Properties

Label 8046.2.a.k.1.8
Level $8046$
Weight $2$
Character 8046.1
Self dual yes
Analytic conductor $64.248$
Analytic rank $0$
Dimension $12$
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] = [8046,2,Mod(1,8046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8046, 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("8046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8046.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: \(64.2476334663\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 31 x^{10} + 82 x^{9} + 334 x^{8} - 684 x^{7} - 1561 x^{6} + 1551 x^{5} + 3573 x^{4} + 345 x^{3} - 1607 x^{2} - 594 x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.761360\) of defining polynomial
Character \(\chi\) \(=\) 8046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +0.761360 q^{5} +1.23946 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +0.761360 q^{5} +1.23946 q^{7} -1.00000 q^{8} -0.761360 q^{10} +4.16498 q^{11} +2.92984 q^{13} -1.23946 q^{14} +1.00000 q^{16} +2.05222 q^{17} +0.666318 q^{19} +0.761360 q^{20} -4.16498 q^{22} +8.48075 q^{23} -4.42033 q^{25} -2.92984 q^{26} +1.23946 q^{28} -1.77307 q^{29} +3.84602 q^{31} -1.00000 q^{32} -2.05222 q^{34} +0.943679 q^{35} +4.44887 q^{37} -0.666318 q^{38} -0.761360 q^{40} -2.52486 q^{41} +5.24402 q^{43} +4.16498 q^{44} -8.48075 q^{46} -3.71322 q^{47} -5.46373 q^{49} +4.42033 q^{50} +2.92984 q^{52} +8.26592 q^{53} +3.17105 q^{55} -1.23946 q^{56} +1.77307 q^{58} +9.37120 q^{59} +10.0229 q^{61} -3.84602 q^{62} +1.00000 q^{64} +2.23066 q^{65} -1.45888 q^{67} +2.05222 q^{68} -0.943679 q^{70} +6.19297 q^{71} -10.9177 q^{73} -4.44887 q^{74} +0.666318 q^{76} +5.16234 q^{77} -9.54970 q^{79} +0.761360 q^{80} +2.52486 q^{82} +5.69162 q^{83} +1.56248 q^{85} -5.24402 q^{86} -4.16498 q^{88} +7.72882 q^{89} +3.63143 q^{91} +8.48075 q^{92} +3.71322 q^{94} +0.507308 q^{95} -18.6599 q^{97} +5.46373 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} + 3 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} + 3 q^{5} - 6 q^{7} - 12 q^{8} - 3 q^{10} + 14 q^{11} - 3 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} - 4 q^{19} + 3 q^{20} - 14 q^{22} + 13 q^{23} + 11 q^{25} + 3 q^{26} - 6 q^{28} + 23 q^{29} - 14 q^{31} - 12 q^{32} - 8 q^{34} + 32 q^{35} - 19 q^{37} + 4 q^{38} - 3 q^{40} + 30 q^{41} - 15 q^{43} + 14 q^{44} - 13 q^{46} - q^{47} + 14 q^{49} - 11 q^{50} - 3 q^{52} + 16 q^{53} - 7 q^{55} + 6 q^{56} - 23 q^{58} + 26 q^{59} - 16 q^{61} + 14 q^{62} + 12 q^{64} + 8 q^{65} - 39 q^{67} + 8 q^{68} - 32 q^{70} + 15 q^{71} - 2 q^{73} + 19 q^{74} - 4 q^{76} + 34 q^{77} - 13 q^{79} + 3 q^{80} - 30 q^{82} + 6 q^{83} - 11 q^{85} + 15 q^{86} - 14 q^{88} + 18 q^{89} - 35 q^{91} + 13 q^{92} + q^{94} + 51 q^{95} + 19 q^{97} - 14 q^{98}+O(q^{100}) \) 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
\(5\) 0.761360 0.340491 0.170245 0.985402i \(-0.445544\pi\)
0.170245 + 0.985402i \(0.445544\pi\)
\(6\) 0 0
\(7\) 1.23946 0.468474 0.234237 0.972180i \(-0.424741\pi\)
0.234237 + 0.972180i \(0.424741\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −0.761360 −0.240763
\(11\) 4.16498 1.25579 0.627894 0.778299i \(-0.283917\pi\)
0.627894 + 0.778299i \(0.283917\pi\)
\(12\) 0 0
\(13\) 2.92984 0.812591 0.406296 0.913742i \(-0.366820\pi\)
0.406296 + 0.913742i \(0.366820\pi\)
\(14\) −1.23946 −0.331261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.05222 0.497737 0.248868 0.968537i \(-0.419941\pi\)
0.248868 + 0.968537i \(0.419941\pi\)
\(18\) 0 0
\(19\) 0.666318 0.152864 0.0764319 0.997075i \(-0.475647\pi\)
0.0764319 + 0.997075i \(0.475647\pi\)
\(20\) 0.761360 0.170245
\(21\) 0 0
\(22\) −4.16498 −0.887976
\(23\) 8.48075 1.76836 0.884179 0.467148i \(-0.154719\pi\)
0.884179 + 0.467148i \(0.154719\pi\)
\(24\) 0 0
\(25\) −4.42033 −0.884066
\(26\) −2.92984 −0.574589
\(27\) 0 0
\(28\) 1.23946 0.234237
\(29\) −1.77307 −0.329250 −0.164625 0.986356i \(-0.552641\pi\)
−0.164625 + 0.986356i \(0.552641\pi\)
\(30\) 0 0
\(31\) 3.84602 0.690765 0.345382 0.938462i \(-0.387749\pi\)
0.345382 + 0.938462i \(0.387749\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.05222 −0.351953
\(35\) 0.943679 0.159511
\(36\) 0 0
\(37\) 4.44887 0.731390 0.365695 0.930735i \(-0.380831\pi\)
0.365695 + 0.930735i \(0.380831\pi\)
\(38\) −0.666318 −0.108091
\(39\) 0 0
\(40\) −0.761360 −0.120382
\(41\) −2.52486 −0.394317 −0.197158 0.980372i \(-0.563171\pi\)
−0.197158 + 0.980372i \(0.563171\pi\)
\(42\) 0 0
\(43\) 5.24402 0.799706 0.399853 0.916579i \(-0.369061\pi\)
0.399853 + 0.916579i \(0.369061\pi\)
\(44\) 4.16498 0.627894
\(45\) 0 0
