Properties

Label 858.2.a.k.1.1
Level $858$
Weight $2$
Character 858.1
Self dual yes
Analytic conductor $6.851$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [858,2,Mod(1,858)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(858, 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("858.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 858 = 2 \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 858.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: \(6.85116449343\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 858.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} +1.00000 q^{22} +3.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -5.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +4.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +10.0000 q^{37} +6.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} -7.00000 q^{41} +1.00000 q^{42} -5.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} +3.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -4.00000 q^{50} +4.00000 q^{51} -1.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} +1.00000 q^{56} +6.00000 q^{57} -5.00000 q^{58} -3.00000 q^{59} -1.00000 q^{60} +13.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} +1.00000 q^{66} -9.00000 q^{67} +4.00000 q^{68} +3.00000 q^{69} -1.00000 q^{70} +2.00000 q^{71} +1.00000 q^{72} -3.00000 q^{73} +10.0000 q^{74} -4.00000 q^{75} +6.00000 q^{76} +1.00000 q^{77} -1.00000 q^{78} +10.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -7.00000 q^{82} -14.0000 q^{83} +1.00000 q^{84} -4.00000 q^{85} -5.00000 q^{86} -5.00000 q^{87} +1.00000 q^{88} -8.00000 q^{89} -1.00000 q^{90} -1.00000 q^{91} +3.00000 q^{92} +4.00000 q^{93} -8.00000 q^{94} -6.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} -6.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) 1.00000 0.213201
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.00000 −0.800000
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) −1.00000 −0.182574
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) 4.00000 0.685994
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 6.00000 0.973329
\(39\) −1.00000 −0.160128
\(40\) −1.00000 −0.158114
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 1.00000 0.154303
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.00000 −0.149071
\(46\) 3.00000 0.442326
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) −4.00000 −0.565685
\(51\) 4.00000 0.560112
\(52\) −1.00000 −0.138675
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.00000 −0.134840
\(56\) 1.00000 0.133631
\(57\) 6.00000 0.794719
\(58\) −5.00000 −0.656532
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) −1.00000 −0.129099
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 4.00000 0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 1.00000 0.123091
\(67\) −9.00000 −1.09952 −0.549762 0.835321i \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) 4.00000 0.485071
\(69\) 3.00000 0.361158
\(70\) −1.00000 −0.119523
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 1.00000 0.117851
\(73\) −3.00000 −0.351123 −0.175562 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) 10.0000 1.16248
\(75\) −4.00000 −0.461880
\(76\) 6.00000 0.688247
\(77\) 1.00000 0.113961
\(78\) −1.00000 −0.113228
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −7.00000 −0.773021
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 1.00000 0.109109
\(85\) −4.00000 −0.433861
\(86\) −5.00000 −0.539164
\(87\) −5.00000 −0.536056
\(88\) 1.00000 0.106600
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) −1.00000 −0.105409
\(91\) −1.00000 −0.104828
\(92\) 3.00000 0.312772
\(93\) 4.00000 0.414781
\(94\) −8.00000 −0.825137
\(95\) −6.00000 −0.615587
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −6.00000 −0.606092
\(99\) 1.00000 0.100504
\(100\) −4.00000 −0.400000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 4.00000 0.396059
\(103\) 13.0000 1.28093 0.640464 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) −1.00000 −0.0980581
\(105\) −1.00000 −0.0975900
\(106\) −2.00000 −0.194257
