Properties

Label 1638.2.a.x.1.2
Level $1638$
Weight $2$
Character 1638.1
Self dual yes
Analytic conductor $13.079$
Analytic rank $0$
Dimension $2$
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] = [1638,2,Mod(1,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, 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("1638.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 1638.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.37228 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.37228 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.37228 q^{10} -3.37228 q^{11} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.37228 q^{17} +7.37228 q^{19} +3.37228 q^{20} -3.37228 q^{22} +9.37228 q^{23} +6.37228 q^{25} -1.00000 q^{26} -1.00000 q^{28} +0.627719 q^{29} +6.74456 q^{31} +1.00000 q^{32} -1.37228 q^{34} -3.37228 q^{35} +1.37228 q^{37} +7.37228 q^{38} +3.37228 q^{40} -2.74456 q^{41} -6.11684 q^{43} -3.37228 q^{44} +9.37228 q^{46} +12.7446 q^{47} +1.00000 q^{49} +6.37228 q^{50} -1.00000 q^{52} -2.74456 q^{53} -11.3723 q^{55} -1.00000 q^{56} +0.627719 q^{58} +2.00000 q^{59} -12.1168 q^{61} +6.74456 q^{62} +1.00000 q^{64} -3.37228 q^{65} -13.4891 q^{67} -1.37228 q^{68} -3.37228 q^{70} +6.74456 q^{71} -2.62772 q^{73} +1.37228 q^{74} +7.37228 q^{76} +3.37228 q^{77} -6.74456 q^{79} +3.37228 q^{80} -2.74456 q^{82} -6.00000 q^{83} -4.62772 q^{85} -6.11684 q^{86} -3.37228 q^{88} -14.7446 q^{89} +1.00000 q^{91} +9.37228 q^{92} +12.7446 q^{94} +24.8614 q^{95} -15.4891 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8} + q^{10} - q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{17} + 9 q^{19} + q^{20} - q^{22} + 13 q^{23} + 7 q^{25} - 2 q^{26} - 2 q^{28} + 7 q^{29} + 2 q^{31} + 2 q^{32} + 3 q^{34} - q^{35} - 3 q^{37} + 9 q^{38} + q^{40} + 6 q^{41} + 5 q^{43} - q^{44} + 13 q^{46} + 14 q^{47} + 2 q^{49} + 7 q^{50} - 2 q^{52} + 6 q^{53} - 17 q^{55} - 2 q^{56} + 7 q^{58} + 4 q^{59} - 7 q^{61} + 2 q^{62} + 2 q^{64} - q^{65} - 4 q^{67} + 3 q^{68} - q^{70} + 2 q^{71} - 11 q^{73} - 3 q^{74} + 9 q^{76} + q^{77} - 2 q^{79} + q^{80} + 6 q^{82} - 12 q^{83} - 15 q^{85} + 5 q^{86} - q^{88} - 18 q^{89} + 2 q^{91} + 13 q^{92} + 14 q^{94} + 21 q^{95} - 8 q^{97} + 2 q^{98}+O(q^{100}) \) 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
\(5\) 3.37228 1.50813 0.754065 0.656800i \(-0.228090\pi\)
0.754065 + 0.656800i \(0.228090\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.37228 1.06641
\(11\) −3.37228 −1.01678 −0.508391 0.861127i \(-0.669759\pi\)
−0.508391 + 0.861127i \(0.669759\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.37228 −0.332827 −0.166414 0.986056i \(-0.553219\pi\)
−0.166414 + 0.986056i \(0.553219\pi\)
\(18\) 0 0
\(19\) 7.37228 1.69132 0.845659 0.533724i \(-0.179208\pi\)
0.845659 + 0.533724i \(0.179208\pi\)
\(20\) 3.37228 0.754065
\(21\) 0 0
\(22\) −3.37228 −0.718973
\(23\) 9.37228 1.95426 0.977128 0.212653i \(-0.0682103\pi\)
0.977128 + 0.212653i \(0.0682103\pi\)
\(24\) 0 0
\(25\) 6.37228 1.27446
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 0.627719 0.116564 0.0582822 0.998300i \(-0.481438\pi\)
0.0582822 + 0.998300i \(0.481438\pi\)
\(30\) 0 0
\(31\) 6.74456 1.21136 0.605680 0.795709i \(-0.292901\pi\)
0.605680 + 0.795709i \(0.292901\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −1.37228 −0.235344
\(35\) −3.37228 −0.570020
\(36\) 0 0
\(37\) 1.37228 0.225602 0.112801 0.993618i \(-0.464018\pi\)
0.112801 + 0.993618i \(0.464018\pi\)
\(38\) 7.37228 1.19594
\(39\) 0 0
\(40\) 3.37228 0.533204
\(41\) −2.74456 −0.428629 −0.214314 0.976765i \(-0.568752\pi\)
−0.214314 + 0.976765i \(0.568752\pi\)
\(42\) 0 0
\(43\) −6.11684 −0.932810 −0.466405 0.884571i \(-0.654451\pi\)
−0.466405 + 0.884571i \(0.654451\pi\)
\(44\) −3.37228 −0.508391
\(45\) 0 0
\(46\) 9.37228 1.38187
\(47\) 12.7446 1.85899 0.929493 0.368840i \(-0.120245\pi\)
0.929493 + 0.368840i \(0.120245\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 6.37228 0.901177
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −2.74456 −0.376995 −0.188497 0.982074i \(-0.560362\pi\)
−0.188497 + 0.982074i \(0.560362\pi\)
\(54\) 0 0
\(55\) −11.3723 −1.53344
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 0.627719 0.0824235
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) −12.1168 −1.55140 −0.775701 0.631100i \(-0.782604\pi\)
−0.775701 + 0.631100i \(0.782604\pi\)