\(46\) −8.48075 −1.25042
\(47\) −3.71322 −0.541630 −0.270815 0.962631i \(-0.587293\pi\)
−0.270815 + 0.962631i \(0.587293\pi\)
\(48\) 0 0
\(49\) −5.46373 −0.780533
\(50\) 4.42033 0.625129
\(51\) 0 0
\(52\) 2.92984 0.406296
\(53\) 8.26592 1.13541 0.567706 0.823231i \(-0.307831\pi\)
0.567706 + 0.823231i \(0.307831\pi\)
\(54\) 0 0
\(55\) 3.17105 0.427584
\(56\) −1.23946 −0.165630
\(57\) 0 0
\(58\) 1.77307 0.232815
\(59\) 9.37120 1.22003 0.610013 0.792391i \(-0.291164\pi\)
0.610013 + 0.792391i \(0.291164\pi\)
\(60\) 0 0
\(61\) 10.0229 1.28330 0.641649 0.766998i \(-0.278251\pi\)
0.641649 + 0.766998i \(0.278251\pi\)
\(62\) −3.84602 −0.488445
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.23066 0.276680
\(66\) 0 0
\(67\) −1.45888 −0.178231 −0.0891155 0.996021i \(-0.528404\pi\)
−0.0891155 + 0.996021i \(0.528404\pi\)
\(68\) 2.05222 0.248868
\(69\) 0 0
\(70\) −0.943679 −0.112791
\(71\) 6.19297 0.734970 0.367485 0.930029i \(-0.380219\pi\)
0.367485 + 0.930029i \(0.380219\pi\)
\(72\) 0 0
\(73\) −10.9177 −1.27782 −0.638910 0.769281i \(-0.720614\pi\)
−0.638910 + 0.769281i \(0.720614\pi\)
\(74\) −4.44887 −0.517171
\(75\) 0 0
\(76\) 0.666318 0.0764319
\(77\) 5.16234 0.588304
\(78\) 0 0
\(79\) −9.54970 −1.07443 −0.537213 0.843447i \(-0.680523\pi\)
−0.537213 + 0.843447i \(0.680523\pi\)
\(80\) 0.761360 0.0851227
\(81\) 0 0
\(82\) 2.52486 0.278824
\(83\) 5.69162 0.624737 0.312368 0.949961i \(-0.398878\pi\)
0.312368 + 0.949961i \(0.398878\pi\)
\(84\) 0 0
\(85\) 1.56248 0.169475
\(86\) −5.24402 −0.565477
\(87\) 0 0
\(88\) −4.16498 −0.443988
\(89\) 7.72882 0.819253 0.409627 0.912253i \(-0.365659\pi\)
0.409627 + 0.912253i \(0.365659\pi\)
\(90\) 0 0
\(91\) 3.63143 0.380678
\(92\) 8.48075 0.884179
\(93\) 0 0
\(94\) 3.71322 0.382990
\(95\) 0.507308 0.0520487
\(96\) 0 0
\(97\) −18.6599 −1.89463 −0.947315 0.320304i \(-0.896215\pi\)
−0.947315 + 0.320304i \(0.896215\pi\)
\(98\) 5.46373 0.551920
\(99\) 0 0
\(100\) −4.42033 −0.442033
\(101\) 5.89482 0.586557 0.293278 0.956027i \(-0.405254\pi\)
0.293278 + 0.956027i \(0.405254\pi\)
\(102\) 0 0
\(103\) −3.77939 −0.372395 −0.186197 0.982512i \(-0.559616\pi\)
−0.186197 + 0.982512i \(0.559616\pi\)
\(104\) −2.92984 −0.287294
\(105\) 0 0
\(106\) −8.26592 −0.802858
\(107\) 7.59349 0.734090 0.367045 0.930203i \(-0.380369\pi\)
0.367045 + 0.930203i \(0.380369\pi\)
\(108\) 0 0
\(109\) 1.05733 0.101274 0.0506371 0.998717i \(-0.483875\pi\)
0.0506371 + 0.998717i \(0.483875\pi\)
\(110\) −3.17105 −0.302348
\(111\) 0 0
\(112\) 1.23946 0.117118
\(113\) −9.52892 −0.896405 −0.448203 0.893932i \(-0.647936\pi\)
−0.448203 + 0.893932i \(0.647936\pi\)
\(114\) 0 0
\(115\) 6.45691 0.602110
\(116\) −1.77307 −0.164625
\(117\) 0 0
\(118\) −9.37120 −0.862689
\(119\) 2.54366 0.233177
\(120\) 0 0
\(121\) 6.34704 0.577004
\(122\) −10.0229 −0.907429
\(123\) 0 0
\(124\) 3.84602 0.345382
\(125\) −7.17227 −0.641507
\(126\) 0 0
\(127\) −15.5448 −1.37938 −0.689689 0.724106i \(-0.742253\pi\)
−0.689689 + 0.724106i \(0.742253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −2.23066 −0.195642
\(131\) 5.51152 0.481544 0.240772 0.970582i \(-0.422599\pi\)
0.240772 + 0.970582i \(0.422599\pi\)
\(132\) 0 0
\(133\) 0.825877 0.0716127
\(134\) 1.45888 0.126028
\(135\) 0 0
\(136\) −2.05222 −0.175977
\(137\) −10.2474 −0.875495 −0.437747 0.899098i \(-0.644224\pi\)
−0.437747 + 0.899098i \(0.644224\pi\)
\(138\) 0 0
\(139\) −9.23075 −0.782942 −0.391471 0.920190i \(-0.628034\pi\)
−0.391471 + 0.920190i \(0.628034\pi\)
\(140\) 0.943679 0.0797555
\(141\) 0 0
\(142\) −6.19297 −0.519702
\(143\) 12.2027 1.02044
\(144\) 0 0
\(145\) −1.34994 −0.112107
\(146\) 10.9177 0.903555
\(147\) 0 0
\(148\) 4.44887 0.365695
\(149\) −1.00000 −0.0819232
\(150\) 0 0
\(151\) 8.58020 0.698247 0.349124 0.937077i \(-0.386479\pi\)
0.349124 + 0.937077i \(0.386479\pi\)
\(152\) −0.666318 −0.0540455
\(153\) 0 0
\(154\) −5.16234 −0.415993
\(155\) 2.92821 0.235199
\(156\) 0 0
\(157\) 9.08853 0.725344 0.362672 0.931917i \(-0.381865\pi\)
0.362672 + 0.931917i \(0.381865\pi\)
\(158\) 9.54970 0.759733
\(159\) 0 0
\(160\) −0.761360 −0.0601908
\(161\) 10.5116 0.828429
\(162\) 0 0
\(163\) 13.3116 1.04264 0.521322 0.853360i \(-0.325439\pi\)
0.521322 + 0.853360i \(0.325439\pi\)
\(164\) −2.52486 −0.197158
\(165\) 0 0
\(166\) −5.69162 −0.441756
\(167\) −6.38977 −0.494455 −0.247228 0.968957i \(-0.579520\pi\)
−0.247228 + 0.968957i \(0.579520\pi\)
\(168\) 0 0
\(169\) −4.41604 −0.339695
\(170\) −1.56248 −0.119837
\(171\) 0 0
\(172\) 5.24402 0.399853
\(173\) 1.23167 0.0936421 0.0468211 0.998903i \(-0.485091\pi\)
0.0468211 + 0.998903i \(0.485091\pi\)
