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Properties

Label 8470.2.a.n.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
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] = [8470,2,Mod(1,8470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8470, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8470.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.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: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8470.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +3.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{18} +3.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} -7.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -3.00000 q^{26} -5.00000 q^{27} -1.00000 q^{28} +8.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{35} -2.00000 q^{36} -6.00000 q^{37} -3.00000 q^{38} +3.00000 q^{39} -1.00000 q^{40} -10.0000 q^{41} +1.00000 q^{42} +6.00000 q^{43} -2.00000 q^{45} +7.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +3.00000 q^{52} -12.0000 q^{53} +5.00000 q^{54} +1.00000 q^{56} +3.00000 q^{57} -8.00000 q^{58} -1.00000 q^{59} +1.00000 q^{60} -10.0000 q^{61} -4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +3.00000 q^{65} -8.00000 q^{67} -7.00000 q^{69} +1.00000 q^{70} +8.00000 q^{71} +2.00000 q^{72} +8.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +3.00000 q^{76} -3.00000 q^{78} -7.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +9.00000 q^{83} -1.00000 q^{84} -6.00000 q^{86} +8.00000 q^{87} +6.00000 q^{89} +2.00000 q^{90} -3.00000 q^{91} -7.00000 q^{92} +4.00000 q^{93} +12.0000 q^{94} +3.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 3.00000 0.832050 0.416025 − 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 2.00000 0.471405
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −7.00000 −1.45960 −0.729800 − 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −3.00000 −0.588348
\(27\) −5.00000 −0.962250
\(28\) −1.00000 −0.188982
\(29\) 8.00000 1.48556 0.742781 − 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) −1.00000 −0.182574
\(31\) 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
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) −2.00000 −0.333333
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −3.00000 −0.486664
\(39\) 3.00000 0.480384
\(40\) −1.00000 −0.158114
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 1.00000 0.154303
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 7.00000 1.03209
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 3.00000 0.416025
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 3.00000 0.397360
\(58\) −8.00000 −1.05045
\(59\) −1.00000 −0.130189 −0.0650945 − 0.997879i \(-0.520735\pi\)
−0.0650945 + 0.997879i \(0.520735\pi\)
\(60\) 1.00000 0.129099
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −4.00000 −0.508001
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −7.00000 −0.842701
\(70\) 1.00000 0.119523
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 2.00000 0.235702
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) 3.00000 0.344124
\(77\) 0 0
\(78\) −3.00000 −0.339683
\(79\) −7.00000 −0.787562 −0.393781 − 0.919204i \(-0.628833\pi\)
−0.393781 + 0.919204i \(0.628833\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 9.00000 0.987878 0.493939 − 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 8.00000 0.857690
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 2.00000 0.210819
\(91\) −3.00000 −0.314485
\(92\) −7.00000 −0.729800
\(93\) 4.00000 0.414781
\(94\) 12.0000 1.23771
\(95\) 3.00000 0.307794
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −3.00000 −0.298511 −0.149256 − 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) −2.00000 −0.197066 −0.0985329 − 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) −3.00000 −0.294174
\(105\) −1.00000 −0.0975900
\(106\) 12.0000 1.16554
\(107\) −18.0000 −1.74013 −0.870063 − 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) −5.00000 −0.481125
\(109\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) −1.00000 −0.0944911
\(113\) −1.00000 −0.0940721 −0.0470360 − 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) −3.00000 −0.280976
