Properties

Label 3870.2.a.bc.1.2
Level $3870$
Weight $2$
Character 3870.1
Self dual yes
Analytic conductor $30.902$
Analytic rank $1$
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] = [3870,2,Mod(1,3870)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3870, 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("3870.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3870 = 2 \cdot 3^{2} \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3870.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: \(30.9021055822\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 430)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3870.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -0.585786 q^{11} -1.82843 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.41421 q^{17} -1.00000 q^{19} -1.00000 q^{20} +0.585786 q^{22} +5.07107 q^{23} +1.00000 q^{25} +1.82843 q^{26} +1.00000 q^{28} -1.24264 q^{29} -0.414214 q^{31} -1.00000 q^{32} +1.41421 q^{34} -1.00000 q^{35} +2.24264 q^{37} +1.00000 q^{38} +1.00000 q^{40} +1.82843 q^{41} +1.00000 q^{43} -0.585786 q^{44} -5.07107 q^{46} -7.07107 q^{47} -6.00000 q^{49} -1.00000 q^{50} -1.82843 q^{52} +5.65685 q^{53} +0.585786 q^{55} -1.00000 q^{56} +1.24264 q^{58} +1.17157 q^{59} +0.0710678 q^{61} +0.414214 q^{62} +1.00000 q^{64} +1.82843 q^{65} -1.24264 q^{67} -1.41421 q^{68} +1.00000 q^{70} -11.8995 q^{71} -9.24264 q^{73} -2.24264 q^{74} -1.00000 q^{76} -0.585786 q^{77} -8.41421 q^{79} -1.00000 q^{80} -1.82843 q^{82} -3.65685 q^{83} +1.41421 q^{85} -1.00000 q^{86} +0.585786 q^{88} +9.07107 q^{89} -1.82843 q^{91} +5.07107 q^{92} +7.07107 q^{94} +1.00000 q^{95} +13.5563 q^{97} +6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{20} + 4 q^{22} - 4 q^{23} + 2 q^{25} - 2 q^{26} + 2 q^{28} + 6 q^{29} + 2 q^{31} - 2 q^{32} - 2 q^{35} - 4 q^{37} + 2 q^{38} + 2 q^{40} - 2 q^{41} + 2 q^{43} - 4 q^{44} + 4 q^{46} - 12 q^{49} - 2 q^{50} + 2 q^{52} + 4 q^{55} - 2 q^{56} - 6 q^{58} + 8 q^{59} - 14 q^{61} - 2 q^{62} + 2 q^{64} - 2 q^{65} + 6 q^{67} + 2 q^{70} - 4 q^{71} - 10 q^{73} + 4 q^{74} - 2 q^{76} - 4 q^{77} - 14 q^{79} - 2 q^{80} + 2 q^{82} + 4 q^{83} - 2 q^{86} + 4 q^{88} + 4 q^{89} + 2 q^{91} - 4 q^{92} + 2 q^{95} - 4 q^{97} + 12 q^{98}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −0.585786 −0.176621 −0.0883106 0.996093i \(-0.528147\pi\)
−0.0883106 + 0.996093i \(0.528147\pi\)
\(12\) 0 0
\(13\) −1.82843 −0.507114 −0.253557 0.967320i \(-0.581601\pi\)
−0.253557 + 0.967320i \(0.581601\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.41421 −0.342997 −0.171499 0.985184i \(-0.554861\pi\)
−0.171499 + 0.985184i \(0.554861\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0.585786 0.124890
\(23\) 5.07107 1.05739 0.528695 0.848812i \(-0.322681\pi\)
0.528695 + 0.848812i \(0.322681\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.82843 0.358584
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −1.24264 −0.230753 −0.115376 0.993322i \(-0.536807\pi\)
−0.115376 + 0.993322i \(0.536807\pi\)
\(30\) 0 0
\(31\) −0.414214 −0.0743950 −0.0371975 0.999308i \(-0.511843\pi\)
−0.0371975 + 0.999308i \(0.511843\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.41421 0.242536
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 2.24264 0.368688 0.184344 0.982862i \(-0.440984\pi\)
0.184344 + 0.982862i \(0.440984\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 1.82843 0.285552 0.142776 0.989755i \(-0.454397\pi\)
0.142776 + 0.989755i \(0.454397\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499
\(44\) −0.585786 −0.0883106
\(45\) 0 0
\(46\) −5.07107 −0.747688
\(47\) −7.07107 −1.03142 −0.515711 0.856763i \(-0.672472\pi\)
−0.515711 + 0.856763i \(0.672472\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −1.82843 −0.253557
\(53\) 5.65685 0.777029 0.388514 0.921443i \(-0.372988\pi\)
0.388514 + 0.921443i \(0.372988\pi\)
\(54\) 0 0
\(55\) 0.585786 0.0789874
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 1.24264 0.163167
\(59\) 1.17157 0.152526 0.0762629 0.997088i \(-0.475701\pi\)
0.0762629 + 0.997088i \(0.475701\pi\)
\(60\) 0 0
\(61\) 0.0710678 0.00909930 0.00454965 0.999990i \(-0.498552\pi\)
0.00454965 + 0.999990i \(0.498552\pi\)
\(62\) 0.414214 0.0526052
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.82843 0.226788
\(66\) 0 0
\(67\) −1.24264 −0.151813 −0.0759064 0.997115i \(-0.524185\pi\)
−0.0759064 + 0.997115i \(0.524185\pi\)
\(68\) −1.41421 −0.171499
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) −11.8995 −1.41221 −0.706105 0.708107i \(-0.749549\pi\)