\(174\) 0 0
\(175\) −5.47884 −0.414162
\(176\) 4.16498 0.313947
\(177\) 0 0
\(178\) −7.72882 −0.579299
\(179\) −10.4034 −0.777585 −0.388792 0.921325i \(-0.627108\pi\)
−0.388792 + 0.921325i \(0.627108\pi\)
\(180\) 0 0
\(181\) 5.03575 0.374304 0.187152 0.982331i \(-0.440074\pi\)
0.187152 + 0.982331i \(0.440074\pi\)
\(182\) −3.63143 −0.269180
\(183\) 0 0
\(184\) −8.48075 −0.625209
\(185\) 3.38720 0.249032
\(186\) 0 0
\(187\) 8.54746 0.625052
\(188\) −3.71322 −0.270815
\(189\) 0 0
\(190\) −0.507308 −0.0368040
\(191\) −14.0973 −1.02004 −0.510022 0.860161i \(-0.670363\pi\)
−0.510022 + 0.860161i \(0.670363\pi\)
\(192\) 0 0
\(193\) 7.50264 0.540052 0.270026 0.962853i \(-0.412968\pi\)
0.270026 + 0.962853i \(0.412968\pi\)
\(194\) 18.6599 1.33971
\(195\) 0 0
\(196\) −5.46373 −0.390266
\(197\) −8.01296 −0.570900 −0.285450 0.958394i \(-0.592143\pi\)
−0.285450 + 0.958394i \(0.592143\pi\)
\(198\) 0 0
\(199\) −14.7857 −1.04813 −0.524064 0.851679i \(-0.675585\pi\)
−0.524064 + 0.851679i \(0.675585\pi\)
\(200\) 4.42033 0.312565
\(201\) 0 0
\(202\) −5.89482 −0.414758
\(203\) −2.19765 −0.154245
\(204\) 0 0
\(205\) −1.92233 −0.134261
\(206\) 3.77939 0.263323
\(207\) 0 0
\(208\) 2.92984 0.203148
\(209\) 2.77520 0.191965
\(210\) 0 0
\(211\) 3.54316 0.243921 0.121961 0.992535i \(-0.461082\pi\)
0.121961 + 0.992535i \(0.461082\pi\)
\(212\) 8.26592 0.567706
\(213\) 0 0
\(214\) −7.59349 −0.519080
\(215\) 3.99259 0.272292
\(216\) 0 0
\(217\) 4.76700 0.323605
\(218\) −1.05733 −0.0716117
\(219\) 0 0
\(220\) 3.17105 0.213792
\(221\) 6.01268 0.404457
\(222\) 0 0
\(223\) 12.5260 0.838806 0.419403 0.907800i \(-0.362239\pi\)
0.419403 + 0.907800i \(0.362239\pi\)
\(224\) −1.23946 −0.0828152
\(225\) 0 0
\(226\) 9.52892 0.633854
\(227\) 14.2435 0.945376 0.472688 0.881230i \(-0.343284\pi\)
0.472688 + 0.881230i \(0.343284\pi\)
\(228\) 0 0
\(229\) −22.3396 −1.47624 −0.738120 0.674670i \(-0.764286\pi\)
−0.738120 + 0.674670i \(0.764286\pi\)
\(230\) −6.45691 −0.425756
\(231\) 0 0
\(232\) 1.77307 0.116407
\(233\) −13.9412 −0.913316 −0.456658 0.889642i \(-0.650954\pi\)
−0.456658 + 0.889642i \(0.650954\pi\)
\(234\) 0 0
\(235\) −2.82710 −0.184420
\(236\) 9.37120 0.610013
\(237\) 0 0
\(238\) −2.54366 −0.164881
\(239\) 15.8580 1.02577 0.512883 0.858458i \(-0.328577\pi\)
0.512883 + 0.858458i \(0.328577\pi\)
\(240\) 0 0
\(241\) −13.2150 −0.851253 −0.425626 0.904899i \(-0.639946\pi\)
−0.425626 + 0.904899i \(0.639946\pi\)
\(242\) −6.34704 −0.408003
\(243\) 0 0
\(244\) 10.0229 0.641649
\(245\) −4.15987 −0.265764
\(246\) 0 0
\(247\) 1.95220 0.124216
\(248\) −3.84602 −0.244222
\(249\) 0 0
\(250\) 7.17227 0.453614
\(251\) 8.73531 0.551368 0.275684 0.961248i \(-0.411096\pi\)
0.275684 + 0.961248i \(0.411096\pi\)
\(252\) 0 0
\(253\) 35.3221 2.22068
\(254\) 15.5448 0.975367
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 17.0182 1.06157 0.530785 0.847507i \(-0.321897\pi\)
0.530785 + 0.847507i \(0.321897\pi\)
\(258\) 0 0
\(259\) 5.51422 0.342637
\(260\) 2.23066 0.138340
\(261\) 0 0
\(262\) −5.51152 −0.340503
\(263\) 1.87735 0.115762 0.0578811 0.998323i \(-0.481566\pi\)
0.0578811 + 0.998323i \(0.481566\pi\)
\(264\) 0 0
\(265\) 6.29335 0.386597
\(266\) −0.825877 −0.0506378
\(267\) 0 0
\(268\) −1.45888 −0.0891155
\(269\) −17.0976 −1.04246 −0.521230 0.853416i \(-0.674527\pi\)
−0.521230 + 0.853416i \(0.674527\pi\)
\(270\) 0 0
\(271\) 2.52171 0.153183 0.0765916 0.997063i \(-0.475596\pi\)
0.0765916 + 0.997063i \(0.475596\pi\)
\(272\) 2.05222 0.124434
\(273\) 0 0
\(274\) 10.2474 0.619068
\(275\) −18.4106 −1.11020
\(276\) 0 0
\(277\) 20.3576 1.22317 0.611586 0.791178i \(-0.290532\pi\)
0.611586 + 0.791178i \(0.290532\pi\)
\(278\) 9.23075 0.553624
\(279\) 0 0
\(280\) −0.943679 −0.0563956
\(281\) −26.7498 −1.59576 −0.797880 0.602816i \(-0.794045\pi\)
−0.797880 + 0.602816i \(0.794045\pi\)
\(282\) 0 0
\(283\) −29.9892 −1.78267 −0.891336 0.453344i \(-0.850231\pi\)
−0.891336 + 0.453344i \(0.850231\pi\)
\(284\) 6.19297 0.367485
\(285\) 0 0
\(286\) −12.2027 −0.721562
\(287\) −3.12947 −0.184727
\(288\) 0 0
\(289\) −12.7884 −0.752258
\(290\) 1.34994 0.0792713
\(291\) 0 0
\(292\) −10.9177 −0.638910
\(293\) 5.78618 0.338032 0.169016 0.985613i \(-0.445941\pi\)
0.169016 + 0.985613i \(0.445941\pi\)
\(294\) 0 0
\(295\) 7.13486 0.415408
\(296\) −4.44887 −0.258586
\(297\) 0 0
\(298\) 1.00000 0.0579284
\(299\) 24.8472 1.43695
\(300\) 0 0
\(301\) 6.49978 0.374641
\(302\) −8.58020 −0.493735
\(303\) 0 0
\(304\) 0.666318 0.0382160
\(305\) 7.63102 0.436951
\(306\) 0 0
\(307\) −15.9386 −0.909665 −0.454832 0.890577i \(-0.650301\pi\)
−0.454832 + 0.890577i \(0.650301\pi\)
