Properties

Label 338.2.a.e.1.1
Level $338$
Weight $2$
Character 338.1
Self dual yes
Analytic conductor $2.699$
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] = [338,2,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(2.69894358832\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 338.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} +4.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -3.00000 q^{18} -1.00000 q^{20} +4.00000 q^{22} -4.00000 q^{23} -4.00000 q^{25} +4.00000 q^{28} -1.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -4.00000 q^{35} -3.00000 q^{36} +3.00000 q^{37} -1.00000 q^{40} -9.00000 q^{41} -8.00000 q^{43} +4.00000 q^{44} +3.00000 q^{45} -4.00000 q^{46} -8.00000 q^{47} +9.00000 q^{49} -4.00000 q^{50} -9.00000 q^{53} -4.00000 q^{55} +4.00000 q^{56} -1.00000 q^{58} -4.00000 q^{59} +7.00000 q^{61} +4.00000 q^{62} -12.0000 q^{63} +1.00000 q^{64} +4.00000 q^{67} +3.00000 q^{68} -4.00000 q^{70} -8.00000 q^{71} -3.00000 q^{72} +11.0000 q^{73} +3.00000 q^{74} +16.0000 q^{77} -4.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} -9.00000 q^{82} -3.00000 q^{85} -8.00000 q^{86} +4.00000 q^{88} -6.00000 q^{89} +3.00000 q^{90} -4.00000 q^{92} -8.00000 q^{94} +2.00000 q^{97} +9.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 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
\(9\) −3.00000 −1.00000
\(10\) −1.00000 −0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) −3.00000 −0.707107
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 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
\(34\) 3.00000 0.514496
\(35\) −4.00000 −0.676123
\(36\) −3.00000 −0.500000
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 4.00000 0.603023
\(45\) 3.00000 0.447214
\(46\) −4.00000 −0.589768
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 4.00000 0.508001
\(63\) −12.0000 −1.51186
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 3.00000 0.363803
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −3.00000 −0.353553
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) 0 0
\(77\) 16.0000 1.82337
\(78\) 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
\(81\) 9.00000 1.00000
\(82\) −9.00000 −0.993884
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 9.00000 0.909137
\(99\) −12.0000 −1.20605
\(100\) −4.00000 −0.400000
\(101\) 7.00000 0.696526 0.348263 0.937397i \(-0.386772\pi\)
0.348263 + 0.937397i \(0.386772\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) −1.00000 −0.0940721 −0.0470360 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 7.00000 0.633750
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 9.00000 0.804984
\(126\) −12.0000 −1.06904
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) −3.00000 −0.250000
\(145\) 1.00000 0.0830455
\(146\) 11.0000 0.910366
\(147\) 0 0
\(148\) 3.00000 0.246598
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) −9.00000 −0.727607
\(154\) 16.0000 1.28932
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 11.0000 0.877896 0.438948 0.898513i \(-0.355351\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −16.0000 −1.26098
\(162\) 9.00000 0.707107
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −3.00000 −0.230089
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) −16.0000 −1.20949
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 3.00000 0.223607
\(181\) −21.0000 −1.56092 −0.780459 0.625207i \(-0.785014\pi\)
−0.780459 + 0.625207i \(0.785014\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −12.0000 −0.852803
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) 7.00000 0.492518
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) 9.00000 0.628587
\(206\) −8.00000 −0.557386
\(207\) 12.0000 0.834058
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −9.00000 −0.618123
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) 0 0
\(222\) 0 0
\(223\) 12.0000 0.803579 0.401790 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) 4.00000 0.267261
\(225\) 12.0000 0.800000
