Properties

Label 4830.2.a.ba.1.1
Level $4830$
Weight $2$
Character 4830.1
Self dual yes
Analytic conductor $38.568$
Analytic rank $1$
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] = [4830,2,Mod(1,4830)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4830, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4830.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.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: \(38.5677441763\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 4830.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} -1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -10.0000 q^{29} -1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -2.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} -1.00000 q^{45} -1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} -2.00000 q^{52} -14.0000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -10.0000 q^{58} +4.00000 q^{59} -1.00000 q^{60} -14.0000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +16.0000 q^{67} -6.00000 q^{68} -1.00000 q^{69} +1.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -2.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -1.00000 q^{84} +6.00000 q^{85} -10.0000 q^{87} -14.0000 q^{89} -1.00000 q^{90} +2.00000 q^{91} -1.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −1.00000 −0.154303
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) −1.00000 −0.147442
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −6.00000 −0.840168
\(52\) −2.00000 −0.277350
\(53\) −14.0000 −1.92305 −0.961524 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −10.0000 −1.31306
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −1.00000 −0.129099
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 8.00000 1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 16.0000 1.95471 0.977356 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) −6.00000 −0.727607
\(69\) −1.00000 −0.120386
\(70\) 1.00000 0.119523
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −1.00000 −0.109109
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) −10.0000 −1.07211
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) −1.00000 −0.105409
\(91\) 2.00000 0.209657
\(92\) −1.00000 −0.104257
\(93\) 8.00000 0.829561
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −6.00000 −0.594089
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −2.00000 −0.196116
\(105\) 1.00000 0.0975900
\(106\) −14.0000 −1.35980
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 0.0962250
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) −1.00000 −0.0944911
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) −10.0000 −0.928477
\(117\) −2.00000 −0.184900
\(118\) 4.00000 0.368230
\(119\) 6.00000 0.550019
\(120\) −1.00000 −0.0912871
\(121\) −11.0000 −1.00000
\(122\) −14.0000 −1.26750
\(123\) −6.00000 −0.541002
\(124\) 8.00000 0.718421
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 16.0000 1.38219
\(135\) −1.00000 −0.0860663
\(136\) −6.00000 −0.514496
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 1.00000 0.0845154
\(141\) 8.00000 0.673722
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 10.0000 0.830455
\(146\) −6.00000 −0.496564
\(147\) 1.00000 0.0824786
\(148\) −6.00000 −0.493197
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 1.00000 0.0816497
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) −2.00000 −0.160128
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) −14.0000 −1.11027
\(160\) −1.00000 −0.0790569
\(161\) 1.00000 0.0788110
\(162\) 1.00000 0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −10.0000 −0.758098
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) −14.0000 −1.04934
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 2.00000 0.148250
\(183\) −14.0000 −1.03491
\(184\) −1.00000 −0.0737210
\(185\) 6.00000 0.441129
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000 0.0721688
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 2.00000 0.143592
\(195\) 2.00000 0.143223
\(196\) 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
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 1.00000 0.0707107
\(201\) 16.0000 1.12855
\(202\) −2.00000 −0.140720
\(203\) 10.0000 0.701862
\(204\) −6.00000 −0.420084
\(205\) 6.00000 0.419058
\(206\) −16.0000 −1.11477
\(207\) −1.00000 −0.0695048
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 1.00000 0.0690066