\(308\) 5.16234 0.294152
\(309\) 0 0
\(310\) −2.92821 −0.166311
\(311\) 18.5036 1.04924 0.524621 0.851336i \(-0.324207\pi\)
0.524621 + 0.851336i \(0.324207\pi\)
\(312\) 0 0
\(313\) −8.94571 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(314\) −9.08853 −0.512896
\(315\) 0 0
\(316\) −9.54970 −0.537213
\(317\) 33.2027 1.86485 0.932424 0.361365i \(-0.117689\pi\)
0.932424 + 0.361365i \(0.117689\pi\)
\(318\) 0 0
\(319\) −7.38478 −0.413468
\(320\) 0.761360 0.0425613
\(321\) 0 0
\(322\) −10.5116 −0.585788
\(323\) 1.36743 0.0760859
\(324\) 0 0
\(325\) −12.9509 −0.718384
\(326\) −13.3116 −0.737260
\(327\) 0 0
\(328\) 2.52486 0.139412
\(329\) −4.60241 −0.253739
\(330\) 0 0
\(331\) −26.3503 −1.44834 −0.724172 0.689620i \(-0.757778\pi\)
−0.724172 + 0.689620i \(0.757778\pi\)
\(332\) 5.69162 0.312368
\(333\) 0 0
\(334\) 6.38977 0.349633
\(335\) −1.11074 −0.0606860
\(336\) 0 0
\(337\) 4.70544 0.256322 0.128161 0.991753i \(-0.459093\pi\)
0.128161 + 0.991753i \(0.459093\pi\)
\(338\) 4.41604 0.240201
\(339\) 0 0
\(340\) 1.56248 0.0847374
\(341\) 16.0186 0.867454
\(342\) 0 0
\(343\) −15.4483 −0.834132
\(344\) −5.24402 −0.282739
\(345\) 0 0
\(346\) −1.23167 −0.0662150
\(347\) 27.0511 1.45218 0.726091 0.687599i \(-0.241335\pi\)
0.726091 + 0.687599i \(0.241335\pi\)
\(348\) 0 0
\(349\) 9.98625 0.534552 0.267276 0.963620i \(-0.413877\pi\)
0.267276 + 0.963620i \(0.413877\pi\)
\(350\) 5.47884 0.292856
\(351\) 0 0
\(352\) −4.16498 −0.221994
\(353\) 9.07685 0.483112 0.241556 0.970387i \(-0.422342\pi\)
0.241556 + 0.970387i \(0.422342\pi\)
\(354\) 0 0
\(355\) 4.71508 0.250251
\(356\) 7.72882 0.409627
\(357\) 0 0
\(358\) 10.4034 0.549836
\(359\) −13.4684 −0.710836 −0.355418 0.934707i \(-0.615661\pi\)
−0.355418 + 0.934707i \(0.615661\pi\)
\(360\) 0 0
\(361\) −18.5560 −0.976633
\(362\) −5.03575 −0.264673
\(363\) 0 0
\(364\) 3.63143 0.190339
\(365\) −8.31230 −0.435086
\(366\) 0 0
\(367\) 18.1866 0.949331 0.474665 0.880166i \(-0.342569\pi\)
0.474665 + 0.880166i \(0.342569\pi\)
\(368\) 8.48075 0.442089
\(369\) 0 0
\(370\) −3.38720 −0.176092
\(371\) 10.2453 0.531911
\(372\) 0 0
\(373\) 15.9422 0.825456 0.412728 0.910854i \(-0.364576\pi\)
0.412728 + 0.910854i \(0.364576\pi\)
\(374\) −8.54746 −0.441978
\(375\) 0 0
\(376\) 3.71322 0.191495
\(377\) −5.19480 −0.267546
\(378\) 0 0
\(379\) −29.0734 −1.49340 −0.746701 0.665160i \(-0.768363\pi\)
−0.746701 + 0.665160i \(0.768363\pi\)
\(380\) 0.507308 0.0260244
\(381\) 0 0
\(382\) 14.0973 0.721280
\(383\) 25.2617 1.29081 0.645407 0.763839i \(-0.276688\pi\)
0.645407 + 0.763839i \(0.276688\pi\)
\(384\) 0 0
\(385\) 3.93040 0.200312
\(386\) −7.50264 −0.381874
\(387\) 0 0
\(388\) −18.6599 −0.947315
\(389\) 1.32764 0.0673138 0.0336569 0.999433i \(-0.489285\pi\)
0.0336569 + 0.999433i \(0.489285\pi\)
\(390\) 0 0
\(391\) 17.4044 0.880177
\(392\) 5.46373 0.275960
\(393\) 0 0
\(394\) 8.01296 0.403687
\(395\) −7.27077 −0.365832
\(396\) 0 0
\(397\) −2.12240 −0.106520 −0.0532601 0.998581i \(-0.516961\pi\)
−0.0532601 + 0.998581i \(0.516961\pi\)
\(398\) 14.7857 0.741138
\(399\) 0 0
\(400\) −4.42033 −0.221017
\(401\) 26.4011 1.31841 0.659204 0.751964i \(-0.270893\pi\)
0.659204 + 0.751964i \(0.270893\pi\)
\(402\) 0 0
\(403\) 11.2682 0.561310
\(404\) 5.89482 0.293278
\(405\) 0 0
\(406\) 2.19765 0.109068
\(407\) 18.5295 0.918471
\(408\) 0 0
\(409\) −10.0056 −0.494744 −0.247372 0.968921i \(-0.579567\pi\)
−0.247372 + 0.968921i \(0.579567\pi\)
\(410\) 1.92233 0.0949370
\(411\) 0 0
\(412\) −3.77939 −0.186197
\(413\) 11.6153 0.571550
\(414\) 0 0
\(415\) 4.33338 0.212717
\(416\) −2.92984 −0.143647
\(417\) 0 0
\(418\) −2.77520 −0.135739
\(419\) −20.0307 −0.978562 −0.489281 0.872126i \(-0.662741\pi\)
−0.489281 + 0.872126i \(0.662741\pi\)
\(420\) 0 0
\(421\) 26.9452 1.31323 0.656614 0.754227i \(-0.271988\pi\)
0.656614 + 0.754227i \(0.271988\pi\)
\(422\) −3.54316 −0.172478
\(423\) 0 0
\(424\) −8.26592 −0.401429
\(425\) −9.07150 −0.440032
\(426\) 0 0
\(427\) 12.4230 0.601191
\(428\) 7.59349 0.367045
\(429\) 0 0
\(430\) −3.99259 −0.192540
\(431\) −16.9357 −0.815763 −0.407881 0.913035i \(-0.633732\pi\)
−0.407881 + 0.913035i \(0.633732\pi\)
\(432\) 0 0
\(433\) −10.1849 −0.489453 −0.244727 0.969592i \(-0.578698\pi\)
−0.244727 + 0.969592i \(0.578698\pi\)
\(434\) −4.76700 −0.228823
\(435\) 0 0
\(436\) 1.05733 0.0506371
\(437\) 5.65087 0.270318
\(438\) 0 0
\(439\) 32.8283 1.56681 0.783405 0.621511i \(-0.213481\pi\)
0.783405 + 0.621511i \(0.213481\pi\)
\(440\) −3.17105 −0.151174
\(441\) 0 0
\(442\) −6.01268 −0.285994
\(443\) −13.1581 −0.625162 −0.312581 0.949891i \(-0.601194\pi\)
