0.686298 0.727321i \(-0.259235\pi\)
0.686298 + 0.727321i \(0.259235\pi\)
\(432\) 0 0
\(433\) −37.5075 −1.80250 −0.901249 0.433302i \(-0.857348\pi\)
−0.901249 + 0.433302i \(0.857348\pi\)
\(434\) 16.8378 0.808243
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 6.80536 0.325544
\(438\) 0 0
\(439\) −32.1300 −1.53348 −0.766740 0.641958i \(-0.778122\pi\)
−0.766740 + 0.641958i \(0.778122\pi\)
\(440\) 1.22713 0.0585013
\(441\) 0 0
\(442\) 6.74037 0.320607
\(443\) −33.0385 −1.56971 −0.784853 0.619681i \(-0.787262\pi\)
−0.784853 + 0.619681i \(0.787262\pi\)
\(444\) 0 0
\(445\) 18.3926 0.871895
\(446\) −1.61072 −0.0762696
\(447\) 0 0
\(448\) 3.40268 0.160761
\(449\) −7.88206 −0.371977 −0.185989 0.982552i \(-0.559549\pi\)
−0.185989 + 0.982552i \(0.559549\pi\)
\(450\) 0 0
\(451\) −2.94841 −0.138835
\(452\) 19.9618 0.938924
\(453\) 0 0
\(454\) −16.7672 −0.786922
\(455\) 5.85695 0.274578
\(456\) 0 0
\(457\) 24.9353 1.16643 0.583213 0.812320i \(-0.301795\pi\)
0.583213 + 0.812320i \(0.301795\pi\)
\(458\) 8.55913 0.399942
\(459\) 0 0
\(460\) 8.35109 0.389372
\(461\) 26.6874 1.24296 0.621478 0.783431i \(-0.286532\pi\)
0.621478 + 0.783431i \(0.286532\pi\)
\(462\) 0 0
\(463\) −32.7022 −1.51980 −0.759900 0.650041i \(-0.774752\pi\)
−0.759900 + 0.650041i \(0.774752\pi\)
\(464\) −8.03249 −0.372899
\(465\) 0 0
\(466\) 5.44255 0.252121
\(467\) −11.3776 −0.526491 −0.263246 0.964729i \(-0.584793\pi\)
−0.263246 + 0.964729i \(0.584793\pi\)
\(468\) 0 0
\(469\) 11.5782 0.534633
\(470\) −6.45427 −0.297713
\(471\) 0 0
\(472\) −14.5193 −0.668303
\(473\) −9.12395 −0.419520
\(474\) 0 0
\(475\) 3.49414 0.160322
\(476\) 16.3511 0.749451
\(477\) 0 0
\(478\) 11.2921 0.516489
\(479\) 12.5973 0.575586 0.287793 0.957693i \(-0.407078\pi\)
0.287793 + 0.957693i \(0.407078\pi\)
\(480\) 0 0
\(481\) −3.29782 −0.150368
\(482\) −11.4101 −0.519714
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −2.88510 −0.131006
\(486\) 0 0
\(487\) 9.69616 0.439375 0.219688 0.975570i \(-0.429496\pi\)
0.219688 + 0.975570i \(0.429496\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) 5.61810 0.253800
\(491\) 11.7538 0.530440 0.265220 0.964188i \(-0.414555\pi\)
0.265220 + 0.964188i \(0.414555\pi\)
\(492\) 0 0
\(493\) −38.5990 −1.73841
\(494\) −1.40268 −0.0631095
\(495\) 0 0
\(496\) −4.94841 −0.222190
\(497\) 25.7863 1.15667
\(498\) 0 0
\(499\) 26.6640 1.19364 0.596822 0.802374i \(-0.296430\pi\)
0.596822 + 0.802374i \(0.296430\pi\)
\(500\) 10.4235 0.466151
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) −11.1622 −0.497696 −0.248848 0.968543i \(-0.580052\pi\)
−0.248848 + 0.968543i \(0.580052\pi\)
\(504\) 0 0
\(505\) 17.2596 0.768043
\(506\) 6.80536 0.302535
\(507\) 0 0
\(508\) 11.7139 0.519720