\(226\) −1.00000 −0.0665190
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 12.0000 0.777844
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 3.00000 0.193247 0.0966235 0.995321i \(-0.469196\pi\)
0.0966235 + 0.995321i \(0.469196\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) 7.00000 0.448129
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 9.00000 0.569210
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −12.0000 −0.755929
\(253\) −16.0000 −1.00591
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 15.0000 0.935674 0.467837 0.883815i \(-0.345033\pi\)
0.467837 + 0.883815i \(0.345033\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 20.0000 1.23560
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) −9.00000 −0.543710
\(275\) −16.0000 −0.964836
\(276\) 0 0
\(277\) −9.00000 −0.540758 −0.270379 0.962754i \(-0.587149\pi\)
−0.270379 + 0.962754i \(0.587149\pi\)
\(278\) 16.0000 0.959616
\(279\) −12.0000 −0.718421
\(280\) −4.00000 −0.239046
\(281\) −5.00000 −0.298275 −0.149137 0.988816i \(-0.547650\pi\)
−0.149137 + 0.988816i \(0.547650\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) −36.0000 −2.12501
\(288\) −3.00000 −0.176777
\(289\) −8.00000 −0.470588
\(290\) 1.00000 0.0587220
\(291\) 0 0
\(292\) 11.0000 0.643726
\(293\) −5.00000 −0.292103 −0.146052 0.989277i \(-0.546657\pi\)
−0.146052 + 0.989277i \(0.546657\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 3.00000 0.174371
\(297\) 0 0
\(298\) 15.0000 0.868927
\(299\) 0 0
\(300\) 0 0
\(301\) −32.0000 −1.84445
\(302\) 12.0000 0.690522
\(303\) 0 0
\(304\) 0 0
\(305\) −7.00000 −0.400819
\(306\) −9.00000 −0.514496
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 16.0000 0.911685
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 11.0000 0.620766
\(315\) 12.0000 0.676123
\(316\) −4.00000 −0.225018
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −16.0000 −0.891645
\(323\) 0 0
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) 0 0
\(328\) −9.00000 −0.496942
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) −9.00000 −0.493197
\(334\) 12.0000 0.656611
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −3.00000 −0.162698
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 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 \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −16.0000 −0.855236
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) 7.00000 0.372572 0.186286 0.982496i \(-0.440355\pi\)
0.186286 + 0.982496i \(0.440355\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −24.0000 −1.26844
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 3.00000 0.158114
\(361\) −19.0000 −1.00000
\(362\) −21.0000 −1.10374
\(363\) 0 0
\(364\) 0 0
\(365\) −11.0000 −0.575766
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) −4.00000 −0.208514
\(369\) 27.0000 1.40556
\(370\) −3.00000 −0.155963
\(371\) −36.0000 −1.86903
\(372\) 0 0
\(373\) −13.0000 −0.673114 −0.336557 0.941663i \(-0.609263\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 0 0
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −20.0000 −1.02329
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) 11.0000 0.559885
\(387\) 24.0000 1.21999
\(388\) 2.00000 0.101535
\(389\) −9.00000 −0.456318 −0.228159 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 4.00000 0.201262
\(396\) −12.0000 −0.603023
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 7.00000 0.348263
\(405\) −9.00000 −0.447214
\(406\) −4.00000 −0.198517
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) 31.0000 1.53285 0.766426 0.642333i \(-0.222033\pi\)
0.766426 + 0.642333i \(0.222033\pi\)
\(410\) 9.00000 0.444478
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) −16.0000 −0.787309
\(414\) 12.0000 0.589768
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −5.00000 −0.243685 −0.121843 0.992549i \(-0.538880\pi\)
−0.121843 + 0.992549i \(0.538880\pi\)
\(422\) 20.0000 0.973585
\(423\) 24.0000 1.16692
\(424\) −9.00000 −0.437079