−0.706105 + 0.708107i \(0.749549\pi\)
\(72\) 0 0
\(73\) −9.24264 −1.08177 −0.540885 0.841097i \(-0.681910\pi\)
−0.540885 + 0.841097i \(0.681910\pi\)
\(74\) −2.24264 −0.260702
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −0.585786 −0.0667566
\(78\) 0 0
\(79\) −8.41421 −0.946673 −0.473336 0.880882i \(-0.656951\pi\)
−0.473336 + 0.880882i \(0.656951\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −1.82843 −0.201916
\(83\) −3.65685 −0.401392 −0.200696 0.979654i \(-0.564320\pi\)
−0.200696 + 0.979654i \(0.564320\pi\)
\(84\) 0 0
\(85\) 1.41421 0.153393
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) 0.585786 0.0624450
\(89\) 9.07107 0.961531 0.480766 0.876849i \(-0.340359\pi\)
0.480766 + 0.876849i \(0.340359\pi\)
\(90\) 0 0
\(91\) −1.82843 −0.191671
\(92\) 5.07107 0.528695
\(93\) 0 0
\(94\) 7.07107 0.729325
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 13.5563 1.37644 0.688219 0.725503i \(-0.258393\pi\)
0.688219 + 0.725503i \(0.258393\pi\)
\(98\) 6.00000 0.606092
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −1.41421 −0.140720 −0.0703598 0.997522i \(-0.522415\pi\)
−0.0703598 + 0.997522i \(0.522415\pi\)
\(102\) 0 0
\(103\) 0.828427 0.0816274 0.0408137 0.999167i \(-0.487005\pi\)
0.0408137 + 0.999167i \(0.487005\pi\)
\(104\) 1.82843 0.179292
\(105\) 0 0
\(106\) −5.65685 −0.549442
\(107\) −1.92893 −0.186477 −0.0932385 0.995644i \(-0.529722\pi\)
−0.0932385 + 0.995644i \(0.529722\pi\)
\(108\) 0 0
\(109\) −10.8284 −1.03718 −0.518588 0.855024i \(-0.673542\pi\)
−0.518588 + 0.855024i \(0.673542\pi\)
\(110\) −0.585786 −0.0558525
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −4.89949 −0.460906 −0.230453 0.973083i \(-0.574021\pi\)
−0.230453 + 0.973083i \(0.574021\pi\)
\(114\) 0 0
\(115\) −5.07107 −0.472880
\(116\) −1.24264 −0.115376
\(117\) 0 0
\(118\) −1.17157 −0.107852
\(119\) −1.41421 −0.129641
\(120\) 0 0
\(121\) −10.6569 −0.968805
\(122\) −0.0710678 −0.00643418
\(123\) 0 0
\(124\) −0.414214 −0.0371975
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 3.51472 0.311881 0.155940 0.987766i \(-0.450159\pi\)
0.155940 + 0.987766i \(0.450159\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −1.82843 −0.160364
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 1.24264 0.107348
\(135\) 0 0
\(136\) 1.41421 0.121268
\(137\) −3.92893 −0.335671 −0.167836 0.985815i \(-0.553678\pi\)
−0.167836 + 0.985815i \(0.553678\pi\)
\(138\) 0 0
\(139\) −6.24264 −0.529494 −0.264747 0.964318i \(-0.585288\pi\)
−0.264747 + 0.964318i \(0.585288\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 11.8995 0.998583
\(143\) 1.07107 0.0895672
\(144\) 0 0
\(145\) 1.24264 0.103196
\(146\) 9.24264 0.764926
\(147\) 0 0
\(148\) 2.24264 0.184344
\(149\) 10.5563 0.864810 0.432405 0.901680i \(-0.357665\pi\)
0.432405 + 0.901680i \(0.357665\pi\)
\(150\) 0 0
\(151\) 10.4853 0.853280 0.426640 0.904422i \(-0.359697\pi\)
0.426640 + 0.904422i \(0.359697\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 0.585786 0.0472040
\(155\) 0.414214 0.0332704
\(156\) 0 0
\(157\) 10.2426 0.817452 0.408726 0.912657i \(-0.365973\pi\)
0.408726 + 0.912657i \(0.365973\pi\)
\(158\) 8.41421 0.669399
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 5.07107 0.399656
\(162\) 0 0
\(163\) 13.6569 1.06969 0.534844 0.844951i \(-0.320370\pi\)
0.534844 + 0.844951i \(0.320370\pi\)
\(164\) 1.82843 0.142776
\(165\) 0 0
\(166\) 3.65685 0.283827
\(167\) −13.5563 −1.04902 −0.524511 0.851404i \(-0.675752\pi\)
−0.524511 + 0.851404i \(0.675752\pi\)
\(168\) 0 0
\(169\) −9.65685 −0.742835
\(170\) −1.41421 −0.108465
\(171\) 0 0
\(172\) 1.00000 0.0762493
\(173\) −0.514719 −0.0391333 −0.0195667 0.999809i \(-0.506229\pi\)
−0.0195667 + 0.999809i \(0.506229\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −0.585786 −0.0441553
\(177\) 0 0
\(178\) −9.07107 −0.679905
\(179\) −17.4853 −1.30691 −0.653456 0.756965i \(-0.726681\pi\)
−0.653456 + 0.756965i \(0.726681\pi\)
\(180\) 0 0
\(181\) −15.0711 −1.12022 −0.560112 0.828417i \(-0.689242\pi\)
−0.560112 + 0.828417i \(0.689242\pi\)
\(182\) 1.82843 0.135532
\(183\) 0 0
\(184\) −5.07107 −0.373844
\(185\) −2.24264 −0.164882
\(186\) 0 0
\(187\) 0.828427 0.0605806
\(188\) −7.07107 −0.515711
\(189\) 0 0
\(190\) −1.00000 −0.0725476
\(191\) −19.8995 −1.43988 −0.719938 0.694038i \(-0.755830\pi\)
−0.719938 + 0.694038i \(0.755830\pi\)
\(192\) 0 0
\(193\) 6.82843 0.491521 0.245760 0.969331i \(-0.420962\pi\)
0.245760 + 0.969331i \(0.420962\pi\)
\(194\) −13.5563 −0.973289