\(62\) 6.74456 0.856560
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.37228 −0.418280
\(66\) 0 0
\(67\) −13.4891 −1.64796 −0.823979 0.566620i \(-0.808251\pi\)
−0.823979 + 0.566620i \(0.808251\pi\)
\(68\) −1.37228 −0.166414
\(69\) 0 0
\(70\) −3.37228 −0.403065
\(71\) 6.74456 0.800432 0.400216 0.916421i \(-0.368935\pi\)
0.400216 + 0.916421i \(0.368935\pi\)
\(72\) 0 0
\(73\) −2.62772 −0.307551 −0.153776 0.988106i \(-0.549143\pi\)
−0.153776 + 0.988106i \(0.549143\pi\)
\(74\) 1.37228 0.159524
\(75\) 0 0
\(76\) 7.37228 0.845659
\(77\) 3.37228 0.384307
\(78\) 0 0
\(79\) −6.74456 −0.758823 −0.379411 0.925228i \(-0.623873\pi\)
−0.379411 + 0.925228i \(0.623873\pi\)
\(80\) 3.37228 0.377033
\(81\) 0 0
\(82\) −2.74456 −0.303086
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −4.62772 −0.501947
\(86\) −6.11684 −0.659596
\(87\) 0 0
\(88\) −3.37228 −0.359486
\(89\) −14.7446 −1.56292 −0.781460 0.623955i \(-0.785525\pi\)
−0.781460 + 0.623955i \(0.785525\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 9.37228 0.977128
\(93\) 0 0
\(94\) 12.7446 1.31450
\(95\) 24.8614 2.55073
\(96\) 0 0
\(97\) −15.4891 −1.57268 −0.786341 0.617792i \(-0.788027\pi\)
−0.786341 + 0.617792i \(0.788027\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 6.37228 0.637228
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 16.8614 1.66140 0.830702 0.556718i \(-0.187939\pi\)
0.830702 + 0.556718i \(0.187939\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −2.74456 −0.266575
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) −9.37228 −0.897702 −0.448851 0.893607i \(-0.648167\pi\)
−0.448851 + 0.893607i \(0.648167\pi\)
\(110\) −11.3723 −1.08430
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 20.0000 1.88144 0.940721 0.339182i \(-0.110150\pi\)
0.940721 + 0.339182i \(0.110150\pi\)
\(114\) 0 0
\(115\) 31.6060 2.94727
\(116\) 0.627719 0.0582822
\(117\) 0 0
\(118\) 2.00000 0.184115
\(119\) 1.37228 0.125797
\(120\) 0 0
\(121\) 0.372281 0.0338438
\(122\) −12.1168 −1.09701
\(123\) 0 0
\(124\) 6.74456 0.605680
\(125\) 4.62772 0.413916
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −3.37228 −0.295769
\(131\) 6.11684 0.534431 0.267216 0.963637i \(-0.413896\pi\)
0.267216 + 0.963637i \(0.413896\pi\)
\(132\) 0 0
\(133\) −7.37228 −0.639258
\(134\) −13.4891 −1.16528
\(135\) 0 0
\(136\) −1.37228 −0.117672
\(137\) −16.1168 −1.37695 −0.688477 0.725258i \(-0.741721\pi\)
−0.688477 + 0.725258i \(0.741721\pi\)
\(138\) 0 0
\(139\) −10.7446 −0.911342 −0.455671 0.890148i \(-0.650601\pi\)
−0.455671 + 0.890148i \(0.650601\pi\)
\(140\) −3.37228 −0.285010
\(141\) 0 0
\(142\) 6.74456 0.565991
\(143\) 3.37228 0.282004
\(144\) 0 0
\(145\) 2.11684 0.175794
\(146\) −2.62772 −0.217472
\(147\) 0 0
\(148\) 1.37228 0.112801
\(149\) 14.2337 1.16607 0.583035 0.812447i \(-0.301865\pi\)
0.583035 + 0.812447i \(0.301865\pi\)
\(150\) 0 0
\(151\) 1.88316 0.153249 0.0766245 0.997060i \(-0.475586\pi\)
0.0766245 + 0.997060i \(0.475586\pi\)
\(152\) 7.37228 0.597971
\(153\) 0 0
\(154\) 3.37228 0.271746
\(155\) 22.7446 1.82689
\(156\) 0 0
\(157\) −14.6277 −1.16742 −0.583710 0.811963i \(-0.698399\pi\)
−0.583710 + 0.811963i \(0.698399\pi\)
\(158\) −6.74456 −0.536569
\(159\) 0 0
\(160\) 3.37228 0.266602
\(161\) −9.37228 −0.738639
\(162\) 0 0
\(163\) 16.2337 1.27152 0.635760 0.771887i \(-0.280687\pi\)
0.635760 + 0.771887i \(0.280687\pi\)
\(164\) −2.74456 −0.214314
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) −4.11684 −0.318571 −0.159285 0.987233i \(-0.550919\pi\)
−0.159285 + 0.987233i \(0.550919\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −4.62772 −0.354930
\(171\) 0 0
\(172\) −6.11684 −0.466405
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 0 0
\(175\) −6.37228 −0.481699
\(176\) −3.37228 −0.254195
\(177\) 0 0
\(178\) −14.7446 −1.10515
\(179\) 8.74456 0.653599 0.326800 0.945094i \(-0.394030\pi\)
0.326800 + 0.945094i \(0.394030\pi\)
\(180\) 0 0
\(181\) −0.510875 −0.0379730 −0.0189865 0.999820i \(-0.506044\pi\)
−0.0189865 + 0.999820i \(0.506044\pi\)
\(182\) 1.00000 0.0741249
\(183\) 0 0
\(184\) 9.37228 0.690934
\(185\) 4.62772 0.340237
\(186\) 0 0
\(187\) 4.62772 0.338412
\(188\) 12.7446 0.929493
\(189\) 0 0
\(190\) 24.8614 1.80364
\(191\) −12.1168 −0.876744 −0.438372 0.898794i \(-0.644445\pi\)
−0.438372 + 0.898794i \(0.644445\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −15.4891 −1.11205
