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Properties

Label 2850.2.a.r.1.1
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
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] = [2850,2,Mod(1,2850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2850, 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("2850.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.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: \(22.7573645761\)
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 \(\chi\) \(=\) 2850.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +2.00000 q^{21} +3.00000 q^{22} -3.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +3.00000 q^{29} -7.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +1.00000 q^{38} +2.00000 q^{39} -6.00000 q^{41} +2.00000 q^{42} +10.0000 q^{43} +3.00000 q^{44} -3.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +6.00000 q^{51} -2.00000 q^{52} +9.00000 q^{53} -1.00000 q^{54} -2.00000 q^{56} -1.00000 q^{57} +3.00000 q^{58} +6.00000 q^{59} -1.00000 q^{61} -7.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} -5.00000 q^{67} -6.00000 q^{68} +3.00000 q^{69} +1.00000 q^{72} +1.00000 q^{73} -2.00000 q^{74} +1.00000 q^{76} -6.00000 q^{77} +2.00000 q^{78} -13.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -15.0000 q^{83} +2.00000 q^{84} +10.0000 q^{86} -3.00000 q^{87} +3.00000 q^{88} -15.0000 q^{89} +4.00000 q^{91} -3.00000 q^{92} +7.00000 q^{93} -12.0000 q^{94} -1.00000 q^{96} -8.00000 q^{97} -3.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 3.00000 0.639602
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 − 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.00000 −0.522233
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 1.00000 0.162221
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 2.00000 0.308607
\(43\) 10.0000 1.52499 0.762493 − 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) −2.00000 −0.277350
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) −1.00000 −0.132453
\(58\) 3.00000 0.393919
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) −7.00000 −0.889001
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.00000 −0.369274
\(67\) −5.00000 −0.610847 −0.305424 − 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) −6.00000 −0.727607
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −6.00000 −0.683763
\(78\) 2.00000 0.226455
\(79\) −13.0000 −1.46261 −0.731307 − 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −15.0000 −1.64646 −0.823232 − 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) −3.00000 −0.321634
\(88\) 3.00000 0.319801
\(89\) −15.0000 −1.59000 −0.794998 − 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −3.00000 −0.312772
\(93\) 7.00000 0.725866
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −3.00000 −0.303046
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 6.00000 0.594089
\(103\) 7.00000 0.689730 0.344865 − 0.938652i \(-0.387925\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −2.00000 −0.188982
\(113\) −9.00000 −0.846649 −0.423324 − 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) −2.00000 −0.184900
\(118\) 6.00000 0.552345
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −1.00000 −0.0905357
\(123\) 6.00000 0.541002
\(124\) −7.00000 −0.628619
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) −11.0000 −0.976092 −0.488046 − 0.872818i \(-0.662290\pi\)
−0.488046 + 0.872818i \(0.662290\pi\)
\(128\) 1.00000 0.0883883
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 − 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) −3.00000 −0.261116
\(133\) −2.00000 −0.173422
\(134\) −5.00000 −0.431934
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 3.00000 0.255377
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 1.00000 0.0827606
\(147\) 3.00000 0.247436
\(148\) −2.00000 −0.164399
\(149\) 12.0000 0.983078 0.491539 − 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 1.00000 0.0811107
\(153\) −6.00000 −0.485071
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −13.0000 −1.03422
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 1.00000 0.0785674
\(163\) 22.0000 1.72317 0.861586 − 0.507611i \(-0.169471\pi\)
0.861586 + 0.507611i \(0.169471\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −15.0000 −1.16423
\(167\) 18.0000 1.39288 0.696441 − 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 2.00000 0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 10.0000 0.762493
\(173\) 3.00000 0.228086 0.114043 − 0.993476i \(-0.463620\pi\)
0.114043 + 0.993476i \(0.463620\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) −6.00000 −0.450988
\(178\) −15.0000 −1.12430
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 4.00000 0.296500