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 1.00000 0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 10.0000 0.949158
\(112\) 1.00000 0.0944911
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 6.00000 0.561951
\(115\) −3.00000 −0.279751
\(116\) −5.00000 −0.464238
\(117\) −1.00000 −0.0924500
\(118\) −3.00000 −0.276172
\(119\) 4.00000 0.366679
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) 13.0000 1.17696
\(123\) −7.00000 −0.631169
\(124\) 4.00000 0.359211
\(125\) 9.00000 0.804984
\(126\) 1.00000 0.0890871
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.00000 −0.440225
\(130\) 1.00000 0.0877058
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 1.00000 0.0870388
\(133\) 6.00000 0.520266
\(134\) −9.00000 −0.777482
\(135\) −1.00000 −0.0860663
\(136\) 4.00000 0.342997
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 3.00000 0.255377
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −8.00000 −0.673722
\(142\) 2.00000 0.167836
\(143\) −1.00000 −0.0836242
\(144\) 1.00000 0.0833333
\(145\) 5.00000 0.415227
\(146\) −3.00000 −0.248282
\(147\) −6.00000 −0.494872
\(148\) 10.0000 0.821995
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) −4.00000 −0.326599
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 6.00000 0.486664
\(153\) 4.00000 0.323381
\(154\) 1.00000 0.0805823
\(155\) −4.00000 −0.321288
\(156\) −1.00000 −0.0800641
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 10.0000 0.795557
\(159\) −2.00000 −0.158610
\(160\) −1.00000 −0.0790569
\(161\) 3.00000 0.236433
\(162\) 1.00000 0.0785674
\(163\) −11.0000 −0.861586 −0.430793 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) −7.00000 −0.546608
\(165\) −1.00000 −0.0778499
\(166\) −14.0000 −1.08661
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 1.00000 0.0771517
\(169\) 1.00000 0.0769231
\(170\) −4.00000 −0.306786
\(171\) 6.00000 0.458831
\(172\) −5.00000 −0.381246
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) −5.00000 −0.379049
\(175\) −4.00000 −0.302372
\(176\) 1.00000 0.0753778
\(177\) −3.00000 −0.225494
\(178\) −8.00000 −0.599625
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 13.0000 0.960988
\(184\) 3.00000 0.221163
\(185\) −10.0000 −0.735215
\(186\) 4.00000 0.293294
\(187\) 4.00000 0.292509
\(188\) −8.00000 −0.583460
\(189\) 1.00000 0.0727393
\(190\) −6.00000 −0.435286
\(191\) −11.0000 −0.795932 −0.397966 0.917400i \(-0.630284\pi\)
−0.397966 + 0.917400i \(0.630284\pi\)
\(192\) 1.00000 0.0721688
\(193\) 26.0000 1.87152 0.935760 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) −14.0000 −1.00514
\(195\) 1.00000 0.0716115
\(196\) −6.00000 −0.428571
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 1.00000 0.0710669
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) −4.00000 −0.282843
\(201\) −9.00000 −0.634811
\(202\) −10.0000 −0.703598
\(203\) −5.00000 −0.350931
\(204\) 4.00000 0.280056
\(205\) 7.00000 0.488901
\(206\) 13.0000 0.905753
\(207\) 3.00000 0.208514
\(208\) −1.00000 −0.0693375
\(209\) 6.00000 0.415029
\(210\) −1.00000 −0.0690066
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −2.00000 −0.137361
\(213\) 2.00000 0.137038
\(214\) 3.00000 0.205076
\(215\) 5.00000 0.340997
\(216\) 1.00000 0.0680414
\(217\) 4.00000 0.271538
\(218\) −16.0000 −1.08366
\(219\) −3.00000 −0.202721
\(220\) −1.00000 −0.0674200
\(221\) −4.00000 −0.269069
\(222\) 10.0000 0.671156
\(223\) −28.0000 −1.87502 −0.937509 0.347960i \(-0.886874\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 1.00000 0.0668153
\(225\) −4.00000 −0.266667
\(226\) 9.00000 0.598671
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 6.00000 0.397360
\(229\) 13.0000 0.859064 0.429532 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) −3.00000 −0.197814
\(231\) 1.00000 0.0657952
\(232\) −5.00000 −0.328266
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 8.00000 0.521862
\(236\) −3.00000 −0.195283
\(237\) 10.0000 0.649570
\(238\) 4.00000 0.259281
\(239\) −5.00000 −0.323423 −0.161712 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) 13.0000 0.832240
\(245\) 6.00000 0.383326
\(246\) −7.00000 −0.446304
\(247\) −6.00000 −0.381771
\(248\) 4.00000 0.254000
\(249\) −14.0000 −0.887214