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −16.8614 −1.19527 −0.597637 0.801767i \(-0.703893\pi\)
−0.597637 + 0.801767i \(0.703893\pi\)
\(200\) 6.37228 0.450588
\(201\) 0 0
\(202\) −2.00000 −0.140720
\(203\) −0.627719 −0.0440572
\(204\) 0 0
\(205\) −9.25544 −0.646428
\(206\) 16.8614 1.17479
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) −24.8614 −1.71970
\(210\) 0 0
\(211\) −23.3723 −1.60901 −0.804507 0.593943i \(-0.797570\pi\)
−0.804507 + 0.593943i \(0.797570\pi\)
\(212\) −2.74456 −0.188497
\(213\) 0 0
\(214\) 2.00000 0.136717
\(215\) −20.6277 −1.40680
\(216\) 0 0
\(217\) −6.74456 −0.457851
\(218\) −9.37228 −0.634771
\(219\) 0 0
\(220\) −11.3723 −0.766719
\(221\) 1.37228 0.0923096
\(222\) 0 0
\(223\) 5.48913 0.367579 0.183790 0.982966i \(-0.441164\pi\)
0.183790 + 0.982966i \(0.441164\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 20.0000 1.33038
\(227\) −11.4891 −0.762560 −0.381280 0.924460i \(-0.624517\pi\)
−0.381280 + 0.924460i \(0.624517\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 31.6060 2.08404
\(231\) 0 0
\(232\) 0.627719 0.0412118
\(233\) 21.4891 1.40780 0.703900 0.710299i \(-0.251440\pi\)
0.703900 + 0.710299i \(0.251440\pi\)
\(234\) 0 0
\(235\) 42.9783 2.80359
\(236\) 2.00000 0.130189
\(237\) 0 0
\(238\) 1.37228 0.0889518
\(239\) 17.4891 1.13128 0.565639 0.824653i \(-0.308630\pi\)
0.565639 + 0.824653i \(0.308630\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 0.372281 0.0239311
\(243\) 0 0
\(244\) −12.1168 −0.775701
\(245\) 3.37228 0.215447
\(246\) 0 0
\(247\) −7.37228 −0.469087
\(248\) 6.74456 0.428280
\(249\) 0 0
\(250\) 4.62772 0.292683
\(251\) −19.3723 −1.22277 −0.611384 0.791334i \(-0.709387\pi\)
−0.611384 + 0.791334i \(0.709387\pi\)
\(252\) 0 0
\(253\) −31.6060 −1.98705
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −11.4891 −0.716672 −0.358336 0.933593i \(-0.616656\pi\)
−0.358336 + 0.933593i \(0.616656\pi\)
\(258\) 0 0
\(259\) −1.37228 −0.0852694
\(260\) −3.37228 −0.209140
\(261\) 0 0
\(262\) 6.11684 0.377900
\(263\) 20.7446 1.27916 0.639582 0.768723i \(-0.279107\pi\)
0.639582 + 0.768723i \(0.279107\pi\)
\(264\) 0 0
\(265\) −9.25544 −0.568557
\(266\) −7.37228 −0.452024
\(267\) 0 0
\(268\) −13.4891 −0.823979
\(269\) 23.4891 1.43216 0.716079 0.698020i \(-0.245935\pi\)
0.716079 + 0.698020i \(0.245935\pi\)
\(270\) 0 0
\(271\) −4.23369 −0.257178 −0.128589 0.991698i \(-0.541045\pi\)
−0.128589 + 0.991698i \(0.541045\pi\)
\(272\) −1.37228 −0.0832068
\(273\) 0 0
\(274\) −16.1168 −0.973654
\(275\) −21.4891 −1.29584
\(276\) 0 0
\(277\) 15.2554 0.916610 0.458305 0.888795i \(-0.348457\pi\)
0.458305 + 0.888795i \(0.348457\pi\)
\(278\) −10.7446 −0.644416
\(279\) 0 0
\(280\) −3.37228 −0.201532
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −30.9783 −1.84147 −0.920733 0.390193i \(-0.872408\pi\)
−0.920733 + 0.390193i \(0.872408\pi\)
\(284\) 6.74456 0.400216
\(285\) 0 0
\(286\) 3.37228 0.199407
\(287\) 2.74456 0.162006
\(288\) 0 0
\(289\) −15.1168 −0.889226
\(290\) 2.11684 0.124305
\(291\) 0 0
\(292\) −2.62772 −0.153776
\(293\) −17.2554 −1.00807 −0.504037 0.863682i \(-0.668152\pi\)
−0.504037 + 0.863682i \(0.668152\pi\)
\(294\) 0 0
\(295\) 6.74456 0.392684
\(296\) 1.37228 0.0797622
\(297\) 0 0
\(298\) 14.2337 0.824535
\(299\) −9.37228 −0.542013
\(300\) 0 0
\(301\) 6.11684 0.352569
\(302\) 1.88316 0.108363
\(303\) 0 0
\(304\) 7.37228 0.422829
\(305\) −40.8614 −2.33972
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 3.37228 0.192154
\(309\) 0 0
\(310\) 22.7446 1.29180
\(311\) −5.48913 −0.311260 −0.155630 0.987815i \(-0.549741\pi\)
−0.155630 + 0.987815i \(0.549741\pi\)
\(312\) 0 0
\(313\) −32.9783 −1.86404 −0.932020 0.362406i \(-0.881956\pi\)
−0.932020 + 0.362406i \(0.881956\pi\)
\(314\) −14.6277 −0.825490
\(315\) 0 0
\(316\) −6.74456 −0.379411
\(317\) 16.9783 0.953594 0.476797 0.879014i \(-0.341798\pi\)
0.476797 + 0.879014i \(0.341798\pi\)
\(318\) 0 0
\(319\) −2.11684 −0.118521
\(320\) 3.37228 0.188516
\(321\) 0 0
\(322\) −9.37228 −0.522297
\(323\) −10.1168 −0.562916
\(324\) 0 0
\(325\) −6.37228 −0.353471
\(326\) 16.2337 0.899101
\(327\) 0 0
\(328\) −2.74456 −0.151543
\(329\) −12.7446 −0.702630
\(330\) 0 0
\(331\) 5.25544 0.288865 0.144432 0.989515i \(-0.453864\pi\)
0.144432 + 0.989515i \(0.453864\pi\)
\(332\) −6.00000 −0.329293
\(333\) 0 0
\(334\) −4.11684 −0.225264
\(335\) −45.4891 −2.48534