\(183\) 1.00000 0.0739221
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) 7.00000 0.513265
\(187\) −18.0000 −1.31629
\(188\) −12.0000 −0.875190
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 − 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −20.0000 −1.43963 −0.719816 − 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 3.00000 0.213201
\(199\) 2.00000 0.141776 0.0708881 − 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 5.00000 0.352673
\(202\) 6.00000 0.422159
\(203\) −6.00000 −0.421117
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 7.00000 0.487713
\(207\) −3.00000 −0.208514
\(208\) −2.00000 −0.138675
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 − 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 9.00000 0.618123
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 14.0000 0.950382
\(218\) 2.00000 0.135457
\(219\) −1.00000 −0.0675737
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 2.00000 0.134231
\(223\) −5.00000 −0.334825 −0.167412 − 0.985887i \(-0.553541\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −9.00000 −0.598671
\(227\) 18.0000 1.19470 0.597351 − 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −13.0000 −0.859064 −0.429532 − 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 3.00000 0.196960
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 13.0000 0.844441
\(238\) 12.0000 0.777844
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −2.00000 −0.128565
\(243\) −1.00000 −0.0641500
\(244\) −1.00000 −0.0640184
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) −2.00000 −0.127257
\(248\) −7.00000 −0.444500
\(249\) 15.0000 0.950586
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 − 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) −2.00000 −0.125988
\(253\) −9.00000 −0.565825
\(254\) −11.0000 −0.690201
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 9.00000 0.561405 0.280702 − 0.959795i \(-0.409433\pi\)
0.280702 + 0.959795i \(0.409433\pi\)
\(258\) −10.0000 −0.622573
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) −15.0000 −0.926703
\(263\) 21.0000 1.29492 0.647458 − 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 15.0000 0.917985
\(268\) −5.00000 −0.305424
\(269\) 30.0000 1.82913 0.914566 − 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −6.00000 −0.363803
\(273\) −4.00000 −0.242091
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 3.00000 0.180579
\(277\) −11.0000 −0.660926 −0.330463 − 0.943819i \(-0.607205\pi\)
−0.330463 + 0.943819i \(0.607205\pi\)
\(278\) −4.00000 −0.239904
\(279\) −7.00000 −0.419079
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 − 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 12.0000 0.714590
\(283\) −8.00000 −0.475551 −0.237775 − 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 12.0000 0.708338
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 1.00000 0.0585206
\(293\) 15.0000 0.876309 0.438155 − 0.898900i \(-0.355632\pi\)
0.438155 + 0.898900i \(0.355632\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) −3.00000 −0.174078
\(298\) 12.0000 0.695141
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) 20.0000 1.15087
\(303\) −6.00000 −0.344691
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 19.0000 1.08439 0.542194 − 0.840254i \(-0.317594\pi\)
0.542194 + 0.840254i \(0.317594\pi\)
\(308\) −6.00000 −0.341882
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 − 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 2.00000 0.113228
\(313\) 13.0000 0.734803 0.367402 − 0.930062i \(-0.380247\pi\)
0.367402 + 0.930062i \(0.380247\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −13.0000 −0.731307
\(317\) 3.00000 0.168497 0.0842484 − 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) −9.00000 −0.504695
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) 0 0
\(322\) 6.00000 0.334367
\(323\) −6.00000 −0.333849
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 22.0000 1.21847
\(327\) −2.00000 −0.110600
\(328\) −6.00000 −0.331295
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 5.00000 0.274825 0.137412 − 0.990514i \(-0.456121\pi\)
0.137412 + 0.990514i \(0.456121\pi\)
\(332\) −15.0000 −0.823232
\(333\) −2.00000 −0.109599
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) −32.0000 −1.74315 −0.871576 − 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) −9.00000 −0.489535
\(339\) 9.00000 0.488813
\(340\) 0 0
\(341\) −21.0000 −1.13721
\(342\) 1.00000 0.0540738
\(343\) 20.0000 1.07990
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) 3.00000 0.161281
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −3.00000 −0.160817
\(349\) −19.0000 −1.01705 −0.508523 − 0.861048i \(-0.669808\pi\)