\(250\) 9.00000 0.569210
\(251\) −14.0000 −0.883672 −0.441836 0.897096i \(-0.645673\pi\)
−0.441836 + 0.897096i \(0.645673\pi\)
\(252\) 1.00000 0.0629941
\(253\) 3.00000 0.188608
\(254\) 2.00000 0.125491
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) 27.0000 1.68421 0.842107 0.539311i \(-0.181315\pi\)
0.842107 + 0.539311i \(0.181315\pi\)
\(258\) −5.00000 −0.311286
\(259\) 10.0000 0.621370
\(260\) 1.00000 0.0620174
\(261\) −5.00000 −0.309492
\(262\) −15.0000 −0.926703
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 1.00000 0.0615457
\(265\) 2.00000 0.122859
\(266\) 6.00000 0.367884
\(267\) −8.00000 −0.489592
\(268\) −9.00000 −0.549762
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 4.00000 0.242536
\(273\) −1.00000 −0.0605228
\(274\) −2.00000 −0.120824
\(275\) −4.00000 −0.241209
\(276\) 3.00000 0.180579
\(277\) 5.00000 0.300421 0.150210 0.988654i \(-0.452005\pi\)
0.150210 + 0.988654i \(0.452005\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) −1.00000 −0.0597614
\(281\) −5.00000 −0.298275 −0.149137 0.988816i \(-0.547650\pi\)
−0.149137 + 0.988816i \(0.547650\pi\)
\(282\) −8.00000 −0.476393
\(283\) −17.0000 −1.01055 −0.505273 0.862960i \(-0.668608\pi\)
−0.505273 + 0.862960i \(0.668608\pi\)
\(284\) 2.00000 0.118678
\(285\) −6.00000 −0.355409
\(286\) −1.00000 −0.0591312
\(287\) −7.00000 −0.413197
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 5.00000 0.293610
\(291\) −14.0000 −0.820695
\(292\) −3.00000 −0.175562
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) −6.00000 −0.349927
\(295\) 3.00000 0.174667
\(296\) 10.0000 0.581238
\(297\) 1.00000 0.0580259
\(298\) −4.00000 −0.231714
\(299\) −3.00000 −0.173494
\(300\) −4.00000 −0.230940
\(301\) −5.00000 −0.288195
\(302\) −16.0000 −0.920697
\(303\) −10.0000 −0.574485
\(304\) 6.00000 0.344124
\(305\) −13.0000 −0.744378
\(306\) 4.00000 0.228665
\(307\) 14.0000 0.799022 0.399511 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) 1.00000 0.0569803
\(309\) 13.0000 0.739544
\(310\) −4.00000 −0.227185
\(311\) 32.0000 1.81455 0.907277 0.420534i \(-0.138157\pi\)
0.907277 + 0.420534i \(0.138157\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 27.0000 1.52613 0.763065 0.646322i \(-0.223694\pi\)
0.763065 + 0.646322i \(0.223694\pi\)
\(314\) 18.0000 1.01580
\(315\) −1.00000 −0.0563436
\(316\) 10.0000 0.562544
\(317\) −11.0000 −0.617822 −0.308911 0.951091i \(-0.599964\pi\)
−0.308911 + 0.951091i \(0.599964\pi\)
\(318\) −2.00000 −0.112154
\(319\) −5.00000 −0.279946
\(320\) −1.00000 −0.0559017
\(321\) 3.00000 0.167444
\(322\) 3.00000 0.167183
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) −11.0000 −0.609234
\(327\) −16.0000 −0.884802
\(328\) −7.00000 −0.386510
\(329\) −8.00000 −0.441054
\(330\) −1.00000 −0.0550482
\(331\) −25.0000 −1.37412 −0.687062 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) −14.0000 −0.768350
\(333\) 10.0000 0.547997
\(334\) 21.0000 1.14907
\(335\) 9.00000 0.491723
\(336\) 1.00000 0.0545545
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 1.00000 0.0543928
\(339\) 9.00000 0.488813
\(340\) −4.00000 −0.216930
\(341\) 4.00000 0.216612
\(342\) 6.00000 0.324443
\(343\) −13.0000 −0.701934
\(344\) −5.00000 −0.269582
\(345\) −3.00000 −0.161515
\(346\) −15.0000 −0.806405
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −5.00000 −0.268028
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) −4.00000 −0.213809
\(351\) −1.00000 −0.0533761
\(352\) 1.00000 0.0533002
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) −3.00000 −0.159448
\(355\) −2.00000 −0.106149
\(356\) −8.00000 −0.423999
\(357\) 4.00000 0.211702
\(358\) 12.0000 0.634220
\(359\) −11.0000 −0.580558 −0.290279 0.956942i \(-0.593748\pi\)
−0.290279 + 0.956942i \(0.593748\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 17.0000 0.894737
\(362\) 14.0000 0.735824
\(363\) 1.00000 0.0524864
\(364\) −1.00000 −0.0524142
\(365\) 3.00000 0.157027
\(366\) 13.0000 0.679521
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 3.00000 0.156386
\(369\) −7.00000 −0.364405
\(370\) −10.0000 −0.519875
\(371\) −2.00000 −0.103835
\(372\) 4.00000 0.207390