\(115\) −7.00000 −0.652753
\(116\) 8.00000 0.742781
\(117\) −6.00000 −0.554700
\(118\) 1.00000 0.0920575
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 0 0
\(122\) 10.0000 0.905357
\(123\) −10.0000 −0.901670
\(124\) 4.00000 0.359211
\(125\) 1.00000 0.0894427
\(126\) −2.00000 −0.178174
\(127\) 17.0000 1.50851 0.754253 − 0.656584i \(-0.227999\pi\)
0.754253 + 0.656584i \(0.227999\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.00000 0.528271
\(130\) −3.00000 −0.263117
\(131\) 7.00000 0.611593 0.305796 − 0.952097i \(-0.401077\pi\)
0.305796 + 0.952097i \(0.401077\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) 8.00000 0.691095
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) −21.0000 −1.79415 −0.897076 − 0.441877i \(-0.854313\pi\)
−0.897076 + 0.441877i \(0.854313\pi\)
\(138\) 7.00000 0.595880
\(139\) −1.00000 −0.0848189 −0.0424094 − 0.999100i \(-0.513503\pi\)
−0.0424094 + 0.999100i \(0.513503\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −12.0000 −1.01058
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) 8.00000 0.664364
\(146\) −8.00000 −0.662085
\(147\) 1.00000 0.0824786
\(148\) −6.00000 −0.493197
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −17.0000 −1.38344 −0.691720 − 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) −3.00000 −0.243332
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 3.00000 0.240192
\(157\) 15.0000 1.19713 0.598565 − 0.801074i \(-0.295738\pi\)
0.598565 + 0.801074i \(0.295738\pi\)
\(158\) 7.00000 0.556890
\(159\) −12.0000 −0.951662
\(160\) −1.00000 −0.0790569
\(161\) 7.00000 0.551677
\(162\) −1.00000 −0.0785674
\(163\) 22.0000 1.72317 0.861586 − 0.507611i \(-0.169471\pi\)
0.861586 + 0.507611i \(0.169471\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 10.0000 0.773823 0.386912 − 0.922117i \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) 1.00000 0.0771517
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 6.00000 0.457496
\(173\) −18.0000 −1.36851 −0.684257 − 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −8.00000 −0.606478
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −1.00000 −0.0751646
\(178\) −6.00000 −0.449719
\(179\) −18.0000 −1.34538 −0.672692 − 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) −2.00000 −0.149071
\(181\) −9.00000 −0.668965 −0.334482 − 0.942402i \(-0.608561\pi\)
−0.334482 + 0.942402i \(0.608561\pi\)
\(182\) 3.00000 0.222375
\(183\) −10.0000 −0.739221
\(184\) 7.00000 0.516047
\(185\) −6.00000 −0.441129
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) −12.0000 −0.875190
\(189\) 5.00000 0.363696
\(190\) −3.00000 −0.217643
\(191\) 25.0000 1.80894 0.904468 − 0.426541i \(-0.140268\pi\)
0.904468 + 0.426541i \(0.140268\pi\)
\(192\) 1.00000 0.0721688
\(193\) 5.00000 0.359908 0.179954 − 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) −8.00000 −0.574367
\(195\) 3.00000 0.214834
\(196\) 1.00000 0.0714286
\(197\) −4.00000 −0.284988 −0.142494 − 0.989796i \(-0.545512\pi\)
−0.142494 + 0.989796i \(0.545512\pi\)
\(198\) 0 0
\(199\) −12.0000 −0.850657 −0.425329 − 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −8.00000 −0.564276
\(202\) 3.00000 0.211079
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) 2.00000 0.139347
\(207\) 14.0000 0.973067
\(208\) 3.00000 0.208013
\(209\) 0 0
\(210\) 1.00000 0.0690066
\(211\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −12.0000 −0.824163
\(213\) 8.00000 0.548151
\(214\) 18.0000 1.23045
\(215\) 6.00000 0.409197
\(216\) 5.00000 0.340207
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) 6.00000 0.402694
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.00000 −0.133333
\(226\) 1.00000 0.0665190
\(227\) 4.00000 0.265489 0.132745 − 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 3.00000 0.198680
\(229\) 2.00000 0.132164 0.0660819 − 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 7.00000 0.461566
\(231\) 0 0
\(232\) −8.00000 −0.525226
\(233\) −3.00000 −0.196537 −0.0982683 − 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) 6.00000 0.392232
\(235\) −12.0000 −0.782794
\(236\) −1.00000 −0.0650945
\(237\) −7.00000 −0.454699
\(238\) 0 0