−0.508523 + 0.861048i \(0.669808\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 3.00000 0.159901
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) −15.0000 −0.794998
\(357\) −12.0000 −0.635107
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −10.0000 −0.525588
\(363\) 2.00000 0.104973
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 1.00000 0.0522708
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −3.00000 −0.156386
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −18.0000 −0.934513
\(372\) 7.00000 0.362933
\(373\) 22.0000 1.13912 0.569558 − 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −18.0000 −0.930758
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) −6.00000 −0.309016
\(378\) 2.00000 0.102869
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 11.0000 0.563547
\(382\) 3.00000 0.153493
\(383\) −12.0000 −0.613171 −0.306586 − 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −20.0000 −1.01797
\(387\) 10.0000 0.508329
\(388\) −8.00000 −0.406138
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) −3.00000 −0.151523
\(393\) 15.0000 0.756650
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) −17.0000 −0.853206 −0.426603 − 0.904439i \(-0.640290\pi\)
−0.426603 + 0.904439i \(0.640290\pi\)
\(398\) 2.00000 0.100251
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) 9.00000 0.449439 0.224719 − 0.974424i \(-0.427853\pi\)
0.224719 + 0.974424i \(0.427853\pi\)
\(402\) 5.00000 0.249377
\(403\) 14.0000 0.697390
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) −6.00000 −0.297409
\(408\) 6.00000 0.297044
\(409\) −34.0000 −1.68119 −0.840596 − 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 7.00000 0.344865
\(413\) −12.0000 −0.590481
\(414\) −3.00000 −0.147442
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 4.00000 0.195881
\(418\) 3.00000 0.146735
\(419\) 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 \(0.578356\pi\)
\(422\) −13.0000 −0.632830
\(423\) −12.0000 −0.583460
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 − 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 16.0000 0.768911 0.384455 − 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 14.0000 0.672022
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −3.00000 −0.143509
\(438\) −1.00000 −0.0477818
\(439\) 35.0000 1.67046 0.835229 − 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 12.0000 0.570782
\(443\) −27.0000 −1.28281 −0.641404 − 0.767203i \(-0.721648\pi\)
−0.641404 + 0.767203i \(0.721648\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −5.00000 −0.236757
\(447\) −12.0000 −0.567581
\(448\) −2.00000 −0.0944911
\(449\) −33.0000 −1.55737 −0.778683 − 0.627417i \(-0.784112\pi\)
−0.778683 + 0.627417i \(0.784112\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) −9.00000 −0.423324
\(453\) −20.0000 −0.939682
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) −2.00000 −0.0935561 −0.0467780 − 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −13.0000 −0.607450
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 − 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 6.00000 0.279145
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 39.0000 1.80470 0.902352 − 0.430999i \(-0.141839\pi\)
0.902352 + 0.430999i \(0.141839\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 10.0000 0.461757
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 6.00000 0.276172
\(473\) 30.0000 1.37940
\(474\) 13.0000 0.597110
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) 9.00000 0.412082
\(478\) 0 0
\(479\) 27.0000 1.23366 0.616831 − 0.787096i \(-0.288416\pi\)
0.616831 + 0.787096i \(0.288416\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −10.0000 −0.455488
\(483\) −6.00000 −0.273009
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 40.0000 1.81257 0.906287 − 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) −1.00000 −0.0452679
\(489\) −22.0000 −0.994874
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 6.00000 0.270501
\(493\) −18.0000 −0.810679
\(494\) −2.00000 −0.0899843
\(495\) 0 0
\(496\) −7.00000 −0.314309
\(497\) 0 0
\(498\) 15.0000 0.672166
\(499\) 14.0000 0.626726 0.313363 − 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) −18.0000 −0.804181
\(502\) 24.0000 1.07117
\(503\) −36.0000 −1.60516 −0.802580 − 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) −9.00000 −0.400099
\(507\) 9.00000 0.399704
\(508\) −11.0000 −0.488046
\(509\) 33.0000 1.46270 0.731350 − 0.682003i \(-0.238891\pi\)
0.731350 + 0.682003i \(0.238891\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 9.00000 0.396973
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) −36.0000 −1.58328
\(518\) 4.00000 0.175750
\(519\) −3.00000 −0.131685
\(520\) 0 0
