Properties

Label 289.2.a.a.1.1
Level $289$
Weight $2$
Character 289.1
Self dual yes
Analytic conductor $2.308$
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] = [289,2,Mod(1,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, 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("289.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 289.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: \(2.30767661842\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -4.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -4.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} -2.00000 q^{10} -2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{16} +3.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -4.00000 q^{23} -1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} -5.00000 q^{32} -8.00000 q^{35} +3.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +6.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} -6.00000 q^{45} +4.00000 q^{46} +9.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +6.00000 q^{53} -12.0000 q^{56} +6.00000 q^{58} -12.0000 q^{59} +10.0000 q^{61} +4.00000 q^{62} +12.0000 q^{63} +7.00000 q^{64} -4.00000 q^{65} +4.00000 q^{67} +8.00000 q^{70} +4.00000 q^{71} -9.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} +4.00000 q^{76} -12.0000 q^{79} -2.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} -4.00000 q^{86} +10.0000 q^{89} +6.00000 q^{90} +8.00000 q^{91} +4.00000 q^{92} -8.00000 q^{95} -2.00000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1.00000 −0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 3.00000 1.06066
\(9\) −3.00000 −1.00000
\(10\) −2.00000 −0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 0 0
\(18\) 3.00000 0.707107
\(19\) −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
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 0 0
\(35\) −8.00000 −1.35225
\(36\) 3.00000 0.500000
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 6.00000 0.948683
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −6.00000 −0.894427
\(46\) 4.00000 0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −12.0000 −1.60357
\(57\) 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
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 4.00000 0.508001
\(63\) 12.0000 1.51186
\(64\) 7.00000 0.875000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 8.00000 0.956183
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) −9.00000 −1.06066
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −2.00000 −0.223607
\(81\) 9.00000 1.00000
\(82\) −6.00000 −0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 6.00000 0.632456
\(91\) 8.00000 0.838628
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 6.00000 0.557086
\(117\) 6.00000 0.554700
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) −12.0000 −1.07331
\(126\) −12.0000 −1.06904
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 8.00000 0.676123
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) 0 0
\(144\) 3.00000 0.250000
\(145\) −12.0000 −0.996546
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −12.0000 −0.973329
\(153\) 0 0
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 12.0000 0.954669
\(159\) 0 0
\(160\) −10.0000 −0.790569
\(161\) 16.0000 1.26098
\(162\) −9.00000 −0.707107
\(163\) −24.0000 −1.87983 −0.939913 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 4.00000 0.309529 0.154765 0.987951i \(-0.450538\pi\)
0.154765 + 0.987951i \(0.450538\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 12.0000 0.917663
\(172\) −4.00000 −0.304997
\(173\) −22.0000 −1.67263 −0.836315 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 6.00000 0.447214
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −8.00000 −0.592999
\(183\) 0 0
\(184\) −12.0000 −0.884652
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) 24.0000 1.68447
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) −8.00000 −0.557386
\(207\) 12.0000 0.834058
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 6.00000 0.406371
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 20.0000 1.33631
\(225\) 3.00000 0.200000
\(226\) −14.0000 −0.931266
\(227\) 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 \(0.436476\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) −18.0000 −1.18176
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) 18.0000 1.14998
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) −12.0000 −0.762001
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −12.0000 −0.755929
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 4.00000 0.248069
\(261\) 18.0000 1.11417
\(262\) 16.0000 0.988483
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) −16.0000 −0.981023
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −22.0000 −1.34136 −0.670682 0.741745i \(-0.733998\pi\)
−0.670682 + 0.741745i \(0.733998\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) −8.00000 −0.479808
\(279\) 12.0000 0.718421