−0.312581 + 0.949891i \(0.601194\pi\)
\(444\) 0 0
\(445\) 5.88442 0.278948
\(446\) −12.5260 −0.593126
\(447\) 0 0
\(448\) 1.23946 0.0585592
\(449\) 1.34071 0.0632721 0.0316361 0.999499i \(-0.489928\pi\)
0.0316361 + 0.999499i \(0.489928\pi\)
\(450\) 0 0
\(451\) −10.5160 −0.495178
\(452\) −9.52892 −0.448203
\(453\) 0 0
\(454\) −14.2435 −0.668482
\(455\) 2.76483 0.129617
\(456\) 0 0
\(457\) 4.66820 0.218369 0.109185 0.994021i \(-0.465176\pi\)
0.109185 + 0.994021i \(0.465176\pi\)
\(458\) 22.3396 1.04386
\(459\) 0 0
\(460\) 6.45691 0.301055
\(461\) −7.86962 −0.366525 −0.183262 0.983064i \(-0.558666\pi\)
−0.183262 + 0.983064i \(0.558666\pi\)
\(462\) 0 0
\(463\) 19.3249 0.898106 0.449053 0.893505i \(-0.351761\pi\)
0.449053 + 0.893505i \(0.351761\pi\)
\(464\) −1.77307 −0.0823125
\(465\) 0 0
\(466\) 13.9412 0.645812
\(467\) −35.3122 −1.63405 −0.817026 0.576601i \(-0.804379\pi\)
−0.817026 + 0.576601i \(0.804379\pi\)
\(468\) 0 0
\(469\) −1.80824 −0.0834965
\(470\) 2.82710 0.130405
\(471\) 0 0
\(472\) −9.37120 −0.431345
\(473\) 21.8412 1.00426
\(474\) 0 0
\(475\) −2.94535 −0.135142
\(476\) 2.54366 0.116588
\(477\) 0 0
\(478\) −15.8580 −0.725326
\(479\) −5.70156 −0.260511 −0.130255 0.991480i \(-0.541580\pi\)
−0.130255 + 0.991480i \(0.541580\pi\)
\(480\) 0 0
\(481\) 13.0345 0.594322
\(482\) 13.2150 0.601927
\(483\) 0 0
\(484\) 6.34704 0.288502
\(485\) −14.2069 −0.645104
\(486\) 0 0
\(487\) −24.4051 −1.10590 −0.552950 0.833214i \(-0.686498\pi\)
−0.552950 + 0.833214i \(0.686498\pi\)
\(488\) −10.0229 −0.453714
\(489\) 0 0
\(490\) 4.15987 0.187924
\(491\) 14.7538 0.665828 0.332914 0.942957i \(-0.391968\pi\)
0.332914 + 0.942957i \(0.391968\pi\)
\(492\) 0 0
\(493\) −3.63872 −0.163880
\(494\) −1.95220 −0.0878338
\(495\) 0 0
\(496\) 3.84602 0.172691
\(497\) 7.67597 0.344314
\(498\) 0 0
\(499\) 3.13631 0.140400 0.0702002 0.997533i \(-0.477636\pi\)
0.0702002 + 0.997533i \(0.477636\pi\)
\(500\) −7.17227 −0.320754
\(501\) 0 0
\(502\) −8.73531 −0.389876
\(503\) −32.1683 −1.43431 −0.717157 0.696912i \(-0.754557\pi\)
−0.717157 + 0.696912i \(0.754557\pi\)
\(504\) 0 0
\(505\) 4.48808 0.199717
\(506\) −35.3221 −1.57026
\(507\) 0 0
\(508\) −15.5448 −0.689689
\(509\) 12.2885 0.544677 0.272339 0.962202i \(-0.412203\pi\)
0.272339 + 0.962202i \(0.412203\pi\)
\(510\) 0 0
\(511\) −13.5321 −0.598625
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −17.0182 −0.750643
\(515\) −2.87748 −0.126797
\(516\) 0 0
\(517\) −15.4655 −0.680172
\(518\) −5.51422 −0.242281
\(519\) 0 0
\(520\) −2.23066 −0.0978211
\(521\) 24.8807 1.09004 0.545022 0.838421i \(-0.316521\pi\)
0.545022 + 0.838421i \(0.316521\pi\)
\(522\) 0 0
\(523\) 5.95819 0.260534 0.130267 0.991479i \(-0.458417\pi\)
0.130267 + 0.991479i \(0.458417\pi\)
\(524\) 5.51152 0.240772
\(525\) 0 0
\(526\) −1.87735 −0.0818562
\(527\) 7.89288 0.343819
\(528\) 0 0
\(529\) 48.9231 2.12709
\(530\) −6.29335 −0.273366
\(531\) 0 0
\(532\) 0.825877 0.0358063
\(533\) −7.39743 −0.320418
\(534\) 0 0
\(535\) 5.78138 0.249951
\(536\) 1.45888 0.0630142
\(537\) 0 0
\(538\) 17.0976 0.737130
\(539\) −22.7563 −0.980183
\(540\) 0 0
\(541\) 19.2533 0.827762 0.413881 0.910331i \(-0.364173\pi\)
0.413881 + 0.910331i \(0.364173\pi\)
\(542\) −2.52171 −0.108317
\(543\) 0 0
\(544\) −2.05222 −0.0879883
\(545\) 0.805012 0.0344829
\(546\) 0 0
\(547\) −21.0064 −0.898171 −0.449085 0.893489i \(-0.648250\pi\)
−0.449085 + 0.893489i \(0.648250\pi\)
\(548\) −10.2474 −0.437747
\(549\) 0 0
\(550\) 18.4106 0.785030
\(551\) −1.18143 −0.0503304
\(552\) 0 0
\(553\) −11.8365 −0.503340
\(554\) −20.3576 −0.864913
\(555\) 0 0
\(556\) −9.23075 −0.391471
\(557\) −6.48989 −0.274986 −0.137493 0.990503i \(-0.543904\pi\)
−0.137493 + 0.990503i \(0.543904\pi\)
\(558\) 0 0
\(559\) 15.3641 0.649834
\(560\) 0.943679 0.0398777
\(561\) 0 0
\(562\) 26.7498 1.12837
\(563\) 44.9514 1.89447 0.947237 0.320533i \(-0.103862\pi\)
0.947237 + 0.320533i \(0.103862\pi\)
\(564\) 0 0
\(565\) −7.25494 −0.305218
\(566\) 29.9892 1.26054
\(567\) 0 0
\(568\) −6.19297 −0.259851
\(569\) −18.5364 −0.777085 −0.388542 0.921431i \(-0.627021\pi\)
−0.388542 + 0.921431i \(0.627021\pi\)
\(570\) 0 0
\(571\) 11.1531 0.466741 0.233370 0.972388i \(-0.425025\pi\)
0.233370 + 0.972388i \(0.425025\pi\)
\(572\) 12.2027 0.510221
\(573\) 0 0
\(574\) 3.12947 0.130622
\(575\) −37.4877 −1.56335
\(576\) 0 0
\(577\) 4.89010 0.203578 0.101789 0.994806i \(-0.467543\pi\)
0.101789 + 0.994806i \(0.467543\pi\)
\(578\) 12.7884 0.531927
\(579\) 0 0
\(580\) −1.34994 −0.0560533
\(581\) 7.05456 0.292673
\(582\) 0 0
\(583\) 34.4274 1.42584
\(584\) 10.9177 0.451778
\(585\) 0 0
