Properties

Label 4998.2.a.d.1.1
Level $4998$
Weight $2$
Character 4998.1
Self dual yes
Analytic conductor $39.909$
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] = [4998,2,Mod(1,4998)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4998, 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("4998.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4998.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: \(39.9092309302\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 102)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4998.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -6.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +10.0000 q^{31} -1.00000 q^{32} -1.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -6.00000 q^{41} -4.00000 q^{43} +6.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} +5.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{57} +12.0000 q^{59} -8.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} -4.00000 q^{67} +1.00000 q^{68} +6.00000 q^{69} +6.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} -8.00000 q^{74} +5.00000 q^{75} +4.00000 q^{76} -2.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} +6.00000 q^{82} -12.0000 q^{83} +4.00000 q^{86} +18.0000 q^{89} -6.00000 q^{92} -10.0000 q^{93} +12.0000 q^{94} +1.00000 q^{96} -14.0000 q^{97} +O(q^{100})\)

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
\(7\) 0 0
\(8\) −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 \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 5.00000 0.707107
\(51\) −1.00000 −0.140028
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −10.0000 −1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 1.00000 0.121268
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −8.00000 −0.929981
\(75\) 5.00000 0.577350
\(76\) 4.00000 0.458831
\(77\) 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
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) −10.0000 −1.03695
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 1.00000 0.0990148
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 8.00000 0.724286
\(123\) 6.00000 0.541002
\(124\) 10.0000 0.898027
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −6.00000 −0.510754
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −5.00000 −0.408248
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −4.00000 −0.324443
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 10.0000 0.795557
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) −18.0000 −1.34916
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) 10.0000 0.733236
\(187\) 0 0
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 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 \(0.331909\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 5.00000 0.353553
\(201\) 4.00000 0.282138
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) −6.00000 −0.417029
\(208\) −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 \(0.411199\pi\)
\(212\) 6.00000 0.412082
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −20.0000 −1.35457
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 8.00000 0.536925
\(223\) 28.0000 1.87502 0.937509 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 6.00000 0.399114
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −4.00000 −0.264906
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −30.0000 −1.96537 −0.982683 0.185296i \(-0.940675\pi\)
−0.982683 + 0.185296i \(0.940675\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 11.0000 0.707107
\(243\) −1.00000 −0.0641500
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −8.00000 −0.509028
\(248\) −10.0000 −0.635001
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) −4.00000 −0.244339
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) 8.00000 0.479808
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) −12.0000 −0.714590
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) −2.00000 −0.117041
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 12.0000 0.693978
\(300\) 5.00000 0.288675
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) 6.00000 0.344691
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −2.00000 −0.113228
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 10.0000 0.554700
\(326\) −20.0000 −1.10770
\(327\) −20.0000 −1.10600
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −12.0000 −0.658586
\(333\) 8.00000 0.438397
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 9.00000 0.489535
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 8.00000 0.420471
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 0 0
\(366\) −8.00000 −0.418167
\(367\) 10.0000 0.521996 0.260998 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) −6.00000 −0.312772
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) −10.0000 −0.518476
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 0 0
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 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
\(387\) −4.00000 −0.203331
\(388\) −14.0000 −0.710742
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 0 0
\(394\) 12.0000 0.604551
\(395\) 0 0
\(396\) 0 0
\(397\) 16.0000 0.803017 0.401508 0.915855i \(-0.368486\pi\)
0.401508 + 0.915855i \(0.368486\pi\)
\(398\) 2.00000 0.100251
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −4.00000 −0.199502
\(403\) −20.0000 −0.996271
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 1.00000 0.0495074