\(521\) −15.0000 −0.657162 −0.328581 − 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 3.00000 0.131306
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −15.0000 −0.655278
\(525\) 0 0
\(526\) 21.0000 0.915644
\(527\) 42.0000 1.82955
\(528\) −3.00000 −0.130558
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) −2.00000 −0.0867110
\(533\) 12.0000 0.519778
\(534\) 15.0000 0.649113
\(535\) 0 0
\(536\) −5.00000 −0.215967
\(537\) 0 0
\(538\) 30.0000 1.29339
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) −43.0000 −1.84871 −0.924357 − 0.381528i \(-0.875398\pi\)
−0.924357 + 0.381528i \(0.875398\pi\)
\(542\) 20.0000 0.859074
\(543\) 10.0000 0.429141
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) −11.0000 −0.470326 −0.235163 − 0.971956i \(-0.575562\pi\)
−0.235163 + 0.971956i \(0.575562\pi\)
\(548\) −12.0000 −0.512615
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) 3.00000 0.127804
\(552\) 3.00000 0.127688
\(553\) 26.0000 1.10563
\(554\) −11.0000 −0.467345
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −7.00000 −0.296334
\(559\) −20.0000 −0.845910
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) 3.00000 0.126547
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) −8.00000 −0.336265
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −6.00000 −0.250873
\(573\) −3.00000 −0.125327
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −23.0000 −0.957503 −0.478751 − 0.877951i \(-0.658910\pi\)
−0.478751 + 0.877951i \(0.658910\pi\)
\(578\) 19.0000 0.790296
\(579\) 20.0000 0.831172
\(580\) 0 0
\(581\) 30.0000 1.24461
\(582\) 8.00000 0.331611
\(583\) 27.0000 1.11823
\(584\) 1.00000 0.0413803
\(585\) 0 0
\(586\) 15.0000 0.619644
\(587\) 45.0000 1.85735 0.928674 − 0.370896i \(-0.120949\pi\)
0.928674 + 0.370896i \(0.120949\pi\)
\(588\) 3.00000 0.123718
\(589\) −7.00000 −0.288430
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) −2.00000 −0.0821995
\(593\) 36.0000 1.47834 0.739171 − 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) −3.00000 −0.123091
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) −2.00000 −0.0818546
\(598\) 6.00000 0.245358
\(599\) 30.0000 1.22577 0.612883 − 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 − 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) −20.0000 −0.815139
\(603\) −5.00000 −0.203616
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) −23.0000 −0.933541 −0.466771 − 0.884378i \(-0.654583\pi\)
−0.466771 + 0.884378i \(0.654583\pi\)
\(608\) 1.00000 0.0405554
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) −6.00000 −0.242536
\(613\) 34.0000 1.37325 0.686624 − 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 19.0000 0.766778
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −7.00000 −0.281581
\(619\) −46.0000 −1.84890 −0.924448 − 0.381308i \(-0.875474\pi\)
−0.924448 + 0.381308i \(0.875474\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) 12.0000 0.481156
\(623\) 30.0000 1.20192
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 13.0000 0.519584
\(627\) −3.00000 −0.119808
\(628\) −14.0000 −0.558661
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 − 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −13.0000 −0.517112
\(633\) 13.0000 0.516704
\(634\) 3.00000 0.119145
\(635\) 0 0
\(636\) −9.00000 −0.356873
\(637\) 6.00000 0.237729
\(638\) 9.00000 0.356313
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 − 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 − 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 15.0000 0.589711 0.294855 − 0.955542i \(-0.404729\pi\)
0.294855 + 0.955542i \(0.404729\pi\)
\(648\) 1.00000 0.0392837
\(649\) 18.0000 0.706562
\(650\) 0 0
\(651\) −14.0000 −0.548703
\(652\) 22.0000 0.861586
\(653\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 1.00000 0.0390137
\(658\) 24.0000 0.935617
\(659\) 6.00000 0.233727 0.116863 − 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 − 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) 5.00000 0.194331
\(663\) −12.0000 −0.466041
\(664\) −15.0000 −0.582113
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −9.00000 −0.348481
\(668\) 18.0000 0.696441
\(669\) 5.00000 0.193311
\(670\) 0 0
\(671\) −3.00000 −0.115814
\(672\) 2.00000 0.0771517
\(673\) 34.0000 1.31060 0.655302 − 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −32.0000 −1.23259
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 33.0000 1.26829 0.634147 − 0.773213i \(-0.281352\pi\)
0.634147 + 0.773213i \(0.281352\pi\)
\(678\) 9.00000 0.345643
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) −21.0000 −0.804132
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 13.0000 0.495981
\(688\) 10.0000 0.381246
\(689\) −18.0000 −0.685745