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −14.0000 −0.961524
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −8.00000 −0.543075
\(218\) −6.00000 −0.406371
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) −6.00000 −0.402694
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 18.0000 1.19734
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 1.00000 0.0659380
\(231\) 0 0
\(232\) −10.0000 −0.656532
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) −8.00000 −0.521862
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) −1.00000 −0.0638877
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 2.00000 0.124035
\(261\) −10.0000 −0.618984
\(262\) −4.00000 −0.247121
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) 14.0000 0.860013
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 16.0000 0.977356
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −6.00000 −0.363803
\(273\) 2.00000 0.121046
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −12.0000 −0.719712
\(279\) 8.00000 0.478947
\(280\) 1.00000 0.0597614
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 8.00000 0.476393
\(283\) −8.00000 −0.475551 −0.237775 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 10.0000 0.587220
\(291\) 2.00000 0.117242
\(292\) −6.00000 −0.351123
\(293\) −22.0000 −1.28525 −0.642627 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) 1.00000 0.0583212
\(295\) −4.00000 −0.232889
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) 2.00000 0.115663
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) −2.00000 −0.114897
\(304\) 0 0
\(305\) 14.0000 0.801638
\(306\) −6.00000 −0.342997
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) −8.00000 −0.454369
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) −2.00000 −0.113228
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −22.0000 −1.24153
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) −14.0000 −0.785081
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) 12.0000 0.664619
\(327\) −6.00000 −0.331801
\(328\) −6.00000 −0.331295
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) −6.00000 −0.328798
\(334\) 0 0
\(335\) −16.0000 −0.874173
\(336\) −1.00000 −0.0545545
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) −9.00000 −0.489535
\(339\) 18.0000 0.977626
\(340\) 6.00000 0.325396
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 6.00000 0.322562
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −10.0000 −0.536056
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 4.00000 0.212598
\(355\) 8.00000 0.424596
\(356\) −14.0000 −0.741999
\(357\) 6.00000 0.317554
\(358\) −20.0000 −1.05703
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −19.0000 −1.00000
\(362\) −6.00000 −0.315353
\(363\) −11.0000 −0.577350
\(364\) 2.00000 0.104828
\(365\) 6.00000 0.314054
\(366\) −14.0000 −0.731792
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −6.00000 −0.312348
\(370\) 6.00000 0.311925
\(371\) 14.0000 0.726844
\(372\) 8.00000 0.414781
\(373\) −38.0000 −1.96757 −0.983783 0.179364i \(-0.942596\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 8.00000 0.412568
\(377\) 20.0000 1.03005
\(378\) −1.00000 −0.0514344
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) −8.00000 −0.409316
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 2.00000 0.101274
\(391\) 6.00000 0.303433
\(392\) 1.00000 0.0505076
\(393\) −4.00000 −0.201773
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) 30.0000 1.50566 0.752828 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 16.0000 0.798007
\(403\) −16.0000 −0.797017
\(404\) −2.00000 −0.0995037
\(405\) −1.00000 −0.0496904
\(406\) 10.0000 0.496292
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 6.00000 0.296319
\(411\) 18.0000 0.887875
\(412\) −16.0000 −0.788263
\(413\) −4.00000 −0.196827
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 1.00000 0.0487950
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 20.0000 0.973585
\(423\) 8.00000 0.388973
\(424\) −14.0000 −0.679900
\(425\) −6.00000 −0.291043
\(426\) −8.00000 −0.387601
\(427\) 14.0000 0.677507
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 1.00000 0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) −8.00000 −0.384012
\(435\) 10.0000 0.479463
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −6.00000 −0.284747