\(239\) −9.00000 −0.582162 −0.291081 − 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 1.00000 0.0645497
\(241\) −4.00000 −0.257663 −0.128831 − 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) −10.0000 −0.640184
\(245\) 1.00000 0.0638877
\(246\) 10.0000 0.637577
\(247\) 9.00000 0.572656
\(248\) −4.00000 −0.254000
\(249\) 9.00000 0.570352
\(250\) −1.00000 −0.0632456
\(251\) −20.0000 −1.26239 −0.631194 − 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) −17.0000 −1.06667
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −20.0000 −1.24757 −0.623783 − 0.781598i \(-0.714405\pi\)
−0.623783 + 0.781598i \(0.714405\pi\)
\(258\) −6.00000 −0.373544
\(259\) 6.00000 0.372822
\(260\) 3.00000 0.186052
\(261\) −16.0000 −0.990375
\(262\) −7.00000 −0.432461
\(263\) −9.00000 −0.554964 −0.277482 − 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 3.00000 0.183942
\(267\) 6.00000 0.367194
\(268\) −8.00000 −0.488678
\(269\) 7.00000 0.426798 0.213399 − 0.976965i \(-0.431547\pi\)
0.213399 + 0.976965i \(0.431547\pi\)
\(270\) 5.00000 0.304290
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) −3.00000 −0.181568
\(274\) 21.0000 1.26866
\(275\) 0 0
\(276\) −7.00000 −0.421350
\(277\) −26.0000 −1.56219 −0.781094 − 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 1.00000 0.0599760
\(279\) −8.00000 −0.478947
\(280\) 1.00000 0.0597614
\(281\) 13.0000 0.775515 0.387757 − 0.921761i \(-0.373250\pi\)
0.387757 + 0.921761i \(0.373250\pi\)
\(282\) 12.0000 0.714590
\(283\) −11.0000 −0.653882 −0.326941 − 0.945045i \(-0.606018\pi\)
−0.326941 + 0.945045i \(0.606018\pi\)
\(284\) 8.00000 0.474713
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) 10.0000 0.590281
\(288\) 2.00000 0.117851
\(289\) −17.0000 −1.00000
\(290\) −8.00000 −0.469776
\(291\) 8.00000 0.468968
\(292\) 8.00000 0.468165
\(293\) −31.0000 −1.81104 −0.905520 − 0.424304i \(-0.860519\pi\)
−0.905520 + 0.424304i \(0.860519\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −1.00000 −0.0582223
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) −21.0000 −1.21446
\(300\) 1.00000 0.0577350
\(301\) −6.00000 −0.345834
\(302\) 17.0000 0.978240
\(303\) −3.00000 −0.172345
\(304\) 3.00000 0.172062
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −2.00000 −0.113776
\(310\) −4.00000 −0.227185
\(311\) −18.0000 −1.02069 −0.510343 − 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −3.00000 −0.169842
\(313\) −34.0000 −1.92179 −0.960897 − 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) −15.0000 −0.846499
\(315\) 2.00000 0.112687
\(316\) −7.00000 −0.393781
\(317\) 4.00000 0.224662 0.112331 − 0.993671i \(-0.464168\pi\)
0.112331 + 0.993671i \(0.464168\pi\)
\(318\) 12.0000 0.672927
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −18.0000 −1.00466
\(322\) −7.00000 −0.390095
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 3.00000 0.166410
\(326\) −22.0000 −1.21847
\(327\) 0 0
\(328\) 10.0000 0.552158
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 − 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 9.00000 0.493939
\(333\) 12.0000 0.657596
\(334\) −10.0000 −0.547176
\(335\) −8.00000 −0.437087
\(336\) −1.00000 −0.0545545
\(337\) 17.0000 0.926049 0.463025 − 0.886345i \(-0.346764\pi\)
0.463025 + 0.886345i \(0.346764\pi\)
\(338\) 4.00000 0.217571
\(339\) −1.00000 −0.0543125
\(340\) 0 0
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) −1.00000 −0.0539949
\(344\) −6.00000 −0.323498
\(345\) −7.00000 −0.376867
\(346\) 18.0000 0.967686
\(347\) 22.0000 1.18102 0.590511 − 0.807030i \(-0.298926\pi\)
0.590511 + 0.807030i \(0.298926\pi\)
\(348\) 8.00000 0.428845
\(349\) −29.0000 −1.55233 −0.776167 − 0.630527i \(-0.782839\pi\)
−0.776167 + 0.630527i \(0.782839\pi\)
\(350\) 1.00000 0.0534522
\(351\) −15.0000 −0.800641
\(352\) 0 0
\(353\) −36.0000 −1.91609 −0.958043 − 0.286623i \(-0.907467\pi\)
−0.958043 + 0.286623i \(0.907467\pi\)
\(354\) 1.00000 0.0531494
\(355\) 8.00000 0.424596
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 18.0000 0.951330
\(359\) −8.00000 −0.422224 −0.211112 − 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 2.00000 0.105409
\(361\) −10.0000 −0.526316
\(362\) 9.00000 0.473029
\(363\) 0 0
\(364\) −3.00000 −0.157243
\(365\) 8.00000 0.418739