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −25.1421 −1.79130 −0.895651 0.444757i \(-0.853290\pi\)
−0.895651 + 0.444757i \(0.853290\pi\)
\(198\) 0 0
\(199\) −0.100505 −0.00712462 −0.00356231 0.999994i \(-0.501134\pi\)
−0.00356231 + 0.999994i \(0.501134\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 1.41421 0.0995037
\(203\) −1.24264 −0.0872163
\(204\) 0 0
\(205\) −1.82843 −0.127703
\(206\) −0.828427 −0.0577193
\(207\) 0 0
\(208\) −1.82843 −0.126779
\(209\) 0.585786 0.0405197
\(210\) 0 0
\(211\) −1.65685 −0.114063 −0.0570313 0.998372i \(-0.518163\pi\)
−0.0570313 + 0.998372i \(0.518163\pi\)
\(212\) 5.65685 0.388514
\(213\) 0 0
\(214\) 1.92893 0.131859
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) −0.414214 −0.0281186
\(218\) 10.8284 0.733394
\(219\) 0 0
\(220\) 0.585786 0.0394937
\(221\) 2.58579 0.173939
\(222\) 0 0
\(223\) 16.1421 1.08096 0.540479 0.841358i \(-0.318243\pi\)
0.540479 + 0.841358i \(0.318243\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 4.89949 0.325910
\(227\) −9.17157 −0.608739 −0.304369 0.952554i \(-0.598446\pi\)
−0.304369 + 0.952554i \(0.598446\pi\)
\(228\) 0 0
\(229\) 0.828427 0.0547440 0.0273720 0.999625i \(-0.491286\pi\)
0.0273720 + 0.999625i \(0.491286\pi\)
\(230\) 5.07107 0.334376
\(231\) 0 0
\(232\) 1.24264 0.0815834
\(233\) −8.48528 −0.555889 −0.277945 0.960597i \(-0.589653\pi\)
−0.277945 + 0.960597i \(0.589653\pi\)
\(234\) 0 0
\(235\) 7.07107 0.461266
\(236\) 1.17157 0.0762629
\(237\) 0 0
\(238\) 1.41421 0.0916698
\(239\) −11.5858 −0.749422 −0.374711 0.927142i \(-0.622258\pi\)
−0.374711 + 0.927142i \(0.622258\pi\)
\(240\) 0 0
\(241\) −21.4142 −1.37941 −0.689705 0.724090i \(-0.742260\pi\)
−0.689705 + 0.724090i \(0.742260\pi\)
\(242\) 10.6569 0.685049
\(243\) 0 0
\(244\) 0.0710678 0.00454965
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) 1.82843 0.116340
\(248\) 0.414214 0.0263026
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 10.9706 0.692456 0.346228 0.938150i \(-0.387462\pi\)
0.346228 + 0.938150i \(0.387462\pi\)
\(252\) 0 0
\(253\) −2.97056 −0.186758
\(254\) −3.51472 −0.220533
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −8.89949 −0.555135 −0.277568 0.960706i \(-0.589528\pi\)
−0.277568 + 0.960706i \(0.589528\pi\)
\(258\) 0 0
\(259\) 2.24264 0.139351
\(260\) 1.82843 0.113394
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) −17.4853 −1.07819 −0.539094 0.842245i \(-0.681233\pi\)
−0.539094 + 0.842245i \(0.681233\pi\)
\(264\) 0 0
\(265\) −5.65685 −0.347498
\(266\) 1.00000 0.0613139
\(267\) 0 0
\(268\) −1.24264 −0.0759064
\(269\) −9.17157 −0.559201 −0.279600 0.960116i \(-0.590202\pi\)
−0.279600 + 0.960116i \(0.590202\pi\)
\(270\) 0 0
\(271\) 8.89949 0.540606 0.270303 0.962775i \(-0.412876\pi\)
0.270303 + 0.962775i \(0.412876\pi\)
\(272\) −1.41421 −0.0857493
\(273\) 0 0
\(274\) 3.92893 0.237355
\(275\) −0.585786 −0.0353243
\(276\) 0 0
\(277\) −0.485281 −0.0291577 −0.0145789 0.999894i \(-0.504641\pi\)
−0.0145789 + 0.999894i \(0.504641\pi\)
\(278\) 6.24264 0.374409
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) 1.34315 0.0801254 0.0400627 0.999197i \(-0.487244\pi\)
0.0400627 + 0.999197i \(0.487244\pi\)
\(282\) 0 0
\(283\) 18.7574 1.11501 0.557505 0.830174i \(-0.311759\pi\)
0.557505 + 0.830174i \(0.311759\pi\)
\(284\) −11.8995 −0.706105
\(285\) 0 0
\(286\) −1.07107 −0.0633336
\(287\) 1.82843 0.107929
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) −1.24264 −0.0729704
\(291\) 0 0
\(292\) −9.24264 −0.540885
\(293\) −10.9706 −0.640907 −0.320454 0.947264i \(-0.603835\pi\)
−0.320454 + 0.947264i \(0.603835\pi\)
\(294\) 0 0
\(295\) −1.17157 −0.0682116
\(296\) −2.24264 −0.130351
\(297\) 0 0
\(298\) −10.5563 −0.611513
\(299\) −9.27208 −0.536218
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) −10.4853 −0.603360
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) −0.0710678 −0.00406933
\(306\) 0 0
\(307\) −5.92893 −0.338382 −0.169191 0.985583i \(-0.554115\pi\)
−0.169191 + 0.985583i \(0.554115\pi\)
\(308\) −0.585786 −0.0333783
\(309\) 0 0
\(310\) −0.414214 −0.0235257
\(311\) 4.07107 0.230849 0.115425 0.993316i \(-0.463177\pi\)
0.115425 + 0.993316i \(0.463177\pi\)
\(312\) 0 0
\(313\) −1.31371 −0.0742552 −0.0371276 0.999311i \(-0.511821\pi\)
−0.0371276 + 0.999311i \(0.511821\pi\)
\(314\) −10.2426 −0.578026
\(315\) 0 0
\(316\) −8.41421 −0.473336
\(317\) −0.857864 −0.0481825 −0.0240912 0.999710i \(-0.507669\pi\)
−0.0240912 + 0.999710i \(0.507669\pi\)
\(318\) 0 0
\(319\) 0.727922 0.0407558
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −5.07107 −0.282600