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) 28.0000 1.35501
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) −5.00000 −0.240285 −0.120142 0.992757i \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) −4.00000 −0.190693
\(441\) −27.0000 −1.28571
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 12.0000 0.568216
\(447\) 0 0
\(448\) 4.00000 0.188982
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 12.0000 0.565685
\(451\) −36.0000 −1.69517
\(452\) −1.00000 −0.0470360
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 0 0
\(457\) 31.0000 1.45012 0.725059 0.688686i \(-0.241812\pi\)
0.725059 + 0.688686i \(0.241812\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) −33.0000 −1.53696 −0.768482 0.639872i \(-0.778987\pi\)
−0.768482 + 0.639872i \(0.778987\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 8.00000 0.369012
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) −32.0000 −1.47136
\(474\) 0 0
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) 27.0000 1.23625
\(478\) −12.0000 −0.548867
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 3.00000 0.136646
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 7.00000 0.316875
\(489\) 0 0
\(490\) −9.00000 −0.406579
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) −3.00000 −0.135113
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) 4.00000 0.179605
\(497\) −32.0000 −1.43540
\(498\) 0 0
\(499\) −32.0000 −1.43252 −0.716258 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 9.00000 0.402492
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −12.0000 −0.534522
\(505\) −7.00000 −0.311496
\(506\) −16.0000 −0.711287
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 43.0000 1.90594 0.952971 0.303062i \(-0.0980090\pi\)
0.952971 + 0.303062i \(0.0980090\pi\)
\(510\) 0 0
\(511\) 44.0000 1.94645
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 15.0000 0.661622
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −32.0000 −1.40736
\(518\) 12.0000 0.527250
\(519\) 0 0
\(520\) 0 0
\(521\) 39.0000 1.70862 0.854311 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) 3.00000 0.131306
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 9.00000 0.390935
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −18.0000 −0.776035
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) 3.00000 0.128624
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) −9.00000 −0.384461
\(549\) −21.0000 −0.896258
\(550\) −16.0000 −0.682242
\(551\) 0 0
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) −9.00000 −0.382373
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) 39.0000 1.65248 0.826242 0.563316i \(-0.190475\pi\)
0.826242 + 0.563316i \(0.190475\pi\)
\(558\) −12.0000 −0.508001
\(559\) 0 0
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) −5.00000 −0.210912
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 1.00000 0.0420703
\(566\) 28.0000 1.17693
\(567\) 36.0000 1.51186
\(568\) −8.00000 −0.335673
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −36.0000 −1.50261
\(575\) 16.0000 0.667246
\(576\) −3.00000 −0.125000
\(577\) 39.0000 1.62359 0.811796 0.583942i \(-0.198490\pi\)
0.811796 + 0.583942i \(0.198490\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) 1.00000 0.0415227
\(581\) 0 0
\(582\) 0 0
\(583\) −36.0000 −1.49097
\(584\) 11.0000 0.455183
\(585\) 0 0
\(586\) −5.00000 −0.206548
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) 3.00000 0.123299
\(593\) −1.00000 −0.0410651 −0.0205325 0.999789i \(-0.506536\pi\)
−0.0205325 + 0.999789i \(0.506536\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 15.0000 0.614424
\(597\) 0 0
\(598\) 0 0
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 0 0
\(601\) 19.0000 0.775026 0.387513 0.921864i \(-0.373334\pi\)
0.387513 + 0.921864i \(0.373334\pi\)
\(602\) −32.0000 −1.30422
\(603\) −12.0000 −0.488678
\(604\) 12.0000 0.488273
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −7.00000 −0.283422
\(611\) 0 0
\(612\) −9.00000 −0.363803
