Properties

Label 2178.2.a.g.1.1
Level $2178$
Weight $2$
Character 2178.1
Self dual yes
Analytic conductor $17.391$
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] = [2178,2,Mod(1,2178)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2178, 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("2178.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2178.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: \(17.3914175602\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2178.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} -2.00000 q^{10} +6.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -4.00000 q^{19} -2.00000 q^{20} -4.00000 q^{23} -1.00000 q^{25} +6.00000 q^{26} +4.00000 q^{28} +6.00000 q^{29} +1.00000 q^{32} +2.00000 q^{34} -8.00000 q^{35} +6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{40} -6.00000 q^{41} -4.00000 q^{43} -4.00000 q^{46} +12.0000 q^{47} +9.00000 q^{49} -1.00000 q^{50} +6.00000 q^{52} -2.00000 q^{53} +4.00000 q^{56} +6.00000 q^{58} -12.0000 q^{59} +14.0000 q^{61} +1.00000 q^{64} -12.0000 q^{65} +4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{70} +12.0000 q^{71} +6.00000 q^{73} +6.00000 q^{74} -4.00000 q^{76} +4.00000 q^{79} -2.00000 q^{80} -6.00000 q^{82} +4.00000 q^{83} -4.00000 q^{85} -4.00000 q^{86} -10.0000 q^{89} +24.0000 q^{91} -4.00000 q^{92} +12.0000 q^{94} +8.00000 q^{95} -14.0000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 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
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 0 0
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −8.00000 −1.35225
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) −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
\(46\) −4.00000 −0.589768
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 6.00000 0.832050
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −8.00000 −0.956183
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 0 0
\(122\) 14.0000 1.26750
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −12.0000 −1.05247
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) −16.0000 −1.38738
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −8.00000 −0.676123
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 24.0000 1.77900
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 0 0
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 14.0000 0.985037
\(203\) 24.0000 1.68447
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 0 0
\(218\) 6.00000 0.406371
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 8.00000 0.518563
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 14.0000 0.896258
\(245\) −18.0000 −1.14998
\(246\) 0 0
\(247\) −24.0000 −1.52708
\(248\) 0 0
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) −12.0000 −0.744208
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 4.00000 0.245718
\(266\) −16.0000 −0.981023
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) −8.00000 −0.478091
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −12.0000 −0.704664
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) 22.0000 1.28525 0.642627 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) −4.00000 −0.230174
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) −28.0000 −1.60328
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) −16.0000 −0.891645
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) −20.0000 −1.10770
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 48.0000 2.64633
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 23.0000 1.25104
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) 0 0
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) −24.0000 −1.27379
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) 24.0000 1.25794
\(365\) −12.0000 −0.628109
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) −12.0000 −0.623850
\(371\) −8.00000 −0.415339
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 36.0000 1.85409
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 38.0000 1.89763 0.948815 0.315833i \(-0.102284\pi\)
0.948815 + 0.315833i \(0.102284\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) 0 0
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 12.0000 0.592638
\(411\) 0 0
\(412\) 0 0
\(413\) −48.0000 −2.36193
\(414\) 0 0
\(415\) −8.00000 −0.392705
\(416\) 6.00000 0.294174
\(417\) 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
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 56.0000 2.71003
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 16.0000 0.765384
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −28.0000 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) 0 0
\(445\) 20.0000 0.948091
\(446\) 0 0
\(447\) 0 0
\(448\) 4.00000 0.188982
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) −48.0000 −2.25027