\(323\) 1.41421 0.0786889
\(324\) 0 0
\(325\) −1.82843 −0.101423
\(326\) −13.6569 −0.756383
\(327\) 0 0
\(328\) −1.82843 −0.100958
\(329\) −7.07107 −0.389841
\(330\) 0 0
\(331\) −25.3137 −1.39137 −0.695684 0.718348i \(-0.744898\pi\)
−0.695684 + 0.718348i \(0.744898\pi\)
\(332\) −3.65685 −0.200696
\(333\) 0 0
\(334\) 13.5563 0.741770
\(335\) 1.24264 0.0678927
\(336\) 0 0
\(337\) −8.34315 −0.454480 −0.227240 0.973839i \(-0.572970\pi\)
−0.227240 + 0.973839i \(0.572970\pi\)
\(338\) 9.65685 0.525264
\(339\) 0 0
\(340\) 1.41421 0.0766965
\(341\) 0.242641 0.0131397
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) 0.514719 0.0276714
\(347\) 22.2426 1.19405 0.597024 0.802224i \(-0.296350\pi\)
0.597024 + 0.802224i \(0.296350\pi\)
\(348\) 0 0
\(349\) 23.4558 1.25556 0.627781 0.778390i \(-0.283963\pi\)
0.627781 + 0.778390i \(0.283963\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 0.585786 0.0312225
\(353\) 32.3848 1.72367 0.861834 0.507191i \(-0.169316\pi\)
0.861834 + 0.507191i \(0.169316\pi\)
\(354\) 0 0
\(355\) 11.8995 0.631560
\(356\) 9.07107 0.480766
\(357\) 0 0
\(358\) 17.4853 0.924126
\(359\) −27.5269 −1.45281 −0.726407 0.687264i \(-0.758811\pi\)
−0.726407 + 0.687264i \(0.758811\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 15.0711 0.792118
\(363\) 0 0
\(364\) −1.82843 −0.0958356
\(365\) 9.24264 0.483782
\(366\) 0 0
\(367\) −31.3137 −1.63456 −0.817281 0.576239i \(-0.804520\pi\)
−0.817281 + 0.576239i \(0.804520\pi\)
\(368\) 5.07107 0.264348
\(369\) 0 0
\(370\) 2.24264 0.116589
\(371\) 5.65685 0.293689
\(372\) 0 0
\(373\) 14.7279 0.762583 0.381291 0.924455i \(-0.375479\pi\)
0.381291 + 0.924455i \(0.375479\pi\)
\(374\) −0.828427 −0.0428369
\(375\) 0 0
\(376\) 7.07107 0.364662
\(377\) 2.27208 0.117018
\(378\) 0 0
\(379\) 4.14214 0.212767 0.106384 0.994325i \(-0.466073\pi\)
0.106384 + 0.994325i \(0.466073\pi\)
\(380\) 1.00000 0.0512989
\(381\) 0 0
\(382\) 19.8995 1.01815
\(383\) 8.31371 0.424811 0.212405 0.977182i \(-0.431870\pi\)
0.212405 + 0.977182i \(0.431870\pi\)
\(384\) 0 0
\(385\) 0.585786 0.0298544
\(386\) −6.82843 −0.347558
\(387\) 0 0
\(388\) 13.5563 0.688219
\(389\) −20.8284 −1.05604 −0.528022 0.849231i \(-0.677066\pi\)
−0.528022 + 0.849231i \(0.677066\pi\)
\(390\) 0 0
\(391\) −7.17157 −0.362682
\(392\) 6.00000 0.303046
\(393\) 0 0
\(394\) 25.1421 1.26664
\(395\) 8.41421 0.423365
\(396\) 0 0
\(397\) −20.4853 −1.02813 −0.514063 0.857752i \(-0.671860\pi\)
−0.514063 + 0.857752i \(0.671860\pi\)
\(398\) 0.100505 0.00503786
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 24.1127 1.20413 0.602065 0.798447i \(-0.294345\pi\)
0.602065 + 0.798447i \(0.294345\pi\)
\(402\) 0 0
\(403\) 0.757359 0.0377268
\(404\) −1.41421 −0.0703598
\(405\) 0 0
\(406\) 1.24264 0.0616712
\(407\) −1.31371 −0.0651181
\(408\) 0 0
\(409\) −27.2132 −1.34561 −0.672803 0.739822i \(-0.734910\pi\)
−0.672803 + 0.739822i \(0.734910\pi\)
\(410\) 1.82843 0.0902996
\(411\) 0 0
\(412\) 0.828427 0.0408137
\(413\) 1.17157 0.0576493
\(414\) 0 0
\(415\) 3.65685 0.179508
\(416\) 1.82843 0.0896460
\(417\) 0 0
\(418\) −0.585786 −0.0286518
\(419\) 12.7990 0.625272 0.312636 0.949873i \(-0.398788\pi\)
0.312636 + 0.949873i \(0.398788\pi\)
\(420\) 0 0
\(421\) 1.24264 0.0605626 0.0302813 0.999541i \(-0.490360\pi\)
0.0302813 + 0.999541i \(0.490360\pi\)
\(422\) 1.65685 0.0806544
\(423\) 0 0
\(424\) −5.65685 −0.274721
\(425\) −1.41421 −0.0685994
\(426\) 0 0
\(427\) 0.0710678 0.00343921
\(428\) −1.92893 −0.0932385
\(429\) 0 0
\(430\) 1.00000 0.0482243
\(431\) −1.02944 −0.0495862 −0.0247931 0.999693i \(-0.507893\pi\)
−0.0247931 + 0.999693i \(0.507893\pi\)
\(432\) 0 0
\(433\) −36.0711 −1.73346 −0.866732 0.498773i \(-0.833784\pi\)
−0.866732 + 0.498773i \(0.833784\pi\)
\(434\) 0.414214 0.0198829
\(435\) 0 0
\(436\) −10.8284 −0.518588
\(437\) −5.07107 −0.242582
\(438\) 0 0
\(439\) −15.3137 −0.730883 −0.365442 0.930834i \(-0.619082\pi\)
−0.365442 + 0.930834i \(0.619082\pi\)
\(440\) −0.585786 −0.0279263
\(441\) 0 0
\(442\) −2.58579 −0.122993
\(443\) 9.92893 0.471738 0.235869 0.971785i \(-0.424206\pi\)
0.235869 + 0.971785i \(0.424206\pi\)
\(444\) 0 0
\(445\) −9.07107 −0.430010
\(446\) −16.1421 −0.764352
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −22.2426 −1.04970 −0.524848 0.851196i \(-0.675878\pi\)
−0.524848 + 0.851196i \(0.675878\pi\)
\(450\) 0 0
\(451\) −1.07107 −0.0504346
\(452\) −4.89949 −0.230453
\(453\) 0 0
\(454\) 9.17157 0.430443
\(455\) 1.82843 0.0857180
\(456\) 0 0
\(457\) −30.4853 −1.42604 −0.713021 0.701143i \(-0.752673\pi\)
