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Properties

Label 3330.2.a.l.1.1
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
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] = [3330,2,Mod(1,3330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3330, 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("3330.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.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: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3330.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} +2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +2.00000 q^{19} +1.00000 q^{20} -2.00000 q^{22} +1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{28} -2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} +4.00000 q^{35} +1.00000 q^{37} -2.00000 q^{38} -1.00000 q^{40} +6.00000 q^{41} -4.00000 q^{43} +2.00000 q^{44} +10.0000 q^{47} +9.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} +2.00000 q^{55} -4.00000 q^{56} +2.00000 q^{58} -4.00000 q^{59} +4.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -12.0000 q^{67} +2.00000 q^{68} -4.00000 q^{70} -8.00000 q^{71} -2.00000 q^{73} -1.00000 q^{74} +2.00000 q^{76} +8.00000 q^{77} -4.00000 q^{79} +1.00000 q^{80} -6.00000 q^{82} -4.00000 q^{83} +2.00000 q^{85} +4.00000 q^{86} -2.00000 q^{88} +6.00000 q^{89} +8.00000 q^{91} -10.0000 q^{94} +2.00000 q^{95} +12.0000 q^{97} -9.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 0 0
\(47\) 10.0000 1.45865 0.729325 − 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) −4.00000 −0.534522
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) −10.0000 −1.03142
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 − 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 8.00000 0.773389 0.386695 − 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 − 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −4.00000 −0.362143
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 − 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −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
\(133\) 8.00000 0.693688
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) −4.00000 −0.327693 −0.163846 − 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) −8.00000 −0.644658
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 20.0000 1.54765 0.773823 − 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 10.0000 0.760286 0.380143 − 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 − 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) −8.00000 −0.592999
\(183\) 0 0
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 10.0000 0.729325
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 12.0000 0.863779 0.431889 − 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 0 0
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) −6.00000 −0.418040
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 − 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 16.0000 1.08366
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 20.0000 1.33930 0.669650 − 0.742677i \(-0.266444\pi\)
0.669650 + 0.742677i \(0.266444\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) −8.00000 −0.524097 −0.262049 − 0.965055i \(-0.584398\pi\)
−0.262049 + 0.965055i \(0.584398\pi\)
\(234\) 0 0
\(235\) 10.0000 0.652328
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) −8.00000 −0.518563
\(239\) −16.0000 −1.03495 −0.517477 − 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) 9.00000 0.574989
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) −26.0000 −1.60323 −0.801614 − 0.597841i \(-0.796025\pi\)
−0.801614 + 0.597841i \(0.796025\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −8.00000 −0.490511
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) 24.0000 1.46331 0.731653 − 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) −4.00000 −0.239046
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) 14.0000 0.817889 0.408944 − 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) 4.00000 0.231714
\(299\) 0 0
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 − 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 8.00000 0.455842
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) −8.00000 −0.453638 −0.226819 − 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −4.00000 −0.226093 −0.113047 − 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 40.0000 2.20527
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 − 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) −20.0000 −1.09435
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 − 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 18.0000 0.946059
\(363\) 0 0
\(364\) 8.00000 0.419314
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 28.0000 1.46159 0.730794 − 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) −10.0000 −0.515711
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 − 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) 0 0
\(383\) −20.0000 −1.02195 −0.510976 − 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) −12.0000 −0.610784
\(387\) 0 0
\(388\) 12.0000 0.609208
\(389\) 18.0000 0.912636 0.456318 − 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −6.00000 −0.301131 −0.150566 − 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 0 0
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) 38.0000 1.87898 0.939490 − 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 6.00000 0.295599
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) −4.00000 −0.195646
\(419\) −6.00000 −0.293119 −0.146560 − 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 − 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) −16.0000 −0.778868
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 16.0000 0.774294
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −8.00000 −0.385346 −0.192673 − 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) 30.0000 1.44171 0.720854 − 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 0 0
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −36.0000 −1.71041 −0.855206 − 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) −20.0000 −0.947027