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 3.00000 0.114043
\(693\) −6.00000 −0.227921
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) 36.0000 1.36360
\(698\) −19.0000 −0.719161
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −24.0000 −0.906467 −0.453234 − 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) 2.00000 0.0754851
\(703\) −2.00000 −0.0754314
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) −12.0000 −0.451306
\(708\) −6.00000 −0.225494
\(709\) 35.0000 1.31445 0.657226 − 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) −13.0000 −0.487538
\(712\) −15.0000 −0.562149
\(713\) 21.0000 0.786456
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 15.0000 0.559406 0.279703 − 0.960087i \(-0.409764\pi\)
0.279703 + 0.960087i \(0.409764\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 1.00000 0.0372161
\(723\) 10.0000 0.371904
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) 34.0000 1.26099 0.630495 − 0.776193i \(-0.282852\pi\)
0.630495 + 0.776193i \(0.282852\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −60.0000 −2.21918
\(732\) 1.00000 0.0369611
\(733\) −5.00000 −0.184679 −0.0923396 − 0.995728i \(-0.529435\pi\)
−0.0923396 + 0.995728i \(0.529435\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) −15.0000 −0.552532
\(738\) −6.00000 −0.220863
\(739\) −40.0000 −1.47142 −0.735712 − 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) −18.0000 −0.660801
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 7.00000 0.256632
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) −15.0000 −0.548821
\(748\) −18.0000 −0.658145
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 − 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −12.0000 −0.437595
\(753\) −24.0000 −0.874609
\(754\) −6.00000 −0.218507
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) −29.0000 −1.05402 −0.527011 − 0.849858i \(-0.676688\pi\)
−0.527011 + 0.849858i \(0.676688\pi\)
\(758\) 20.0000 0.726433
\(759\) 9.00000 0.326679
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 − 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 11.0000 0.398488
\(763\) −4.00000 −0.144810
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) −12.0000 −0.433295
\(768\) −1.00000 −0.0360844
\(769\) 11.0000 0.396670 0.198335 − 0.980134i \(-0.436447\pi\)
0.198335 + 0.980134i \(0.436447\pi\)
\(770\) 0 0
\(771\) −9.00000 −0.324127
\(772\) −20.0000 −0.719816
\(773\) 18.0000 0.647415 0.323708 − 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 10.0000 0.359443
\(775\) 0 0
\(776\) −8.00000 −0.287183
\(777\) −4.00000 −0.143499
\(778\) −6.00000 −0.215110
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) 18.0000 0.643679
\(783\) −3.00000 −0.107211
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 15.0000 0.535032
\(787\) 49.0000 1.74666 0.873331 − 0.487128i \(-0.161955\pi\)
0.873331 + 0.487128i \(0.161955\pi\)
\(788\) −6.00000 −0.213741
\(789\) −21.0000 −0.747620
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 3.00000 0.106600
\(793\) 2.00000 0.0710221
\(794\) −17.0000 −0.603307
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 2.00000 0.0707992
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) −15.0000 −0.529999
\(802\) 9.00000 0.317801
\(803\) 3.00000 0.105868
\(804\) 5.00000 0.176336
\(805\) 0 0
\(806\) 14.0000 0.493129
\(807\) −30.0000 −1.05605
\(808\) 6.00000 0.211079
\(809\) −30.0000 −1.05474 −0.527372 − 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 41.0000 1.43970 0.719852 − 0.694127i \(-0.244209\pi\)
0.719852 + 0.694127i \(0.244209\pi\)
\(812\) −6.00000 −0.210559
\(813\) −20.0000 −0.701431
\(814\) −6.00000 −0.210300
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 10.0000 0.349856
\(818\) −34.0000 −1.18878
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 36.0000 1.25641 0.628204 − 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) 12.0000 0.418548
\(823\) −26.0000 −0.906303 −0.453152 − 0.891434i \(-0.649700\pi\)
−0.453152 + 0.891434i \(0.649700\pi\)
\(824\) 7.00000 0.243857
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 18.0000 0.625921 0.312961 − 0.949766i \(-0.398679\pi\)
0.312961 + 0.949766i \(0.398679\pi\)
\(828\) −3.00000 −0.104257
\(829\) −4.00000 −0.138926 −0.0694629 − 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) 11.0000 0.381586
\(832\) −2.00000 −0.0693375
\(833\) 18.0000 0.623663
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 3.00000 0.103757
\(837\) 7.00000 0.241955
\(838\) 12.0000 0.414533
\(839\) −6.00000 −0.207143 −0.103572 − 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −10.0000 −0.344623
\(843\) −3.00000 −0.103325
\(844\) −13.0000 −0.447478
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 4.00000 0.137442