−0.713021 + 0.701143i \(0.752673\pi\)
\(458\) −0.828427 −0.0387099
\(459\) 0 0
\(460\) −5.07107 −0.236440
\(461\) 15.0711 0.701930 0.350965 0.936389i \(-0.385854\pi\)
0.350965 + 0.936389i \(0.385854\pi\)
\(462\) 0 0
\(463\) 30.3137 1.40880 0.704399 0.709804i \(-0.251217\pi\)
0.704399 + 0.709804i \(0.251217\pi\)
\(464\) −1.24264 −0.0576881
\(465\) 0 0
\(466\) 8.48528 0.393073
\(467\) 20.9289 0.968475 0.484238 0.874936i \(-0.339097\pi\)
0.484238 + 0.874936i \(0.339097\pi\)
\(468\) 0 0
\(469\) −1.24264 −0.0573798
\(470\) −7.07107 −0.326164
\(471\) 0 0
\(472\) −1.17157 −0.0539260
\(473\) −0.585786 −0.0269345
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) −1.41421 −0.0648204
\(477\) 0 0
\(478\) 11.5858 0.529922
\(479\) −9.17157 −0.419060 −0.209530 0.977802i \(-0.567193\pi\)
−0.209530 + 0.977802i \(0.567193\pi\)
\(480\) 0 0
\(481\) −4.10051 −0.186967
\(482\) 21.4142 0.975391
\(483\) 0 0
\(484\) −10.6569 −0.484402
\(485\) −13.5563 −0.615562
\(486\) 0 0
\(487\) −33.2132 −1.50503 −0.752517 0.658573i \(-0.771160\pi\)
−0.752517 + 0.658573i \(0.771160\pi\)
\(488\) −0.0710678 −0.00321709
\(489\) 0 0
\(490\) −6.00000 −0.271052
\(491\) −16.6274 −0.750385 −0.375192 0.926947i \(-0.622423\pi\)
−0.375192 + 0.926947i \(0.622423\pi\)
\(492\) 0 0
\(493\) 1.75736 0.0791475
\(494\) −1.82843 −0.0822648
\(495\) 0 0
\(496\) −0.414214 −0.0185987
\(497\) −11.8995 −0.533765
\(498\) 0 0
\(499\) −34.4558 −1.54246 −0.771228 0.636559i \(-0.780357\pi\)
−0.771228 + 0.636559i \(0.780357\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −10.9706 −0.489640
\(503\) 31.6569 1.41151 0.705755 0.708456i \(-0.250608\pi\)
0.705755 + 0.708456i \(0.250608\pi\)
\(504\) 0 0
\(505\) 1.41421 0.0629317
\(506\) 2.97056 0.132058
\(507\) 0 0
\(508\) 3.51472 0.155940
\(509\) 20.4853 0.907994 0.453997 0.891003i \(-0.349998\pi\)
0.453997 + 0.891003i \(0.349998\pi\)
\(510\) 0 0
\(511\) −9.24264 −0.408870
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 8.89949 0.392540
\(515\) −0.828427 −0.0365049
\(516\) 0 0
\(517\) 4.14214 0.182171
\(518\) −2.24264 −0.0985360
\(519\) 0 0
\(520\) −1.82843 −0.0801818
\(521\) 11.0711 0.485032 0.242516 0.970147i \(-0.422027\pi\)
0.242516 + 0.970147i \(0.422027\pi\)
\(522\) 0 0
\(523\) 10.7279 0.469099 0.234550 0.972104i \(-0.424638\pi\)
0.234550 + 0.972104i \(0.424638\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) 17.4853 0.762394
\(527\) 0.585786 0.0255173
\(528\) 0 0
\(529\) 2.71573 0.118075
\(530\) 5.65685 0.245718
\(531\) 0 0
\(532\) −1.00000 −0.0433555
\(533\) −3.34315 −0.144808
\(534\) 0 0
\(535\) 1.92893 0.0833950
\(536\) 1.24264 0.0536739
\(537\) 0 0
\(538\) 9.17157 0.395415
\(539\) 3.51472 0.151390
\(540\) 0 0
\(541\) 7.85786 0.337836 0.168918 0.985630i \(-0.445973\pi\)
0.168918 + 0.985630i \(0.445973\pi\)
\(542\) −8.89949 −0.382266
\(543\) 0 0
\(544\) 1.41421 0.0606339
\(545\) 10.8284 0.463839
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) −3.92893 −0.167836
\(549\) 0 0
\(550\) 0.585786 0.0249780
\(551\) 1.24264 0.0529383
\(552\) 0 0
\(553\) −8.41421 −0.357809
\(554\) 0.485281 0.0206176
\(555\) 0 0
\(556\) −6.24264 −0.264747
\(557\) −41.8284 −1.77233 −0.886164 0.463372i \(-0.846639\pi\)
−0.886164 + 0.463372i \(0.846639\pi\)
\(558\) 0 0
\(559\) −1.82843 −0.0773342
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −1.34315 −0.0566572
\(563\) 1.92893 0.0812948 0.0406474 0.999174i \(-0.487058\pi\)
0.0406474 + 0.999174i \(0.487058\pi\)
\(564\) 0 0
\(565\) 4.89949 0.206123
\(566\) −18.7574 −0.788431
\(567\) 0 0
\(568\) 11.8995 0.499292
\(569\) −22.7990 −0.955783 −0.477892 0.878419i \(-0.658599\pi\)
−0.477892 + 0.878419i \(0.658599\pi\)
\(570\) 0 0
\(571\) 33.0000 1.38101 0.690504 0.723329i \(-0.257389\pi\)
0.690504 + 0.723329i \(0.257389\pi\)
\(572\) 1.07107 0.0447836
\(573\) 0 0
\(574\) −1.82843 −0.0763171
\(575\) 5.07107 0.211478
\(576\) 0 0
\(577\) −5.58579 −0.232539 −0.116270 0.993218i \(-0.537094\pi\)
−0.116270 + 0.993218i \(0.537094\pi\)
\(578\) 15.0000 0.623918
\(579\) 0 0
\(580\) 1.24264 0.0515978
\(581\) −3.65685 −0.151712
\(582\) 0 0
\(583\) −3.31371 −0.137240
\(584\) 9.24264 0.382463
\(585\) 0 0
\(586\) 10.9706 0.453190
\(587\) 27.5563 1.13737 0.568686 0.822555i \(-0.307452\pi\)
0.568686 + 0.822555i \(0.307452\pi\)
\(588\) 0 0
\(589\) 0.414214 0.0170674
\(590\) 1.17157 0.0482329
\(591\) 0 0
\(592\) 2.24264 0.0921720
\(593\) 7.24264 0.297420 0.148710 0.988881i \(-0.452488\pi\)
0.148710 + 0.988881i \(0.452488\pi\)
\(594\) 0 0
\(595\) 1.41421 0.0579771
\(596\) 10.5563 0.432405
\(597\) 0 0
\(598\) 9.27208 0.379163