\(366\) 10.0000 0.522708
\(367\) 2.00000 0.104399 0.0521996 − 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) −7.00000 −0.364900
\(369\) 20.0000 1.04116
\(370\) 6.00000 0.311925
\(371\) 12.0000 0.623009
\(372\) 4.00000 0.207390
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 12.0000 0.618853
\(377\) 24.0000 1.23606
\(378\) −5.00000 −0.257172
\(379\) 2.00000 0.102733 0.0513665 − 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 3.00000 0.153897
\(381\) 17.0000 0.870936
\(382\) −25.0000 −1.27911
\(383\) −38.0000 −1.94171 −0.970855 − 0.239669i \(-0.922961\pi\)
−0.970855 + 0.239669i \(0.922961\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −5.00000 −0.254493
\(387\) −12.0000 −0.609994
\(388\) 8.00000 0.406138
\(389\) 4.00000 0.202808 0.101404 − 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) −3.00000 −0.151911
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 7.00000 0.353103
\(394\) 4.00000 0.201517
\(395\) −7.00000 −0.352208
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 − 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 12.0000 0.601506
\(399\) −3.00000 −0.150188
\(400\) 1.00000 0.0500000
\(401\) 6.00000 0.299626 0.149813 − 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 8.00000 0.399004
\(403\) 12.0000 0.597763
\(404\) −3.00000 −0.149256
\(405\) 1.00000 0.0496904
\(406\) 8.00000 0.397033
\(407\) 0 0
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 − 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 10.0000 0.493865
\(411\) −21.0000 −1.03585
\(412\) −2.00000 −0.0985329
\(413\) 1.00000 0.0492068
\(414\) −14.0000 −0.688062
\(415\) 9.00000 0.441793
\(416\) −3.00000 −0.147087
\(417\) −1.00000 −0.0489702
\(418\) 0 0
\(419\) 7.00000 0.341972 0.170986 − 0.985273i \(-0.445305\pi\)
0.170986 + 0.985273i \(0.445305\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) 24.0000 1.16692
\(424\) 12.0000 0.582772
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 10.0000 0.483934
\(428\) −18.0000 −0.870063
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) −1.00000 −0.0481683 −0.0240842 − 0.999710i \(-0.507667\pi\)
−0.0240842 + 0.999710i \(0.507667\pi\)
\(432\) −5.00000 −0.240563
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 4.00000 0.192006
\(435\) 8.00000 0.383571
\(436\) 0 0
\(437\) −21.0000 −1.00457
\(438\) −8.00000 −0.382255
\(439\) 22.0000 1.05000 0.525001 − 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 28.0000 1.33032 0.665160 − 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) −6.00000 −0.284747
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) −1.00000 −0.0472456
\(449\) 41.0000 1.93491 0.967455 − 0.253044i \(-0.0814317\pi\)
0.967455 + 0.253044i \(0.0814317\pi\)
\(450\) 2.00000 0.0942809
\(451\) 0 0
\(452\) −1.00000 −0.0470360
\(453\) −17.0000 −0.798730
\(454\) −4.00000 −0.187729
\(455\) −3.00000 −0.140642
\(456\) −3.00000 −0.140488
\(457\) −31.0000 −1.45012 −0.725059 − 0.688686i \(-0.758188\pi\)
−0.725059 + 0.688686i \(0.758188\pi\)
\(458\) −2.00000 −0.0934539
\(459\) 0 0
\(460\) −7.00000 −0.326377
\(461\) 34.0000 1.58354 0.791769 − 0.610821i \(-0.209160\pi\)
0.791769 + 0.610821i \(0.209160\pi\)
\(462\) 0 0
\(463\) −23.0000 −1.06890 −0.534450 − 0.845200i \(-0.679481\pi\)
−0.534450 + 0.845200i \(0.679481\pi\)
\(464\) 8.00000 0.371391
\(465\) 4.00000 0.185496
\(466\) 3.00000 0.138972
\(467\) 33.0000 1.52706 0.763529 − 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) −6.00000 −0.277350
\(469\) 8.00000 0.369406
\(470\) 12.0000 0.553519
\(471\) 15.0000 0.691164
\(472\) 1.00000 0.0460287
\(473\) 0 0
\(474\) 7.00000 0.321521
\(475\) 3.00000 0.137649
\(476\) 0 0
\(477\) 24.0000 1.09888
\(478\) 9.00000 0.411650
\(479\) 26.0000 1.18797 0.593985 − 0.804476i \(-0.297554\pi\)
0.593985 + 0.804476i \(0.297554\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −18.0000 −0.820729
\(482\) 4.00000 0.182195
\(483\) 7.00000 0.318511
\(484\) 0 0
\(485\) 8.00000 0.363261
\(486\) −16.0000 −0.725775
\(487\) −37.0000 −1.67663 −0.838315 − 0.545186i \(-0.816459\pi\)
−0.838315 + 0.545186i \(0.816459\pi\)
\(488\) 10.0000 0.452679
\(489\) 22.0000 0.994874
\(490\) −1.00000 −0.0451754