\(445\) 14.0000 0.663664
\(446\) 16.0000 0.757622
\(447\) 18.0000 0.851371
\(448\) −1.00000 −0.0472456
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 18.0000 0.846649
\(453\) 16.0000 0.751746
\(454\) −8.00000 −0.375459
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 10.0000 0.467269
\(459\) −6.00000 −0.280056
\(460\) 1.00000 0.0466252
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −10.0000 −0.464238
\(465\) −8.00000 −0.370991
\(466\) −6.00000 −0.277945
\(467\) 32.0000 1.48078 0.740392 0.672176i \(-0.234640\pi\)
0.740392 + 0.672176i \(0.234640\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −16.0000 −0.738811
\(470\) −8.00000 −0.369012
\(471\) −22.0000 −1.01371
\(472\) 4.00000 0.184115
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) −14.0000 −0.641016
\(478\) −8.00000 −0.365911
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 12.0000 0.547153
\(482\) −6.00000 −0.273293
\(483\) 1.00000 0.0455016
\(484\) −11.0000 −0.500000
\(485\) −2.00000 −0.0908153
\(486\) 1.00000 0.0453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −14.0000 −0.633750
\(489\) 12.0000 0.542659
\(490\) −1.00000 −0.0451754
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −6.00000 −0.270501
\(493\) 60.0000 2.70226
\(494\) 0 0
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −16.0000 −0.714115
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 6.00000 0.265684
\(511\) 6.00000 0.265424
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 0 0
\(518\) 6.00000 0.263625
\(519\) 6.00000 0.263371
\(520\) 2.00000 0.0877058
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −10.0000 −0.437688
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −4.00000 −0.174741
\(525\) −1.00000 −0.0436436
\(526\) −8.00000 −0.348817
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 14.0000 0.608121
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) −14.0000 −0.605839
\(535\) 0 0
\(536\) 16.0000 0.691095
\(537\) −20.0000 −0.863064
\(538\) −18.0000 −0.776035
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 0 0
\(543\) −6.00000 −0.257485
\(544\) −6.00000 −0.257248
\(545\) 6.00000 0.257012
\(546\) 2.00000 0.0855921
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 18.0000 0.768922
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) 6.00000 0.254686
\(556\) −12.0000 −0.508913
\(557\) 34.0000 1.44063 0.720313 0.693649i \(-0.243998\pi\)
0.720313 + 0.693649i \(0.243998\pi\)
\(558\) 8.00000 0.338667
\(559\) 0 0
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −14.0000 −0.590554
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 8.00000 0.336861
\(565\) −18.0000 −0.757266
\(566\) −8.00000 −0.336265
\(567\) −1.00000 −0.0419961
\(568\) −8.00000 −0.335673
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 6.00000 0.250435
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 19.0000 0.790296
\(579\) 18.0000 0.748054
\(580\) 10.0000 0.415227
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) 2.00000 0.0826898
\(586\) −22.0000 −0.908812
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) −4.00000 −0.164677
\(591\) 6.00000 0.246807
\(592\) −6.00000 −0.246598
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) −6.00000 −0.245976
\(596\) 18.0000 0.737309
\(597\) −8.00000 −0.327418
\(598\) 2.00000 0.0817861
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 1.00000 0.0408248
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 16.0000 0.651570
\(604\) 16.0000 0.651031
\(605\) 11.0000 0.447214
\(606\) −2.00000 −0.0812444
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 10.0000 0.405220
\(610\) 14.0000 0.566843
\(611\) −16.0000 −0.647291
\(612\) −6.00000 −0.242536
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 4.00000 0.161427
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −16.0000 −0.643614
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) −8.00000 −0.321288
\(621\) −1.00000 −0.0401286
\(622\) 16.0000 0.641542
\(623\) 14.0000 0.560898
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) 36.0000 1.43541
\(630\) 1.00000 0.0398410
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 14.0000 0.556011
\(635\) −8.00000 −0.317470
\(636\) −14.0000 −0.555136
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) −1.00000 −0.0395285
\(641\) 34.0000 1.34292 0.671460 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) −8.00000 −0.313545