\(599\) −9.85786 −0.402781 −0.201391 0.979511i \(-0.564546\pi\)
−0.201391 + 0.979511i \(0.564546\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) −1.00000 −0.0407570
\(603\) 0 0
\(604\) 10.4853 0.426640
\(605\) 10.6569 0.433263
\(606\) 0 0
\(607\) 6.00000 0.243532 0.121766 0.992559i \(-0.461144\pi\)
0.121766 + 0.992559i \(0.461144\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0.0710678 0.00287745
\(611\) 12.9289 0.523049
\(612\) 0 0
\(613\) 38.1716 1.54174 0.770868 0.636995i \(-0.219823\pi\)
0.770868 + 0.636995i \(0.219823\pi\)
\(614\) 5.92893 0.239272
\(615\) 0 0
\(616\) 0.585786 0.0236020
\(617\) 16.2843 0.655580 0.327790 0.944751i \(-0.393696\pi\)
0.327790 + 0.944751i \(0.393696\pi\)
\(618\) 0 0
\(619\) −0.100505 −0.00403964 −0.00201982 0.999998i \(-0.500643\pi\)
−0.00201982 + 0.999998i \(0.500643\pi\)
\(620\) 0.414214 0.0166352
\(621\) 0 0
\(622\) −4.07107 −0.163235
\(623\) 9.07107 0.363425
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 1.31371 0.0525064
\(627\) 0 0
\(628\) 10.2426 0.408726
\(629\) −3.17157 −0.126459
\(630\) 0 0
\(631\) −13.6985 −0.545328 −0.272664 0.962109i \(-0.587905\pi\)
−0.272664 + 0.962109i \(0.587905\pi\)
\(632\) 8.41421 0.334699
\(633\) 0 0
\(634\) 0.857864 0.0340701
\(635\) −3.51472 −0.139477
\(636\) 0 0
\(637\) 10.9706 0.434670
\(638\) −0.727922 −0.0288187
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −42.7696 −1.68930 −0.844648 0.535322i \(-0.820190\pi\)
−0.844648 + 0.535322i \(0.820190\pi\)
\(642\) 0 0
\(643\) 39.7279 1.56672 0.783358 0.621571i \(-0.213505\pi\)
0.783358 + 0.621571i \(0.213505\pi\)
\(644\) 5.07107 0.199828
\(645\) 0 0
\(646\) −1.41421 −0.0556415
\(647\) −5.20101 −0.204473 −0.102236 0.994760i \(-0.532600\pi\)
−0.102236 + 0.994760i \(0.532600\pi\)
\(648\) 0 0
\(649\) −0.686292 −0.0269393
\(650\) 1.82843 0.0717168
\(651\) 0 0
\(652\) 13.6569 0.534844
\(653\) 6.58579 0.257722 0.128861 0.991663i \(-0.458868\pi\)
0.128861 + 0.991663i \(0.458868\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 1.82843 0.0713881
\(657\) 0 0
\(658\) 7.07107 0.275659
\(659\) −24.3848 −0.949896 −0.474948 0.880014i \(-0.657533\pi\)
−0.474948 + 0.880014i \(0.657533\pi\)
\(660\) 0 0
\(661\) 24.8701 0.967333 0.483667 0.875252i \(-0.339305\pi\)
0.483667 + 0.875252i \(0.339305\pi\)
\(662\) 25.3137 0.983845
\(663\) 0 0
\(664\) 3.65685 0.141913
\(665\) 1.00000 0.0387783
\(666\) 0 0
\(667\) −6.30152 −0.243996
\(668\) −13.5563 −0.524511
\(669\) 0 0
\(670\) −1.24264 −0.0480074
\(671\) −0.0416306 −0.00160713
\(672\) 0 0
\(673\) 20.4142 0.786910 0.393455 0.919344i \(-0.371280\pi\)
0.393455 + 0.919344i \(0.371280\pi\)
\(674\) 8.34315 0.321366
\(675\) 0 0
\(676\) −9.65685 −0.371417
\(677\) −43.4558 −1.67014 −0.835072 0.550141i \(-0.814574\pi\)
−0.835072 + 0.550141i \(0.814574\pi\)
\(678\) 0 0
\(679\) 13.5563 0.520245
\(680\) −1.41421 −0.0542326
\(681\) 0 0
\(682\) −0.242641 −0.00929119
\(683\) 3.51472 0.134487 0.0672435 0.997737i \(-0.478580\pi\)
0.0672435 + 0.997737i \(0.478580\pi\)
\(684\) 0 0
\(685\) 3.92893 0.150117
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) 1.00000 0.0381246
\(689\) −10.3431 −0.394042
\(690\) 0 0
\(691\) 18.4853 0.703213 0.351607 0.936148i \(-0.385635\pi\)
0.351607 + 0.936148i \(0.385635\pi\)
\(692\) −0.514719 −0.0195667
\(693\) 0 0
\(694\) −22.2426 −0.844319
\(695\) 6.24264 0.236797
\(696\) 0 0
\(697\) −2.58579 −0.0979436
\(698\) −23.4558 −0.887817
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −11.2721 −0.425741 −0.212870 0.977080i \(-0.568281\pi\)
−0.212870 + 0.977080i \(0.568281\pi\)
\(702\) 0 0
\(703\) −2.24264 −0.0845828
\(704\) −0.585786 −0.0220777
\(705\) 0 0
\(706\) −32.3848 −1.21882
\(707\) −1.41421 −0.0531870
\(708\) 0 0
\(709\) −7.21320 −0.270898 −0.135449 0.990784i \(-0.543248\pi\)
−0.135449 + 0.990784i \(0.543248\pi\)
\(710\) −11.8995 −0.446580
\(711\) 0 0
\(712\) −9.07107 −0.339953
\(713\) −2.10051 −0.0786645
\(714\) 0 0
\(715\) −1.07107 −0.0400557
\(716\) −17.4853 −0.653456
\(717\) 0 0
\(718\) 27.5269 1.02730
\(719\) 36.6274 1.36597 0.682986 0.730431i \(-0.260681\pi\)
0.682986 + 0.730431i \(0.260681\pi\)
\(720\) 0 0
\(721\) 0.828427 0.0308522
\(722\) 18.0000 0.669891
\(723\) 0 0
\(724\) −15.0711 −0.560112
\(725\) −1.24264 −0.0461505
\(726\) 0 0
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 1.82843 0.0677660
\(729\) 0 0
\(730\) −9.24264 −0.342085
\(731\) −1.41421 −0.0523066
\(732\) 0 0
\(733\) 27.7574 1.02524 0.512621 0.858615i \(-0.328675\pi\)
0.512621 + 0.858615i \(0.328675\pi\)
\(734\) 31.3137 1.15581