\(586\) −5.78618 −0.239025
\(587\) −37.4879 −1.54729 −0.773646 0.633618i \(-0.781569\pi\)
−0.773646 + 0.633618i \(0.781569\pi\)
\(588\) 0 0
\(589\) 2.56267 0.105593
\(590\) −7.13486 −0.293738
\(591\) 0 0
\(592\) 4.44887 0.182848
\(593\) −11.4627 −0.470719 −0.235359 0.971908i \(-0.575627\pi\)
−0.235359 + 0.971908i \(0.575627\pi\)
\(594\) 0 0
\(595\) 1.93664 0.0793944
\(596\) −1.00000 −0.0409616
\(597\) 0 0
\(598\) −24.8472 −1.01608
\(599\) −4.91928 −0.200996 −0.100498 0.994937i \(-0.532044\pi\)
−0.100498 + 0.994937i \(0.532044\pi\)
\(600\) 0 0
\(601\) 3.31114 0.135064 0.0675321 0.997717i \(-0.478487\pi\)
0.0675321 + 0.997717i \(0.478487\pi\)
\(602\) −6.49978 −0.264911
\(603\) 0 0
\(604\) 8.58020 0.349124
\(605\) 4.83239 0.196464
\(606\) 0 0
\(607\) −36.6659 −1.48822 −0.744112 0.668055i \(-0.767127\pi\)
−0.744112 + 0.668055i \(0.767127\pi\)
\(608\) −0.666318 −0.0270228
\(609\) 0 0
\(610\) −7.63102 −0.308971
\(611\) −10.8792 −0.440123
\(612\) 0 0
\(613\) 23.5176 0.949865 0.474932 0.880022i \(-0.342472\pi\)
0.474932 + 0.880022i \(0.342472\pi\)
\(614\) 15.9386 0.643230
\(615\) 0 0
\(616\) −5.16234 −0.207997
\(617\) 9.64068 0.388119 0.194060 0.980990i \(-0.437834\pi\)
0.194060 + 0.980990i \(0.437834\pi\)
\(618\) 0 0
\(619\) 11.9779 0.481431 0.240715 0.970596i \(-0.422618\pi\)
0.240715 + 0.970596i \(0.422618\pi\)
\(620\) 2.92821 0.117600
\(621\) 0 0
\(622\) −18.5036 −0.741927
\(623\) 9.57960 0.383798
\(624\) 0 0
\(625\) 16.6410 0.665639
\(626\) 8.94571 0.357542
\(627\) 0 0
\(628\) 9.08853 0.362672
\(629\) 9.13007 0.364040
\(630\) 0 0
\(631\) −26.2849 −1.04638 −0.523192 0.852215i \(-0.675259\pi\)
−0.523192 + 0.852215i \(0.675259\pi\)
\(632\) 9.54970 0.379867
\(633\) 0 0
\(634\) −33.2027 −1.31865
\(635\) −11.8352 −0.469665
\(636\) 0 0
\(637\) −16.0078 −0.634254
\(638\) 7.38478 0.292366
\(639\) 0 0
\(640\) −0.761360 −0.0300954
\(641\) −3.50080 −0.138273 −0.0691366 0.997607i \(-0.522024\pi\)
−0.0691366 + 0.997607i \(0.522024\pi\)
\(642\) 0 0
\(643\) 4.61292 0.181916 0.0909578 0.995855i \(-0.471007\pi\)
0.0909578 + 0.995855i \(0.471007\pi\)
\(644\) 10.5116 0.414214
\(645\) 0 0
\(646\) −1.36743 −0.0538009
\(647\) −17.2644 −0.678735 −0.339368 0.940654i \(-0.610213\pi\)
−0.339368 + 0.940654i \(0.610213\pi\)
\(648\) 0 0
\(649\) 39.0309 1.53210
\(650\) 12.9509 0.507974
\(651\) 0 0
\(652\) 13.3116 0.521322
\(653\) 7.86506 0.307784 0.153892 0.988088i \(-0.450819\pi\)
0.153892 + 0.988088i \(0.450819\pi\)
\(654\) 0 0
\(655\) 4.19625 0.163961
\(656\) −2.52486 −0.0985792
\(657\) 0 0
\(658\) 4.60241 0.179421
\(659\) 25.2923 0.985247 0.492623 0.870243i \(-0.336038\pi\)
0.492623 + 0.870243i \(0.336038\pi\)
\(660\) 0 0
\(661\) 18.4738 0.718546 0.359273 0.933233i \(-0.383025\pi\)
0.359273 + 0.933233i \(0.383025\pi\)
\(662\) 26.3503 1.02413
\(663\) 0 0
\(664\) −5.69162 −0.220878
\(665\) 0.628790 0.0243834
\(666\) 0 0
\(667\) −15.0369 −0.582232
\(668\) −6.38977 −0.247228
\(669\) 0 0
\(670\) 1.11074 0.0429115
\(671\) 41.7451 1.61155
\(672\) 0 0
\(673\) 38.6055 1.48813 0.744067 0.668105i \(-0.232895\pi\)
0.744067 + 0.668105i \(0.232895\pi\)
\(674\) −4.70544 −0.181247
\(675\) 0 0
\(676\) −4.41604 −0.169848
\(677\) 8.94930 0.343950 0.171975 0.985101i \(-0.444985\pi\)
0.171975 + 0.985101i \(0.444985\pi\)
\(678\) 0 0
\(679\) −23.1283 −0.887584
\(680\) −1.56248 −0.0599184
\(681\) 0 0
\(682\) −16.0186 −0.613383
\(683\) 24.7539 0.947182 0.473591 0.880745i \(-0.342958\pi\)
0.473591 + 0.880745i \(0.342958\pi\)
\(684\) 0 0
\(685\) −7.80197 −0.298098
\(686\) 15.4483 0.589821
\(687\) 0 0
\(688\) 5.24402 0.199926
\(689\) 24.2178 0.922626
\(690\) 0 0
\(691\) 17.9206 0.681733 0.340867 0.940112i \(-0.389280\pi\)
0.340867 + 0.940112i \(0.389280\pi\)
\(692\) 1.23167 0.0468211
\(693\) 0 0
\(694\) −27.0511 −1.02685
\(695\) −7.02793 −0.266585
\(696\) 0 0
\(697\) −5.18157 −0.196266
\(698\) −9.98625 −0.377985
\(699\) 0 0
\(700\) −5.47884 −0.207081
\(701\) 29.8120 1.12598 0.562992 0.826463i \(-0.309650\pi\)
0.562992 + 0.826463i \(0.309650\pi\)
\(702\) 0 0
\(703\) 2.96436 0.111803
\(704\) 4.16498 0.156974
\(705\) 0 0
\(706\) −9.07685 −0.341612
\(707\) 7.30642 0.274786
\(708\) 0 0
\(709\) 17.8874 0.671776 0.335888 0.941902i \(-0.390964\pi\)
0.335888 + 0.941902i \(0.390964\pi\)
\(710\) −4.71508 −0.176954
\(711\) 0 0
\(712\) −7.72882 −0.289650
\(713\) 32.6171 1.22152
\(714\) 0 0
\(715\) 9.29067 0.347451
\(716\) −10.4034 −0.388792
\(717\) 0 0
\(718\) 13.4684 0.502637
\(719\) 31.1012 1.15988 0.579939 0.814660i \(-0.303076\pi\)
0.579939 + 0.814660i \(0.303076\pi\)
\(720\) 0 0
\(721\) −4.68442 −0.174457
\(722\) 18.5560 0.690584