\(409\) −38.0000 −1.87898 −0.939490 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) −8.00000 −0.389434
\(423\) −12.0000 −0.583460
\(424\) −6.00000 −0.291386
\(425\) −5.00000 −0.242536
\(426\) 6.00000 0.290701
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 20.0000 0.957826
\(437\) −24.0000 −1.14808
\(438\) −2.00000 −0.0955637
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) −28.0000 −1.32584
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 5.00000 0.235702
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) −8.00000 −0.375873
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 14.0000 0.654177
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 30.0000 1.38972
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) −10.0000 −0.459315
\(475\) −20.0000 −0.917663
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 12.0000 0.548867
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −10.0000 −0.453143 −0.226572 0.973995i \(-0.572752\pi\)
−0.226572 + 0.973995i \(0.572752\pi\)
\(488\) 8.00000 0.362143
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 6.00000 0.270501
\(493\) 0 0
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 10.0000 0.449013
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 12.0000 0.535586
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 8.00000 0.354943
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 10.0000 0.435607
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 18.0000 0.778936
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 12.0000 0.517838
\(538\) 24.0000 1.03471
\(539\) 0 0
\(540\) 0 0
\(541\) −16.0000 −0.687894 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) −16.0000 −0.687259
\(543\) 8.00000 0.343313
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −6.00000 −0.256307
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) 0 0
\(552\) −6.00000 −0.255377
\(553\) 0 0
\(554\) 4.00000 0.169944
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) −42.0000 −1.77960 −0.889799 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) −10.0000 −0.423334
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) −30.0000 −1.26547
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) 30.0000 1.25109
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) −14.0000 −0.580319
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 40.0000 1.64817
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 8.00000 0.328798
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 2.00000 0.0818546
\(598\) −12.0000 −0.490716
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −5.00000 −0.204124
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) 10.0000 0.405887 0.202944 0.979190i \(-0.434949\pi\)
0.202944 + 0.979190i \(0.434949\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 1.00000 0.0404226
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 4.00000 0.160904
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) 18.0000 0.721734
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 25.0000 1.00000
\(626\) 14.0000 0.559553
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 10.0000 0.397779
\(633\) −8.00000 −0.317971
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) −36.0000 −1.41531 −0.707653 0.706560i \(-0.750246\pi\)
−0.707653 + 0.706560i \(0.750246\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −10.0000 −0.392232
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 20.0000 0.782062
\(655\) 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
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −20.0000 −0.777322
\(663\) 2.00000 0.0776736
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) 0 0
\(668\) −18.0000 −0.696441
\(669\) −28.0000 −1.08254
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 22.0000 0.847408
\(675\) 5.00000 0.192450
\(676\) −9.00000 −0.346154
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) 26.0000 0.984115
\(699\) 30.0000 1.13470
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 32.0000 1.20690
\(704\) 0 0
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) 20.0000 0.751116 0.375558 0.926799i \(-0.377451\pi\)
0.375558 + 0.926799i \(0.377451\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) −18.0000 −0.674579
\(713\) −60.0000 −2.24702
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 12.0000 0.448148
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −10.0000 −0.371904
\(724\) −8.00000 −0.297318
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 8.00000 0.295689
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 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
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) 42.0000 1.54083 0.770415 0.637542i \(-0.220049\pi\)
0.770415 + 0.637542i \(0.220049\pi\)
\(744\) 10.0000 0.366618
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.00000 0.0729810 0.0364905 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) −12.0000 −0.437595
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 28.0000 1.01701
\(759\) 0 0
\(760\) 0 0
\(761\) 54.0000 1.95750 0.978749 0.205061i \(-0.0657392\pi\)
0.978749 + 0.205061i \(0.0657392\pi\)
\(762\) 8.00000 0.289809
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) −24.0000 −0.866590
\(768\) −1.00000 −0.0360844
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 14.0000 0.503871
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 4.00000 0.143777
\(775\) −50.0000 −1.79605