\(373\) −11.0000 −0.569558 −0.284779 0.958593i \(-0.591920\pi\)
−0.284779 + 0.958593i \(0.591920\pi\)
\(374\) 4.00000 0.206835
\(375\) 9.00000 0.464758
\(376\) −8.00000 −0.412568
\(377\) 5.00000 0.257513
\(378\) 1.00000 0.0514344
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −6.00000 −0.307794
\(381\) 2.00000 0.102463
\(382\) −11.0000 −0.562809
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 1.00000 0.0510310
\(385\) −1.00000 −0.0509647
\(386\) 26.0000 1.32337
\(387\) −5.00000 −0.254164
\(388\) −14.0000 −0.710742
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 1.00000 0.0506370
\(391\) 12.0000 0.606866
\(392\) −6.00000 −0.303046
\(393\) −15.0000 −0.756650
\(394\) −12.0000 −0.604551
\(395\) −10.0000 −0.503155
\(396\) 1.00000 0.0502519
\(397\) −15.0000 −0.752828 −0.376414 0.926451i \(-0.622843\pi\)
−0.376414 + 0.926451i \(0.622843\pi\)
\(398\) 11.0000 0.551380
\(399\) 6.00000 0.300376
\(400\) −4.00000 −0.200000
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) −9.00000 −0.448879
\(403\) −4.00000 −0.199254
\(404\) −10.0000 −0.497519
\(405\) −1.00000 −0.0496904
\(406\) −5.00000 −0.248146
\(407\) 10.0000 0.495682
\(408\) 4.00000 0.198030
\(409\) 11.0000 0.543915 0.271957 0.962309i \(-0.412329\pi\)
0.271957 + 0.962309i \(0.412329\pi\)
\(410\) 7.00000 0.345705
\(411\) −2.00000 −0.0986527
\(412\) 13.0000 0.640464
\(413\) −3.00000 −0.147620
\(414\) 3.00000 0.147442
\(415\) 14.0000 0.687233
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 9.00000 0.438633 0.219317 0.975654i \(-0.429617\pi\)
0.219317 + 0.975654i \(0.429617\pi\)
\(422\) −12.0000 −0.584151
\(423\) −8.00000 −0.388973
\(424\) −2.00000 −0.0971286
\(425\) −16.0000 −0.776114
\(426\) 2.00000 0.0969003
\(427\) 13.0000 0.629114
\(428\) 3.00000 0.145010
\(429\) −1.00000 −0.0482805
\(430\) 5.00000 0.241121
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 1.00000 0.0481125
\(433\) 21.0000 1.00920 0.504598 0.863355i \(-0.331641\pi\)
0.504598 + 0.863355i \(0.331641\pi\)
\(434\) 4.00000 0.192006
\(435\) 5.00000 0.239732
\(436\) −16.0000 −0.766261
\(437\) 18.0000 0.861057
\(438\) −3.00000 −0.143346
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) −1.00000 −0.0476731
\(441\) −6.00000 −0.285714
\(442\) −4.00000 −0.190261
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 10.0000 0.474579
\(445\) 8.00000 0.379236
\(446\) −28.0000 −1.32584
\(447\) −4.00000 −0.189194
\(448\) 1.00000 0.0472456
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) −4.00000 −0.188562
\(451\) −7.00000 −0.329617
\(452\) 9.00000 0.423324
\(453\) −16.0000 −0.751746
\(454\) −24.0000 −1.12638
\(455\) 1.00000 0.0468807
\(456\) 6.00000 0.280976
\(457\) 31.0000 1.45012 0.725059 0.688686i \(-0.241812\pi\)
0.725059 + 0.688686i \(0.241812\pi\)
\(458\) 13.0000 0.607450
\(459\) 4.00000 0.186704
\(460\) −3.00000 −0.139876
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 1.00000 0.0465242
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) −5.00000 −0.232119
\(465\) −4.00000 −0.185496
\(466\) −18.0000 −0.833834
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −9.00000 −0.415581
\(470\) 8.00000 0.369012
\(471\) 18.0000 0.829396
\(472\) −3.00000 −0.138086
\(473\) −5.00000 −0.229900
\(474\) 10.0000 0.459315
\(475\) −24.0000 −1.10120
\(476\) 4.00000 0.183340
\(477\) −2.00000 −0.0915737
\(478\) −5.00000 −0.228695
\(479\) 25.0000 1.14228 0.571140 0.820853i \(-0.306501\pi\)
0.571140 + 0.820853i \(0.306501\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −10.0000 −0.455961
\(482\) 18.0000 0.819878
\(483\) 3.00000 0.136505
\(484\) 1.00000 0.0454545
\(485\) 14.0000 0.635707
\(486\) 1.00000 0.0453609
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 13.0000 0.588482
\(489\) −11.0000 −0.497437
\(490\) 6.00000 0.271052
\(491\) 9.00000 0.406164 0.203082 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) −7.00000 −0.315584
\(493\) −20.0000 −0.900755
\(494\) −6.00000 −0.269953
\(495\) −1.00000 −0.0449467
\(496\) 4.00000 0.179605
\(497\) 2.00000 0.0897123
\(498\) −14.0000 −0.627355
\(499\) −11.0000 −0.492428 −0.246214 0.969216i \(-0.579187\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) 9.00000 0.402492
\(501\) 21.0000 0.938211
