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Properties

Label 8670.2.a.p.1.1
Level $8670$
Weight $2$
Character 8670.1
Self dual yes
Analytic conductor $69.230$
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] = [8670,2,Mod(1,8670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8670, 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("8670.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8670.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: \(69.2302985525\)
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 \(\chi\) \(=\) 8670.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} +5.00000 q^{22} -7.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -4.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} -5.00000 q^{33} +1.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} +6.00000 q^{43} +5.00000 q^{44} -1.00000 q^{45} -7.00000 q^{46} -7.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} +13.0000 q^{53} -1.00000 q^{54} -5.00000 q^{55} -1.00000 q^{56} +4.00000 q^{57} -4.00000 q^{58} +9.00000 q^{59} +1.00000 q^{60} -10.0000 q^{61} -8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} -5.00000 q^{66} +4.00000 q^{67} +7.00000 q^{69} +1.00000 q^{70} -10.0000 q^{71} +1.00000 q^{72} -8.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -5.00000 q^{77} -1.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -10.0000 q^{83} +1.00000 q^{84} +6.00000 q^{86} +4.00000 q^{87} +5.00000 q^{88} +7.00000 q^{89} -1.00000 q^{90} -1.00000 q^{91} -7.00000 q^{92} +8.00000 q^{93} -7.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} -12.0000 q^{97} -6.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 5.00000 1.50756 0.753778 − 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −1.00000 −0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) 5.00000 1.06600
\(23\) −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
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 1.00000 0.182574
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.00000 −0.870388
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) −1.00000 −0.160128
\(40\) −1.00000 −0.158114
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 1.00000 0.154303
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 5.00000 0.753778
\(45\) −1.00000 −0.149071
\(46\) −7.00000 −1.03209
\(47\) −7.00000 −1.02105 −0.510527 − 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.00000 −0.857143
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 13.0000 1.78569 0.892844 − 0.450367i \(-0.148707\pi\)
0.892844 + 0.450367i \(0.148707\pi\)
\(54\) −1.00000 −0.136083
\(55\) −5.00000 −0.674200
\(56\) −1.00000 −0.133631
\(57\) 4.00000 0.529813
\(58\) −4.00000 −0.525226
\(59\) 9.00000 1.17170 0.585850 − 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 1.00000 0.129099
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −8.00000 −1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) −5.00000 −0.615457
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 7.00000 0.842701
\(70\) 1.00000 0.119523
\(71\) −10.0000 −1.18678 −0.593391 − 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 1.00000 0.117851
\(73\) −8.00000 −0.936329 −0.468165 − 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 6.00000 0.697486
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) −5.00000 −0.569803
\(78\) −1.00000 −0.113228
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −10.0000 −1.09764 −0.548821 − 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) 4.00000 0.428845
\(88\) 5.00000 0.533002
\(89\) 7.00000 0.741999 0.370999 − 0.928633i \(-0.379015\pi\)
0.370999 + 0.928633i \(0.379015\pi\)
\(90\) −1.00000 −0.105409
\(91\) −1.00000 −0.104828