\(456\) 0 0
\(457\) −34.0000 −1.59045 −0.795226 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) −24.0000 −1.10704
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 8.00000 0.366679
\(477\) 0 0
\(478\) 8.00000 0.365911
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 36.0000 1.64146
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 0 0
\(485\) 28.0000 1.27141
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 14.0000 0.633750
\(489\) 0 0
\(490\) −18.0000 −0.813157
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) 0 0
\(497\) 48.0000 2.15309
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) 32.0000 1.42681 0.713405 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) 0 0
\(505\) −28.0000 −1.24598
\(506\) 0 0
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) 22.0000 0.975133 0.487566 0.873086i \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 24.0000 1.05450
\(519\) 0 0
\(520\) −12.0000 −0.526235
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 4.00000 0.173749
\(531\) 0 0
\(532\) −16.0000 −0.693688
\(533\) −36.0000 −1.55933
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −26.0000 −1.12094
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 0 0
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) −8.00000 −0.338062
\(561\) 0 0
\(562\) −22.0000 −0.928014
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) 4.00000 0.168281
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −46.0000 −1.91501 −0.957503 0.288425i \(-0.906868\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) −12.0000 −0.498273
\(581\) 16.0000 0.663792
\(582\) 0 0
\(583\) 0 0
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 24.0000 0.988064
\(591\) 0 0
\(592\) 6.00000 0.246598
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) −24.0000 −0.981433
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −16.0000 −0.652111
\(603\) 0 0
\(604\) −4.00000 −0.162758
\(605\) 0 0
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) −28.0000 −1.13369
\(611\) 72.0000 2.91281
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −4.00000 −0.160385
\(623\) −40.0000 −1.60257
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −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
\(634\) −18.0000 −0.714871
\(635\) 24.0000 0.952411
\(636\) 0 0
\(637\) 54.0000 2.13956
\(638\) 0 0
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) −20.0000 −0.783260
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 48.0000 1.87123
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 32.0000 1.24091
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 0 0
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 46.0000 1.76792 0.883962 0.467559i \(-0.154866\pi\)
0.883962 + 0.467559i \(0.154866\pi\)
\(678\) 0 0
\(679\) −56.0000 −2.14908
\(680\) −4.00000 −0.153393
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 4.00000 0.152832
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 6.00000 0.227103
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 56.0000 2.10610
\(708\) 0 0
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) −24.0000 −0.900704
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 0 0
\(718\) 0 0
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 24.0000 0.889499
\(729\) 0 0
\(730\) −12.0000 −0.444140
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 0 0
\(738\) 0 0
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) −8.00000 −0.293689
\(743\) −32.0000 −1.17397 −0.586983 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 0 0
\(745\) 20.0000 0.732743
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 0 0
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 36.0000 1.31104
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 24.0000 0.868858
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) −72.0000 −2.59977
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) −8.00000 −0.286079
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 20.0000 0.713831
\(786\) 0 0
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −8.00000 −0.284447
\(792\) 0 0
\(793\) 84.0000 2.98293
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 54.0000 1.91278 0.956389 0.292096i \(-0.0943526\pi\)
0.956389 + 0.292096i \(0.0943526\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 38.0000 1.34183
\(803\) 0 0
\(804\) 0 0
\(805\) 32.0000 1.12785
\(806\) 0 0
\(807\) 0 0
\(808\) 14.0000 0.492518
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 24.0000 0.842235
\(813\) 0 0
\(814\) 0 0
\(815\) 40.0000 1.40114
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) 12.0000 0.419058
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) −52.0000 −1.80822 −0.904109 0.427303i \(-0.859464\pi\)
−0.904109 + 0.427303i \(0.859464\pi\)