\(509\) 35.9618 1.59398 0.796989 0.603993i \(-0.206425\pi\)
0.796989 + 0.603993i \(0.206425\pi\)
\(510\) 0 0
\(511\) −34.2479 −1.51504
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 24.2331 1.06888
\(515\) 19.1166 0.842377
\(516\) 0 0
\(517\) −5.25963 −0.231318
\(518\) −8.00000 −0.351500
\(519\) 0 0
\(520\) −1.72128 −0.0754829
\(521\) −10.1300 −0.443802 −0.221901 0.975069i \(-0.571226\pi\)
−0.221901 + 0.975069i \(0.571226\pi\)
\(522\) 0 0
\(523\) −7.93502 −0.346974 −0.173487 0.984836i \(-0.555503\pi\)
−0.173487 + 0.984836i \(0.555503\pi\)
\(524\) −1.85695 −0.0811211
\(525\) 0 0
\(526\) 20.5665 0.896742
\(527\) −23.7789 −1.03582
\(528\) 0 0
\(529\) 23.3129 1.01360
\(530\) 5.46599 0.237427
\(531\) 0 0
\(532\) −3.40268 −0.147525
\(533\) 4.13567 0.179136
\(534\) 0 0
\(535\) −5.33937 −0.230841
\(536\) −3.40268 −0.146973
\(537\) 0 0
\(538\) −1.36281 −0.0587548
\(539\) 4.57822 0.197198
\(540\) 0 0
\(541\) 10.8436 0.466201 0.233100 0.972453i \(-0.425113\pi\)
0.233100 + 0.972453i \(0.425113\pi\)
\(542\) 23.1963 0.996367
\(543\) 0 0
\(544\) −4.80536 −0.206028
\(545\) −7.36281 −0.315388
\(546\) 0 0
\(547\) −10.3129 −0.440947 −0.220474 0.975393i \(-0.570760\pi\)
−0.220474 + 0.975393i \(0.570760\pi\)
\(548\) −10.5973 −0.452695
\(549\) 0 0
\(550\) 3.49414 0.148991
\(551\) 8.03249 0.342196
\(552\) 0 0
\(553\) −12.6372 −0.537388
\(554\) −31.0385 −1.31870
\(555\) 0 0
\(556\) 14.7347 0.624889
\(557\) 14.4811 0.613582 0.306791 0.951777i \(-0.400745\pi\)
0.306791 + 0.951777i \(0.400745\pi\)
\(558\) 0 0
\(559\) 12.7980 0.541297
\(560\) −4.17554 −0.176449
\(561\) 0 0
\(562\) −2.77287 −0.116966
\(563\) −10.5340 −0.443956 −0.221978 0.975052i \(-0.571251\pi\)
−0.221978 + 0.975052i \(0.571251\pi\)
\(564\) 0 0
\(565\) −24.4958 −1.03055
\(566\) 9.23451 0.388156
\(567\) 0 0
\(568\) −7.57822 −0.317975
\(569\) −10.3762 −0.434993 −0.217496 0.976061i \(-0.569789\pi\)
−0.217496 + 0.976061i \(0.569789\pi\)
\(570\) 0 0
\(571\) 33.2847 1.39292 0.696461 0.717594i \(-0.254757\pi\)
0.696461 + 0.717594i \(0.254757\pi\)
\(572\) −1.40268 −0.0586489
\(573\) 0 0
\(574\) 10.0325 0.418748
\(575\) 23.7789 0.991648
\(576\) 0 0
\(577\) −2.93365 −0.122129 −0.0610647 0.998134i \(-0.519450\pi\)
−0.0610647 + 0.998134i \(0.519450\pi\)
\(578\) −6.09146 −0.253371
\(579\) 0 0
\(580\) 9.85695 0.409287
\(581\) −36.3055 −1.50621
\(582\) 0 0
\(583\) 4.45427 0.184477
\(584\) 10.0650 0.416492
\(585\) 0 0
\(586\) 17.7390 0.732792
\(587\) −8.90854 −0.367695 −0.183847 0.982955i \(-0.558855\pi\)
−0.183847 + 0.982955i \(0.558855\pi\)
\(588\) 0 0
\(589\) 4.94841 0.203896
\(590\) 17.8171 0.733517
\(591\) 0 0
\(592\) 2.35109 0.0966292
\(593\) −16.3129 −0.669890 −0.334945 0.942238i \(-0.608718\pi\)
−0.334945 + 0.942238i \(0.608718\pi\)
