Properties

Label 138.2.a.b.1.1
Level $138$
Weight $2$
Character 138.1
Self dual yes
Analytic conductor $1.102$
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] = [138,2,Mod(1,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("138.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 138.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: \(1.10193554789\)
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 \(\chi\) \(=\) 138.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -1.00000 q^{18} +2.00000 q^{19} +2.00000 q^{21} -1.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{36} -10.0000 q^{37} -2.00000 q^{38} +2.00000 q^{39} -6.00000 q^{41} -2.00000 q^{42} +2.00000 q^{43} +1.00000 q^{46} +1.00000 q^{48} -3.00000 q^{49} +5.00000 q^{50} +2.00000 q^{52} +12.0000 q^{53} -1.00000 q^{54} -2.00000 q^{56} +2.00000 q^{57} +6.00000 q^{58} +12.0000 q^{59} -10.0000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +14.0000 q^{67} -1.00000 q^{69} -1.00000 q^{72} +2.00000 q^{73} +10.0000 q^{74} -5.00000 q^{75} +2.00000 q^{76} -2.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +2.00000 q^{84} -2.00000 q^{86} -6.00000 q^{87} +12.0000 q^{89} +4.00000 q^{91} -1.00000 q^{92} -4.00000 q^{93} -1.00000 q^{96} -10.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 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 \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −2.00000 −0.324443
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −2.00000 −0.308607
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 5.00000 0.707107
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 2.00000 0.264906
\(58\) 6.00000 0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 4.00000 0.508001
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 10.0000 1.16248
\(75\) −5.00000 −0.577350
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −1.00000 −0.104257
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 2.00000 0.188982
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 10.0000 0.905357
\(123\) −6.00000 −0.541002
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −14.0000 −1.20942
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 1.00000 0.0851257
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) −3.00000 −0.247436
\(148\) −10.0000 −0.821995
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 5.00000 0.408248
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 10.0000 0.795557
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) −1.00000 −0.0785674
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) −2.00000 −0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 2.00000 0.152499
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 6.00000 0.454859
\(175\) −10.0000 −0.755929
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) −12.0000 −0.899438
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −4.00000 −0.296500
\(183\) −10.0000 −0.739221
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 5.00000 0.353553
\(201\) 14.0000 0.987484
\(202\) 18.0000 1.26648
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) −1.00000 −0.0695048
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −8.00000 −0.543075
\(218\) −2.00000 −0.135457
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 10.0000 0.671156
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −2.00000 −0.133631
\(225\) −5.00000 −0.333333
\(226\) −12.0000 −0.798228
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 2.00000 0.132453
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 11.0000 0.707107
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 4.00000 0.254514
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −2.00000 −0.124515
\(259\) −20.0000 −1.24274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 12.0000 0.741362
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) 12.0000 0.734388
\(268\) 14.0000 0.855186
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −20.0000 −1.19952
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) −10.0000 −0.594438 −0.297219 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 2.00000 0.117041
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) −5.00000 −0.288675
\(301\) 4.00000 0.230556
\(302\) −8.00000 −0.460348
\(303\) −18.0000 −1.03407
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −2.00000 −0.113228
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −12.0000 −0.672927
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 2.00000 0.111456
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −10.0000 −0.554700
\(326\) −8.00000 −0.443079
\(327\) 2.00000 0.110600
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) 0 0
\(333\) −10.0000 −0.547997
\(334\) 24.0000 1.31322
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 9.00000 0.489535
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) −2.00000 −0.108148
\(343\) −20.0000 −1.07990
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −6.00000 −0.321634
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 10.0000 0.534522
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 12.0000 0.635999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −2.00000 −0.105118
\(363\) −11.0000 −0.577350
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 10.0000 0.522708
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) −4.00000 −0.207390
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) −2.00000 −0.102869
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 12.0000 0.613973
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 2.00000 0.101666