\(502\) −14.0000 −0.624851
\(503\) 28.0000 1.24846 0.624229 0.781241i \(-0.285413\pi\)
0.624229 + 0.781241i \(0.285413\pi\)
\(504\) 1.00000 0.0445435
\(505\) 10.0000 0.444994
\(506\) 3.00000 0.133366
\(507\) 1.00000 0.0444116
\(508\) 2.00000 0.0887357
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) −4.00000 −0.177123
\(511\) −3.00000 −0.132712
\(512\) 1.00000 0.0441942
\(513\) 6.00000 0.264906
\(514\) 27.0000 1.19092
\(515\) −13.0000 −0.572848
\(516\) −5.00000 −0.220113
\(517\) −8.00000 −0.351840
\(518\) 10.0000 0.439375
\(519\) −15.0000 −0.658427
\(520\) 1.00000 0.0438529
\(521\) 39.0000 1.70862 0.854311 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) −5.00000 −0.218844
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −15.0000 −0.655278
\(525\) −4.00000 −0.174574
\(526\) −16.0000 −0.697633
\(527\) 16.0000 0.696971
\(528\) 1.00000 0.0435194
\(529\) −14.0000 −0.608696
\(530\) 2.00000 0.0868744
\(531\) −3.00000 −0.130189
\(532\) 6.00000 0.260133
\(533\) 7.00000 0.303204
\(534\) −8.00000 −0.346194
\(535\) −3.00000 −0.129701
\(536\) −9.00000 −0.388741
\(537\) 12.0000 0.517838
\(538\) 4.00000 0.172452
\(539\) −6.00000 −0.258438
\(540\) −1.00000 −0.0430331
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −8.00000 −0.343629
\(543\) 14.0000 0.600798
\(544\) 4.00000 0.171499
\(545\) 16.0000 0.685365
\(546\) −1.00000 −0.0427960
\(547\) −5.00000 −0.213785 −0.106892 0.994271i \(-0.534090\pi\)
−0.106892 + 0.994271i \(0.534090\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 13.0000 0.554826
\(550\) −4.00000 −0.170561
\(551\) −30.0000 −1.27804
\(552\) 3.00000 0.127688
\(553\) 10.0000 0.425243
\(554\) 5.00000 0.212430
\(555\) −10.0000 −0.424476
\(556\) 0 0
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) 4.00000 0.169334
\(559\) 5.00000 0.211477
\(560\) −1.00000 −0.0422577
\(561\) 4.00000 0.168880
\(562\) −5.00000 −0.210912
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) −8.00000 −0.336861
\(565\) −9.00000 −0.378633
\(566\) −17.0000 −0.714563
\(567\) 1.00000 0.0419961
\(568\) 2.00000 0.0839181
\(569\) 38.0000 1.59304 0.796521 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) −6.00000 −0.251312
\(571\) −37.0000 −1.54840 −0.774201 0.632940i \(-0.781848\pi\)
−0.774201 + 0.632940i \(0.781848\pi\)
\(572\) −1.00000 −0.0418121
\(573\) −11.0000 −0.459532
\(574\) −7.00000 −0.292174
\(575\) −12.0000 −0.500435
\(576\) 1.00000 0.0416667
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 26.0000 1.08052
\(580\) 5.00000 0.207614
\(581\) −14.0000 −0.580818
\(582\) −14.0000 −0.580319
\(583\) −2.00000 −0.0828315
\(584\) −3.00000 −0.124141
\(585\) 1.00000 0.0413449
\(586\) 14.0000 0.578335
\(587\) 35.0000 1.44460 0.722302 0.691577i \(-0.243084\pi\)
0.722302 + 0.691577i \(0.243084\pi\)
\(588\) −6.00000 −0.247436
\(589\) 24.0000 0.988903
\(590\) 3.00000 0.123508
\(591\) −12.0000 −0.493614
\(592\) 10.0000 0.410997
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 1.00000 0.0410305
\(595\) −4.00000 −0.163984
\(596\) −4.00000 −0.163846
\(597\) 11.0000 0.450200
\(598\) −3.00000 −0.122679
\(599\) −9.00000 −0.367730 −0.183865 0.982952i \(-0.558861\pi\)
−0.183865 + 0.982952i \(0.558861\pi\)
\(600\) −4.00000 −0.163299
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) −5.00000 −0.203785
\(603\) −9.00000 −0.366508
\(604\) −16.0000 −0.651031
\(605\) −1.00000 −0.0406558
\(606\) −10.0000 −0.406222
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 6.00000 0.243332
\(609\) −5.00000 −0.202610
\(610\) −13.0000 −0.526355
\(611\) 8.00000 0.323645
\(612\) 4.00000 0.161690
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 14.0000 0.564994
\(615\) 7.00000 0.282267
\(616\) 1.00000 0.0402911
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 13.0000 0.522937
\(619\) −17.0000 −0.683288 −0.341644 0.939829i \(-0.610984\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) −4.00000 −0.160644
\(621\) 3.00000 0.120386
\(622\) 32.0000 1.28308
\(623\) −8.00000 −0.320513
\(624\) −1.00000 −0.0400320
\(625\) 11.0000 0.440000
\(626\) 27.0000 1.07914
\(627\) 6.00000 0.239617
\(628\) 18.0000 0.718278
\(629\) 40.0000 1.59490