\(594\) 0 0
\(595\) −20.0650 −0.822584
\(596\) −18.3511 −0.751690
\(597\) 0 0
\(598\) −9.54573 −0.390354
\(599\) −40.9826 −1.67450 −0.837251 0.546818i \(-0.815839\pi\)
−0.837251 + 0.546818i \(0.815839\pi\)
\(600\) 0 0
\(601\) 0.349413 0.0142528 0.00712642 0.999975i \(-0.497732\pi\)
0.00712642 + 0.999975i \(0.497732\pi\)
\(602\) 31.0459 1.26534
\(603\) 0 0
\(604\) −7.64891 −0.311230
\(605\) −1.22713 −0.0498901
\(606\) 0 0
\(607\) 5.56049 0.225693 0.112847 0.993612i \(-0.464003\pi\)
0.112847 + 0.993612i \(0.464003\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 2.45427 0.0993704
\(611\) 7.37757 0.298464
\(612\) 0 0
\(613\) −19.7407 −0.797319 −0.398659 0.917099i \(-0.630524\pi\)
−0.398659 + 0.917099i \(0.630524\pi\)
\(614\) −20.9735 −0.846422
\(615\) 0 0
\(616\) −3.40268 −0.137098
\(617\) −10.5973 −0.426632 −0.213316 0.976983i \(-0.568426\pi\)
−0.213316 + 0.976983i \(0.568426\pi\)
\(618\) 0 0
\(619\) 4.22112 0.169661 0.0848306 0.996395i \(-0.472965\pi\)
0.0848306 + 0.996395i \(0.472965\pi\)
\(620\) 6.07236 0.243872
\(621\) 0 0
\(622\) −3.36281 −0.134836
\(623\) −51.0003 −2.04328
\(624\) 0 0
\(625\) 4.67973 0.187189
\(626\) 4.94103 0.197483
\(627\) 0 0
\(628\) 8.38358 0.334541
\(629\) 11.2978 0.450474
\(630\) 0 0
\(631\) 13.6905 0.545009 0.272504 0.962155i \(-0.412148\pi\)
0.272504 + 0.962155i \(0.412148\pi\)
\(632\) 3.71390 0.147731
\(633\) 0 0
\(634\) 14.2861 0.567374
\(635\) −14.3745 −0.570436
\(636\) 0 0
\(637\) −6.42178 −0.254440
\(638\) 8.03249 0.318009
\(639\) 0 0
\(640\) 1.22713 0.0485067
\(641\) 42.9883 1.69794 0.848968 0.528445i \(-0.177225\pi\)
0.848968 + 0.528445i \(0.177225\pi\)
\(642\) 0 0
\(643\) 16.3658 0.645406 0.322703 0.946500i \(-0.395408\pi\)
0.322703 + 0.946500i \(0.395408\pi\)
\(644\) −23.1564 −0.912492
\(645\) 0 0
\(646\) 4.80536 0.189064
\(647\) −39.9886 −1.57211 −0.786057 0.618154i \(-0.787881\pi\)
−0.786057 + 0.618154i \(0.787881\pi\)
\(648\) 0 0
\(649\) 14.5193 0.569931
\(650\) −4.90116 −0.192239
\(651\) 0 0
\(652\) −14.2479 −0.557991
\(653\) 37.5976 1.47131 0.735655 0.677357i \(-0.236875\pi\)
0.735655 + 0.677357i \(0.236875\pi\)
\(654\) 0 0
\(655\) 2.27872 0.0890371
\(656\) −2.94841 −0.115116
\(657\) 0 0
\(658\) 17.8968 0.697691
\(659\) 11.5075 0.448270 0.224135 0.974558i \(-0.428044\pi\)
0.224135 + 0.974558i \(0.428044\pi\)
\(660\) 0 0
\(661\) −29.2362 −1.13716 −0.568578 0.822629i \(-0.692506\pi\)
−0.568578 + 0.822629i \(0.692506\pi\)
\(662\) 31.9293 1.24097
\(663\) 0 0
\(664\) 10.6697 0.414064
\(665\) 4.17554 0.161921
\(666\) 0 0
\(667\) 54.6640 2.11660
\(668\) 10.8703 0.420586
\(669\) 0 0
\(670\) 4.17554 0.161315
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) −47.0784 −1.81474 −0.907369 0.420335i \(-0.861913\pi\)