\(652\) 12.0000 0.469956
\(653\) 38.0000 1.48705 0.743527 0.668705i \(-0.233151\pi\)
0.743527 + 0.668705i \(0.233151\pi\)
\(654\) −6.00000 −0.234619
\(655\) 4.00000 0.156293
\(656\) −6.00000 −0.234261
\(657\) −6.00000 −0.234082
\(658\) −8.00000 −0.311872
\(659\) −32.0000 −1.24654 −0.623272 0.782006i \(-0.714197\pi\)
−0.623272 + 0.782006i \(0.714197\pi\)
\(660\) 0 0
\(661\) −6.00000 −0.233373 −0.116686 0.993169i \(-0.537227\pi\)
−0.116686 + 0.993169i \(0.537227\pi\)
\(662\) −20.0000 −0.777322
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 10.0000 0.387202
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) −16.0000 −0.618134
\(671\) 0 0
\(672\) −1.00000 −0.0385758
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 26.0000 1.00148
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 18.0000 0.691286
\(679\) −2.00000 −0.0767530
\(680\) 6.00000 0.230089
\(681\) −8.00000 −0.306561
\(682\) 0 0
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) −1.00000 −0.0381802
\(687\) 10.0000 0.381524
\(688\) 0 0
\(689\) 28.0000 1.06672
\(690\) 1.00000 0.0380693
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 12.0000 0.455186
\(696\) −10.0000 −0.379049
\(697\) 36.0000 1.36360
\(698\) 14.0000 0.529908
\(699\) −6.00000 −0.226941
\(700\) −1.00000 −0.0377964
\(701\) −46.0000 −1.73740 −0.868698 0.495342i \(-0.835043\pi\)
−0.868698 + 0.495342i \(0.835043\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) −14.0000 −0.526897
\(707\) 2.00000 0.0752177
\(708\) 4.00000 0.150329
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 8.00000 0.300235
\(711\) 0 0
\(712\) −14.0000 −0.524672
\(713\) −8.00000 −0.299602
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) −8.00000 −0.298765
\(718\) 16.0000 0.597115
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 16.0000 0.595871
\(722\) −19.0000 −0.707107
\(723\) −6.00000 −0.223142
\(724\) −6.00000 −0.222988
\(725\) −10.0000 −0.371391
\(726\) −11.0000 −0.408248
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 6.00000 0.222070
\(731\) 0 0
\(732\) −14.0000 −0.517455
\(733\) −38.0000 −1.40356 −0.701781 0.712393i \(-0.747612\pi\)
−0.701781 + 0.712393i \(0.747612\pi\)
\(734\) 16.0000 0.590571
\(735\) −1.00000 −0.0368856
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) 14.0000 0.513956
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 8.00000 0.293294
\(745\) −18.0000 −0.659469
\(746\) −38.0000 −1.39128
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 8.00000 0.291730
\(753\) −16.0000 −0.583072
\(754\) 20.0000 0.728357
\(755\) −16.0000 −0.582300
\(756\) −1.00000 −0.0363696
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 8.00000 0.289809
\(763\) 6.00000 0.217215
\(764\) −8.00000 −0.289430
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 1.00000 0.0360844
\(769\) −54.0000 −1.94729 −0.973645 0.228069i \(-0.926759\pi\)
−0.973645 + 0.228069i \(0.926759\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) 18.0000 0.647834
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 2.00000 0.0717958
\(777\) 6.00000 0.215249
\(778\) −6.00000 −0.215110
\(779\) 0 0
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) 6.00000 0.214560
\(783\) −10.0000 −0.357371
\(784\) 1.00000 0.0357143
\(785\) 22.0000 0.785214
\(786\) −4.00000 −0.142675
\(787\) 48.0000 1.71102 0.855508 0.517790i \(-0.173245\pi\)
0.855508 + 0.517790i \(0.173245\pi\)
\(788\) 6.00000 0.213741
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) 28.0000 0.994309
\(794\) 30.0000 1.06466
\(795\) 14.0000 0.496529
\(796\) −8.00000 −0.283552
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 1.00000 0.0353553
\(801\) −14.0000 −0.494666
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 16.0000 0.564276
\(805\) −1.00000 −0.0352454
\(806\) −16.0000 −0.563576
\(807\) −18.0000 −0.633630
\(808\) −2.00000 −0.0703598
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 10.0000 0.350931
\(813\) 0 0
\(814\) 0 0
\(815\) −12.0000 −0.420342
\(816\) −6.00000 −0.210042
\(817\) 0 0
\(818\) −6.00000 −0.209785
\(819\) 2.00000 0.0698857
\(820\) 6.00000 0.209529
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 18.0000 0.627822
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) 32.0000 1.11275 0.556375 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) −2.00000 −0.0693375
\(833\) −6.00000 −0.207888