\(491\) −24.0000 −1.08310 −0.541552 − 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) −10.0000 −0.450835
\(493\) 0 0
\(494\) −9.00000 −0.404929
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −8.00000 −0.358849
\(498\) −9.00000 −0.403300
\(499\) 14.0000 0.626726 0.313363 − 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 1.00000 0.0447214
\(501\) 10.0000 0.446767
\(502\) 20.0000 0.892644
\(503\) −30.0000 −1.33763 −0.668817 − 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −3.00000 −0.133498
\(506\) 0 0
\(507\) −4.00000 −0.177646
\(508\) 17.0000 0.754253
\(509\) 15.0000 0.664863 0.332432 − 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −1.00000 −0.0441942
\(513\) −15.0000 −0.662266
\(514\) 20.0000 0.882162
\(515\) −2.00000 −0.0881305
\(516\) 6.00000 0.264135
\(517\) 0 0
\(518\) −6.00000 −0.263625
\(519\) −18.0000 −0.790112
\(520\) −3.00000 −0.131559
\(521\) −40.0000 −1.75243 −0.876216 − 0.481919i \(-0.839940\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(522\) 16.0000 0.700301
\(523\) −11.0000 −0.480996 −0.240498 − 0.970650i \(-0.577311\pi\)
−0.240498 + 0.970650i \(0.577311\pi\)
\(524\) 7.00000 0.305796
\(525\) −1.00000 −0.0436436
\(526\) 9.00000 0.392419
\(527\) 0 0
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 12.0000 0.521247
\(531\) 2.00000 0.0867926
\(532\) −3.00000 −0.130066
\(533\) −30.0000 −1.29944
\(534\) −6.00000 −0.259645
\(535\) −18.0000 −0.778208
\(536\) 8.00000 0.345547
\(537\) −18.0000 −0.776757
\(538\) −7.00000 −0.301791
\(539\) 0 0
\(540\) −5.00000 −0.215166
\(541\) 14.0000 0.601907 0.300954 − 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 16.0000 0.687259
\(543\) −9.00000 −0.386227
\(544\) 0 0
\(545\) 0 0
\(546\) 3.00000 0.128388
\(547\) −14.0000 −0.598597 −0.299298 − 0.954160i \(-0.596753\pi\)
−0.299298 + 0.954160i \(0.596753\pi\)
\(548\) −21.0000 −0.897076
\(549\) 20.0000 0.853579
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 7.00000 0.297940
\(553\) 7.00000 0.297670
\(554\) 26.0000 1.10463
\(555\) −6.00000 −0.254686
\(556\) −1.00000 −0.0424094
\(557\) 12.0000 0.508456 0.254228 − 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 8.00000 0.338667
\(559\) 18.0000 0.761319
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −13.0000 −0.548372
\(563\) −19.0000 −0.800755 −0.400377 − 0.916350i \(-0.631121\pi\)
−0.400377 + 0.916350i \(0.631121\pi\)
\(564\) −12.0000 −0.505291
\(565\) −1.00000 −0.0420703
\(566\) 11.0000 0.462364
\(567\) −1.00000 −0.0419961
\(568\) −8.00000 −0.335673
\(569\) 39.0000 1.63497 0.817483 − 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(570\) −3.00000 −0.125656
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 25.0000 1.04439
\(574\) −10.0000 −0.417392
\(575\) −7.00000 −0.291920
\(576\) −2.00000 −0.0833333
\(577\) −14.0000 −0.582828 −0.291414 − 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 17.0000 0.707107
\(579\) 5.00000 0.207793
\(580\) 8.00000 0.332182
\(581\) −9.00000 −0.373383
\(582\) −8.00000 −0.331611
\(583\) 0 0
\(584\) −8.00000 −0.331042
\(585\) −6.00000 −0.248069
\(586\) 31.0000 1.28060
\(587\) −1.00000 −0.0412744 −0.0206372 − 0.999787i \(-0.506569\pi\)
−0.0206372 + 0.999787i \(0.506569\pi\)
\(588\) 1.00000 0.0412393
\(589\) 12.0000 0.494451
\(590\) 1.00000 0.0411693
\(591\) −4.00000 −0.164538
\(592\) −6.00000 −0.246598
\(593\) −12.0000 −0.492781 −0.246390 − 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) −12.0000 −0.491127
\(598\) 21.0000 0.858754
\(599\) 17.0000 0.694601 0.347301 − 0.937754i \(-0.387098\pi\)
0.347301 + 0.937754i \(0.387098\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 16.0000 0.652654 0.326327 − 0.945257i \(-0.394189\pi\)
0.326327 + 0.945257i \(0.394189\pi\)
\(602\) 6.00000 0.244542
\(603\) 16.0000 0.651570
\(604\) −17.0000 −0.691720
\(605\) 0 0
\(606\) 3.00000 0.121867
\(607\) 18.0000 0.730597 0.365299 − 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) −3.00000 −0.121666
\(609\) −8.00000 −0.324176
\(610\) 10.0000 0.404888
\(611\) −36.0000 −1.45640
\(612\) 0 0
\(613\) 18.0000 0.727013 0.363507 − 0.931592i \(-0.381579\pi\)
0.363507 + 0.931592i \(0.381579\pi\)
\(614\) −4.00000 −0.161427
\(615\) −10.0000 −0.403239
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 2.00000 0.0804518