−0.907369 + 0.420335i \(0.861913\pi\)
\(674\) 8.84523 0.340706
\(675\) 0 0
\(676\) −11.0325 −0.424327
\(677\) −43.2539 −1.66238 −0.831192 0.555986i \(-0.812341\pi\)
−0.831192 + 0.555986i \(0.812341\pi\)
\(678\) 0 0
\(679\) 8.00000 0.307012
\(680\) 5.89682 0.226133
\(681\) 0 0
\(682\) 4.94841 0.189484
\(683\) −34.8556 −1.33371 −0.666856 0.745187i \(-0.732360\pi\)
−0.666856 + 0.745187i \(0.732360\pi\)
\(684\) 0 0
\(685\) 13.0043 0.496870
\(686\) 8.24053 0.314625
\(687\) 0 0
\(688\) −9.12395 −0.347847
\(689\) −6.24791 −0.238026
\(690\) 0 0
\(691\) 28.4811 1.08347 0.541735 0.840549i \(-0.317768\pi\)
0.541735 + 0.840549i \(0.317768\pi\)
\(692\) −3.39530 −0.129070
\(693\) 0 0
\(694\) 5.27439 0.200213
\(695\) −18.0814 −0.685867
\(696\) 0 0
\(697\) −14.1682 −0.536657
\(698\) 10.9085 0.412894
\(699\) 0 0
\(700\) −11.8894 −0.449379
\(701\) −12.8703 −0.486106 −0.243053 0.970013i \(-0.578149\pi\)
−0.243053 + 0.970013i \(0.578149\pi\)
\(702\) 0 0
\(703\) −2.35109 −0.0886730
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −22.1300 −0.832872
\(707\) −47.8586 −1.79991
\(708\) 0 0
\(709\) 1.41744 0.0532330 0.0266165 0.999646i \(-0.491527\pi\)
0.0266165 + 0.999646i \(0.491527\pi\)
\(710\) 9.29950 0.349004
\(711\) 0 0
\(712\) 14.9883 0.561710
\(713\) 33.6757 1.26116
\(714\) 0 0
\(715\) 1.72128 0.0643721
\(716\) 0.311217 0.0116307
\(717\) 0 0
\(718\) −16.3910 −0.611705
\(719\) 31.1564 1.16194 0.580970 0.813925i \(-0.302673\pi\)
0.580970 + 0.813925i \(0.302673\pi\)
\(720\) 0 0
\(721\) −53.0077 −1.97411
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) −15.3246 −0.569535
\(725\) 28.0667 1.04237
\(726\) 0 0
\(727\) −8.57221 −0.317926 −0.158963 0.987285i \(-0.550815\pi\)
−0.158963 + 0.987285i \(0.550815\pi\)
\(728\) 4.77287 0.176894
\(729\) 0 0
\(730\) −12.3511 −0.457134
\(731\) −43.8439 −1.62162
\(732\) 0 0
\(733\) 44.6787 1.65025 0.825123 0.564952i \(-0.191105\pi\)
0.825123 + 0.564952i \(0.191105\pi\)
\(734\) 32.3511 1.19410
\(735\) 0 0
\(736\) 6.80536 0.250849
\(737\) 3.40268 0.125339
\(738\) 0 0
\(739\) 29.4898 1.08480 0.542400 0.840120i \(-0.317516\pi\)
0.542400 + 0.840120i \(0.317516\pi\)
\(740\) −2.88510 −0.106058
\(741\) 0 0
\(742\) −15.1564 −0.556411
\(743\) 13.2596 0.486449 0.243224 0.969970i \(-0.421795\pi\)
0.243224 + 0.969970i \(0.421795\pi\)
\(744\) 0 0
\(745\) 22.5193 0.825042
\(746\) −17.5782 −0.643584
\(747\) 0 0
\(748\) 4.80536 0.175701
\(749\) 14.8054 0.540976
\(750\) 0 0
\(751\) −10.1829 −0.371580 −0.185790 0.982589i \(-0.559484\pi\)
−0.185790 + 0.982589i \(0.559484\pi\)
\(752\) −5.25963 −0.191799
\(753\) 0 0
\(754\) −11.2670 −0.410320
\(755\) 9.38624 0.341600
\(756\) 0 0
\(757\) 40.1122 1.45790 0.728952 0.684565i \(-0.240008\pi\)