\(280\) −24.0000 −1.43427
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) 15.0000 0.883883
\(289\) 0 0
\(290\) 12.0000 0.704664
\(291\) 0 0
\(292\) −6.00000 −0.351123
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 20.0000 1.14520
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 2.00000 0.112867
\(315\) 24.0000 1.35225
\(316\) 12.0000 0.675053
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 14.0000 0.782624
\(321\) 0 0
\(322\) −16.0000 −0.891645
\(323\) 0 0
\(324\) −9.00000 −0.500000
\(325\) 2.00000 0.110940
\(326\) 24.0000 1.32924
\(327\) 0 0
\(328\) 18.0000 0.993884
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 4.00000 0.219529
\(333\) −6.00000 −0.328798
\(334\) −4.00000 −0.218870
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) −12.0000 −0.648886
\(343\) −8.00000 −0.431959
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −18.0000 −0.948683
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) 12.0000 0.628109
\(366\) 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
\(369\) −18.0000 −0.937043
\(370\) −4.00000 −0.207950
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) 16.0000 0.818631
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) −12.0000 −0.609994
\(388\) 2.00000 0.101535
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 27.0000 1.36371
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) −24.0000 −1.20757
\(396\) 0 0
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 14.0000 0.699127 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 10.0000 0.497519
\(405\) 18.0000 0.894427
\(406\) −24.0000 −1.19110
\(407\) 0 0
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) −12.0000 −0.592638
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 48.0000 2.36193
\(414\) −12.0000 −0.589768
\(415\) −8.00000 −0.392705
\(416\) 10.0000 0.490290
\(417\) 0 0
\(418\) 0 0
\(419\) −8.00000 −0.390826 −0.195413 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) 0 0
\(426\) 0 0
\(427\) −40.0000 −1.93574
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 16.0000 0.765384
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −27.0000 −1.28571
\(442\) 0 0
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 0 0
\(445\) 20.0000 0.948091
\(446\) −24.0000 −1.13643
\(447\) 0 0
\(448\) −28.0000 −1.32288
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) −3.00000 −0.141421
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) −24.0000 −1.12638
\(455\) 16.0000 0.750092
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −6.00000 −0.277350
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) −36.0000 −1.65703
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −18.0000 −0.824163
\(478\) 16.0000 0.731823
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 30.0000 1.35804
\(489\) 0 0
\(490\) −18.0000 −0.813157
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 36.0000 1.60357
\(505\) −20.0000 −0.889988
\(506\) 0 0
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) −24.0000 −1.06170
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 0 0
\(518\) 8.00000 0.351500
\(519\) 0 0
\(520\) −12.0000 −0.526235
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) −18.0000 −0.787839
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 16.0000 0.698963
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −12.0000 −0.521247
\(531\) 36.0000 1.56227
\(532\) −16.0000 −0.693688
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −16.0000 −0.691740
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) 22.0000 0.948487
\(539\) 0 0
\(540\) 0 0
\(541\) −6.00000 −0.257960 −0.128980 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) 16.0000 0.687259
\(543\) 0 0
\(544\) 0 0
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) 32.0000 1.36822 0.684111 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) 6.00000 0.256307
\(549\) −30.0000 −1.28037
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 48.0000 2.04117
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) −12.0000 −0.508001
\(559\) −8.00000 −0.338364
\(560\) 8.00000 0.338062
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 28.0000 1.17797
\(566\) −16.0000 −0.672530
\(567\) −36.0000 −1.51186
\(568\) 12.0000 0.503509
\(569\) −38.0000 −1.59304 −0.796521 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 24.0000 1.00174
\(575\) 4.00000 0.166812
\(576\) −21.0000 −0.875000
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 12.0000 0.498273
\(581\) 16.0000 0.663792
\(582\) 0 0
\(583\) 0 0
\(584\) 18.0000 0.744845
\(585\) 12.0000 0.496139
\(586\) −6.00000 −0.247858
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 24.0000 0.988064
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 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
\(603\) −12.0000 −0.488678
\(604\) 16.0000 0.651031
\(605\) −22.0000 −0.894427
\(606\) 0 0
\(607\) −20.0000 −0.811775 −0.405887 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) 20.0000 0.811107
\(609\) 0 0