\(92\) −7.00000 −0.729800
\(93\) 8.00000 0.829561
\(94\) −7.00000 −0.721995
\(95\) 4.00000 0.410391
\(96\) −1.00000 −0.102062
\(97\) −12.0000 −1.21842 −0.609208 − 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) −6.00000 −0.606092
\(99\) 5.00000 0.502519
\(100\) 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
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 1.00000 0.0980581
\(105\) −1.00000 −0.0975900
\(106\) 13.0000 1.26267
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 16.0000 1.53252 0.766261 − 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) −5.00000 −0.476731
\(111\) −6.00000 −0.569495
\(112\) −1.00000 −0.0944911
\(113\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 4.00000 0.374634
\(115\) 7.00000 0.652753
\(116\) −4.00000 −0.371391
\(117\) 1.00000 0.0924500
\(118\) 9.00000 0.828517
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 14.0000 1.27273
\(122\) −10.0000 −0.905357
\(123\) −2.00000 −0.180334
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −4.00000 −0.354943 −0.177471 − 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.00000 −0.528271
\(130\) −1.00000 −0.0877058
\(131\) −7.00000 −0.611593 −0.305796 − 0.952097i \(-0.598923\pi\)
−0.305796 + 0.952097i \(0.598923\pi\)
\(132\) −5.00000 −0.435194
\(133\) 4.00000 0.346844
\(134\) 4.00000 0.345547
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 7.00000 0.595880
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 1.00000 0.0845154
\(141\) 7.00000 0.589506
\(142\) −10.0000 −0.839181
\(143\) 5.00000 0.418121
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) −8.00000 −0.662085
\(147\) 6.00000 0.494872
\(148\) 6.00000 0.493197
\(149\) 4.00000 0.327693 0.163846 − 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) −5.00000 −0.402911
\(155\) 8.00000 0.642575
\(156\) −1.00000 −0.0800641
\(157\) 17.0000 1.35675 0.678374 − 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 0 0
\(159\) −13.0000 −1.03097
\(160\) −1.00000 −0.0790569
\(161\) 7.00000 0.551677
\(162\) 1.00000 0.0785674
\(163\) −2.00000 −0.156652 −0.0783260 − 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 2.00000 0.156174
\(165\) 5.00000 0.389249
\(166\) −10.0000 −0.776151
\(167\) 15.0000 1.16073 0.580367 − 0.814355i \(-0.302909\pi\)
0.580367 + 0.814355i \(0.302909\pi\)
\(168\) 1.00000 0.0771517
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 6.00000 0.457496
\(173\) 13.0000 0.988372 0.494186 − 0.869356i \(-0.335466\pi\)
0.494186 + 0.869356i \(0.335466\pi\)
\(174\) 4.00000 0.303239
\(175\) −1.00000 −0.0755929
\(176\) 5.00000 0.376889
\(177\) −9.00000 −0.676481
\(178\) 7.00000 0.524672
\(179\) 21.0000 1.56961 0.784807 − 0.619740i \(-0.212762\pi\)
0.784807 + 0.619740i \(0.212762\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −22.0000 −1.63525 −0.817624 − 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 10.0000 0.739221
\(184\) −7.00000 −0.516047
\(185\) −6.00000 −0.441129
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) −7.00000 −0.510527
\(189\) 1.00000 0.0727393
\(190\) 4.00000 0.290191
\(191\) 4.00000 0.289430 0.144715 − 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) −12.0000 −0.861550
\(195\) 1.00000 0.0716115
\(196\) −6.00000 −0.428571
\(197\) 7.00000 0.498729 0.249365 − 0.968410i \(-0.419778\pi\)
0.249365 + 0.968410i \(0.419778\pi\)
\(198\) 5.00000 0.355335
\(199\) −20.0000 −1.41776 −0.708881 − 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) −10.0000 −0.703598
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) −7.00000 −0.486534
\(208\) 1.00000 0.0693375
\(209\) −20.0000 −1.38343
\(210\) −1.00000 −0.0690066
\(211\) −23.0000 −1.58339 −0.791693 − 0.610920i \(-0.790800\pi\)
−0.791693 + 0.610920i \(0.790800\pi\)