\(613\) 11.0000 0.444286 0.222143 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 16.0000 0.644658
\(617\) −29.0000 −1.16750 −0.583748 0.811935i \(-0.698414\pi\)
−0.583748 + 0.811935i \(0.698414\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) −24.0000 −0.961540
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) 11.0000 0.438948
\(629\) 9.00000 0.358854
\(630\) 12.0000 0.478091
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) 3.00000 0.119145
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) 0 0
\(638\) −4.00000 −0.158362
\(639\) 24.0000 0.949425
\(640\) −1.00000 −0.0395285
\(641\) 19.0000 0.750455 0.375227 0.926933i \(-0.377565\pi\)
0.375227 + 0.926933i \(0.377565\pi\)
\(642\) 0 0
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) 0 0
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 9.00000 0.353553
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) −20.0000 −0.781465
\(656\) −9.00000 −0.351391
\(657\) −33.0000 −1.28745
\(658\) −32.0000 −1.24749
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) −49.0000 −1.90588 −0.952940 0.303160i \(-0.901958\pi\)
−0.952940 + 0.303160i \(0.901958\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −9.00000 −0.348743
\(667\) 4.00000 0.154881
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) 28.0000 1.08093
\(672\) 0 0
\(673\) −29.0000 −1.11787 −0.558934 0.829212i \(-0.688789\pi\)
−0.558934 + 0.829212i \(0.688789\pi\)
\(674\) 23.0000 0.885927
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 8.00000 0.307012
\(680\) −3.00000 −0.115045
\(681\) 0 0
\(682\) 16.0000 0.612672
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 0 0
\(685\) 9.00000 0.343872
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 14.0000 0.532200
\(693\) −48.0000 −1.82337
\(694\) 12.0000 0.455514
\(695\) −16.0000 −0.606915
\(696\) 0 0
\(697\) −27.0000 −1.02270
\(698\) −2.00000 −0.0757011
\(699\) 0 0
\(700\) −16.0000 −0.604743
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 7.00000 0.263448
\(707\) 28.0000 1.05305
\(708\) 0 0
\(709\) −13.0000 −0.488225 −0.244113 0.969747i \(-0.578497\pi\)
−0.244113 + 0.969747i \(0.578497\pi\)
\(710\) 8.00000 0.300235
\(711\) 12.0000 0.450035
\(712\) −6.00000 −0.224860
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) −28.0000 −1.04422 −0.522112 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) 3.00000 0.111803
\(721\) −32.0000 −1.19174
\(722\) −19.0000 −0.707107
\(723\) 0 0
\(724\) −21.0000 −0.780459
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) −11.0000 −0.407128
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) −1.00000 −0.0369358 −0.0184679 0.999829i \(-0.505879\pi\)
−0.0184679 + 0.999829i \(0.505879\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 16.0000 0.589368
\(738\) 27.0000 0.993884
\(739\) −24.0000 −0.882854 −0.441427 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) −3.00000 −0.110282
\(741\) 0 0
\(742\) −36.0000 −1.32160
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) 0 0
\(745\) −15.0000 −0.549557
\(746\) −13.0000 −0.475964
\(747\) 0 0
\(748\) 12.0000 0.438763
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) 6.00000 0.218074 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) −8.00000 −0.290573
\(759\) 0 0
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) −8.00000 −0.289619
\(764\) −20.0000 −0.723575
\(765\) 9.00000 0.325396
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) −16.0000 −0.576600
\(771\) 0 0
\(772\) 11.0000 0.395899
\(773\) 38.0000 1.36677 0.683383 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 24.0000 0.862662
\(775\) −16.0000 −0.574737
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −9.00000 −0.322666
\(779\) 0 0
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) −12.0000 −0.429119
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) −11.0000 −0.392607
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 4.00000 0.142314
\(791\) −4.00000 −0.142224
\(792\) −12.0000 −0.426401
\(793\) 0 0