\(447\) 0 0
\(448\) 4.00000 0.188982
\(449\) 30.0000 1.41579 0.707894 − 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) −20.0000 −0.935561 −0.467780 − 0.883845i \(-0.654946\pi\)
−0.467780 + 0.883845i \(0.654946\pi\)
\(458\) 22.0000 1.02799
\(459\) 0 0
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 − 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 6.00000 0.278844 0.139422 − 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 8.00000 0.370593
\(467\) 20.0000 0.925490 0.462745 − 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) −48.0000 −2.21643
\(470\) −10.0000 −0.461266
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 8.00000 0.366679
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) −40.0000 −1.82765 −0.913823 − 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 12.0000 0.544892
\(486\) 0 0
\(487\) 42.0000 1.90320 0.951601 − 0.307337i \(-0.0994378\pi\)
0.951601 + 0.307337i \(0.0994378\pi\)
\(488\) −4.00000 −0.181071
\(489\) 0 0
\(490\) −9.00000 −0.406579
\(491\) 22.0000 0.992846 0.496423 − 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −32.0000 −1.43540
\(498\) 0 0
\(499\) 22.0000 0.984855 0.492428 − 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) −24.0000 −1.06378 −0.531891 − 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 6.00000 0.264392
\(516\) 0 0
\(517\) 20.0000 0.879599
\(518\) −4.00000 −0.175750
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) 18.0000 0.788594 0.394297 − 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 26.0000 1.13365
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) −24.0000 −1.03471
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −4.00000 −0.171973 −0.0859867 − 0.996296i \(-0.527404\pi\)
−0.0859867 + 0.996296i \(0.527404\pi\)
\(542\) −16.0000 −0.687259
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) −16.0000 −0.685365
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 − 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) −12.0000 −0.512615
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 34.0000 1.44063 0.720313 − 0.693649i \(-0.243998\pi\)
0.720313 + 0.693649i \(0.243998\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −4.00000 −0.168580 −0.0842900 − 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) 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 \(0.266474\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 0 0
\(577\) 8.00000 0.333044 0.166522 − 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) −2.00000 −0.0830455
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) −28.0000 −1.15568 −0.577842 − 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −44.0000 −1.80686 −0.903432 − 0.428732i \(-0.858960\pi\)
−0.903432 + 0.428732i \(0.858960\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) −4.00000 −0.163846
\(597\) 0 0
\(598\) 0 0
\(599\) 4.00000 0.163436 0.0817178 − 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 − 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 16.0000 0.652111
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 38.0000 1.54237 0.771186 − 0.636610i \(-0.219664\pi\)
0.771186 + 0.636610i \(0.219664\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) 20.0000 0.809113
\(612\) 0 0
\(613\) −30.0000 −1.21169 −0.605844 − 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) −8.00000 −0.322329
\(617\) −28.0000 −1.12724 −0.563619 − 0.826035i \(-0.690591\pi\)
−0.563619 + 0.826035i \(0.690591\pi\)
\(618\) 0 0
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) 8.00000 0.320771
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 4.00000 0.159872
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 2.00000 0.0797452
\(630\) 0 0
\(631\) −48.0000 −1.91085 −0.955425 − 0.295234i \(-0.904602\pi\)
−0.955425 + 0.295234i \(0.904602\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) −12.0000 −0.476205
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −34.0000 −1.34292 −0.671460 − 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) 8.00000 0.315489 0.157745 − 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) −8.00000 −0.314512 −0.157256 − 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −14.0000 −0.547862 −0.273931 − 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) −40.0000 −1.55936
\(659\) −30.0000 −1.16863 −0.584317 − 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) −8.00000 −0.311164 −0.155582 − 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) 10.0000 0.388661
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 0 0
\(668\) 20.0000 0.773823
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 − 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 14.0000 0.538064 0.269032 − 0.963131i \(-0.413296\pi\)
0.269032 + 0.963131i \(0.413296\pi\)
\(678\) 0 0
\(679\) 48.0000 1.84207
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) −8.00000 −0.305441
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 − 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 10.0000 0.380143
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −20.0000 −0.758643
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) −2.00000 −0.0757011
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) −30.0000 −1.13308 −0.566542 − 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 0 0
\(709\) −8.00000 −0.300446 −0.150223 − 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) 8.00000 0.300235
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) 20.0000 0.745874 0.372937 − 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 15.0000 0.558242
\(723\) 0 0
\(724\) −18.0000 −0.668965
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 6.00000 0.222528 0.111264 − 0.993791i \(-0.464510\pi\)
0.111264 + 0.993791i \(0.464510\pi\)
\(728\) −8.00000 −0.296500
\(729\) 0 0
\(730\) 2.00000 0.0740233
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −46.0000 −1.69905 −0.849524 − 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) 0 0
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) −32.0000 −1.17714 −0.588570 − 0.808447i \(-0.700309\pi\)