\(336\) 0 0
\(337\) 26.8614 1.46323 0.731617 0.681716i \(-0.238766\pi\)
0.731617 + 0.681716i \(0.238766\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) −4.62772 −0.250973
\(341\) −22.7446 −1.23169
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −6.11684 −0.329798
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) −31.7228 −1.70297 −0.851485 0.524379i \(-0.824297\pi\)
−0.851485 + 0.524379i \(0.824297\pi\)
\(348\) 0 0
\(349\) −11.2554 −0.602490 −0.301245 0.953547i \(-0.597402\pi\)
−0.301245 + 0.953547i \(0.597402\pi\)
\(350\) −6.37228 −0.340613
\(351\) 0 0
\(352\) −3.37228 −0.179743
\(353\) 6.74456 0.358977 0.179488 0.983760i \(-0.442556\pi\)
0.179488 + 0.983760i \(0.442556\pi\)
\(354\) 0 0
\(355\) 22.7446 1.20716
\(356\) −14.7446 −0.781460
\(357\) 0 0
\(358\) 8.74456 0.462164
\(359\) −14.7446 −0.778188 −0.389094 0.921198i \(-0.627212\pi\)
−0.389094 + 0.921198i \(0.627212\pi\)
\(360\) 0 0
\(361\) 35.3505 1.86055
\(362\) −0.510875 −0.0268510
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) −8.86141 −0.463827
\(366\) 0 0
\(367\) 25.4891 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(368\) 9.37228 0.488564
\(369\) 0 0
\(370\) 4.62772 0.240584
\(371\) 2.74456 0.142491
\(372\) 0 0
\(373\) 7.48913 0.387772 0.193886 0.981024i \(-0.437891\pi\)
0.193886 + 0.981024i \(0.437891\pi\)
\(374\) 4.62772 0.239294
\(375\) 0 0
\(376\) 12.7446 0.657251
\(377\) −0.627719 −0.0323292
\(378\) 0 0
\(379\) −17.2554 −0.886352 −0.443176 0.896435i \(-0.646148\pi\)
−0.443176 + 0.896435i \(0.646148\pi\)
\(380\) 24.8614 1.27536
\(381\) 0 0
\(382\) −12.1168 −0.619952
\(383\) −33.6060 −1.71718 −0.858592 0.512659i \(-0.828661\pi\)
−0.858592 + 0.512659i \(0.828661\pi\)
\(384\) 0 0
\(385\) 11.3723 0.579585
\(386\) 2.00000 0.101797
\(387\) 0 0
\(388\) −15.4891 −0.786341
\(389\) 14.7446 0.747579 0.373790 0.927514i \(-0.378058\pi\)
0.373790 + 0.927514i \(0.378058\pi\)
\(390\) 0 0
\(391\) −12.8614 −0.650429
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −22.7446 −1.14440
\(396\) 0 0
\(397\) 16.7446 0.840386 0.420193 0.907435i \(-0.361962\pi\)
0.420193 + 0.907435i \(0.361962\pi\)
\(398\) −16.8614 −0.845186
\(399\) 0 0
\(400\) 6.37228 0.318614
\(401\) −12.5109 −0.624763 −0.312382 0.949957i \(-0.601127\pi\)
−0.312382 + 0.949957i \(0.601127\pi\)
\(402\) 0 0
\(403\) −6.74456 −0.335971
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) −0.627719 −0.0311532
\(407\) −4.62772 −0.229387
\(408\) 0 0
\(409\) −2.86141 −0.141487 −0.0707437 0.997495i \(-0.522537\pi\)
−0.0707437 + 0.997495i \(0.522537\pi\)
\(410\) −9.25544 −0.457093
\(411\) 0 0
\(412\) 16.8614 0.830702
\(413\) −2.00000 −0.0984136
\(414\) 0 0
\(415\) −20.2337 −0.993233
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) −24.8614 −1.21601
\(419\) 24.8614 1.21456 0.607280 0.794488i \(-0.292261\pi\)
0.607280 + 0.794488i \(0.292261\pi\)
\(420\) 0 0
\(421\) 15.4891 0.754894 0.377447 0.926031i \(-0.376802\pi\)
0.377447 + 0.926031i \(0.376802\pi\)
\(422\) −23.3723 −1.13774
\(423\) 0 0
\(424\) −2.74456 −0.133288
\(425\) −8.74456 −0.424174
\(426\) 0 0
\(427\) 12.1168 0.586375
\(428\) 2.00000 0.0966736
\(429\) 0 0
\(430\) −20.6277 −0.994757
\(431\) −21.2554 −1.02384 −0.511919 0.859034i \(-0.671065\pi\)
−0.511919 + 0.859034i \(0.671065\pi\)
\(432\) 0 0
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) −6.74456 −0.323749
\(435\) 0 0
\(436\) −9.37228 −0.448851
\(437\) 69.0951 3.30527
\(438\) 0 0
\(439\) 15.3723 0.733679 0.366839 0.930284i \(-0.380440\pi\)
0.366839 + 0.930284i \(0.380440\pi\)
\(440\) −11.3723 −0.542152
\(441\) 0 0
\(442\) 1.37228 0.0652728
\(443\) −28.9783 −1.37680 −0.688399 0.725332i \(-0.741686\pi\)
−0.688399 + 0.725332i \(0.741686\pi\)
\(444\) 0 0
\(445\) −49.7228 −2.35709
\(446\) 5.48913 0.259918
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 5.37228 0.253534 0.126767 0.991933i \(-0.459540\pi\)
0.126767 + 0.991933i \(0.459540\pi\)
\(450\) 0 0
\(451\) 9.25544 0.435822
\(452\) 20.0000 0.940721
\(453\) 0 0
\(454\) −11.4891 −0.539211
\(455\) 3.37228 0.158095
\(456\) 0 0
\(457\) −22.2337 −1.04005 −0.520024 0.854152i \(-0.674077\pi\)
−0.520024 + 0.854152i \(0.674077\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) 31.6060 1.47364
\(461\) 3.60597 0.167947 0.0839734 0.996468i \(-0.473239\pi\)
0.0839734 + 0.996468i \(0.473239\pi\)
\(462\) 0 0
\(463\) 34.3505 1.59640 0.798202 0.602389i \(-0.205785\pi\)