0.728952 + 0.684565i \(0.240008\pi\)
\(758\) −8.93365 −0.324485
\(759\) 0 0
\(760\) −1.22713 −0.0445128
\(761\) 38.7819 1.40584 0.702922 0.711267i \(-0.251878\pi\)
0.702922 + 0.711267i \(0.251878\pi\)
\(762\) 0 0
\(763\) 20.4161 0.739111
\(764\) −26.8703 −0.972135
\(765\) 0 0
\(766\) −16.0399 −0.579544
\(767\) −20.3658 −0.735368
\(768\) 0 0
\(769\) 8.15340 0.294019 0.147010 0.989135i \(-0.453035\pi\)
0.147010 + 0.989135i \(0.453035\pi\)
\(770\) 4.17554 0.150476
\(771\) 0 0
\(772\) 18.5665 0.668223
\(773\) 24.3893 0.877222 0.438611 0.898677i \(-0.355471\pi\)
0.438611 + 0.898677i \(0.355471\pi\)
\(774\) 0 0
\(775\) 17.2904 0.621091
\(776\) −2.35109 −0.0843992
\(777\) 0 0
\(778\) 17.2271 0.617623
\(779\) 2.94841 0.105638
\(780\) 0 0
\(781\) 7.57822 0.271170
\(782\) 32.7022 1.16943
\(783\) 0 0
\(784\) 4.57822 0.163508
\(785\) −10.2878 −0.367187
\(786\) 0 0
\(787\) −46.3779 −1.65319 −0.826596 0.562795i \(-0.809726\pi\)
−0.826596 + 0.562795i \(0.809726\pi\)
\(788\) −14.9085 −0.531095
\(789\) 0 0
\(790\) −4.55745 −0.162147
\(791\) 67.9236 2.41509
\(792\) 0 0
\(793\) −2.80536 −0.0996212
\(794\) −15.0784 −0.535112
\(795\) 0 0
\(796\) 15.1564 0.537206
\(797\) −3.89682 −0.138032 −0.0690162 0.997616i \(-0.521986\pi\)
−0.0690162 + 0.997616i \(0.521986\pi\)
\(798\) 0 0
\(799\) −25.2744 −0.894144
\(800\) 3.49414 0.123537
\(801\) 0 0
\(802\) −13.1564 −0.464570
\(803\) −10.0650 −0.355186
\(804\) 0 0
\(805\) 28.4161 1.00153
\(806\) −6.94103 −0.244487
\(807\) 0 0
\(808\) 14.0650 0.494804
\(809\) 43.3896 1.52550 0.762748 0.646695i \(-0.223849\pi\)
0.762748 + 0.646695i \(0.223849\pi\)
\(810\) 0 0
\(811\) −12.9233 −0.453798 −0.226899 0.973918i \(-0.572859\pi\)
−0.226899 + 0.973918i \(0.572859\pi\)
\(812\) −27.3320 −0.959165
\(813\) 0 0
\(814\) −2.35109 −0.0824056
\(815\) 17.4841 0.612441
\(816\) 0 0
\(817\) 9.12395 0.319207
\(818\) 16.3836 0.572838
\(819\) 0 0
\(820\) 3.61810 0.126349
\(821\) 36.5842 1.27680 0.638399 0.769705i \(-0.279597\pi\)
0.638399 + 0.769705i \(0.279597\pi\)
\(822\) 0 0
\(823\) −21.6107 −0.753302 −0.376651 0.926355i \(-0.622924\pi\)
−0.376651 + 0.926355i \(0.622924\pi\)
\(824\) 15.5782 0.542693
\(825\) 0 0
\(826\) −49.4044 −1.71900
\(827\) 34.8851 1.21307 0.606537 0.795055i \(-0.292558\pi\)
0.606537 + 0.795055i \(0.292558\pi\)
\(828\) 0 0
\(829\) 31.7407 1.10240 0.551200 0.834373i \(-0.314170\pi\)
0.551200 + 0.834373i \(0.314170\pi\)
\(830\) −13.0931 −0.454469
\(831\) 0 0
\(832\) −1.40268 −0.0486291
\(833\) 22.0000 0.762255
\(834\) 0 0
\(835\) −13.3394 −0.461628
\(836\) −1.00000 −0.0345857
\(837\) 0 0
\(838\) 1.12966 0.0390234
\(839\) −13.4603 −0.464701 −0.232350 0.972632i \(-0.574642\pi\)
−0.232350 + 0.972632i \(0.574642\pi\)
\(840\) 0 0