\(619\) 29.0000 1.16561 0.582804 − 0.812613i \(-0.301955\pi\)
0.582804 + 0.812613i \(0.301955\pi\)
\(620\) 4.00000 0.160644
\(621\) 35.0000 1.40450
\(622\) 18.0000 0.721734
\(623\) −6.00000 −0.240385
\(624\) 3.00000 0.120096
\(625\) 1.00000 0.0400000
\(626\) 34.0000 1.35891
\(627\) 0 0
\(628\) 15.0000 0.598565
\(629\) 0 0
\(630\) −2.00000 −0.0796819
\(631\) −8.00000 −0.318475 −0.159237 − 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 7.00000 0.278445
\(633\) 0 0
\(634\) −4.00000 −0.158860
\(635\) 17.0000 0.674624
\(636\) −12.0000 −0.475831
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) −1.00000 −0.0395285
\(641\) −17.0000 −0.671460 −0.335730 − 0.941958i \(-0.608983\pi\)
−0.335730 + 0.941958i \(0.608983\pi\)
\(642\) 18.0000 0.710403
\(643\) −32.0000 −1.26196 −0.630978 − 0.775800i \(-0.717346\pi\)
−0.630978 + 0.775800i \(0.717346\pi\)
\(644\) 7.00000 0.275839
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 − 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −3.00000 −0.117670
\(651\) −4.00000 −0.156772
\(652\) 22.0000 0.861586
\(653\) −26.0000 −1.01746 −0.508729 − 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 0 0
\(655\) 7.00000 0.273513
\(656\) −10.0000 −0.390434
\(657\) −16.0000 −0.624219
\(658\) −12.0000 −0.467809
\(659\) 40.0000 1.55818 0.779089 − 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 − 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) −10.0000 −0.388661
\(663\) 0 0
\(664\) −9.00000 −0.349268
\(665\) −3.00000 −0.116335
\(666\) −12.0000 −0.464991
\(667\) −56.0000 −2.16833
\(668\) 10.0000 0.386912
\(669\) 0 0
\(670\) 8.00000 0.309067
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) −1.00000 −0.0385472 −0.0192736 − 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −17.0000 −0.654816
\(675\) −5.00000 −0.192450
\(676\) −4.00000 −0.153846
\(677\) 27.0000 1.03769 0.518847 − 0.854867i \(-0.326361\pi\)
0.518847 + 0.854867i \(0.326361\pi\)
\(678\) 1.00000 0.0384048
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 − 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) −6.00000 −0.229416
\(685\) −21.0000 −0.802369
\(686\) 1.00000 0.0381802
\(687\) 2.00000 0.0763048
\(688\) 6.00000 0.228748
\(689\) −36.0000 −1.37149
\(690\) 7.00000 0.266485
\(691\) −12.0000 −0.456502 −0.228251 − 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) −22.0000 −0.835109
\(695\) −1.00000 −0.0379322
\(696\) −8.00000 −0.303239
\(697\) 0 0
\(698\) 29.0000 1.09767
\(699\) −3.00000 −0.113470
\(700\) −1.00000 −0.0377964
\(701\) −20.0000 −0.755390 −0.377695 − 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) 15.0000 0.566139
\(703\) −18.0000 −0.678883
\(704\) 0 0
\(705\) −12.0000 −0.451946
\(706\) 36.0000 1.35488
\(707\) 3.00000 0.112827
\(708\) −1.00000 −0.0375823
\(709\) 6.00000 0.225335 0.112667 − 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −8.00000 −0.300235
\(711\) 14.0000 0.525041
\(712\) −6.00000 −0.224860
\(713\) −28.0000 −1.04861
\(714\) 0 0
\(715\) 0 0
\(716\) −18.0000 −0.672692
\(717\) −9.00000 −0.336111
\(718\) 8.00000 0.298557
\(719\) −28.0000 −1.04422 −0.522112 − 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 2.00000 0.0744839
\(722\) 10.0000 0.372161
\(723\) −4.00000 −0.148762
\(724\) −9.00000 −0.334482
\(725\) 8.00000 0.297113
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 − 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 3.00000 0.111187
\(729\) 13.0000 0.481481
\(730\) −8.00000 −0.296093
\(731\) 0 0
\(732\) −10.0000 −0.369611
\(733\) −11.0000 −0.406294 −0.203147 − 0.979148i \(-0.565117\pi\)
−0.203147 + 0.979148i \(0.565117\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 1.00000 0.0368856
\(736\) 7.00000 0.258023
\(737\) 0 0
\(738\) −20.0000 −0.736210
\(739\) 24.0000 0.882854 0.441427 − 0.897297i \(-0.354472\pi\)
0.441427 + 0.897297i \(0.354472\pi\)
\(740\) −6.00000 −0.220564
\(741\) 9.00000 0.330623
\(742\) −12.0000 −0.440534
\(743\) 28.0000 1.02722 0.513610 − 0.858024i \(-0.328308\pi\)
0.513610 + 0.858024i \(0.328308\pi\)
\(744\) −4.00000 −0.146647
\(745\) 6.00000 0.219823
\(746\) −14.0000 −0.512576
\(747\) −18.0000 −0.658586
\(748\) 0 0
\(749\) 18.0000 0.657706