\(735\) 0 0
\(736\) −5.07107 −0.186922
\(737\) 0.727922 0.0268134
\(738\) 0 0
\(739\) 8.02944 0.295368 0.147684 0.989035i \(-0.452818\pi\)
0.147684 + 0.989035i \(0.452818\pi\)
\(740\) −2.24264 −0.0824411
\(741\) 0 0
\(742\) −5.65685 −0.207670
\(743\) −2.31371 −0.0848817 −0.0424409 0.999099i \(-0.513513\pi\)
−0.0424409 + 0.999099i \(0.513513\pi\)
\(744\) 0 0
\(745\) −10.5563 −0.386755
\(746\) −14.7279 −0.539228
\(747\) 0 0
\(748\) 0.828427 0.0302903
\(749\) −1.92893 −0.0704816
\(750\) 0 0
\(751\) −8.72792 −0.318486 −0.159243 0.987239i \(-0.550905\pi\)
−0.159243 + 0.987239i \(0.550905\pi\)
\(752\) −7.07107 −0.257855
\(753\) 0 0
\(754\) −2.27208 −0.0827442
\(755\) −10.4853 −0.381598
\(756\) 0 0
\(757\) 36.4264 1.32394 0.661970 0.749530i \(-0.269721\pi\)
0.661970 + 0.749530i \(0.269721\pi\)
\(758\) −4.14214 −0.150449
\(759\) 0 0
\(760\) −1.00000 −0.0362738
\(761\) 44.8284 1.62503 0.812515 0.582941i \(-0.198098\pi\)
0.812515 + 0.582941i \(0.198098\pi\)
\(762\) 0 0
\(763\) −10.8284 −0.392015
\(764\) −19.8995 −0.719938
\(765\) 0 0
\(766\) −8.31371 −0.300386
\(767\) −2.14214 −0.0773480
\(768\) 0 0
\(769\) 21.0000 0.757279 0.378640 0.925544i \(-0.376392\pi\)
0.378640 + 0.925544i \(0.376392\pi\)
\(770\) −0.585786 −0.0211103
\(771\) 0 0
\(772\) 6.82843 0.245760
\(773\) 25.0711 0.901744 0.450872 0.892589i \(-0.351113\pi\)
0.450872 + 0.892589i \(0.351113\pi\)
\(774\) 0 0
\(775\) −0.414214 −0.0148790
\(776\) −13.5563 −0.486645
\(777\) 0 0
\(778\) 20.8284 0.746735
\(779\) −1.82843 −0.0655102
\(780\) 0 0
\(781\) 6.97056 0.249426
\(782\) 7.17157 0.256455
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) −10.2426 −0.365576
\(786\) 0 0
\(787\) 31.2426 1.11368 0.556840 0.830620i \(-0.312014\pi\)
0.556840 + 0.830620i \(0.312014\pi\)
\(788\) −25.1421 −0.895651
\(789\) 0 0
\(790\) −8.41421 −0.299364
\(791\) −4.89949 −0.174206
\(792\) 0 0
\(793\) −0.129942 −0.00461439
\(794\) 20.4853 0.726995
\(795\) 0 0
\(796\) −0.100505 −0.00356231
\(797\) 42.9411 1.52105 0.760526 0.649307i \(-0.224941\pi\)
0.760526 + 0.649307i \(0.224941\pi\)
\(798\) 0 0
\(799\) 10.0000 0.353775
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −24.1127 −0.851449
\(803\) 5.41421 0.191063
\(804\) 0 0
\(805\) −5.07107 −0.178732
\(806\) −0.757359 −0.0266768
\(807\) 0 0
\(808\) 1.41421 0.0497519
\(809\) 28.7990 1.01252 0.506259 0.862381i \(-0.331028\pi\)
0.506259 + 0.862381i \(0.331028\pi\)
\(810\) 0 0
\(811\) −48.1127 −1.68947 −0.844733 0.535188i \(-0.820241\pi\)
−0.844733 + 0.535188i \(0.820241\pi\)
\(812\) −1.24264 −0.0436081
\(813\) 0 0
\(814\) 1.31371 0.0460455
\(815\) −13.6569 −0.478379
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 27.2132 0.951487
\(819\) 0 0
\(820\) −1.82843 −0.0638514
\(821\) 20.1421 0.702965 0.351483 0.936194i \(-0.385678\pi\)
0.351483 + 0.936194i \(0.385678\pi\)
\(822\) 0 0
\(823\) 13.5147 0.471093 0.235547 0.971863i \(-0.424312\pi\)
0.235547 + 0.971863i \(0.424312\pi\)
\(824\) −0.828427 −0.0288596
\(825\) 0 0
\(826\) −1.17157 −0.0407642
\(827\) 43.5269 1.51358 0.756790 0.653659i \(-0.226767\pi\)
0.756790 + 0.653659i \(0.226767\pi\)
\(828\) 0 0
\(829\) 3.24264 0.112622 0.0563108 0.998413i \(-0.482066\pi\)
0.0563108 + 0.998413i \(0.482066\pi\)
\(830\) −3.65685 −0.126931
\(831\) 0 0
\(832\) −1.82843 −0.0633893
\(833\) 8.48528 0.293998
\(834\) 0 0
\(835\) 13.5563 0.469137
\(836\) 0.585786 0.0202598
\(837\) 0 0
\(838\) −12.7990 −0.442134
\(839\) 51.0122 1.76114 0.880568 0.473919i \(-0.157161\pi\)
0.880568 + 0.473919i \(0.157161\pi\)
\(840\) 0 0
\(841\) −27.4558 −0.946753
\(842\) −1.24264 −0.0428242
\(843\) 0 0
\(844\) −1.65685 −0.0570313
\(845\) 9.65685 0.332206
\(846\) 0 0
\(847\) −10.6569 −0.366174
\(848\) 5.65685 0.194257
\(849\) 0 0
\(850\) 1.41421 0.0485071
\(851\) 11.3726 0.389847
\(852\) 0 0
\(853\) −35.9411 −1.23060 −0.615300 0.788293i \(-0.710965\pi\)
−0.615300 + 0.788293i \(0.710965\pi\)
\(854\) −0.0710678 −0.00243189
\(855\) 0 0
\(856\) 1.92893 0.0659295
\(857\) −47.0711 −1.60792 −0.803959 0.594685i \(-0.797277\pi\)
−0.803959 + 0.594685i \(0.797277\pi\)
\(858\) 0 0
\(859\) −16.5147 −0.563475 −0.281737 0.959492i \(-0.590911\pi\)
−0.281737 + 0.959492i \(0.590911\pi\)
\(860\) −1.00000 −0.0340997
\(861\) 0 0
\(862\) 1.02944 0.0350628
\(863\) 40.8284 1.38982 0.694908 0.719099i \(-0.255445\pi\)
0.694908 + 0.719099i \(0.255445\pi\)
\(864\) 0 0
\(865\) 0.514719 0.0175010
\(866\) 36.0711 1.22574
\(867\) 0 0
\(868\) −0.414214 −0.0140593
\(869\) 4.92893 0.167203
\(870\) 0 0
\(871\) 2.27208 0.0769864
\(872\) 10.8284 0.366697