\(610\) −20.0000 −0.809776
\(611\) 0 0
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 48.0000 1.92928 0.964641 0.263566i \(-0.0848986\pi\)
0.964641 + 0.263566i \(0.0848986\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 28.0000 1.12270
\(623\) −40.0000 −1.60257
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 0 0
\(630\) −24.0000 −0.956183
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −36.0000 −1.43200
\(633\) 0 0
\(634\) −10.0000 −0.397151
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 6.00000 0.237171
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) −32.0000 −1.26196 −0.630978 0.775800i \(-0.717346\pi\)
−0.630978 + 0.775800i \(0.717346\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 27.0000 1.06066
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 24.0000 0.939913
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) −32.0000 −1.25034
\(656\) −6.00000 −0.234261
\(657\) −18.0000 −0.702247
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 32.0000 1.24091
\(666\) 6.00000 0.232495
\(667\) 24.0000 0.929284
\(668\) −4.00000 −0.154765
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 0 0
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 0 0
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 40.0000 1.53056 0.765279 0.643699i \(-0.222601\pi\)
0.765279 + 0.643699i \(0.222601\pi\)
\(684\) −12.0000 −0.458831
\(685\) −12.0000 −0.458496
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 22.0000 0.836315
\(693\) 0 0
\(694\) 32.0000 1.21470
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 0 0
\(698\) 18.0000 0.681310
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 40.0000 1.50435
\(708\) 0 0
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) −8.00000 −0.300235
\(711\) 36.0000 1.35011
\(712\) 30.0000 1.12430
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) −4.00000 −0.149175 −0.0745874 0.997214i \(-0.523764\pi\)
−0.0745874 + 0.997214i \(0.523764\pi\)
\(720\) 6.00000 0.223607
\(721\) −32.0000 −1.19174
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 24.0000 0.889499
\(729\) −27.0000 −1.00000
\(730\) −12.0000 −0.444140
\(731\) 0 0
\(732\) 0 0
\(733\) −50.0000 −1.84679 −0.923396 0.383849i \(-0.874598\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) 20.0000 0.737210
\(737\) 0 0
\(738\) 18.0000 0.662589
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 0 0
\(745\) −20.0000 −0.732743
\(746\) −6.00000 −0.219676
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −8.00000 −0.290573
\(759\) 0 0
\(760\) −24.0000 −0.870572
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) 24.0000 0.868858
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.00000 0.0719816
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 12.0000 0.431331
\(775\) 4.00000 0.143684
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) 24.0000 0.853882
\(791\) −56.0000 −1.99113
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) −50.0000 −1.77109 −0.885545 0.464553i \(-0.846215\pi\)
−0.885545 + 0.464553i \(0.846215\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) −30.0000 −1.06000
\(802\) −14.0000 −0.494357
\(803\) 0 0
\(804\) 0 0
\(805\) 32.0000 1.12785
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) −30.0000 −1.05540
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) −18.0000 −0.632456
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) −24.0000 −0.842235
\(813\) 0 0
\(814\) 0 0
\(815\) −48.0000 −1.68137
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) −26.0000 −0.909069
\(819\) −24.0000 −0.838628
\(820\) −12.0000 −0.419058
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) 48.0000 1.66912 0.834562 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) −12.0000 −0.417029
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 8.00000 0.277684
\(831\) 0 0
\(832\) −14.0000 −0.485363
\(833\) 0 0
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) 8.00000 0.276355
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) −18.0000 −0.619219
\(846\) 0 0
\(847\) 44.0000 1.51186
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 40.0000 1.36877
\(855\) 24.0000 0.820783
\(856\) −24.0000 −0.820303
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 0 0
\(859\) 52.0000 1.77422 0.887109 0.461561i \(-0.152710\pi\)
0.887109 + 0.461561i \(0.152710\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 16.0000 0.544646 0.272323 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 0 0
\(865\) −44.0000 −1.49604
\(866\) −2.00000 −0.0679628
\(867\) 0 0
\(868\) −16.0000 −0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) −18.0000 −0.609557
\(873\) 6.00000 0.203069
\(874\) −16.0000 −0.541208
\(875\) 48.0000 1.62270
\(876\) 0 0
\(877\) −6.00000 −0.202606 −0.101303 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) −20.0000 −0.674967
\(879\) 0 0
\(880\) 0 0
\(881\) 46.0000 1.54978 0.774890 0.632096i \(-0.217805\pi\)
0.774890 + 0.632096i \(0.217805\pi\)
\(882\) 27.0000 0.909137