\(630\) −1.00000 −0.0398410
\(631\) 30.0000 1.19428 0.597141 0.802137i \(-0.296303\pi\)
0.597141 + 0.802137i \(0.296303\pi\)
\(632\) 10.0000 0.397779
\(633\) −12.0000 −0.476957
\(634\) −11.0000 −0.436866
\(635\) −2.00000 −0.0793676
\(636\) −2.00000 −0.0793052
\(637\) 6.00000 0.237729
\(638\) −5.00000 −0.197952
\(639\) 2.00000 0.0791188
\(640\) −1.00000 −0.0395285
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 3.00000 0.118401
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 3.00000 0.118217
\(645\) 5.00000 0.196875
\(646\) 24.0000 0.944267
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 1.00000 0.0392837
\(649\) −3.00000 −0.117760
\(650\) 4.00000 0.156893
\(651\) 4.00000 0.156772
\(652\) −11.0000 −0.430793
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −16.0000 −0.625650
\(655\) 15.0000 0.586098
\(656\) −7.00000 −0.273304
\(657\) −3.00000 −0.117041
\(658\) −8.00000 −0.311872
\(659\) 16.0000 0.623272 0.311636 0.950202i \(-0.399123\pi\)
0.311636 + 0.950202i \(0.399123\pi\)
\(660\) −1.00000 −0.0389249
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) −25.0000 −0.971653
\(663\) −4.00000 −0.155347
\(664\) −14.0000 −0.543305
\(665\) −6.00000 −0.232670
\(666\) 10.0000 0.387492
\(667\) −15.0000 −0.580802
\(668\) 21.0000 0.812514
\(669\) −28.0000 −1.08254
\(670\) 9.00000 0.347700
\(671\) 13.0000 0.501859
\(672\) 1.00000 0.0385758
\(673\) −40.0000 −1.54189 −0.770943 0.636904i \(-0.780215\pi\)
−0.770943 + 0.636904i \(0.780215\pi\)
\(674\) 2.00000 0.0770371
\(675\) −4.00000 −0.153960
\(676\) 1.00000 0.0384615
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 9.00000 0.345643
\(679\) −14.0000 −0.537271
\(680\) −4.00000 −0.153393
\(681\) −24.0000 −0.919682
\(682\) 4.00000 0.153168
\(683\) 5.00000 0.191320 0.0956598 0.995414i \(-0.469504\pi\)
0.0956598 + 0.995414i \(0.469504\pi\)
\(684\) 6.00000 0.229416
\(685\) 2.00000 0.0764161
\(686\) −13.0000 −0.496342
\(687\) 13.0000 0.495981
\(688\) −5.00000 −0.190623
\(689\) 2.00000 0.0761939
\(690\) −3.00000 −0.114208
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) −15.0000 −0.570214
\(693\) 1.00000 0.0379869
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −5.00000 −0.189525
\(697\) −28.0000 −1.06058
\(698\) 28.0000 1.05982
\(699\) −18.0000 −0.680823
\(700\) −4.00000 −0.151186
\(701\) −33.0000 −1.24639 −0.623196 0.782065i \(-0.714166\pi\)
−0.623196 + 0.782065i \(0.714166\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 60.0000 2.26294
\(704\) 1.00000 0.0376889
\(705\) 8.00000 0.301297
\(706\) 18.0000 0.677439
\(707\) −10.0000 −0.376089
\(708\) −3.00000 −0.112747
\(709\) −11.0000 −0.413114 −0.206557 0.978435i \(-0.566226\pi\)
−0.206557 + 0.978435i \(0.566226\pi\)
\(710\) −2.00000 −0.0750587
\(711\) 10.0000 0.375029
\(712\) −8.00000 −0.299813
\(713\) 12.0000 0.449404
\(714\) 4.00000 0.149696
\(715\) 1.00000 0.0373979
\(716\) 12.0000 0.448461
\(717\) −5.00000 −0.186728
\(718\) −11.0000 −0.410516
\(719\) 13.0000 0.484818 0.242409 0.970174i \(-0.422062\pi\)
0.242409 + 0.970174i \(0.422062\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 13.0000 0.484145
\(722\) 17.0000 0.632674
\(723\) 18.0000 0.669427
\(724\) 14.0000 0.520306
\(725\) 20.0000 0.742781
\(726\) 1.00000 0.0371135
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 1.00000 0.0370370
\(730\) 3.00000 0.111035
\(731\) −20.0000 −0.739727
\(732\) 13.0000 0.480494
\(733\) −50.0000 −1.84679 −0.923396 0.383849i \(-0.874598\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) 32.0000 1.18114
\(735\) 6.00000 0.221313
\(736\) 3.00000 0.110581
\(737\) −9.00000 −0.331519
\(738\) −7.00000 −0.257674
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −10.0000 −0.367607
\(741\) −6.00000 −0.220416
\(742\) −2.00000 −0.0734223
\(743\) −19.0000 −0.697042 −0.348521 0.937301i \(-0.613316\pi\)
−0.348521 + 0.937301i \(0.613316\pi\)
\(744\) 4.00000 0.146647
\(745\) 4.00000 0.146549
\(746\) −11.0000 −0.402739
\(747\) −14.0000 −0.512233
\(748\) 4.00000 0.146254
\(749\) 3.00000 0.109618
\(750\) 9.00000 0.328634
\(751\) −53.0000 −1.93400 −0.966999 0.254781i \(-0.917997\pi\)
−0.966999 + 0.254781i \(0.917997\pi\)
\(752\) −8.00000 −0.291730