\(212\) 13.0000 0.892844
\(213\) 10.0000 0.685189
\(214\) −12.0000 −0.820303
\(215\) −6.00000 −0.409197
\(216\) −1.00000 −0.0680414
\(217\) 8.00000 0.543075
\(218\) 16.0000 1.08366
\(219\) 8.00000 0.540590
\(220\) −5.00000 −0.337100
\(221\) 0 0
\(222\) −6.00000 −0.402694
\(223\) 19.0000 1.27233 0.636167 − 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 10.0000 0.663723 0.331862 − 0.943328i \(-0.392323\pi\)
0.331862 + 0.943328i \(0.392323\pi\)
\(228\) 4.00000 0.264906
\(229\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 7.00000 0.461566
\(231\) 5.00000 0.328976
\(232\) −4.00000 −0.262613
\(233\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 1.00000 0.0653720
\(235\) 7.00000 0.456630
\(236\) 9.00000 0.585850
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 1.00000 0.0645497
\(241\) 7.00000 0.450910 0.225455 − 0.974254i \(-0.427613\pi\)
0.225455 + 0.974254i \(0.427613\pi\)
\(242\) 14.0000 0.899954
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 6.00000 0.383326
\(246\) −2.00000 −0.127515
\(247\) −4.00000 −0.254514
\(248\) −8.00000 −0.508001
\(249\) 10.0000 0.633724
\(250\) −1.00000 −0.0632456
\(251\) −31.0000 −1.95670 −0.978351 − 0.206951i \(-0.933646\pi\)
−0.978351 + 0.206951i \(0.933646\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −35.0000 −2.20043
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −6.00000 −0.373544
\(259\) −6.00000 −0.372822
\(260\) −1.00000 −0.0620174
\(261\) −4.00000 −0.247594
\(262\) −7.00000 −0.432461
\(263\) −31.0000 −1.91154 −0.955771 − 0.294112i \(-0.904976\pi\)
−0.955771 + 0.294112i \(0.904976\pi\)
\(264\) −5.00000 −0.307729
\(265\) −13.0000 −0.798584
\(266\) 4.00000 0.245256
\(267\) −7.00000 −0.428393
\(268\) 4.00000 0.244339
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 1.00000 0.0608581
\(271\) −18.0000 −1.09342 −0.546711 − 0.837321i \(-0.684120\pi\)
−0.546711 + 0.837321i \(0.684120\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) −12.0000 −0.724947
\(275\) 5.00000 0.301511
\(276\) 7.00000 0.421350
\(277\) −1.00000 −0.0600842 −0.0300421 − 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) 4.00000 0.239904
\(279\) −8.00000 −0.478947
\(280\) 1.00000 0.0597614
\(281\) −17.0000 −1.01413 −0.507067 − 0.861906i \(-0.669271\pi\)
−0.507067 + 0.861906i \(0.669271\pi\)
\(282\) 7.00000 0.416844
\(283\) 10.0000 0.594438 0.297219 − 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) −10.0000 −0.593391
\(285\) −4.00000 −0.236940
\(286\) 5.00000 0.295656
\(287\) −2.00000 −0.118056
\(288\) 1.00000 0.0589256
\(289\) 0 0
\(290\) 4.00000 0.234888
\(291\) 12.0000 0.703452
\(292\) −8.00000 −0.468165
\(293\) −29.0000 −1.69420 −0.847099 − 0.531435i \(-0.821653\pi\)
−0.847099 + 0.531435i \(0.821653\pi\)
\(294\) 6.00000 0.349927
\(295\) −9.00000 −0.524000
\(296\) 6.00000 0.348743
\(297\) −5.00000 −0.290129
\(298\) 4.00000 0.231714
\(299\) −7.00000 −0.404820
\(300\) −1.00000 −0.0577350
\(301\) −6.00000 −0.345834
\(302\) −8.00000 −0.460348
\(303\) 10.0000 0.574485
\(304\) −4.00000 −0.229416
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) −10.0000 −0.570730 −0.285365 − 0.958419i \(-0.592115\pi\)
−0.285365 + 0.958419i \(0.592115\pi\)
\(308\) −5.00000 −0.284901
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) −28.0000 −1.58773 −0.793867 − 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −8.00000 −0.452187 −0.226093 − 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 17.0000 0.959366
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) −13.0000 −0.729004
\(319\) −20.0000 −1.11979
\(320\) −1.00000 −0.0559017
\(321\) 12.0000 0.669775
\(322\) 7.00000 0.390095
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 1.00000 0.0554700