0.798202 + 0.602389i \(0.205785\pi\)
\(464\) 0.627719 0.0291411
\(465\) 0 0
\(466\) 21.4891 0.995465
\(467\) 29.0951 1.34636 0.673180 0.739478i \(-0.264928\pi\)
0.673180 + 0.739478i \(0.264928\pi\)
\(468\) 0 0
\(469\) 13.4891 0.622870
\(470\) 42.9783 1.98244
\(471\) 0 0
\(472\) 2.00000 0.0920575
\(473\) 20.6277 0.948464
\(474\) 0 0
\(475\) 46.9783 2.15551
\(476\) 1.37228 0.0628984
\(477\) 0 0
\(478\) 17.4891 0.799934
\(479\) −12.3505 −0.564310 −0.282155 0.959369i \(-0.591049\pi\)
−0.282155 + 0.959369i \(0.591049\pi\)
\(480\) 0 0
\(481\) −1.37228 −0.0625706
\(482\) −6.00000 −0.273293
\(483\) 0 0
\(484\) 0.372281 0.0169219
\(485\) −52.2337 −2.37181
\(486\) 0 0
\(487\) 36.4674 1.65249 0.826247 0.563308i \(-0.190471\pi\)
0.826247 + 0.563308i \(0.190471\pi\)
\(488\) −12.1168 −0.548504
\(489\) 0 0
\(490\) 3.37228 0.152344
\(491\) 36.7446 1.65826 0.829129 0.559057i \(-0.188837\pi\)
0.829129 + 0.559057i \(0.188837\pi\)
\(492\) 0 0
\(493\) −0.861407 −0.0387958
\(494\) −7.37228 −0.331695
\(495\) 0 0
\(496\) 6.74456 0.302840
\(497\) −6.74456 −0.302535
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 4.62772 0.206958
\(501\) 0 0
\(502\) −19.3723 −0.864627
\(503\) −1.48913 −0.0663968 −0.0331984 0.999449i \(-0.510569\pi\)
−0.0331984 + 0.999449i \(0.510569\pi\)
\(504\) 0 0
\(505\) −6.74456 −0.300129
\(506\) −31.6060 −1.40506
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 37.0951 1.64421 0.822106 0.569335i \(-0.192799\pi\)
0.822106 + 0.569335i \(0.192799\pi\)
\(510\) 0 0
\(511\) 2.62772 0.116243
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −11.4891 −0.506764
\(515\) 56.8614 2.50561
\(516\) 0 0
\(517\) −42.9783 −1.89018
\(518\) −1.37228 −0.0602946
\(519\) 0 0
\(520\) −3.37228 −0.147884
\(521\) −22.6277 −0.991338 −0.495669 0.868511i \(-0.665077\pi\)
−0.495669 + 0.868511i \(0.665077\pi\)
\(522\) 0 0
\(523\) −32.2337 −1.40948 −0.704740 0.709465i \(-0.748937\pi\)
−0.704740 + 0.709465i \(0.748937\pi\)
\(524\) 6.11684 0.267216
\(525\) 0 0
\(526\) 20.7446 0.904506
\(527\) −9.25544 −0.403173
\(528\) 0 0
\(529\) 64.8397 2.81912
\(530\) −9.25544 −0.402031
\(531\) 0 0
\(532\) −7.37228 −0.319629
\(533\) 2.74456 0.118880
\(534\) 0 0
\(535\) 6.74456 0.291593
\(536\) −13.4891 −0.582641
\(537\) 0 0
\(538\) 23.4891 1.01269
\(539\) −3.37228 −0.145254
\(540\) 0 0
\(541\) −28.3505 −1.21888 −0.609442 0.792830i \(-0.708607\pi\)
−0.609442 + 0.792830i \(0.708607\pi\)
\(542\) −4.23369 −0.181852
\(543\) 0 0
\(544\) −1.37228 −0.0588361
\(545\) −31.6060 −1.35385
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −16.1168 −0.688477
\(549\) 0 0
\(550\) −21.4891 −0.916299
\(551\) 4.62772 0.197147
\(552\) 0 0
\(553\) 6.74456 0.286808
\(554\) 15.2554 0.648141
\(555\) 0 0
\(556\) −10.7446 −0.455671
\(557\) −11.4891 −0.486810 −0.243405 0.969925i \(-0.578264\pi\)
−0.243405 + 0.969925i \(0.578264\pi\)
\(558\) 0 0
\(559\) 6.11684 0.258715
\(560\) −3.37228 −0.142505
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) 6.11684 0.257794 0.128897 0.991658i \(-0.458856\pi\)
0.128897 + 0.991658i \(0.458856\pi\)
\(564\) 0 0
\(565\) 67.4456 2.83746
\(566\) −30.9783 −1.30211
\(567\) 0 0
\(568\) 6.74456 0.282996
\(569\) 1.25544 0.0526307 0.0263153 0.999654i \(-0.491623\pi\)
0.0263153 + 0.999654i \(0.491623\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 3.37228 0.141002
\(573\) 0 0
\(574\) 2.74456 0.114556
\(575\) 59.7228 2.49061
\(576\) 0 0
\(577\) −23.4891 −0.977865 −0.488933 0.872322i \(-0.662614\pi\)
−0.488933 + 0.872322i \(0.662614\pi\)
\(578\) −15.1168 −0.628778
\(579\) 0 0
\(580\) 2.11684 0.0878972
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 9.25544 0.383321
\(584\) −2.62772 −0.108736
\(585\) 0 0
\(586\) −17.2554 −0.712816
\(587\) −4.51087 −0.186184 −0.0930919 0.995658i \(-0.529675\pi\)
−0.0930919 + 0.995658i \(0.529675\pi\)
\(588\) 0 0
\(589\) 49.7228 2.04879
\(590\) 6.74456 0.277669
\(591\) 0 0
\(592\) 1.37228 0.0564004
\(593\) −4.00000 −0.164260 −0.0821302 0.996622i \(-0.526172\pi\)
−0.0821302 + 0.996622i \(0.526172\pi\)
\(594\) 0 0
\(595\) 4.62772 0.189718
\(596\) 14.2337 0.583035
\(597\) 0 0
\(598\) −9.37228 −0.383261
\(599\) −3.88316 −0.158661 −0.0793307 0.996848i \(-0.525278\pi\)
−0.0793307 + 0.996848i \(0.525278\pi\)
\(600\) 0 0
\(601\) 26.4674 1.07963 0.539813 0.841785i \(-0.318495\pi\)
0.539813 + 0.841785i \(0.318495\pi\)
\(602\) 6.11684 0.249304