\(723\) 0 0
\(724\) 5.03575 0.187152
\(725\) 7.83754 0.291079
\(726\) 0 0
\(727\) 39.7921 1.47581 0.737904 0.674906i \(-0.235816\pi\)
0.737904 + 0.674906i \(0.235816\pi\)
\(728\) −3.63143 −0.134590
\(729\) 0 0
\(730\) 8.31230 0.307652
\(731\) 10.7619 0.398043
\(732\) 0 0
\(733\) −28.9878 −1.07069 −0.535344 0.844634i \(-0.679818\pi\)
−0.535344 + 0.844634i \(0.679818\pi\)
\(734\) −18.1866 −0.671278
\(735\) 0 0
\(736\) −8.48075 −0.312604
\(737\) −6.07622 −0.223820
\(738\) 0 0
\(739\) −9.23444 −0.339694 −0.169847 0.985470i \(-0.554327\pi\)
−0.169847 + 0.985470i \(0.554327\pi\)
\(740\) 3.38720 0.124516
\(741\) 0 0
\(742\) −10.2453 −0.376118
\(743\) 4.29294 0.157493 0.0787463 0.996895i \(-0.474908\pi\)
0.0787463 + 0.996895i \(0.474908\pi\)
\(744\) 0 0
\(745\) −0.761360 −0.0278941
\(746\) −15.9422 −0.583685
\(747\) 0 0
\(748\) 8.54746 0.312526
\(749\) 9.41186 0.343902
\(750\) 0 0
\(751\) 33.4787 1.22165 0.610827 0.791764i \(-0.290837\pi\)
0.610827 + 0.791764i \(0.290837\pi\)
\(752\) −3.71322 −0.135407
\(753\) 0 0
\(754\) 5.19480 0.189183
\(755\) 6.53263 0.237747
\(756\) 0 0
\(757\) −40.1690 −1.45997 −0.729984 0.683464i \(-0.760473\pi\)
−0.729984 + 0.683464i \(0.760473\pi\)
\(758\) 29.0734 1.05599
\(759\) 0 0
\(760\) −0.507308 −0.0184020
\(761\) −27.1823 −0.985356 −0.492678 0.870212i \(-0.663982\pi\)
−0.492678 + 0.870212i \(0.663982\pi\)
\(762\) 0 0
\(763\) 1.31053 0.0474443
\(764\) −14.0973 −0.510022
\(765\) 0 0
\(766\) −25.2617 −0.912744
\(767\) 27.4561 0.991383
\(768\) 0 0
\(769\) 27.1059 0.977464 0.488732 0.872434i \(-0.337460\pi\)
0.488732 + 0.872434i \(0.337460\pi\)
\(770\) −3.93040 −0.141642
\(771\) 0 0
\(772\) 7.50264 0.270026
\(773\) −30.4154 −1.09397 −0.546984 0.837143i \(-0.684224\pi\)
−0.546984 + 0.837143i \(0.684224\pi\)
\(774\) 0 0
\(775\) −17.0007 −0.610682
\(776\) 18.6599 0.669853
\(777\) 0 0
\(778\) −1.32764 −0.0475980
\(779\) −1.68236 −0.0602768
\(780\) 0 0
\(781\) 25.7936 0.922967
\(782\) −17.4044 −0.622379
\(783\) 0 0
\(784\) −5.46373 −0.195133
\(785\) 6.91965 0.246973
\(786\) 0 0
\(787\) −17.2434 −0.614660 −0.307330 0.951603i \(-0.599436\pi\)
−0.307330 + 0.951603i \(0.599436\pi\)
\(788\) −8.01296 −0.285450
\(789\) 0 0
\(790\) 7.27077 0.258682
\(791\) −11.8108 −0.419942
\(792\) 0 0
\(793\) 29.3654 1.04280
\(794\) 2.12240 0.0753211
\(795\) 0 0
\(796\) −14.7857 −0.524064
\(797\) −53.3978 −1.89145 −0.945724 0.324971i \(-0.894645\pi\)
−0.945724 + 0.324971i \(0.894645\pi\)
\(798\) 0 0
\(799\) −7.62036 −0.269589
\(800\) 4.42033 0.156282
\(801\) 0 0
\(802\) −26.4011 −0.932256
\(803\) −45.4720 −1.60467
\(804\) 0 0
\(805\) 8.00311 0.282072
\(806\) −11.2682 −0.396906
\(807\) 0 0
\(808\) −5.89482 −0.207379
\(809\) 33.8973 1.19177 0.595883 0.803071i \(-0.296802\pi\)
0.595883 + 0.803071i \(0.296802\pi\)
\(810\) 0 0
\(811\) 8.67149 0.304497 0.152249 0.988342i \(-0.451349\pi\)
0.152249 + 0.988342i \(0.451349\pi\)
\(812\) −2.19765 −0.0771225
\(813\) 0 0
\(814\) −18.5295 −0.649457
\(815\) 10.1349 0.355010
\(816\) 0 0
\(817\) 3.49418 0.122246
\(818\) 10.0056 0.349837
\(819\) 0 0
\(820\) −1.92233 −0.0671306
\(821\) −12.9393 −0.451585 −0.225793 0.974175i \(-0.572497\pi\)
−0.225793 + 0.974175i \(0.572497\pi\)
\(822\) 0 0
\(823\) 2.76391 0.0963437 0.0481719 0.998839i \(-0.484660\pi\)
0.0481719 + 0.998839i \(0.484660\pi\)
\(824\) 3.77939 0.131661
\(825\) 0 0
\(826\) −11.6153 −0.404147
\(827\) −10.9901 −0.382163 −0.191082 0.981574i \(-0.561200\pi\)
−0.191082 + 0.981574i \(0.561200\pi\)
\(828\) 0 0
\(829\) −9.82219 −0.341139 −0.170569 0.985346i \(-0.554561\pi\)
−0.170569 + 0.985346i \(0.554561\pi\)
\(830\) −4.33338 −0.150414
\(831\) 0 0
\(832\) 2.92984 0.101574
\(833\) −11.2128 −0.388500
\(834\) 0 0
\(835\) −4.86492 −0.168357
\(836\) 2.77520 0.0959823
\(837\) 0 0
\(838\) 20.0307 0.691948
\(839\) 4.03589 0.139334 0.0696671 0.997570i \(-0.477806\pi\)
0.0696671 + 0.997570i \(0.477806\pi\)
\(840\) 0 0
\(841\) −25.8562 −0.891594
\(842\) −26.9452 −0.928592
\(843\) 0 0
\(844\) 3.54316 0.121961
\(845\) −3.36220 −0.115663
\(846\) 0 0
\(847\) 7.86693 0.270311
\(848\) 8.26592 0.283853
\(849\) 0 0
\(850\) 9.07150 0.311150
\(851\) 37.7298 1.29336
\(852\) 0 0
\(853\) 11.3792 0.389616 0.194808 0.980841i \(-0.437592\pi\)
0.194808 + 0.980841i \(0.437592\pi\)
\(854\) −12.4230 −0.425106
\(855\) 0 0
\(856\) −7.59349 −0.259540
\(857\) 28.3238 0.967522 0.483761 0.875200i \(-0.339270\pi\)
0.483761 + 0.875200i \(0.339270\pi\)
\(858\) 0 0
\(859\) 52.8590 1.80352 0.901762 0.432233i \(-0.142274\pi\)
0.901762 + 0.432233i \(0.142274\pi\)
\(860\) 3.99259 0.136146
\(861\) 0 0
\(862\) 16.9357 0.576831