\(753\) −14.0000 −0.510188
\(754\) 5.00000 0.182089
\(755\) 16.0000 0.582300
\(756\) 1.00000 0.0363696
\(757\) 30.0000 1.09037 0.545184 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) 16.0000 0.581146
\(759\) 3.00000 0.108893
\(760\) −6.00000 −0.217643
\(761\) −29.0000 −1.05125 −0.525625 0.850717i \(-0.676168\pi\)
−0.525625 + 0.850717i \(0.676168\pi\)
\(762\) 2.00000 0.0724524
\(763\) −16.0000 −0.579239
\(764\) −11.0000 −0.397966
\(765\) −4.00000 −0.144620
\(766\) 6.00000 0.216789
\(767\) 3.00000 0.108324
\(768\) 1.00000 0.0360844
\(769\) −35.0000 −1.26213 −0.631066 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 27.0000 0.972381
\(772\) 26.0000 0.935760
\(773\) 46.0000 1.65451 0.827253 0.561830i \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) −5.00000 −0.179721
\(775\) −16.0000 −0.574737
\(776\) −14.0000 −0.502571
\(777\) 10.0000 0.358748
\(778\) −2.00000 −0.0717035
\(779\) −42.0000 −1.50481
\(780\) 1.00000 0.0358057
\(781\) 2.00000 0.0715656
\(782\) 12.0000 0.429119
\(783\) −5.00000 −0.178685
\(784\) −6.00000 −0.214286
\(785\) −18.0000 −0.642448
\(786\) −15.0000 −0.535032
\(787\) −52.0000 −1.85360 −0.926800 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) −12.0000 −0.427482
\(789\) −16.0000 −0.569615
\(790\) −10.0000 −0.355784
\(791\) 9.00000 0.320003
\(792\) 1.00000 0.0355335
\(793\) −13.0000 −0.461644
\(794\) −15.0000 −0.532330
\(795\) 2.00000 0.0709327
\(796\) 11.0000 0.389885
\(797\) 28.0000 0.991811 0.495905 0.868377i \(-0.334836\pi\)
0.495905 + 0.868377i \(0.334836\pi\)
\(798\) 6.00000 0.212398
\(799\) −32.0000 −1.13208
\(800\) −4.00000 −0.141421
\(801\) −8.00000 −0.282666
\(802\) 24.0000 0.847469
\(803\) −3.00000 −0.105868
\(804\) −9.00000 −0.317406
\(805\) −3.00000 −0.105736
\(806\) −4.00000 −0.140894
\(807\) 4.00000 0.140807
\(808\) −10.0000 −0.351799
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) −5.00000 −0.175466
\(813\) −8.00000 −0.280572
\(814\) 10.0000 0.350500
\(815\) 11.0000 0.385313
\(816\) 4.00000 0.140028
\(817\) −30.0000 −1.04957
\(818\) 11.0000 0.384606
\(819\) −1.00000 −0.0349428
\(820\) 7.00000 0.244451
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −9.00000 −0.313720 −0.156860 0.987621i \(-0.550137\pi\)
−0.156860 + 0.987621i \(0.550137\pi\)
\(824\) 13.0000 0.452876
\(825\) −4.00000 −0.139262
\(826\) −3.00000 −0.104383
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 3.00000 0.104257
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 14.0000 0.485947
\(831\) 5.00000 0.173448
\(832\) −1.00000 −0.0346688
\(833\) −24.0000 −0.831551
\(834\) 0 0
\(835\) −21.0000 −0.726735
\(836\) 6.00000 0.207514
\(837\) 4.00000 0.138260
\(838\) 0 0
\(839\) −42.0000 −1.45000 −0.725001 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −4.00000 −0.137931
\(842\) 9.00000 0.310160
\(843\) −5.00000 −0.172209
\(844\) −12.0000 −0.413057
\(845\) −1.00000 −0.0344010
\(846\) −8.00000 −0.275046
\(847\) 1.00000 0.0343604
\(848\) −2.00000 −0.0686803
\(849\) −17.0000 −0.583438
\(850\) −16.0000 −0.548795
\(851\) 30.0000 1.02839
\(852\) 2.00000 0.0685189
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 13.0000 0.444851
\(855\) −6.00000 −0.205196
\(856\) 3.00000 0.102538
\(857\) −52.0000 −1.77629 −0.888143 0.459567i \(-0.848005\pi\)
−0.888143 + 0.459567i \(0.848005\pi\)
\(858\) −1.00000 −0.0341394
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 5.00000 0.170499
\(861\) −7.00000 −0.238559
\(862\) −16.0000 −0.544962
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 1.00000 0.0340207
\(865\) 15.0000 0.510015
\(866\) 21.0000 0.713609
\(867\) −1.00000 −0.0339618
\(868\) 4.00000 0.135769
\(869\) 10.0000 0.339227
\(870\) 5.00000 0.169516
\(871\) 9.00000 0.304953
\(872\) −16.0000 −0.541828
\(873\) −14.0000 −0.473828
\(874\) 18.0000 0.608859
\(875\) 9.00000 0.304256
\(876\) −3.00000 −0.101361
\(877\) 52.0000 1.75592 0.877958 0.478738i \(-0.158906\pi\)
0.877958 + 0.478738i \(0.158906\pi\)
\(878\) −8.00000 −0.269987
\(879\) 14.0000 0.472208
\(880\) −1.00000 −0.0337100
\(881\) −35.0000 −1.17918 −0.589590 0.807703i \(-0.700711\pi\)
−0.589590 + 0.807703i \(0.700711\pi\)
\(882\) −6.00000 −0.202031