\(603\) 0 0
\(604\) 1.88316 0.0766245
\(605\) 1.25544 0.0510408
\(606\) 0 0
\(607\) 1.88316 0.0764349 0.0382175 0.999269i \(-0.487832\pi\)
0.0382175 + 0.999269i \(0.487832\pi\)
\(608\) 7.37228 0.298985
\(609\) 0 0
\(610\) −40.8614 −1.65443
\(611\) −12.7446 −0.515590
\(612\) 0 0
\(613\) −34.8614 −1.40804 −0.704019 0.710181i \(-0.748613\pi\)
−0.704019 + 0.710181i \(0.748613\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 3.37228 0.135873
\(617\) 49.6060 1.99706 0.998531 0.0541915i \(-0.0172581\pi\)
0.998531 + 0.0541915i \(0.0172581\pi\)
\(618\) 0 0
\(619\) −6.11684 −0.245857 −0.122928 0.992416i \(-0.539229\pi\)
−0.122928 + 0.992416i \(0.539229\pi\)
\(620\) 22.7446 0.913444
\(621\) 0 0
\(622\) −5.48913 −0.220094
\(623\) 14.7446 0.590728
\(624\) 0 0
\(625\) −16.2554 −0.650217
\(626\) −32.9783 −1.31808
\(627\) 0 0
\(628\) −14.6277 −0.583710
\(629\) −1.88316 −0.0750863
\(630\) 0 0
\(631\) −23.6060 −0.939739 −0.469869 0.882736i \(-0.655699\pi\)
−0.469869 + 0.882736i \(0.655699\pi\)
\(632\) −6.74456 −0.268284
\(633\) 0 0
\(634\) 16.9783 0.674292
\(635\) −26.9783 −1.07060
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) −2.11684 −0.0838067
\(639\) 0 0
\(640\) 3.37228 0.133301
\(641\) −16.2337 −0.641192 −0.320596 0.947216i \(-0.603883\pi\)
−0.320596 + 0.947216i \(0.603883\pi\)
\(642\) 0 0
\(643\) −28.8614 −1.13818 −0.569091 0.822274i \(-0.692705\pi\)
−0.569091 + 0.822274i \(0.692705\pi\)
\(644\) −9.37228 −0.369320
\(645\) 0 0
\(646\) −10.1168 −0.398042
\(647\) −1.48913 −0.0585436 −0.0292718 0.999571i \(-0.509319\pi\)
−0.0292718 + 0.999571i \(0.509319\pi\)
\(648\) 0 0
\(649\) −6.74456 −0.264747
\(650\) −6.37228 −0.249941
\(651\) 0 0
\(652\) 16.2337 0.635760
\(653\) 36.8614 1.44250 0.721249 0.692676i \(-0.243568\pi\)
0.721249 + 0.692676i \(0.243568\pi\)
\(654\) 0 0
\(655\) 20.6277 0.805992
\(656\) −2.74456 −0.107157
\(657\) 0 0
\(658\) −12.7446 −0.496835
\(659\) 20.7446 0.808093 0.404047 0.914738i \(-0.367603\pi\)
0.404047 + 0.914738i \(0.367603\pi\)
\(660\) 0 0
\(661\) −0.744563 −0.0289601 −0.0144801 0.999895i \(-0.504609\pi\)
−0.0144801 + 0.999895i \(0.504609\pi\)
\(662\) 5.25544 0.204258
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) −24.8614 −0.964084
\(666\) 0 0
\(667\) 5.88316 0.227797
\(668\) −4.11684 −0.159285
\(669\) 0 0
\(670\) −45.4891 −1.75740
\(671\) 40.8614 1.57744
\(672\) 0 0
\(673\) −47.0951 −1.81538 −0.907691 0.419639i \(-0.862157\pi\)
−0.907691 + 0.419639i \(0.862157\pi\)
\(674\) 26.8614 1.03466
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 12.5109 0.480832 0.240416 0.970670i \(-0.422716\pi\)
0.240416 + 0.970670i \(0.422716\pi\)
\(678\) 0 0
\(679\) 15.4891 0.594418
\(680\) −4.62772 −0.177465
\(681\) 0 0
\(682\) −22.7446 −0.870934
\(683\) −6.11684 −0.234055 −0.117027 0.993129i \(-0.537336\pi\)
−0.117027 + 0.993129i \(0.537336\pi\)
\(684\) 0 0
\(685\) −54.3505 −2.07663
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −6.11684 −0.233202
\(689\) 2.74456 0.104560
\(690\) 0 0
\(691\) 22.5109 0.856354 0.428177 0.903695i \(-0.359156\pi\)
0.428177 + 0.903695i \(0.359156\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) −31.7228 −1.20418
\(695\) −36.2337 −1.37442
\(696\) 0 0
\(697\) 3.76631 0.142659
\(698\) −11.2554 −0.426025
\(699\) 0 0
\(700\) −6.37228 −0.240850
\(701\) −25.7228 −0.971537 −0.485769 0.874087i \(-0.661460\pi\)
−0.485769 + 0.874087i \(0.661460\pi\)
\(702\) 0 0
\(703\) 10.1168 0.381564
\(704\) −3.37228 −0.127098
\(705\) 0 0
\(706\) 6.74456 0.253835
\(707\) 2.00000 0.0752177
\(708\) 0 0
\(709\) −51.4891 −1.93371 −0.966857 0.255317i \(-0.917820\pi\)
−0.966857 + 0.255317i \(0.917820\pi\)
\(710\) 22.7446 0.853588
\(711\) 0 0
\(712\) −14.7446 −0.552576
\(713\) 63.2119 2.36731
\(714\) 0 0
\(715\) 11.3723 0.425299
\(716\) 8.74456 0.326800
\(717\) 0 0
\(718\) −14.7446 −0.550262
\(719\) −22.5109 −0.839514 −0.419757 0.907637i \(-0.637885\pi\)
−0.419757 + 0.907637i \(0.637885\pi\)
\(720\) 0 0
\(721\) −16.8614 −0.627952
\(722\) 35.3505 1.31561
\(723\) 0 0
\(724\) −0.510875 −0.0189865
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) −17.0951 −0.634022 −0.317011 0.948422i \(-0.602679\pi\)
−0.317011 + 0.948422i \(0.602679\pi\)
\(728\) 1.00000 0.0370625
\(729\) 0 0
\(730\) −8.86141 −0.327975
\(731\) 8.39403 0.310464
\(732\) 0 0
\(733\) −19.2554 −0.711216 −0.355608 0.934635i \(-0.615726\pi\)
−0.355608 + 0.934635i \(0.615726\pi\)