−0.588570 + 0.808447i \(0.700309\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) 6.00000 0.220119 0.110059 − 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 0 0
\(745\) −4.00000 −0.146549
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) 4.00000 0.146254
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 10.0000 0.364662
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) 30.0000 1.09037 0.545184 − 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) −2.00000 −0.0725476
\(761\) 50.0000 1.81250 0.906249 − 0.422744i \(-0.138933\pi\)
0.906249 + 0.422744i \(0.138933\pi\)
\(762\) 0 0
\(763\) −64.0000 −2.31696
\(764\) 0 0
\(765\) 0 0
\(766\) 20.0000 0.722629
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) −8.00000 −0.288300
\(771\) 0 0
\(772\) 12.0000 0.431889
\(773\) −54.0000 −1.94225 −0.971123 − 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) −12.0000 −0.430775
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 − 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 4.00000 0.142314
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −30.0000 −1.06265 −0.531327 − 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 20.0000 0.707549
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 30.0000 1.05934
\(803\) −4.00000 −0.141157
\(804\) 0 0
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 − 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −8.00000 −0.280745
\(813\) 0 0
\(814\) −2.00000 −0.0701000
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) −38.0000 −1.32864
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) −16.0000 −0.558404 −0.279202 − 0.960232i \(-0.590070\pi\)
−0.279202 + 0.960232i \(0.590070\pi\)
\(822\) 0 0
\(823\) −56.0000 −1.95204 −0.976019 − 0.217687i \(-0.930149\pi\)
−0.976019 + 0.217687i \(0.930149\pi\)
\(824\) −6.00000 −0.209020
\(825\) 0 0
\(826\) 16.0000 0.556711
\(827\) 4.00000 0.139094 0.0695468 − 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 0 0
\(829\) −48.0000 −1.66711 −0.833554 − 0.552437i \(-0.813698\pi\)
−0.833554 + 0.552437i \(0.813698\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 20.0000 0.692129
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) 6.00000 0.207267
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 20.0000 0.689246
\(843\) 0 0
\(844\) 16.0000 0.550743
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −28.0000 −0.962091
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 0 0
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 − 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) −16.0000 −0.547509
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) −10.0000 −0.341593 −0.170797 − 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 − 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 8.00000 0.272481
\(863\) 26.0000 0.885050 0.442525 − 0.896756i \(-0.354083\pi\)
0.442525 + 0.896756i \(0.354083\pi\)
\(864\) 0 0
\(865\) 10.0000 0.340010
\(866\) −30.0000 −1.01944
\(867\) 0 0
\(868\) 16.0000 0.543075
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 16.0000 0.541828
\(873\) 0 0
\(874\) 0 0
\(875\) 4.00000 0.135225
\(876\) 0 0
\(877\) −14.0000 −0.472746 −0.236373 − 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) −14.0000 −0.471672 −0.235836 − 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) 32.0000 1.07689 0.538443 − 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 22.0000 0.738688 0.369344 − 0.929293i \(-0.379582\pi\)
0.369344 + 0.929293i \(0.379582\pi\)
\(888\) 0 0
\(889\) −48.0000 −1.60987
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) 20.0000 0.669650
\(893\) 20.0000 0.669274
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) −12.0000 −0.399556
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 − 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) −8.00000 −0.265197
\(911\) 40.0000 1.32526 0.662630 − 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) 20.0000 0.661541
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) −16.0000 −0.528367
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 − 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 14.0000 0.461065
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) −6.00000 −0.197172
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) 10.0000 0.328089 0.164045 − 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) −8.00000 −0.262049
\(933\) 0 0
\(934\) −20.0000 −0.654420
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 − 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 48.0000 1.56726
\(939\) 0 0
\(940\) 10.0000 0.326164
\(941\) −8.00000 −0.260793 −0.130396 − 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) −2.00000 −0.0648886
\(951\) 0 0
\(952\) −8.00000 −0.259281
\(953\) 36.0000 1.16615 0.583077 − 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) 40.0000 1.29234
\(959\) −48.0000 −1.55000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −2.00000 −0.0644826
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) −46.0000 −1.47926 −0.739630 − 0.673014i \(-0.765000\pi\)
−0.739630 + 0.673014i \(0.765000\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) −12.0000 −0.385297
\(971\) 6.00000 0.192549 0.0962746 − 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 0 0
\(973\) −80.0000 −2.56468
\(974\) −42.0000 −1.34577
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) 10.0000 0.319928 0.159964 − 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 9.00000 0.287494
\(981\) 0 0
\(982\) −22.0000 −0.702048
\(983\) 14.0000 0.446531 0.223265 − 0.974758i \(-0.428328\pi\)
0.223265 + 0.974758i \(0.428328\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 − 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) 32.0000 1.01498
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) −30.0000 −0.950110 −0.475055 − 0.879956i \(-0.657572\pi\)
−0.475055 + 0.879956i \(0.657572\pi\)
\(998\) −22.0000 −0.696398
\(999\) 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 3330.2.a.l.1.1 1
3.2 odd 2 1110.2.a.o.1.1 ✓ 1
12.11 even 2 8880.2.a.b.1.1 1
15.14 odd 2 5550.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.o.1.1 ✓ 1 3.2 odd 2
3330.2.a.l.1.1 1 1.1 even 1 trivial
5550.2.a.b.1.1 1 15.14 odd 2
8880.2.a.b.1.1 1 12.11 even 2