\(388\) −10.0000 −0.507673
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) −12.0000 −0.605320
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 10.0000 0.501255
\(399\) 4.00000 0.200250
\(400\) −5.00000 −0.250000
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) −14.0000 −0.698257
\(403\) −8.00000 −0.398508
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 14.0000 0.689730
\(413\) 24.0000 1.18096
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 0 0
\(427\) −20.0000 −0.967868
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000 0.0481125
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −2.00000 −0.0956730
\(438\) −2.00000 −0.0955637
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 5.00000 0.235702
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) 8.00000 0.375873
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −34.0000 −1.59045 −0.795226 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) −26.0000 −1.21490
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 2.00000 0.0924500
\(469\) 28.0000 1.29292
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 10.0000 0.459315
\(475\) −10.0000 −0.458831
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 24.0000 1.09773
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) −14.0000 −0.637683
\(483\) −2.00000 −0.0910032
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 10.0000 0.452679
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −6.00000 −0.270501
\(493\) 0 0
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) 12.0000 0.535586
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) −4.00000 −0.177471
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) 0 0
\(518\) 20.0000 0.878750
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 6.00000 0.262613
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) −12.0000 −0.524222
\(525\) −10.0000 −0.436436
\(526\) −12.0000 −0.523225
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 4.00000 0.173422
\(533\) −12.0000 −0.519778
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) −14.0000 −0.604708
\(537\) 12.0000 0.517838
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) −20.0000 −0.859074
\(543\) 2.00000 0.0858282
\(544\) 0 0
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 12.0000 0.512615
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 1.00000 0.0425628
\(553\) −20.0000 −0.850487
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) −36.0000 −1.52537 −0.762684 0.646771i \(-0.776119\pi\)
−0.762684 + 0.646771i \(0.776119\pi\)
\(558\) 4.00000 0.169334
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 12.0000 0.506189
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 10.0000 0.420331
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 38.0000 1.59025 0.795125 0.606445i \(-0.207405\pi\)
0.795125 + 0.606445i \(0.207405\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 12.0000 0.500870
\(575\) 5.00000 0.208514
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 17.0000 0.707107
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) 10.0000 0.414513
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) −3.00000 −0.123718
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) −10.0000 −0.410997
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −10.0000 −0.409273
\(598\) 2.00000 0.0817861
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 5.00000 0.204124
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) −4.00000 −0.163028
\(603\) 14.0000 0.570124
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −2.00000 −0.0811107
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) −14.0000 −0.563163
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 24.0000 0.962312
\(623\) 24.0000 0.961540
\(624\) 2.00000 0.0800641
\(625\) 25.0000 1.00000
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 0 0
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) 10.0000 0.397779
\(633\) 8.00000 0.317971
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −48.0000 −1.89589 −0.947943 0.318440i \(-0.896841\pi\)
−0.947943 + 0.318440i \(0.896841\pi\)
\(642\) 0 0
\(643\) −22.0000 −0.867595 −0.433798 0.901010i \(-0.642827\pi\)
−0.433798 + 0.901010i \(0.642827\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) 0 0
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 10.0000 0.392232
\(651\) −8.00000 −0.313545
\(652\) 8.00000 0.313304
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −32.0000 −1.24372
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) 6.00000 0.232321
\(668\) −24.0000 −0.928588
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) 38.0000 1.46479 0.732396 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(674\) 10.0000 0.385186
\(675\) −5.00000 −0.192450
\(676\) −9.00000 −0.346154
\(677\) −48.0000 −1.84479 −0.922395 0.386248i \(-0.873771\pi\)
−0.922395 + 0.386248i \(0.873771\pi\)
\(678\) −12.0000 −0.460857
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 26.0000 0.991962
\(688\) 2.00000 0.0762493
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 0 0
\(698\) −2.00000 −0.0757011
\(699\) −6.00000 −0.226941
\(700\) −10.0000 −0.377964
\(701\) −48.0000 −1.81293 −0.906467 0.422276i \(-0.861231\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −20.0000 −0.754314
\(704\) 0 0
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) −36.0000 −1.35392
\(708\) 12.0000 0.450988
\(709\) 50.0000 1.87779 0.938895 0.344204i \(-0.111851\pi\)
0.938895 + 0.344204i \(0.111851\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) −12.0000 −0.449719