\(873\) 0 0
\(874\) 5.07107 0.171531
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −18.6274 −0.629003 −0.314502 0.949257i \(-0.601837\pi\)
−0.314502 + 0.949257i \(0.601837\pi\)
\(878\) 15.3137 0.516813
\(879\) 0 0
\(880\) 0.585786 0.0197469
\(881\) 1.20101 0.0404631 0.0202315 0.999795i \(-0.493560\pi\)
0.0202315 + 0.999795i \(0.493560\pi\)
\(882\) 0 0
\(883\) 26.2132 0.882145 0.441072 0.897472i \(-0.354598\pi\)
0.441072 + 0.897472i \(0.354598\pi\)
\(884\) 2.58579 0.0869694
\(885\) 0 0
\(886\) −9.92893 −0.333569
\(887\) 56.1127 1.88408 0.942040 0.335501i \(-0.108905\pi\)
0.942040 + 0.335501i \(0.108905\pi\)
\(888\) 0 0
\(889\) 3.51472 0.117880
\(890\) 9.07107 0.304063
\(891\) 0 0
\(892\) 16.1421 0.540479
\(893\) 7.07107 0.236624
\(894\) 0 0
\(895\) 17.4853 0.584468
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 22.2426 0.742247
\(899\) 0.514719 0.0171668
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) 1.07107 0.0356627
\(903\) 0 0
\(904\) 4.89949 0.162955
\(905\) 15.0711 0.500979
\(906\) 0 0
\(907\) 14.8995 0.494730 0.247365 0.968922i \(-0.420435\pi\)
0.247365 + 0.968922i \(0.420435\pi\)
\(908\) −9.17157 −0.304369
\(909\) 0 0
\(910\) −1.82843 −0.0606118
\(911\) 20.3848 0.675378 0.337689 0.941258i \(-0.390355\pi\)
0.337689 + 0.941258i \(0.390355\pi\)
\(912\) 0 0
\(913\) 2.14214 0.0708943
\(914\) 30.4853 1.00836
\(915\) 0 0
\(916\) 0.828427 0.0273720
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) −34.8995 −1.15123 −0.575614 0.817722i \(-0.695237\pi\)
−0.575614 + 0.817722i \(0.695237\pi\)
\(920\) 5.07107 0.167188
\(921\) 0 0
\(922\) −15.0711 −0.496339
\(923\) 21.7574 0.716152
\(924\) 0 0
\(925\) 2.24264 0.0737376
\(926\) −30.3137 −0.996170
\(927\) 0 0
\(928\) 1.24264 0.0407917
\(929\) 13.7990 0.452730 0.226365 0.974043i \(-0.427316\pi\)
0.226365 + 0.974043i \(0.427316\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) −8.48528 −0.277945
\(933\) 0 0
\(934\) −20.9289 −0.684816
\(935\) −0.828427 −0.0270925
\(936\) 0 0
\(937\) −34.4264 −1.12466 −0.562331 0.826912i \(-0.690095\pi\)
−0.562331 + 0.826912i \(0.690095\pi\)
\(938\) 1.24264 0.0405737
\(939\) 0 0
\(940\) 7.07107 0.230633
\(941\) 53.0711 1.73007 0.865034 0.501714i \(-0.167297\pi\)
0.865034 + 0.501714i \(0.167297\pi\)
\(942\) 0 0
\(943\) 9.27208 0.301940
\(944\) 1.17157 0.0381314
\(945\) 0 0
\(946\) 0.585786 0.0190456
\(947\) −59.1838 −1.92321 −0.961607 0.274430i \(-0.911511\pi\)
−0.961607 + 0.274430i \(0.911511\pi\)
\(948\) 0 0
\(949\) 16.8995 0.548581
\(950\) 1.00000 0.0324443
\(951\) 0 0
\(952\) 1.41421 0.0458349
\(953\) 9.10051 0.294794 0.147397 0.989077i \(-0.452910\pi\)
0.147397 + 0.989077i \(0.452910\pi\)
\(954\) 0 0
\(955\) 19.8995 0.643933
\(956\) −11.5858 −0.374711
\(957\) 0 0
\(958\) 9.17157 0.296320
\(959\) −3.92893 −0.126872
\(960\) 0 0
\(961\) −30.8284 −0.994465
\(962\) 4.10051 0.132206
\(963\) 0 0
\(964\) −21.4142 −0.689705
\(965\) −6.82843 −0.219815
\(966\) 0 0
\(967\) 36.2426 1.16548 0.582742 0.812657i \(-0.301980\pi\)
0.582742 + 0.812657i \(0.301980\pi\)
\(968\) 10.6569 0.342524
\(969\) 0 0
\(970\) 13.5563 0.435268
\(971\) −21.0711 −0.676203 −0.338101 0.941110i \(-0.609785\pi\)
−0.338101 + 0.941110i \(0.609785\pi\)
\(972\) 0 0
\(973\) −6.24264 −0.200130
\(974\) 33.2132 1.06422
\(975\) 0 0
\(976\) 0.0710678 0.00227483
\(977\) 17.0711 0.546152 0.273076 0.961992i \(-0.411959\pi\)
0.273076 + 0.961992i \(0.411959\pi\)
\(978\) 0 0
\(979\) −5.31371 −0.169827
\(980\) 6.00000 0.191663
\(981\) 0 0
\(982\) 16.6274 0.530602
\(983\) 27.3431 0.872111 0.436055 0.899920i \(-0.356375\pi\)
0.436055 + 0.899920i \(0.356375\pi\)
\(984\) 0 0
\(985\) 25.1421 0.801095
\(986\) −1.75736 −0.0559657
\(987\) 0 0
\(988\) 1.82843 0.0581700
\(989\) 5.07107 0.161251
\(990\) 0 0
\(991\) 41.2548 1.31050 0.655251 0.755411i \(-0.272563\pi\)
0.655251 + 0.755411i \(0.272563\pi\)
\(992\) 0.414214 0.0131513
\(993\) 0 0
\(994\) 11.8995 0.377429
\(995\) 0.100505 0.00318622
\(996\) 0 0
\(997\) 37.1716 1.17724 0.588618 0.808411i \(-0.299672\pi\)
0.588618 + 0.808411i \(0.299672\pi\)
\(998\) 34.4558 1.09068
\(999\) 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 3870.2.a.bc.1.2 2
3.2 odd 2 430.2.a.g.1.2 2
12.11 even 2 3440.2.a.j.1.1 2
15.2 even 4 2150.2.b.o.1549.3 4
15.8 even 4 2150.2.b.o.1549.2 4
15.14 odd 2 2150.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
430.2.a.g.1.2 2 3.2 odd 2
2150.2.a.v.1.1 2 15.14 odd 2
2150.2.b.o.1549.2 4 15.8 even 4
2150.2.b.o.1549.3 4 15.2 even 4
3440.2.a.j.1.1 2 12.11 even 2
3870.2.a.bc.1.2 2 1.1 even 1 trivial