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −4.00000 −0.141421
\(801\) 18.0000 0.635999
\(802\) 3.00000 0.105934
\(803\) 44.0000 1.55273
\(804\) 0 0
\(805\) 16.0000 0.563926
\(806\) 0 0
\(807\) 0 0
\(808\) 7.00000 0.246259
\(809\) 39.0000 1.37117 0.685583 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) −9.00000 −0.316228
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) −4.00000 −0.140372
\(813\) 0 0
\(814\) 12.0000 0.420600
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 0 0
\(818\) 31.0000 1.08389
\(819\) 0 0
\(820\) 9.00000 0.314294
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) 0 0
\(823\) 12.0000 0.418294 0.209147 0.977884i \(-0.432931\pi\)
0.209147 + 0.977884i \(0.432931\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 12.0000 0.417029
\(829\) 7.00000 0.243120 0.121560 0.992584i \(-0.461210\pi\)
0.121560 + 0.992584i \(0.461210\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 27.0000 0.935495
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) 0 0
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −5.00000 −0.172311
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 24.0000 0.825137
\(847\) 20.0000 0.687208
\(848\) −9.00000 −0.309061
\(849\) 0 0
\(850\) −12.0000 −0.411597
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 7.00000 0.239675 0.119838 0.992793i \(-0.461763\pi\)
0.119838 + 0.992793i \(0.461763\pi\)
\(854\) 28.0000 0.958140
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) −17.0000 −0.580709 −0.290354 0.956919i \(-0.593773\pi\)
−0.290354 + 0.956919i \(0.593773\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 52.0000 1.77010 0.885050 0.465495i \(-0.154124\pi\)
0.885050 + 0.465495i \(0.154124\pi\)
\(864\) 0 0
\(865\) −14.0000 −0.476014
\(866\) −5.00000 −0.169907
\(867\) 0 0
\(868\) 16.0000 0.543075
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) −2.00000 −0.0677285
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 36.0000 1.21702
\(876\) 0 0
\(877\) 15.0000 0.506514 0.253257 0.967399i \(-0.418498\pi\)
0.253257 + 0.967399i \(0.418498\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) −4.00000 −0.134840
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) −27.0000 −0.909137
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) −20.0000 −0.671534 −0.335767 0.941945i \(-0.608996\pi\)
−0.335767 + 0.941945i \(0.608996\pi\)
\(888\) 0 0
\(889\) −32.0000 −1.07325
\(890\) 6.00000 0.201120
\(891\) 36.0000 1.20605
\(892\) 12.0000 0.401790
\(893\) 0 0
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) 34.0000 1.13459
\(899\) −4.00000 −0.133407
\(900\) 12.0000 0.400000
\(901\) −27.0000 −0.899500
\(902\) −36.0000 −1.19867
\(903\) 0 0
\(904\) −1.00000 −0.0332595
\(905\) 21.0000 0.698064
\(906\) 0 0
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) 24.0000 0.796468
\(909\) −21.0000 −0.696526
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 31.0000 1.02539
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 80.0000 2.64183
\(918\) 0 0
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 4.00000 0.131876
\(921\) 0 0
\(922\) −33.0000 −1.08680
\(923\) 0 0
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) 16.0000 0.525793
\(927\) 24.0000 0.788263
\(928\) −1.00000 −0.0328266
\(929\) −21.0000 −0.688988 −0.344494 0.938789i \(-0.611949\pi\)
−0.344494 + 0.938789i \(0.611949\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) 20.0000 0.654420
\(935\) −12.0000 −0.392442
\(936\) 0 0
\(937\) −21.0000 −0.686040 −0.343020 0.939328i \(-0.611450\pi\)
−0.343020 + 0.939328i \(0.611450\pi\)
\(938\) 16.0000 0.522419
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 0 0
\(943\) 36.0000 1.17232
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −32.0000 −1.04041
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 12.0000 0.388922
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 27.0000 0.874157
\(955\) 20.0000 0.647185
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 0 0