\(883\) 2.00000 0.0673054 0.0336527 0.999434i \(-0.489286\pi\)
0.0336527 + 0.999434i \(0.489286\pi\)
\(884\) −4.00000 −0.134535
\(885\) 3.00000 0.100844
\(886\) 24.0000 0.806296
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 10.0000 0.335578
\(889\) 2.00000 0.0670778
\(890\) 8.00000 0.268161
\(891\) 1.00000 0.0335013
\(892\) −28.0000 −0.937509
\(893\) −48.0000 −1.60626
\(894\) −4.00000 −0.133780
\(895\) −12.0000 −0.401116
\(896\) 1.00000 0.0334077
\(897\) −3.00000 −0.100167
\(898\) −12.0000 −0.400445
\(899\) −20.0000 −0.667037
\(900\) −4.00000 −0.133333
\(901\) −8.00000 −0.266519
\(902\) −7.00000 −0.233075
\(903\) −5.00000 −0.166390
\(904\) 9.00000 0.299336
\(905\) −14.0000 −0.465376
\(906\) −16.0000 −0.531564
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −24.0000 −0.796468
\(909\) −10.0000 −0.331679
\(910\) 1.00000 0.0331497
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 6.00000 0.198680
\(913\) −14.0000 −0.463332
\(914\) 31.0000 1.02539
\(915\) −13.0000 −0.429767
\(916\) 13.0000 0.429532
\(917\) −15.0000 −0.495344
\(918\) 4.00000 0.132020
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 14.0000 0.461316
\(922\) 0 0
\(923\) −2.00000 −0.0658308
\(924\) 1.00000 0.0328976
\(925\) −40.0000 −1.31519
\(926\) −26.0000 −0.854413
\(927\) 13.0000 0.426976
\(928\) −5.00000 −0.164133
\(929\) −36.0000 −1.18112 −0.590561 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) −4.00000 −0.131165
\(931\) −36.0000 −1.17985
\(932\) −18.0000 −0.589610
\(933\) 32.0000 1.04763
\(934\) −8.00000 −0.261768
\(935\) −4.00000 −0.130814
\(936\) −1.00000 −0.0326860
\(937\) 28.0000 0.914720 0.457360 0.889282i \(-0.348795\pi\)
0.457360 + 0.889282i \(0.348795\pi\)
\(938\) −9.00000 −0.293860
\(939\) 27.0000 0.881112
\(940\) 8.00000 0.260931
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 18.0000 0.586472
\(943\) −21.0000 −0.683854
\(944\) −3.00000 −0.0976417
\(945\) −1.00000 −0.0325300
\(946\) −5.00000 −0.162564
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 10.0000 0.324785
\(949\) 3.00000 0.0973841
\(950\) −24.0000 −0.778663
\(951\) −11.0000 −0.356699
\(952\) 4.00000 0.129641
\(953\) −12.0000 −0.388718 −0.194359 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 11.0000 0.355952
\(956\) −5.00000 −0.161712
\(957\) −5.00000 −0.161627
\(958\) 25.0000 0.807713
\(959\) −2.00000 −0.0645834
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) −10.0000 −0.322413
\(963\) 3.00000 0.0966736
\(964\) 18.0000 0.579741
\(965\) −26.0000 −0.836970
\(966\) 3.00000 0.0965234
\(967\) 23.0000 0.739630 0.369815 0.929105i \(-0.379421\pi\)
0.369815 + 0.929105i \(0.379421\pi\)
\(968\) 1.00000 0.0321412
\(969\) 24.0000 0.770991
\(970\) 14.0000 0.449513
\(971\) −22.0000 −0.706014 −0.353007 0.935621i \(-0.614841\pi\)
−0.353007 + 0.935621i \(0.614841\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −2.00000 −0.0640841
\(975\) 4.00000 0.128103
\(976\) 13.0000 0.416120
\(977\) −16.0000 −0.511885 −0.255943 0.966692i \(-0.582386\pi\)
−0.255943 + 0.966692i \(0.582386\pi\)
\(978\) −11.0000 −0.351741
\(979\) −8.00000 −0.255681
\(980\) 6.00000 0.191663
\(981\) −16.0000 −0.510841
\(982\) 9.00000 0.287202
\(983\) −10.0000 −0.318950 −0.159475 0.987202i \(-0.550980\pi\)
−0.159475 + 0.987202i \(0.550980\pi\)
\(984\) −7.00000 −0.223152
\(985\) 12.0000 0.382352
\(986\) −20.0000 −0.636930
\(987\) −8.00000 −0.254643
\(988\) −6.00000 −0.190885
\(989\) −15.0000 −0.476972
\(990\) −1.00000 −0.0317821
\(991\) −23.0000 −0.730619 −0.365310 0.930886i \(-0.619037\pi\)
−0.365310 + 0.930886i \(0.619037\pi\)
\(992\) 4.00000 0.127000
\(993\) −25.0000 −0.793351
\(994\) 2.00000 0.0634361
\(995\) −11.0000 −0.348723
\(996\) −14.0000 −0.443607
\(997\) 11.0000 0.348373 0.174187 0.984713i \(-0.444270\pi\)
0.174187 + 0.984713i \(0.444270\pi\)
\(998\) −11.0000 −0.348199
\(999\) 10.0000 0.316386
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 858.2.a.k.1.1 1
3.2 odd 2 2574.2.a.j.1.1 1
4.3 odd 2 6864.2.a.g.1.1 1
11.10 odd 2 9438.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
858.2.a.k.1.1 1 1.1 even 1 trivial
2574.2.a.j.1.1 1 3.2 odd 2
6864.2.a.g.1.1 1 4.3 odd 2
9438.2.a.m.1.1 1 11.10 odd 2