\(863\) −19.7991 −0.673968 −0.336984 0.941510i \(-0.609407\pi\)
−0.336984 + 0.941510i \(0.609407\pi\)
\(864\) 0 0
\(865\) 0.937745 0.0318843
\(866\) 10.1849 0.346096
\(867\) 0 0
\(868\) 4.76700 0.161803
\(869\) −39.7743 −1.34925
\(870\) 0 0
\(871\) −4.27430 −0.144829
\(872\) −1.05733 −0.0358059
\(873\) 0 0
\(874\) −5.65087 −0.191144
\(875\) −8.88977 −0.300529
\(876\) 0 0
\(877\) −13.9418 −0.470781 −0.235391 0.971901i \(-0.575637\pi\)
−0.235391 + 0.971901i \(0.575637\pi\)
\(878\) −32.8283 −1.10790
\(879\) 0 0
\(880\) 3.17105 0.106896
\(881\) 35.1454 1.18408 0.592039 0.805910i \(-0.298323\pi\)
0.592039 + 0.805910i \(0.298323\pi\)
\(882\) 0 0
\(883\) 22.9553 0.772506 0.386253 0.922393i \(-0.373769\pi\)
0.386253 + 0.922393i \(0.373769\pi\)
\(884\) 6.01268 0.202228
\(885\) 0 0
\(886\) 13.1581 0.442056
\(887\) −4.29374 −0.144170 −0.0720849 0.997398i \(-0.522965\pi\)
−0.0720849 + 0.997398i \(0.522965\pi\)
\(888\) 0 0
\(889\) −19.2672 −0.646202
\(890\) −5.88442 −0.197246
\(891\) 0 0
\(892\) 12.5260 0.419403
\(893\) −2.47419 −0.0827956
\(894\) 0 0
\(895\) −7.92072 −0.264760
\(896\) −1.23946 −0.0414076
\(897\) 0 0
\(898\) −1.34071 −0.0447401
\(899\) −6.81924 −0.227434
\(900\) 0 0
\(901\) 16.9635 0.565136
\(902\) 10.5160 0.350144
\(903\) 0 0
\(904\) 9.52892 0.316927
\(905\) 3.83402 0.127447
\(906\) 0 0
\(907\) −16.0955 −0.534442 −0.267221 0.963635i \(-0.586105\pi\)
−0.267221 + 0.963635i \(0.586105\pi\)
\(908\) 14.2435 0.472688
\(909\) 0 0
\(910\) −2.76483 −0.0916532
\(911\) 29.8829 0.990063 0.495032 0.868875i \(-0.335157\pi\)
0.495032 + 0.868875i \(0.335157\pi\)
\(912\) 0 0
\(913\) 23.7055 0.784537
\(914\) −4.66820 −0.154410
\(915\) 0 0
\(916\) −22.3396 −0.738120
\(917\) 6.83134 0.225591
\(918\) 0 0
\(919\) 32.2582 1.06410 0.532050 0.846713i \(-0.321422\pi\)
0.532050 + 0.846713i \(0.321422\pi\)
\(920\) −6.45691 −0.212878
\(921\) 0 0
\(922\) 7.86962 0.259172
\(923\) 18.1444 0.597230
\(924\) 0 0
\(925\) −19.6655 −0.646597
\(926\) −19.3249 −0.635057
\(927\) 0 0
\(928\) 1.77307 0.0582037
\(929\) 3.07707 0.100955 0.0504777 0.998725i \(-0.483926\pi\)
0.0504777 + 0.998725i \(0.483926\pi\)
\(930\) 0 0
\(931\) −3.64058 −0.119315
\(932\) −13.9412 −0.456658
\(933\) 0 0
\(934\) 35.3122 1.15545
\(935\) 6.50770 0.212824
\(936\) 0 0
\(937\) 22.4776 0.734311 0.367156 0.930160i \(-0.380332\pi\)
0.367156 + 0.930160i \(0.380332\pi\)
\(938\) 1.80824 0.0590410
\(939\) 0 0
\(940\) −2.82710 −0.0922099
\(941\) 22.0470 0.718712 0.359356 0.933201i \(-0.382996\pi\)
0.359356 + 0.933201i \(0.382996\pi\)
\(942\) 0 0
\(943\) −21.4127 −0.697293
\(944\) 9.37120 0.305007
\(945\) 0 0
\(946\) −21.8412 −0.710120
\(947\) −5.17219 −0.168073 −0.0840367 0.996463i \(-0.526781\pi\)
−0.0840367 + 0.996463i \(0.526781\pi\)
\(948\) 0 0
\(949\) −31.9871 −1.03835
\(950\) 2.94535 0.0955596
\(951\) 0 0
\(952\) −2.54366 −0.0824403
\(953\) −32.0348 −1.03771 −0.518855 0.854862i \(-0.673642\pi\)
−0.518855 + 0.854862i \(0.673642\pi\)
\(954\) 0 0
\(955\) −10.7331 −0.347316
\(956\) 15.8580 0.512883
\(957\) 0 0
\(958\) 5.70156 0.184209
\(959\) −12.7013 −0.410146
\(960\) 0 0
\(961\) −16.2082 −0.522844
\(962\) −13.0345 −0.420249
\(963\) 0 0
\(964\) −13.2150 −0.425626
\(965\) 5.71221 0.183883
\(966\) 0 0
\(967\) −10.9279 −0.351419 −0.175709 0.984442i \(-0.556222\pi\)
−0.175709 + 0.984442i \(0.556222\pi\)
\(968\) −6.34704 −0.204002
\(969\) 0 0
\(970\) 14.2069 0.456157
\(971\) −1.27203 −0.0408214 −0.0204107 0.999792i \(-0.506497\pi\)
−0.0204107 + 0.999792i \(0.506497\pi\)
\(972\) 0 0
\(973\) −11.4412 −0.366788
\(974\) 24.4051 0.781990
\(975\) 0 0
\(976\) 10.0229 0.320825
\(977\) 59.6830 1.90943 0.954714 0.297526i \(-0.0961615\pi\)
0.954714 + 0.297526i \(0.0961615\pi\)
\(978\) 0 0
\(979\) 32.1904 1.02881
\(980\) −4.15987 −0.132882
\(981\) 0 0
\(982\) −14.7538 −0.470811
\(983\) −39.9669 −1.27475 −0.637374 0.770555i \(-0.719979\pi\)
−0.637374 + 0.770555i \(0.719979\pi\)
\(984\) 0 0
\(985\) −6.10075 −0.194386
\(986\) 3.63872 0.115881
\(987\) 0 0
\(988\) 1.95220 0.0621079
\(989\) 44.4732 1.41417
\(990\) 0 0
\(991\) −18.7853 −0.596735 −0.298368 0.954451i \(-0.596442\pi\)
−0.298368 + 0.954451i \(0.596442\pi\)
\(992\) −3.84602 −0.122111
\(993\) 0 0
\(994\) −7.67597 −0.243467
\(995\) −11.2572 −0.356878
\(996\) 0 0
\(997\) 42.9444 1.36006 0.680032 0.733183i \(-0.261966\pi\)
0.680032 + 0.733183i \(0.261966\pi\)
\(998\) −3.13631 −0.0992781
\(999\) 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 8046.2.a.k.1.8 12
3.2 odd 2 8046.2.a.n.1.5 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8046.2.a.k.1.8 12 1.1 even 1 trivial
8046.2.a.n.1.5 yes 12 3.2 odd 2