\(750\) −1.00000 −0.0365148
\(751\) −47.0000 −1.71505 −0.857527 − 0.514439i \(-0.828000\pi\)
−0.857527 + 0.514439i \(0.828000\pi\)
\(752\) −12.0000 −0.437595
\(753\) −20.0000 −0.728841
\(754\) −24.0000 −0.874028
\(755\) −17.0000 −0.618693
\(756\) 5.00000 0.181848
\(757\) 34.0000 1.23575 0.617876 − 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) −2.00000 −0.0726433
\(759\) 0 0
\(760\) −3.00000 −0.108821
\(761\) −36.0000 −1.30500 −0.652499 − 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) −17.0000 −0.615845
\(763\) 0 0
\(764\) 25.0000 0.904468
\(765\) 0 0
\(766\) 38.0000 1.37300
\(767\) −3.00000 −0.108324
\(768\) 1.00000 0.0360844
\(769\) 54.0000 1.94729 0.973645 − 0.228069i \(-0.0732413\pi\)
0.973645 + 0.228069i \(0.0732413\pi\)
\(770\) 0 0
\(771\) −20.0000 −0.720282
\(772\) 5.00000 0.179954
\(773\) 21.0000 0.755318 0.377659 − 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) 12.0000 0.431331
\(775\) 4.00000 0.143684
\(776\) −8.00000 −0.287183
\(777\) 6.00000 0.215249
\(778\) −4.00000 −0.143407
\(779\) −30.0000 −1.07486
\(780\) 3.00000 0.107417
\(781\) 0 0
\(782\) 0 0
\(783\) −40.0000 −1.42948
\(784\) 1.00000 0.0357143
\(785\) 15.0000 0.535373
\(786\) −7.00000 −0.249682
\(787\) −12.0000 −0.427754 −0.213877 − 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) −4.00000 −0.142494
\(789\) −9.00000 −0.320408
\(790\) 7.00000 0.249049
\(791\) 1.00000 0.0355559
\(792\) 0 0
\(793\) −30.0000 −1.06533
\(794\) 22.0000 0.780751
\(795\) −12.0000 −0.425596
\(796\) −12.0000 −0.425329
\(797\) 3.00000 0.106265 0.0531327 − 0.998587i \(-0.483079\pi\)
0.0531327 + 0.998587i \(0.483079\pi\)
\(798\) 3.00000 0.106199
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −12.0000 −0.423999
\(802\) −6.00000 −0.211867
\(803\) 0 0
\(804\) −8.00000 −0.282138
\(805\) 7.00000 0.246718
\(806\) −12.0000 −0.422682
\(807\) 7.00000 0.246412
\(808\) 3.00000 0.105540
\(809\) −42.0000 −1.47664 −0.738321 − 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −20.0000 −0.702295 −0.351147 − 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) −8.00000 −0.280745
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 22.0000 0.770626
\(816\) 0 0
\(817\) 18.0000 0.629740
\(818\) 20.0000 0.699284
\(819\) 6.00000 0.209657
\(820\) −10.0000 −0.349215
\(821\) −10.0000 −0.349002 −0.174501 − 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 21.0000 0.732459
\(823\) −40.0000 −1.39431 −0.697156 − 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 2.00000 0.0696733
\(825\) 0 0
\(826\) −1.00000 −0.0347945
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 14.0000 0.486534
\(829\) −19.0000 −0.659897 −0.329949 − 0.943999i \(-0.607031\pi\)
−0.329949 + 0.943999i \(0.607031\pi\)
\(830\) −9.00000 −0.312395
\(831\) −26.0000 −0.901930
\(832\) 3.00000 0.104006
\(833\) 0 0
\(834\) 1.00000 0.0346272
\(835\) 10.0000 0.346064
\(836\) 0 0
\(837\) −20.0000 −0.691301
\(838\) −7.00000 −0.241811
\(839\) 14.0000 0.483334 0.241667 − 0.970359i \(-0.422306\pi\)
0.241667 + 0.970359i \(0.422306\pi\)
\(840\) 1.00000 0.0345033
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) 13.0000 0.447744
\(844\) 0 0
\(845\) −4.00000 −0.137604
\(846\) −24.0000 −0.825137
\(847\) 0 0
\(848\) −12.0000 −0.412082
\(849\) −11.0000 −0.377519
\(850\) 0 0
\(851\) 42.0000 1.43974
\(852\) 8.00000 0.274075
\(853\) 19.0000 0.650548 0.325274 − 0.945620i \(-0.394544\pi\)
0.325274 + 0.945620i \(0.394544\pi\)
\(854\) −10.0000 −0.342193
\(855\) −6.00000 −0.205196
\(856\) 18.0000 0.615227
\(857\) −24.0000 −0.819824 −0.409912 − 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 6.00000 0.204598
\(861\) 10.0000 0.340799
\(862\) 1.00000 0.0340601
\(863\) −32.0000 −1.08929 −0.544646 − 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 5.00000 0.170103
\(865\) −18.0000 −0.612018
\(866\) −14.0000 −0.475739
\(867\) −17.0000 −0.577350
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) −8.00000 −0.271225
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) −16.0000 −0.541518
\(874\) 21.0000 0.710336
\(875\) −1.00000 −0.0338062
\(876\) 8.00000 0.270295
\(877\) −2.00000 −0.0675352 −0.0337676 − 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) −22.0000 −0.742464