\(848\) 9.00000 0.309061
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 − 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 6.00000 0.204837
\(859\) −22.0000 −0.750630 −0.375315 − 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 18.0000 0.613082
\(863\) 12.0000 0.408485 0.204242 − 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) −19.0000 −0.645274
\(868\) 14.0000 0.475191
\(869\) −39.0000 −1.32298
\(870\) 0 0
\(871\) 10.0000 0.338837
\(872\) 2.00000 0.0677285
\(873\) −8.00000 −0.270759
\(874\) −3.00000 −0.101477
\(875\) 0 0
\(876\) −1.00000 −0.0337869
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 35.0000 1.18119
\(879\) −15.0000 −0.505937
\(880\) 0 0
\(881\) −24.0000 −0.808581 −0.404290 − 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) −3.00000 −0.101015
\(883\) −56.0000 −1.88455 −0.942275 − 0.334840i \(-0.891318\pi\)
−0.942275 + 0.334840i \(0.891318\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −27.0000 −0.907083
\(887\) −12.0000 −0.402921 −0.201460 − 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 2.00000 0.0671156
\(889\) 22.0000 0.737856
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) −5.00000 −0.167412
\(893\) −12.0000 −0.401565
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) −6.00000 −0.200334
\(898\) −33.0000 −1.10122
\(899\) −21.0000 −0.700389
\(900\) 0 0
\(901\) −54.0000 −1.79900
\(902\) −18.0000 −0.599334
\(903\) 20.0000 0.665558
\(904\) −9.00000 −0.299336
\(905\) 0 0
\(906\) −20.0000 −0.664455
\(907\) −8.00000 −0.265636 −0.132818 − 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) 18.0000 0.597351
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 − 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) −1.00000 −0.0331133
\(913\) −45.0000 −1.48928
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −13.0000 −0.429532
\(917\) 30.0000 0.990687
\(918\) 6.00000 0.198030
\(919\) −40.0000 −1.31948 −0.659739 − 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) −19.0000 −0.626071
\(922\) −12.0000 −0.395199
\(923\) 0 0
\(924\) 6.00000 0.197386
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 7.00000 0.229910
\(928\) 3.00000 0.0984798
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) −6.00000 −0.196537
\(933\) −12.0000 −0.392862
\(934\) 39.0000 1.27612
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −2.00000 −0.0653372 −0.0326686 − 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 10.0000 0.326512
\(939\) −13.0000 −0.424239
\(940\) 0 0
\(941\) −51.0000 −1.66255 −0.831276 − 0.555860i \(-0.812389\pi\)
−0.831276 + 0.555860i \(0.812389\pi\)
\(942\) 14.0000 0.456145
\(943\) 18.0000 0.586161
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 30.0000 0.975384
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 13.0000 0.422220
\(949\) −2.00000 −0.0649227
\(950\) 0 0
\(951\) −3.00000 −0.0972817
\(952\) 12.0000 0.388922
\(953\) 33.0000 1.06897 0.534487 − 0.845176i \(-0.320505\pi\)
0.534487 + 0.845176i \(0.320505\pi\)
\(954\) 9.00000 0.291386
\(955\) 0 0
\(956\) 0 0
\(957\) −9.00000 −0.290929
\(958\) 27.0000 0.872330
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 4.00000 0.128965
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) −6.00000 −0.193047
\(967\) −38.0000 −1.22200 −0.610999 − 0.791632i \(-0.709232\pi\)
−0.610999 + 0.791632i \(0.709232\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 − 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 8.00000 0.256468
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) 54.0000 1.72761 0.863807 − 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) −22.0000 −0.703482
\(979\) −45.0000 −1.43821
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) −12.0000 −0.382935
\(983\) −36.0000 −1.14822 −0.574111 − 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) −18.0000 −0.573237
\(987\) −24.0000 −0.763928
\(988\) −2.00000 −0.0636285
\(989\) −30.0000 −0.953945
\(990\) 0 0
\(991\) −25.0000 −0.794151 −0.397076 − 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) −7.00000 −0.222250
\(993\) −5.00000 −0.158670
\(994\) 0 0
\(995\) 0 0
\(996\) 15.0000 0.475293
\(997\) 19.0000 0.601736 0.300868 − 0.953666i \(-0.402724\pi\)
0.300868 + 0.953666i \(0.402724\pi\)
\(998\) 14.0000 0.443162
\(999\) 2.00000 0.0632772
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 2850.2.a.r.1.1 yes 1
3.2 odd 2 8550.2.a.d.1.1 1
5.2 odd 4 2850.2.d.h.799.2 2
5.3 odd 4 2850.2.d.h.799.1 2
5.4 even 2 2850.2.a.o.1.1 ✓ 1
15.14 odd 2 8550.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.o.1.1 ✓ 1 5.4 even 2
2850.2.a.r.1.1 yes 1 1.1 even 1 trivial
2850.2.d.h.799.1 2 5.3 odd 4
2850.2.d.h.799.2 2 5.2 odd 4
8550.2.a.d.1.1 1 3.2 odd 2
8550.2.a.be.1.1 1 15.14 odd 2