\(841\) 35.5209 1.22486
\(842\) 2.63719 0.0908837
\(843\) 0 0
\(844\) 11.1564 0.384021
\(845\) 13.5384 0.465733
\(846\) 0 0
\(847\) 3.40268 0.116917
\(848\) 4.45427 0.152960
\(849\) 0 0
\(850\) 16.7906 0.575913
\(851\) −16.0000 −0.548473
\(852\) 0 0
\(853\) 36.2981 1.24282 0.621412 0.783484i \(-0.286559\pi\)
0.621412 + 0.783484i \(0.286559\pi\)
\(854\) −6.80536 −0.232875
\(855\) 0 0
\(856\) −4.35109 −0.148717
\(857\) 43.8569 1.49812 0.749062 0.662499i \(-0.230504\pi\)
0.749062 + 0.662499i \(0.230504\pi\)
\(858\) 0 0
\(859\) −41.1980 −1.40566 −0.702829 0.711359i \(-0.748080\pi\)
−0.702829 + 0.711359i \(0.748080\pi\)
\(860\) 11.1963 0.381791
\(861\) 0 0
\(862\) −28.4958 −0.970571
\(863\) 24.8720 0.846653 0.423327 0.905977i \(-0.360862\pi\)
0.423327 + 0.905977i \(0.360862\pi\)
\(864\) 0 0
\(865\) 4.16649 0.141665
\(866\) 37.5075 1.27456
\(867\) 0 0
\(868\) −16.8378 −0.571514
\(869\) −3.71390 −0.125985
\(870\) 0 0
\(871\) −4.77287 −0.161722
\(872\) −6.00000 −0.203186
\(873\) 0 0
\(874\) −6.80536 −0.230195
\(875\) 35.4677 1.19903
\(876\) 0 0
\(877\) 39.8911 1.34703 0.673514 0.739175i \(-0.264784\pi\)
0.673514 + 0.739175i \(0.264784\pi\)
\(878\) 32.1300 1.08433
\(879\) 0 0
\(880\) −1.22713 −0.0413667
\(881\) 2.88343 0.0971451 0.0485725 0.998820i \(-0.484533\pi\)
0.0485725 + 0.998820i \(0.484533\pi\)
\(882\) 0 0
\(883\) −4.27134 −0.143742 −0.0718711 0.997414i \(-0.522897\pi\)
−0.0718711 + 0.997414i \(0.522897\pi\)
\(884\) −6.74037 −0.226703
\(885\) 0 0
\(886\) 33.0385 1.10995
\(887\) −22.3129 −0.749194 −0.374597 0.927188i \(-0.622219\pi\)
−0.374597 + 0.927188i \(0.622219\pi\)
\(888\) 0 0
\(889\) 39.8586 1.33682
\(890\) −18.3926 −0.616523
\(891\) 0 0
\(892\) 1.61072 0.0539307
\(893\) 5.25963 0.176007
\(894\) 0 0
\(895\) −0.381905 −0.0127657
\(896\) −3.40268 −0.113676
\(897\) 0 0
\(898\) 7.88206 0.263028
\(899\) 39.7481 1.32567
\(900\) 0 0
\(901\) 21.4044 0.713082
\(902\) 2.94841 0.0981713
\(903\) 0 0
\(904\) −19.9618 −0.663920
\(905\) 18.8054 0.625111
\(906\) 0 0
\(907\) −35.5578 −1.18068 −0.590338 0.807156i \(-0.701006\pi\)
−0.590338 + 0.807156i \(0.701006\pi\)
\(908\) 16.7672 0.556438
\(909\) 0 0
\(910\) −5.85695 −0.194156
\(911\) 49.1980 1.63000 0.815001 0.579459i \(-0.196736\pi\)
0.815001 + 0.579459i \(0.196736\pi\)
\(912\) 0 0
\(913\) −10.6697 −0.353115
\(914\) −24.9353 −0.824787
\(915\) 0 0
\(916\) −8.55913 −0.282801
\(917\) −6.31860 −0.208658
\(918\) 0 0
\(919\) −8.47200 −0.279466 −0.139733 0.990189i \(-0.544624\pi\)
−0.139733 + 0.990189i \(0.544624\pi\)
\(920\) −8.35109 −0.275327
\(921\) 0 0
\(922\) −26.6874 −0.878903
\(923\) −10.6298 −0.349885
\(924\) 0 0
\(925\) −8.21504 −0.270109
\(926\) 32.7022 1.07466
\(927\) 0 0
\(928\) 8.03249 0.263679