\(734\) 25.4891 0.940821
\(735\) 0 0
\(736\) 9.37228 0.345467
\(737\) 45.4891 1.67561
\(738\) 0 0
\(739\) −25.7228 −0.946229 −0.473114 0.881001i \(-0.656870\pi\)
−0.473114 + 0.881001i \(0.656870\pi\)
\(740\) 4.62772 0.170118
\(741\) 0 0
\(742\) 2.74456 0.100756
\(743\) −24.4674 −0.897621 −0.448810 0.893627i \(-0.648152\pi\)
−0.448810 + 0.893627i \(0.648152\pi\)
\(744\) 0 0
\(745\) 48.0000 1.75858
\(746\) 7.48913 0.274196
\(747\) 0 0
\(748\) 4.62772 0.169206
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) 34.9783 1.27637 0.638187 0.769881i \(-0.279685\pi\)
0.638187 + 0.769881i \(0.279685\pi\)
\(752\) 12.7446 0.464746
\(753\) 0 0
\(754\) −0.627719 −0.0228602
\(755\) 6.35053 0.231120
\(756\) 0 0
\(757\) 37.2119 1.35249 0.676245 0.736676i \(-0.263606\pi\)
0.676245 + 0.736676i \(0.263606\pi\)
\(758\) −17.2554 −0.626746
\(759\) 0 0
\(760\) 24.8614 0.901818
\(761\) −1.02175 −0.0370384 −0.0185192 0.999829i \(-0.505895\pi\)
−0.0185192 + 0.999829i \(0.505895\pi\)
\(762\) 0 0
\(763\) 9.37228 0.339299
\(764\) −12.1168 −0.438372
\(765\) 0 0
\(766\) −33.6060 −1.21423
\(767\) −2.00000 −0.0722158
\(768\) 0 0
\(769\) −25.3723 −0.914948 −0.457474 0.889223i \(-0.651246\pi\)
−0.457474 + 0.889223i \(0.651246\pi\)
\(770\) 11.3723 0.409829
\(771\) 0 0
\(772\) 2.00000 0.0719816
\(773\) 29.0951 1.04648 0.523239 0.852186i \(-0.324724\pi\)
0.523239 + 0.852186i \(0.324724\pi\)
\(774\) 0 0
\(775\) 42.9783 1.54382
\(776\) −15.4891 −0.556027
\(777\) 0 0
\(778\) 14.7446 0.528618
\(779\) −20.2337 −0.724947
\(780\) 0 0
\(781\) −22.7446 −0.813864
\(782\) −12.8614 −0.459923
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −49.3288 −1.76062
\(786\) 0 0
\(787\) 9.88316 0.352296 0.176148 0.984364i \(-0.443636\pi\)
0.176148 + 0.984364i \(0.443636\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −22.7446 −0.809215
\(791\) −20.0000 −0.711118
\(792\) 0 0
\(793\) 12.1168 0.430282
\(794\) 16.7446 0.594242
\(795\) 0 0
\(796\) −16.8614 −0.597637
\(797\) −7.72281 −0.273556 −0.136778 0.990602i \(-0.543675\pi\)
−0.136778 + 0.990602i \(0.543675\pi\)
\(798\) 0 0
\(799\) −17.4891 −0.618721
\(800\) 6.37228 0.225294
\(801\) 0 0
\(802\) −12.5109 −0.441774
\(803\) 8.86141 0.312712
\(804\) 0 0
\(805\) −31.6060 −1.11396
\(806\) −6.74456 −0.237567
\(807\) 0 0
\(808\) −2.00000 −0.0703598
\(809\) 21.2554 0.747301 0.373651 0.927569i \(-0.378106\pi\)
0.373651 + 0.927569i \(0.378106\pi\)
\(810\) 0 0
\(811\) −14.1168 −0.495709 −0.247855 0.968797i \(-0.579726\pi\)
−0.247855 + 0.968797i \(0.579726\pi\)
\(812\) −0.627719 −0.0220286
\(813\) 0 0
\(814\) −4.62772 −0.162201
\(815\) 54.7446 1.91762
\(816\) 0 0
\(817\) −45.0951 −1.57768
\(818\) −2.86141 −0.100047
\(819\) 0 0
\(820\) −9.25544 −0.323214
\(821\) 3.72281 0.129927 0.0649635 0.997888i \(-0.479307\pi\)
0.0649635 + 0.997888i \(0.479307\pi\)
\(822\) 0 0
\(823\) 25.7228 0.896641 0.448320 0.893873i \(-0.352022\pi\)
0.448320 + 0.893873i \(0.352022\pi\)
\(824\) 16.8614 0.587395
\(825\) 0 0
\(826\) −2.00000 −0.0695889
\(827\) 28.6277 0.995483 0.497742 0.867325i \(-0.334163\pi\)
0.497742 + 0.867325i \(0.334163\pi\)
\(828\) 0 0
\(829\) −18.6277 −0.646967 −0.323484 0.946234i \(-0.604854\pi\)
−0.323484 + 0.946234i \(0.604854\pi\)
\(830\) −20.2337 −0.702322
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) −1.37228 −0.0475467
\(834\) 0 0
\(835\) −13.8832 −0.480446
\(836\) −24.8614 −0.859850
\(837\) 0 0
\(838\) 24.8614 0.858823
\(839\) −12.7446 −0.439991 −0.219996 0.975501i \(-0.570604\pi\)
−0.219996 + 0.975501i \(0.570604\pi\)
\(840\) 0 0
\(841\) −28.6060 −0.986413
\(842\) 15.4891 0.533791
\(843\) 0 0
\(844\) −23.3723 −0.804507
\(845\) 3.37228 0.116010
\(846\) 0 0
\(847\) −0.372281 −0.0127917
\(848\) −2.74456 −0.0942487
\(849\) 0 0
\(850\) −8.74456 −0.299936
\(851\) 12.8614 0.440883
\(852\) 0 0
\(853\) 41.2119 1.41107 0.705535 0.708675i \(-0.250707\pi\)
0.705535 + 0.708675i \(0.250707\pi\)
\(854\) 12.1168 0.414630
\(855\) 0 0
\(856\) 2.00000 0.0683586
\(857\) 44.9783 1.53643 0.768214 0.640193i \(-0.221146\pi\)
0.768214 + 0.640193i \(0.221146\pi\)
\(858\) 0 0
\(859\) 10.7446 0.366600 0.183300 0.983057i \(-0.441322\pi\)
0.183300 + 0.983057i \(0.441322\pi\)
\(860\) −20.6277 −0.703399
\(861\) 0 0
\(862\) −21.2554 −0.723963
\(863\) 26.9783 0.918350 0.459175 0.888346i \(-0.348145\pi\)
0.459175 + 0.888346i \(0.348145\pi\)
\(864\) 0 0