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −24.0000 −0.896296
\(718\) −36.0000 −1.34351
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 15.0000 0.558242
\(723\) 14.0000 0.520666
\(724\) 2.00000 0.0743294
\(725\) 30.0000 1.11417
\(726\) 11.0000 0.408248
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −10.0000 −0.369611
\(733\) −46.0000 −1.69905 −0.849524 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) 10.0000 0.369107
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) −24.0000 −0.881068
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −22.0000 −0.802791 −0.401396 0.915905i \(-0.631475\pi\)
−0.401396 + 0.915905i \(0.631475\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −26.0000 −0.944363
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 4.00000 0.144905
\(763\) 4.00000 0.144810
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 24.0000 0.866590
\(768\) 1.00000 0.0360844
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 14.0000 0.503871
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 20.0000 0.718421
\(776\) 10.0000 0.358979
\(777\) −20.0000 −0.717496
\(778\) −12.0000 −0.430221
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 18.0000 0.641223
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) 12.0000 0.423999
\(802\) −24.0000 −0.847469
\(803\) 0 0
\(804\) 14.0000 0.493742
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) −6.00000 −0.211210
\(808\) 18.0000 0.633238
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) −12.0000 −0.421117
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 10.0000 0.349642
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −12.0000 −0.418548
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) −24.0000 −0.835067
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 14.0000 0.485655
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −2.00000 −0.0689246
\(843\) −12.0000 −0.413302
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) −22.0000 −0.755929
\(848\) 12.0000 0.412082
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) 10.0000 0.342796
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 20.0000 0.684386
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) −24.0000 −0.817443
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −26.0000 −0.883516
\(867\) −17.0000 −0.577350
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) 0 0
\(871\) 28.0000 0.948744
\(872\) −2.00000 −0.0677285
\(873\) −10.0000 −0.338449
\(874\) 2.00000 0.0676510
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −8.00000 −0.269987
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) 12.0000 0.404290 0.202145 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) 3.00000 0.101015
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 10.0000 0.335578
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) −2.00000 −0.0667781
\(898\) −18.0000 −0.600668
\(899\) 24.0000 0.800445
\(900\) −5.00000 −0.166667
\(901\) 0 0
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) −34.0000 −1.12895 −0.564476 0.825450i \(-0.690922\pi\)
−0.564476 + 0.825450i \(0.690922\pi\)
\(908\) 12.0000 0.398234
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 2.00000 0.0662266
\(913\) 0 0
\(914\) 34.0000 1.12462
\(915\) 0 0
\(916\) 26.0000 0.859064
\(917\) −24.0000 −0.792550
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) 50.0000 1.64399
\(926\) −8.00000 −0.262896
\(927\) 14.0000 0.459820
\(928\) 6.00000 0.196960
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −6.00000 −0.196537
\(933\) −24.0000 −0.785725
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −46.0000 −1.50275 −0.751377 0.659873i \(-0.770610\pi\)
−0.751377 + 0.659873i \(0.770610\pi\)
\(938\) −28.0000 −0.914232
\(939\) −22.0000 −0.717943
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) −14.0000 −0.456145
\(943\) 6.00000 0.195387
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −10.0000 −0.324785
\(949\) 4.00000 0.129845
\(950\) 10.0000 0.324443
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 0 0
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 20.0000 0.644826
\(963\) 0 0
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) 2.00000 0.0643489
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 1.00000 0.0320750
\(973\) 40.0000 1.28234
\(974\) −8.00000 −0.256337
\(975\) −10.0000 −0.320256
\(976\) −10.0000 −0.320092
\(977\) 36.0000 1.15174 0.575871 0.817541i \(-0.304663\pi\)
0.575871 + 0.817541i \(0.304663\pi\)
\(978\) −8.00000 −0.255812
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) −12.0000 −0.382935
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) −2.00000 −0.0635963
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 4.00000 0.127000
\(993\) 32.0000 1.01549
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 40.0000 1.26618
\(999\) −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 138.2.a.b.1.1 1
3.2 odd 2 414.2.a.c.1.1 1
4.3 odd 2 1104.2.a.b.1.1 1
5.2 odd 4 3450.2.d.g.2899.1 2
5.3 odd 4 3450.2.d.g.2899.2 2
5.4 even 2 3450.2.a.o.1.1 1
7.6 odd 2 6762.2.a.g.1.1 1
8.3 odd 2 4416.2.a.t.1.1 1
8.5 even 2 4416.2.a.i.1.1 1
12.11 even 2 3312.2.a.h.1.1 1
23.22 odd 2 3174.2.a.d.1.1 1
69.68 even 2 9522.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.b.1.1 1 1.1 even 1 trivial
414.2.a.c.1.1 1 3.2 odd 2
1104.2.a.b.1.1 1 4.3 odd 2
3174.2.a.d.1.1 1 23.22 odd 2
3312.2.a.h.1.1 1 12.11 even 2
3450.2.a.o.1.1 1 5.4 even 2
3450.2.d.g.2899.1 2 5.2 odd 4
3450.2.d.g.2899.2 2 5.3 odd 4
4416.2.a.i.1.1 1 8.5 even 2
4416.2.a.t.1.1 1 8.3 odd 2
6762.2.a.g.1.1 1 7.6 odd 2
9522.2.a.k.1.1 1 69.68 even 2