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 3.00000 0.0966235
\(965\) −11.0000 −0.354103
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 64.0000 2.05175
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 7.00000 0.224065
\(977\) −45.0000 −1.43968 −0.719839 0.694141i \(-0.755784\pi\)
−0.719839 + 0.694141i \(0.755784\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) −9.00000 −0.287494
\(981\) 6.00000 0.191565
\(982\) −8.00000 −0.255290
\(983\) −52.0000 −1.65854 −0.829271 0.558846i \(-0.811244\pi\)
−0.829271 + 0.558846i \(0.811244\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) −3.00000 −0.0955395
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0000 1.01754
\(990\) 12.0000 0.381385
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) −32.0000 −1.01498
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) −25.0000 −0.791758 −0.395879 0.918303i \(-0.629560\pi\)
−0.395879 + 0.918303i \(0.629560\pi\)
\(998\) −32.0000 −1.01294
\(999\) 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 338.2.a.e.1.1 1
3.2 odd 2 3042.2.a.e.1.1 1
4.3 odd 2 2704.2.a.h.1.1 1
5.4 even 2 8450.2.a.f.1.1 1
13.2 odd 12 338.2.e.b.147.2 4
13.3 even 3 26.2.c.a.9.1 yes 2
13.4 even 6 338.2.c.e.315.1 2
13.5 odd 4 338.2.b.b.337.1 2
13.6 odd 12 338.2.e.b.23.1 4
13.7 odd 12 338.2.e.b.23.2 4
13.8 odd 4 338.2.b.b.337.2 2
13.9 even 3 26.2.c.a.3.1 2
13.10 even 6 338.2.c.e.191.1 2
13.11 odd 12 338.2.e.b.147.1 4
13.12 even 2 338.2.a.c.1.1 1
39.5 even 4 3042.2.b.e.1351.2 2
39.8 even 4 3042.2.b.e.1351.1 2
39.29 odd 6 234.2.h.c.217.1 2
39.35 odd 6 234.2.h.c.55.1 2
39.38 odd 2 3042.2.a.k.1.1 1
52.3 odd 6 208.2.i.b.113.1 2
52.31 even 4 2704.2.f.g.337.2 2
52.35 odd 6 208.2.i.b.81.1 2
52.47 even 4 2704.2.f.g.337.1 2
52.51 odd 2 2704.2.a.i.1.1 1
65.3 odd 12 650.2.o.c.399.2 4
65.9 even 6 650.2.e.c.601.1 2
65.22 odd 12 650.2.o.c.549.2 4
65.29 even 6 650.2.e.c.451.1 2
65.42 odd 12 650.2.o.c.399.1 4
65.48 odd 12 650.2.o.c.549.1 4
65.64 even 2 8450.2.a.s.1.1 1
91.3 odd 6 1274.2.h.a.373.1 2
91.9 even 3 1274.2.h.b.263.1 2
91.16 even 3 1274.2.e.n.165.1 2
91.48 odd 6 1274.2.g.a.393.1 2
91.55 odd 6 1274.2.g.a.295.1 2
91.61 odd 6 1274.2.h.a.263.1 2
91.68 odd 6 1274.2.e.m.165.1 2
91.74 even 3 1274.2.e.n.471.1 2
91.81 even 3 1274.2.h.b.373.1 2
91.87 odd 6 1274.2.e.m.471.1 2
104.3 odd 6 832.2.i.f.321.1 2
104.29 even 6 832.2.i.e.321.1 2
104.35 odd 6 832.2.i.f.705.1 2
104.61 even 6 832.2.i.e.705.1 2
156.35 even 6 1872.2.t.k.289.1 2
156.107 even 6 1872.2.t.k.1153.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.c.a.3.1 2 13.9 even 3
26.2.c.a.9.1 yes 2 13.3 even 3
208.2.i.b.81.1 2 52.35 odd 6
208.2.i.b.113.1 2 52.3 odd 6
234.2.h.c.55.1 2 39.35 odd 6
234.2.h.c.217.1 2 39.29 odd 6
338.2.a.c.1.1 1 13.12 even 2
338.2.a.e.1.1 1 1.1 even 1 trivial
338.2.b.b.337.1 2 13.5 odd 4
338.2.b.b.337.2 2 13.8 odd 4
338.2.c.e.191.1 2 13.10 even 6
338.2.c.e.315.1 2 13.4 even 6
338.2.e.b.23.1 4 13.6 odd 12
338.2.e.b.23.2 4 13.7 odd 12
338.2.e.b.147.1 4 13.11 odd 12
338.2.e.b.147.2 4 13.2 odd 12
650.2.e.c.451.1 2 65.29 even 6
650.2.e.c.601.1 2 65.9 even 6
650.2.o.c.399.1 4 65.42 odd 12
650.2.o.c.399.2 4 65.3 odd 12
650.2.o.c.549.1 4 65.48 odd 12
650.2.o.c.549.2 4 65.22 odd 12
832.2.i.e.321.1 2 104.29 even 6
832.2.i.e.705.1 2 104.61 even 6
832.2.i.f.321.1 2 104.3 odd 6
832.2.i.f.705.1 2 104.35 odd 6
1274.2.e.m.165.1 2 91.68 odd 6
1274.2.e.m.471.1 2 91.87 odd 6
1274.2.e.n.165.1 2 91.16 even 3
1274.2.e.n.471.1 2 91.74 even 3
1274.2.g.a.295.1 2 91.55 odd 6
1274.2.g.a.393.1 2 91.48 odd 6
1274.2.h.a.263.1 2 91.61 odd 6
1274.2.h.a.373.1 2 91.3 odd 6
1274.2.h.b.263.1 2 91.9 even 3
1274.2.h.b.373.1 2 91.81 even 3
1872.2.t.k.289.1 2 156.35 even 6
1872.2.t.k.1153.1 2 156.107 even 6
2704.2.a.h.1.1 1 4.3 odd 2
2704.2.a.i.1.1 1 52.51 odd 2
2704.2.f.g.337.1 2 52.47 even 4
2704.2.f.g.337.2 2 52.31 even 4
3042.2.a.e.1.1 1 3.2 odd 2
3042.2.a.k.1.1 1 39.38 odd 2
3042.2.b.e.1351.1 2 39.8 even 4
3042.2.b.e.1351.2 2 39.5 even 4
8450.2.a.f.1.1 1 5.4 even 2
8450.2.a.s.1.1 1 65.64 even 2