\(865\) −33.7228 −1.14661
\(866\) 10.0000 0.339814
\(867\) 0 0
\(868\) −6.74456 −0.228925
\(869\) 22.7446 0.771556
\(870\) 0 0
\(871\) 13.4891 0.457062
\(872\) −9.37228 −0.317385
\(873\) 0 0
\(874\) 69.0951 2.33718
\(875\) −4.62772 −0.156445
\(876\) 0 0
\(877\) −7.02175 −0.237108 −0.118554 0.992948i \(-0.537826\pi\)
−0.118554 + 0.992948i \(0.537826\pi\)
\(878\) 15.3723 0.518789
\(879\) 0 0
\(880\) −11.3723 −0.383360
\(881\) −42.8614 −1.44404 −0.722019 0.691873i \(-0.756786\pi\)
−0.722019 + 0.691873i \(0.756786\pi\)
\(882\) 0 0
\(883\) −24.6277 −0.828789 −0.414394 0.910097i \(-0.636007\pi\)
−0.414394 + 0.910097i \(0.636007\pi\)
\(884\) 1.37228 0.0461548
\(885\) 0 0
\(886\) −28.9783 −0.973543
\(887\) −27.2119 −0.913687 −0.456844 0.889547i \(-0.651020\pi\)
−0.456844 + 0.889547i \(0.651020\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) −49.7228 −1.66671
\(891\) 0 0
\(892\) 5.48913 0.183790
\(893\) 93.9565 3.14413
\(894\) 0 0
\(895\) 29.4891 0.985713
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 5.37228 0.179275
\(899\) 4.23369 0.141201
\(900\) 0 0
\(901\) 3.76631 0.125474
\(902\) 9.25544 0.308172
\(903\) 0 0
\(904\) 20.0000 0.665190
\(905\) −1.72281 −0.0572682
\(906\) 0 0
\(907\) 33.9565 1.12751 0.563754 0.825943i \(-0.309357\pi\)
0.563754 + 0.825943i \(0.309357\pi\)
\(908\) −11.4891 −0.381280
\(909\) 0 0
\(910\) 3.37228 0.111790
\(911\) 40.1168 1.32913 0.664565 0.747230i \(-0.268617\pi\)
0.664565 + 0.747230i \(0.268617\pi\)
\(912\) 0 0
\(913\) 20.2337 0.669637
\(914\) −22.2337 −0.735425
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) −6.11684 −0.201996
\(918\) 0 0
\(919\) 31.2119 1.02959 0.514793 0.857314i \(-0.327869\pi\)
0.514793 + 0.857314i \(0.327869\pi\)
\(920\) 31.6060 1.04202
\(921\) 0 0
\(922\) 3.60597 0.118756
\(923\) −6.74456 −0.222000
\(924\) 0 0
\(925\) 8.74456 0.287519
\(926\) 34.3505 1.12883
\(927\) 0 0
\(928\) 0.627719 0.0206059
\(929\) 26.7446 0.877461 0.438730 0.898619i \(-0.355428\pi\)
0.438730 + 0.898619i \(0.355428\pi\)
\(930\) 0 0
\(931\) 7.37228 0.241617
\(932\) 21.4891 0.703900
\(933\) 0 0
\(934\) 29.0951 0.952021
\(935\) 15.6060 0.510370
\(936\) 0 0
\(937\) 47.4891 1.55140 0.775701 0.631101i \(-0.217396\pi\)
0.775701 + 0.631101i \(0.217396\pi\)
\(938\) 13.4891 0.440436
\(939\) 0 0
\(940\) 42.9783 1.40180
\(941\) 13.2554 0.432115 0.216057 0.976381i \(-0.430680\pi\)
0.216057 + 0.976381i \(0.430680\pi\)
\(942\) 0 0
\(943\) −25.7228 −0.837650
\(944\) 2.00000 0.0650945
\(945\) 0 0
\(946\) 20.6277 0.670665
\(947\) −36.8614 −1.19783 −0.598917 0.800811i \(-0.704402\pi\)
−0.598917 + 0.800811i \(0.704402\pi\)
\(948\) 0 0
\(949\) 2.62772 0.0852994
\(950\) 46.9783 1.52418
\(951\) 0 0
\(952\) 1.37228 0.0444759
\(953\) −22.5109 −0.729199 −0.364599 0.931164i \(-0.618794\pi\)
−0.364599 + 0.931164i \(0.618794\pi\)
\(954\) 0 0
\(955\) −40.8614 −1.32224
\(956\) 17.4891 0.565639
\(957\) 0 0
\(958\) −12.3505 −0.399028
\(959\) 16.1168 0.520440
\(960\) 0 0
\(961\) 14.4891 0.467391
\(962\) −1.37228 −0.0442441
\(963\) 0 0
\(964\) −6.00000 −0.193247
\(965\) 6.74456 0.217115
\(966\) 0 0
\(967\) 42.1168 1.35439 0.677193 0.735805i \(-0.263196\pi\)
0.677193 + 0.735805i \(0.263196\pi\)
\(968\) 0.372281 0.0119656
\(969\) 0 0
\(970\) −52.2337 −1.67712
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 10.7446 0.344455
\(974\) 36.4674 1.16849
\(975\) 0 0
\(976\) −12.1168 −0.387851
\(977\) −7.88316 −0.252205 −0.126102 0.992017i \(-0.540247\pi\)
−0.126102 + 0.992017i \(0.540247\pi\)
\(978\) 0 0
\(979\) 49.7228 1.58915
\(980\) 3.37228 0.107724
\(981\) 0 0
\(982\) 36.7446 1.17257
\(983\) −16.3505 −0.521501 −0.260750 0.965406i \(-0.583970\pi\)
−0.260750 + 0.965406i \(0.583970\pi\)
\(984\) 0 0
\(985\) 20.2337 0.644699
\(986\) −0.861407 −0.0274328
\(987\) 0 0
\(988\) −7.37228 −0.234544
\(989\) −57.3288 −1.82295
\(990\) 0 0
\(991\) −20.2337 −0.642744 −0.321372 0.946953i \(-0.604144\pi\)
−0.321372 + 0.946953i \(0.604144\pi\)
\(992\) 6.74456 0.214140
\(993\) 0 0
\(994\) −6.74456 −0.213925
\(995\) −56.8614 −1.80263
\(996\) 0 0
\(997\) 43.4891 1.37731 0.688657 0.725087i \(-0.258201\pi\)
0.688657 + 0.725087i \(0.258201\pi\)
\(998\) −20.0000 −0.633089
\(999\) 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 1638.2.a.x.1.2 yes 2
3.2 odd 2 1638.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1638.2.a.v.1.1 2 3.2 odd 2
1638.2.a.x.1.2 yes 2 1.1 even 1 trivial