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 6.00000 0.214560
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) −12.0000 −0.427482
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) −16.0000 −0.567819
\(795\) 0 0
\(796\) −2.00000 −0.0708881
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 5.00000 0.176777
\(801\) 18.0000 0.635999
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) 24.0000 0.844840
\(808\) 6.00000 0.211079
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) −16.0000 −0.559769
\(818\) 38.0000 1.32864
\(819\) 0 0
\(820\) 0 0
\(821\) −36.0000 −1.25641 −0.628204 0.778048i \(-0.716210\pi\)
−0.628204 + 0.778048i \(0.716210\pi\)
\(822\) −6.00000 −0.209274
\(823\) −34.0000 −1.18517 −0.592583 0.805510i \(-0.701892\pi\)
−0.592583 + 0.805510i \(0.701892\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 48.0000 1.66912 0.834562 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) −6.00000 −0.208514
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) 4.00000 0.138758
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) −36.0000 −1.24360
\(839\) 18.0000 0.621429 0.310715 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 22.0000 0.758170
\(843\) −30.0000 −1.03325
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −16.0000 −0.549119
\(850\) 5.00000 0.171499
\(851\) −48.0000 −1.64542
\(852\) −6.00000 −0.205557
\(853\) 28.0000 0.958702 0.479351 0.877623i \(-0.340872\pi\)
0.479351 + 0.877623i \(0.340872\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 30.0000 1.02180
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 14.0000 0.475739
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −20.0000 −0.677285
\(873\) −14.0000 −0.473828
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) −4.00000 −0.135070 −0.0675352 0.997717i \(-0.521513\pi\)
−0.0675352 + 0.997717i \(0.521513\pi\)
\(878\) 26.0000 0.877457
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 54.0000 1.81931 0.909653 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 18.0000 0.604381 0.302190 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) 8.00000 0.268462
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 28.0000 0.937509
\(893\) −48.0000 −1.60626
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) 0 0
\(897\) −12.0000 −0.400668
\(898\) −6.00000 −0.200223
\(899\) 0 0
\(900\) −5.00000 −0.166667
\(901\) 6.00000 0.199889
\(902\) 0 0
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) −40.0000 −1.32818 −0.664089 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) −12.0000 −0.398234
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) −4.00000 −0.132453
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 1.00000 0.0330049
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 30.0000 0.987997
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −40.0000 −1.31519
\(926\) −20.0000 −0.657241
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −30.0000 −0.982683
\(933\) 18.0000 0.589294
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) −12.0000 −0.391189 −0.195594 0.980685i \(-0.562664\pi\)
−0.195594 + 0.980685i \(0.562664\pi\)
\(942\) 10.0000 0.325818
\(943\) 36.0000 1.17232
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 10.0000 0.324785
\(949\) 4.00000 0.129845
\(950\) 20.0000 0.648886
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 6.00000 0.193851
\(959\) 0 0
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 16.0000 0.515861
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 11.0000 0.353553
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 10.0000 0.320421
\(975\) −10.0000 −0.320256
\(976\) −8.00000 −0.256074
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 20.0000 0.639529
\(979\) 0 0
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) 36.0000 1.14881
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) −10.0000 −0.317500
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) −8.00000 −0.253363 −0.126681 0.991943i \(-0.540433\pi\)
−0.126681 + 0.991943i \(0.540433\pi\)
\(998\) −32.0000 −1.01294
\(999\) −8.00000 −0.253109
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 4998.2.a.d.1.1 1
7.6 odd 2 102.2.a.b.1.1 1
21.20 even 2 306.2.a.c.1.1 1
28.27 even 2 816.2.a.d.1.1 1
35.13 even 4 2550.2.d.g.2449.2 2
35.27 even 4 2550.2.d.g.2449.1 2
35.34 odd 2 2550.2.a.u.1.1 1
56.13 odd 2 3264.2.a.i.1.1 1
56.27 even 2 3264.2.a.w.1.1 1
84.83 odd 2 2448.2.a.i.1.1 1
105.104 even 2 7650.2.a.j.1.1 1
119.13 odd 4 1734.2.b.f.577.2 2
119.55 odd 4 1734.2.b.f.577.1 2
119.76 odd 8 1734.2.f.b.829.1 4
119.83 odd 8 1734.2.f.b.1483.1 4
119.104 odd 8 1734.2.f.b.1483.2 4
119.111 odd 8 1734.2.f.b.829.2 4
119.118 odd 2 1734.2.a.b.1.1 1
168.83 odd 2 9792.2.a.ba.1.1 1
168.125 even 2 9792.2.a.bg.1.1 1
357.356 even 2 5202.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.b.1.1 1 7.6 odd 2
306.2.a.c.1.1 1 21.20 even 2
816.2.a.d.1.1 1 28.27 even 2
1734.2.a.b.1.1 1 119.118 odd 2
1734.2.b.f.577.1 2 119.55 odd 4
1734.2.b.f.577.2 2 119.13 odd 4
1734.2.f.b.829.1 4 119.76 odd 8
1734.2.f.b.829.2 4 119.111 odd 8
1734.2.f.b.1483.1 4 119.83 odd 8
1734.2.f.b.1483.2 4 119.104 odd 8
2448.2.a.i.1.1 1 84.83 odd 2
2550.2.a.u.1.1 1 35.34 odd 2
2550.2.d.g.2449.1 2 35.27 even 4
2550.2.d.g.2449.2 2 35.13 even 4
3264.2.a.i.1.1 1 56.13 odd 2
3264.2.a.w.1.1 1 56.27 even 2
4998.2.a.d.1.1 1 1.1 even 1 trivial
5202.2.a.j.1.1 1 357.356 even 2
7650.2.a.j.1.1 1 105.104 even 2
9792.2.a.ba.1.1 1 168.83 odd 2
9792.2.a.bg.1.1 1 168.125 even 2