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) −24.0000 −0.829066
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 1.00000 0.0345033
\(841\) 71.0000 2.44828
\(842\) 26.0000 0.896019
\(843\) −14.0000 −0.482186
\(844\) 20.0000 0.688428
\(845\) 9.00000 0.309609
\(846\) 8.00000 0.275046
\(847\) 11.0000 0.377964
\(848\) −14.0000 −0.480762
\(849\) −8.00000 −0.274559
\(850\) −6.00000 −0.205798
\(851\) 6.00000 0.205677
\(852\) −8.00000 −0.274075
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 14.0000 0.479070
\(855\) 0 0
\(856\) 0 0
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 8.00000 0.272481
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.00000 −0.204006
\(866\) −14.0000 −0.475739
\(867\) 19.0000 0.645274
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) 10.0000 0.339032
\(871\) −32.0000 −1.08428
\(872\) −6.00000 −0.203186
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) −6.00000 −0.202721
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 40.0000 1.34993
\(879\) −22.0000 −0.742042
\(880\) 0 0
\(881\) 10.0000 0.336909 0.168454 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) 1.00000 0.0336718
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 12.0000 0.403604
\(885\) −4.00000 −0.134459
\(886\) 4.00000 0.134383
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) −6.00000 −0.201347
\(889\) −8.00000 −0.268311
\(890\) 14.0000 0.469281
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) 20.0000 0.668526
\(896\) −1.00000 −0.0334077
\(897\) 2.00000 0.0667781
\(898\) 18.0000 0.600668
\(899\) −80.0000 −2.66815
\(900\) 1.00000 0.0333333
\(901\) 84.0000 2.79845
\(902\) 0 0
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 6.00000 0.199447
\(906\) 16.0000 0.531564
\(907\) 16.0000 0.531271 0.265636 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) −8.00000 −0.265489
\(909\) −2.00000 −0.0663358
\(910\) −2.00000 −0.0662994
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 14.0000 0.462826
\(916\) 10.0000 0.330409
\(917\) 4.00000 0.132092
\(918\) −6.00000 −0.198030
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 1.00000 0.0329690
\(921\) 4.00000 0.131804
\(922\) −2.00000 −0.0658665
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −24.0000 −0.788689
\(927\) −16.0000 −0.525509
\(928\) −10.0000 −0.328266
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) −8.00000 −0.262330
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 16.0000 0.523816
\(934\) 32.0000 1.04707
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −46.0000 −1.50275 −0.751377 0.659873i \(-0.770610\pi\)
−0.751377 + 0.659873i \(0.770610\pi\)
\(938\) −16.0000 −0.522419
\(939\) 26.0000 0.848478
\(940\) −8.00000 −0.260931
\(941\) −46.0000 −1.49956 −0.749779 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) −22.0000 −0.716799
\(943\) 6.00000 0.195387
\(944\) 4.00000 0.130189
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 14.0000 0.453981
\(952\) 6.00000 0.194461
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) −14.0000 −0.453267
\(955\) 8.00000 0.258874
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 16.0000 0.516937
\(959\) −18.0000 −0.581250
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) 12.0000 0.386896
\(963\) 0 0
\(964\) −6.00000 −0.193247
\(965\) −18.0000 −0.579441
\(966\) 1.00000 0.0321745
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −11.0000 −0.353553
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) 56.0000 1.79713 0.898563 0.438845i \(-0.144612\pi\)
0.898563 + 0.438845i \(0.144612\pi\)
\(972\) 1.00000 0.0320750
\(973\) 12.0000 0.384702
\(974\) 16.0000 0.512673
\(975\) −2.00000 −0.0640513
\(976\) −14.0000 −0.448129
\(977\) −22.0000 −0.703842 −0.351921 0.936030i \(-0.614471\pi\)
−0.351921 + 0.936030i \(0.614471\pi\)
\(978\) 12.0000 0.383718
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −6.00000 −0.191565
\(982\) −12.0000 −0.382935
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −6.00000 −0.191273
\(985\) −6.00000 −0.191176
\(986\) 60.0000 1.91079
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 8.00000 0.254000
\(993\) −20.0000 −0.634681
\(994\) 8.00000 0.253745
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) −4.00000 −0.126618
\(999\) −6.00000 −0.189832
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 4830.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.ba.1.1 1 1.1 even 1 trivial