\(828\) 0 0
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) 6.00000 0.208013
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) −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
\(842\) −10.0000 −0.344623
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) −46.0000 −1.58245
\(846\) 0 0
\(847\) 0 0
\(848\) −2.00000 −0.0686803
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 56.0000 1.91628
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) 26.0000 0.888143 0.444072 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 0 0
\(863\) −20.0000 −0.680808 −0.340404 0.940279i \(-0.610564\pi\)
−0.340404 + 0.940279i \(0.610564\pi\)
\(864\) 0 0
\(865\) 20.0000 0.680020
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) 16.0000 0.541208
\(875\) 48.0000 1.62270
\(876\) 0 0
\(877\) −42.0000 −1.41824 −0.709120 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) −4.00000 −0.134993
\(879\) 0 0
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −28.0000 −0.940678
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −48.0000 −1.60987
\(890\) 20.0000 0.670402
\(891\) 0 0
\(892\) 0 0
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) 40.0000 1.33705
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) 22.0000 0.734150
\(899\) 0 0
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) 4.00000 0.132964
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) −48.0000 −1.59118
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −34.0000 −1.12462
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 8.00000 0.263752
\(921\) 0 0
\(922\) 14.0000 0.461065
\(923\) 72.0000 2.36991
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 16.0000 0.522419
\(939\) 0 0
\(940\) −24.0000 −0.782794
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 36.0000 1.16861
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 8.00000 0.259281
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) 0 0
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 36.0000 1.16069
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 20.0000 0.643823
\(966\) 0 0
\(967\) 44.0000 1.41494 0.707472 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 28.0000 0.899026
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 16.0000 0.512936
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) −26.0000 −0.831814 −0.415907 0.909407i \(-0.636536\pi\)
−0.415907 + 0.909407i \(0.636536\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −18.0000 −0.574989
\(981\) 0 0
\(982\) −28.0000 −0.893516
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) 4.00000 0.127451
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 48.0000 1.52247
\(995\) 32.0000 1.01447
\(996\) 0 0
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) −4.00000 −0.126618
\(999\) 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 2178.2.a.g.1.1 1
3.2 odd 2 726.2.a.c.1.1 1
11.10 odd 2 198.2.a.a.1.1 1
12.11 even 2 5808.2.a.bc.1.1 1
33.2 even 10 726.2.e.g.565.1 4
33.5 odd 10 726.2.e.o.487.1 4
33.8 even 10 726.2.e.g.493.1 4
33.14 odd 10 726.2.e.o.493.1 4
33.17 even 10 726.2.e.g.487.1 4
33.20 odd 10 726.2.e.o.565.1 4
33.26 odd 10 726.2.e.o.511.1 4
33.29 even 10 726.2.e.g.511.1 4
33.32 even 2 66.2.a.b.1.1 1
44.43 even 2 1584.2.a.f.1.1 1
55.32 even 4 4950.2.c.p.199.1 2
55.43 even 4 4950.2.c.p.199.2 2
55.54 odd 2 4950.2.a.bu.1.1 1
77.76 even 2 9702.2.a.x.1.1 1
88.21 odd 2 6336.2.a.bw.1.1 1
88.43 even 2 6336.2.a.cj.1.1 1
99.32 even 6 1782.2.e.e.1189.1 2
99.43 odd 6 1782.2.e.v.595.1 2
99.65 even 6 1782.2.e.e.595.1 2
99.76 odd 6 1782.2.e.v.1189.1 2
132.131 odd 2 528.2.a.j.1.1 1
165.32 odd 4 1650.2.c.e.199.2 2
165.98 odd 4 1650.2.c.e.199.1 2
165.164 even 2 1650.2.a.k.1.1 1
231.230 odd 2 3234.2.a.t.1.1 1
264.131 odd 2 2112.2.a.e.1.1 1
264.197 even 2 2112.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.b.1.1 1 33.32 even 2
198.2.a.a.1.1 1 11.10 odd 2
528.2.a.j.1.1 1 132.131 odd 2
726.2.a.c.1.1 1 3.2 odd 2
726.2.e.g.487.1 4 33.17 even 10
726.2.e.g.493.1 4 33.8 even 10
726.2.e.g.511.1 4 33.29 even 10
726.2.e.g.565.1 4 33.2 even 10
726.2.e.o.487.1 4 33.5 odd 10
726.2.e.o.493.1 4 33.14 odd 10
726.2.e.o.511.1 4 33.26 odd 10
726.2.e.o.565.1 4 33.20 odd 10
1584.2.a.f.1.1 1 44.43 even 2
1650.2.a.k.1.1 1 165.164 even 2
1650.2.c.e.199.1 2 165.98 odd 4
1650.2.c.e.199.2 2 165.32 odd 4
1782.2.e.e.595.1 2 99.65 even 6
1782.2.e.e.1189.1 2 99.32 even 6
1782.2.e.v.595.1 2 99.43 odd 6
1782.2.e.v.1189.1 2 99.76 odd 6
2112.2.a.e.1.1 1 264.131 odd 2
2112.2.a.r.1.1 1 264.197 even 2
2178.2.a.g.1.1 1 1.1 even 1 trivial
3234.2.a.t.1.1 1 231.230 odd 2
4950.2.a.bu.1.1 1 55.54 odd 2
4950.2.c.p.199.1 2 55.32 even 4
4950.2.c.p.199.2 2 55.43 even 4
5808.2.a.bc.1.1 1 12.11 even 2
6336.2.a.bw.1.1 1 88.21 odd 2
6336.2.a.cj.1.1 1 88.43 even 2
9702.2.a.x.1.1 1 77.76 even 2