\(883\) −12.0000 −0.403832 −0.201916 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −28.0000 −0.940678
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) −32.0000 −1.07325
\(890\) −20.0000 −0.670402
\(891\) 0 0
\(892\) −24.0000 −0.803579
\(893\) 0 0
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) −12.0000 −0.400892
\(897\) 0 0
\(898\) 34.0000 1.13459
\(899\) 24.0000 0.800445
\(900\) −3.00000 −0.100000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 42.0000 1.39690
\(905\) 4.00000 0.132964
\(906\) 0 0
\(907\) −32.0000 −1.06254 −0.531271 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) −24.0000 −0.796468
\(909\) 30.0000 0.995037
\(910\) −16.0000 −0.530395
\(911\) 4.00000 0.132526 0.0662630 0.997802i \(-0.478892\pi\)
0.0662630 + 0.997802i \(0.478892\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 64.0000 2.11347
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) −24.0000 −0.791257
\(921\) 0 0
\(922\) 2.00000 0.0658665
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −32.0000 −1.05159
\(927\) −24.0000 −0.788263
\(928\) 30.0000 0.984798
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 18.0000 0.588348
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 16.0000 0.522419
\(939\) 0 0
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) −32.0000 −1.03986 −0.519930 0.854209i \(-0.674042\pi\)
−0.519930 + 0.854209i \(0.674042\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 18.0000 0.582772
\(955\) −32.0000 −1.03550
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) 36.0000 1.16311
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 4.00000 0.128965
\(963\) 24.0000 0.773389
\(964\) 18.0000 0.579741
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −33.0000 −1.06066
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) −32.0000 −1.02587
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −18.0000 −0.574989
\(981\) 18.0000 0.574696
\(982\) −20.0000 −0.638226
\(983\) −12.0000 −0.382741 −0.191370 0.981518i \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 0 0
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) 12.0000 0.381193 0.190596 0.981669i \(-0.438958\pi\)
0.190596 + 0.981669i \(0.438958\pi\)
\(992\) 20.0000 0.635001
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) 40.0000 1.26809
\(996\) 0 0
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) −40.0000 −1.26618
\(999\) 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 289.2.a.a.1.1 1
3.2 odd 2 2601.2.a.g.1.1 1
4.3 odd 2 4624.2.a.d.1.1 1
5.4 even 2 7225.2.a.g.1.1 1
17.2 even 8 289.2.c.a.38.1 4
17.3 odd 16 289.2.d.d.179.1 8
17.4 even 4 289.2.b.a.288.1 2
17.5 odd 16 289.2.d.d.110.1 8
17.6 odd 16 289.2.d.d.155.1 8
17.7 odd 16 289.2.d.d.134.1 8
17.8 even 8 289.2.c.a.251.2 4
17.9 even 8 289.2.c.a.251.1 4
17.10 odd 16 289.2.d.d.134.2 8
17.11 odd 16 289.2.d.d.155.2 8
17.12 odd 16 289.2.d.d.110.2 8
17.13 even 4 289.2.b.a.288.2 2
17.14 odd 16 289.2.d.d.179.2 8
17.15 even 8 289.2.c.a.38.2 4
17.16 even 2 17.2.a.a.1.1 1
51.50 odd 2 153.2.a.c.1.1 1
68.67 odd 2 272.2.a.b.1.1 1
85.33 odd 4 425.2.b.b.324.2 2
85.67 odd 4 425.2.b.b.324.1 2
85.84 even 2 425.2.a.d.1.1 1
119.16 even 6 833.2.e.b.18.1 2
119.33 odd 6 833.2.e.a.18.1 2
119.67 even 6 833.2.e.b.324.1 2
119.101 odd 6 833.2.e.a.324.1 2
119.118 odd 2 833.2.a.a.1.1 1
136.67 odd 2 1088.2.a.h.1.1 1
136.101 even 2 1088.2.a.i.1.1 1
187.186 odd 2 2057.2.a.e.1.1 1
204.203 even 2 2448.2.a.o.1.1 1
221.220 even 2 2873.2.a.c.1.1 1
255.254 odd 2 3825.2.a.d.1.1 1
323.322 odd 2 6137.2.a.b.1.1 1
340.339 odd 2 6800.2.a.n.1.1 1
357.356 even 2 7497.2.a.l.1.1 1
391.390 odd 2 8993.2.a.a.1.1 1
408.101 odd 2 9792.2.a.n.1.1 1
408.203 even 2 9792.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.2.a.a.1.1 1 17.16 even 2
153.2.a.c.1.1 1 51.50 odd 2
272.2.a.b.1.1 1 68.67 odd 2
289.2.a.a.1.1 1 1.1 even 1 trivial
289.2.b.a.288.1 2 17.4 even 4
289.2.b.a.288.2 2 17.13 even 4
289.2.c.a.38.1 4 17.2 even 8
289.2.c.a.38.2 4 17.15 even 8
289.2.c.a.251.1 4 17.9 even 8
289.2.c.a.251.2 4 17.8 even 8
289.2.d.d.110.1 8 17.5 odd 16
289.2.d.d.110.2 8 17.12 odd 16
289.2.d.d.134.1 8 17.7 odd 16
289.2.d.d.134.2 8 17.10 odd 16
289.2.d.d.155.1 8 17.6 odd 16
289.2.d.d.155.2 8 17.11 odd 16
289.2.d.d.179.1 8 17.3 odd 16
289.2.d.d.179.2 8 17.14 odd 16
425.2.a.d.1.1 1 85.84 even 2
425.2.b.b.324.1 2 85.67 odd 4
425.2.b.b.324.2 2 85.33 odd 4
833.2.a.a.1.1 1 119.118 odd 2
833.2.e.a.18.1 2 119.33 odd 6
833.2.e.a.324.1 2 119.101 odd 6
833.2.e.b.18.1 2 119.16 even 6
833.2.e.b.324.1 2 119.67 even 6
1088.2.a.h.1.1 1 136.67 odd 2
1088.2.a.i.1.1 1 136.101 even 2
2057.2.a.e.1.1 1 187.186 odd 2
2448.2.a.o.1.1 1 204.203 even 2
2601.2.a.g.1.1 1 3.2 odd 2
2873.2.a.c.1.1 1 221.220 even 2
3825.2.a.d.1.1 1 255.254 odd 2
4624.2.a.d.1.1 1 4.3 odd 2
6137.2.a.b.1.1 1 323.322 odd 2
6800.2.a.n.1.1 1 340.339 odd 2
7225.2.a.g.1.1 1 5.4 even 2
7497.2.a.l.1.1 1 357.356 even 2
8993.2.a.a.1.1 1 391.390 odd 2
9792.2.a.i.1.1 1 408.203 even 2
9792.2.a.n.1.1 1 408.101 odd 2