\(879\) −31.0000 −1.04560
\(880\) 0 0
\(881\) −26.0000 −0.875962 −0.437981 − 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 2.00000 0.0673435
\(883\) 46.0000 1.54802 0.774012 − 0.633171i \(-0.218247\pi\)
0.774012 + 0.633171i \(0.218247\pi\)
\(884\) 0 0
\(885\) −1.00000 −0.0336146
\(886\) −28.0000 −0.940678
\(887\) 14.0000 0.470074 0.235037 − 0.971986i \(-0.424479\pi\)
0.235037 + 0.971986i \(0.424479\pi\)
\(888\) 6.00000 0.201347
\(889\) −17.0000 −0.570162
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) 0 0
\(893\) −36.0000 −1.20469
\(894\) −6.00000 −0.200670
\(895\) −18.0000 −0.601674
\(896\) 1.00000 0.0334077
\(897\) −21.0000 −0.701170
\(898\) −41.0000 −1.36819
\(899\) 32.0000 1.06726
\(900\) −2.00000 −0.0666667
\(901\) 0 0
\(902\) 0 0
\(903\) −6.00000 −0.199667
\(904\) 1.00000 0.0332595
\(905\) −9.00000 −0.299170
\(906\) 17.0000 0.564787
\(907\) −24.0000 −0.796907 −0.398453 − 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) 4.00000 0.132745
\(909\) 6.00000 0.199007
\(910\) 3.00000 0.0994490
\(911\) 19.0000 0.629498 0.314749 − 0.949175i \(-0.398080\pi\)
0.314749 + 0.949175i \(0.398080\pi\)
\(912\) 3.00000 0.0993399
\(913\) 0 0
\(914\) 31.0000 1.02539
\(915\) −10.0000 −0.330590
\(916\) 2.00000 0.0660819
\(917\) −7.00000 −0.231160
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 − 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 7.00000 0.230783
\(921\) 4.00000 0.131804
\(922\) −34.0000 −1.11973
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 23.0000 0.755827
\(927\) 4.00000 0.131377
\(928\) −8.00000 −0.262613
\(929\) −36.0000 −1.18112 −0.590561 − 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) −4.00000 −0.131165
\(931\) 3.00000 0.0983210
\(932\) −3.00000 −0.0982683
\(933\) −18.0000 −0.589294
\(934\) −33.0000 −1.07979
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) −12.0000 −0.392023 −0.196011 − 0.980602i \(-0.562799\pi\)
−0.196011 + 0.980602i \(0.562799\pi\)
\(938\) −8.00000 −0.261209
\(939\) −34.0000 −1.10955
\(940\) −12.0000 −0.391397
\(941\) −42.0000 −1.36916 −0.684580 − 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) −15.0000 −0.488726
\(943\) 70.0000 2.27951
\(944\) −1.00000 −0.0325472
\(945\) 5.00000 0.162650
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −7.00000 −0.227349
\(949\) 24.0000 0.779073
\(950\) −3.00000 −0.0973329
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) −39.0000 −1.26333 −0.631667 − 0.775240i \(-0.717629\pi\)
−0.631667 + 0.775240i \(0.717629\pi\)
\(954\) −24.0000 −0.777029
\(955\) 25.0000 0.808981
\(956\) −9.00000 −0.291081
\(957\) 0 0
\(958\) −26.0000 −0.840022
\(959\) 21.0000 0.678125
\(960\) 1.00000 0.0322749
\(961\) −15.0000 −0.483871
\(962\) 18.0000 0.580343
\(963\) 36.0000 1.16008
\(964\) −4.00000 −0.128831
\(965\) 5.00000 0.160956
\(966\) −7.00000 −0.225221
\(967\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −8.00000 −0.256865
\(971\) −43.0000 −1.37994 −0.689968 − 0.723840i \(-0.742375\pi\)
−0.689968 + 0.723840i \(0.742375\pi\)
\(972\) 16.0000 0.513200
\(973\) 1.00000 0.0320585
\(974\) 37.0000 1.18556
\(975\) 3.00000 0.0960769
\(976\) −10.0000 −0.320092
\(977\) −9.00000 −0.287936 −0.143968 − 0.989582i \(-0.545986\pi\)
−0.143968 + 0.989582i \(0.545986\pi\)
\(978\) −22.0000 −0.703482
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 24.0000 0.765871
\(983\) −58.0000 −1.84991 −0.924956 − 0.380073i \(-0.875899\pi\)
−0.924956 + 0.380073i \(0.875899\pi\)
\(984\) 10.0000 0.318788
\(985\) −4.00000 −0.127451
\(986\) 0 0
\(987\) 12.0000 0.381964
\(988\) 9.00000 0.286328
\(989\) −42.0000 −1.33552
\(990\) 0 0
\(991\) 41.0000 1.30241 0.651204 − 0.758903i \(-0.274264\pi\)
0.651204 + 0.758903i \(0.274264\pi\)
\(992\) −4.00000 −0.127000
\(993\) 10.0000 0.317340
\(994\) 8.00000 0.253745
\(995\) −12.0000 −0.380426
\(996\) 9.00000 0.285176
\(997\) 45.0000 1.42516 0.712582 − 0.701589i \(-0.247526\pi\)
0.712582 + 0.701589i \(0.247526\pi\)
\(998\) −14.0000 −0.443162
\(999\) 30.0000 0.949158
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 8470.2.a.n.1.1 ✓ 1
11.10 odd 2 8470.2.a.bf.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.n.1.1 ✓ 1 1.1 even 1 trivial
8470.2.a.bf.1.1 yes 1 11.10 odd 2