\(929\) 16.0017 0.524998 0.262499 0.964932i \(-0.415453\pi\)
0.262499 + 0.964932i \(0.415453\pi\)
\(930\) 0 0
\(931\) −4.57822 −0.150045
\(932\) −5.44255 −0.178277
\(933\) 0 0
\(934\) 11.3776 0.372285
\(935\) −5.89682 −0.192847
\(936\) 0 0
\(937\) −15.9766 −0.521932 −0.260966 0.965348i \(-0.584041\pi\)
−0.260966 + 0.965348i \(0.584041\pi\)
\(938\) −11.5782 −0.378042
\(939\) 0 0
\(940\) 6.45427 0.210515
\(941\) 21.0915 0.687562 0.343781 0.939050i \(-0.388292\pi\)
0.343781 + 0.939050i \(0.388292\pi\)
\(942\) 0 0
\(943\) 20.0650 0.653406
\(944\) 14.5193 0.472561
\(945\) 0 0
\(946\) 9.12395 0.296645
\(947\) −20.9735 −0.681548 −0.340774 0.940145i \(-0.610689\pi\)
−0.340774 + 0.940145i \(0.610689\pi\)
\(948\) 0 0
\(949\) 14.1179 0.458288
\(950\) −3.49414 −0.113365
\(951\) 0 0
\(952\) −16.3511 −0.529942
\(953\) −3.27439 −0.106068 −0.0530339 0.998593i \(-0.516889\pi\)
−0.0530339 + 0.998593i \(0.516889\pi\)
\(954\) 0 0
\(955\) 32.9735 1.06700
\(956\) −11.2921 −0.365213
\(957\) 0 0
\(958\) −12.5973 −0.407001
\(959\) −36.0593 −1.16441
\(960\) 0 0
\(961\) −6.51324 −0.210104
\(962\) 3.29782 0.106326
\(963\) 0 0
\(964\) 11.4101 0.367493
\(965\) −22.7836 −0.733430
\(966\) 0 0
\(967\) −29.2744 −0.941401 −0.470700 0.882293i \(-0.655999\pi\)
−0.470700 + 0.882293i \(0.655999\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 2.88510 0.0926350
\(971\) 38.3836 1.23179 0.615894 0.787829i \(-0.288795\pi\)
0.615894 + 0.787829i \(0.288795\pi\)
\(972\) 0 0
\(973\) 50.1373 1.60733
\(974\) −9.69616 −0.310685
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −10.9233 −0.349467 −0.174734 0.984616i \(-0.555906\pi\)
−0.174734 + 0.984616i \(0.555906\pi\)
\(978\) 0 0
\(979\) −14.9883 −0.479028
\(980\) −5.61810 −0.179463
\(981\) 0 0
\(982\) −11.7538 −0.375078
\(983\) 32.1196 1.02446 0.512228 0.858849i \(-0.328820\pi\)
0.512228 + 0.858849i \(0.328820\pi\)
\(984\) 0 0
\(985\) 18.2948 0.582920
\(986\) 38.5990 1.22924
\(987\) 0 0
\(988\) 1.40268 0.0446252
\(989\) 62.0918 1.97440
\(990\) 0 0
\(991\) 4.37620 0.139015 0.0695073 0.997581i \(-0.477857\pi\)
0.0695073 + 0.997581i \(0.477857\pi\)
\(992\) 4.94841 0.157112
\(993\) 0 0
\(994\) −25.7863 −0.817890
\(995\) −18.5990 −0.589628
\(996\) 0 0
\(997\) −8.37788 −0.265330 −0.132665 0.991161i \(-0.542353\pi\)
−0.132665 + 0.991161i \(0.542353\pi\)
\(998\) −26.6640 −0.844034
\(999\) 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 3762.2.a.bd.1.2 3
3.2 odd 2 418.2.a.h.1.2 3
12.11 even 2 3344.2.a.p.1.2 3
33.32 even 2 4598.2.a.bm.1.2 3
57.56 even 2 7942.2.a.bc.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.h.1.2 3 3.2 odd 2
3344.2.a.p.1.2 3 12.11 even 2
3762.2.a.bd.1.2 3 1.1 even 1 trivial
4598.2.a.bm.1.2 3 33.32 even 2
7942.2.a.bc.1.2 3 57.56 even 2