\(326\) −2.00000 −0.110770
\(327\) −16.0000 −0.884802
\(328\) 2.00000 0.110432
\(329\) 7.00000 0.385922
\(330\) 5.00000 0.275241
\(331\) 25.0000 1.37412 0.687062 − 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) −10.0000 −0.548821
\(333\) 6.00000 0.328798
\(334\) 15.0000 0.820763
\(335\) −4.00000 −0.218543
\(336\) 1.00000 0.0545545
\(337\) 8.00000 0.435788 0.217894 − 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) −12.0000 −0.652714
\(339\) 0 0
\(340\) 0 0
\(341\) −40.0000 −2.16612
\(342\) −4.00000 −0.216295
\(343\) 13.0000 0.701934
\(344\) 6.00000 0.323498
\(345\) −7.00000 −0.376867
\(346\) 13.0000 0.698884
\(347\) 2.00000 0.107366 0.0536828 − 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) 4.00000 0.214423
\(349\) −8.00000 −0.428230 −0.214115 − 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −1.00000 −0.0533761
\(352\) 5.00000 0.266501
\(353\) −28.0000 −1.49029 −0.745145 − 0.666903i \(-0.767620\pi\)
−0.745145 + 0.666903i \(0.767620\pi\)
\(354\) −9.00000 −0.478345
\(355\) 10.0000 0.530745
\(356\) 7.00000 0.370999
\(357\) 0 0
\(358\) 21.0000 1.10988
\(359\) −6.00000 −0.316668 −0.158334 − 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) −22.0000 −1.15629
\(363\) −14.0000 −0.734809
\(364\) −1.00000 −0.0524142
\(365\) 8.00000 0.418739
\(366\) 10.0000 0.522708
\(367\) −11.0000 −0.574195 −0.287098 − 0.957901i \(-0.592690\pi\)
−0.287098 + 0.957901i \(0.592690\pi\)
\(368\) −7.00000 −0.364900
\(369\) 2.00000 0.104116
\(370\) −6.00000 −0.311925
\(371\) −13.0000 −0.674926
\(372\) 8.00000 0.414781
\(373\) −7.00000 −0.362446 −0.181223 − 0.983442i \(-0.558006\pi\)
−0.181223 + 0.983442i \(0.558006\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) −7.00000 −0.360997
\(377\) −4.00000 −0.206010
\(378\) 1.00000 0.0514344
\(379\) −28.0000 −1.43826 −0.719132 − 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 4.00000 0.205196
\(381\) 4.00000 0.204926
\(382\) 4.00000 0.204658
\(383\) −16.0000 −0.817562 −0.408781 − 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 5.00000 0.254824
\(386\) 0 0
\(387\) 6.00000 0.304997
\(388\) −12.0000 −0.609208
\(389\) −36.0000 −1.82527 −0.912636 − 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 1.00000 0.0506370
\(391\) 0 0
\(392\) −6.00000 −0.303046
\(393\) 7.00000 0.353103
\(394\) 7.00000 0.352655
\(395\) 0 0
\(396\) 5.00000 0.251259
\(397\) −9.00000 −0.451697 −0.225849 − 0.974162i \(-0.572515\pi\)
−0.225849 + 0.974162i \(0.572515\pi\)
\(398\) −20.0000 −1.00251
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) 27.0000 1.34832 0.674158 − 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) −4.00000 −0.199502
\(403\) −8.00000 −0.398508
\(404\) −10.0000 −0.497519
\(405\) −1.00000 −0.0496904
\(406\) 4.00000 0.198517
\(407\) 30.0000 1.48704
\(408\) 0 0
\(409\) −11.0000 −0.543915 −0.271957 − 0.962309i \(-0.587671\pi\)
−0.271957 + 0.962309i \(0.587671\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) −9.00000 −0.442861
\(414\) −7.00000 −0.344031
\(415\) 10.0000 0.490881
\(416\) 1.00000 0.0490290
\(417\) −4.00000 −0.195881
\(418\) −20.0000 −0.978232
\(419\) 15.0000 0.732798 0.366399 − 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −12.0000 −0.584844 −0.292422 − 0.956289i \(-0.594461\pi\)
−0.292422 + 0.956289i \(0.594461\pi\)
\(422\) −23.0000 −1.11962
\(423\) −7.00000 −0.340352
\(424\) 13.0000 0.631336
\(425\) 0 0
\(426\) 10.0000 0.484502
\(427\) 10.0000 0.483934
\(428\) −12.0000 −0.580042
\(429\) −5.00000 −0.241402
\(430\) −6.00000 −0.289346
\(431\) −40.0000 −1.92673 −0.963366 − 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 16.0000 0.768911 0.384455 − 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 8.00000 0.384012
\(435\) −4.00000 −0.191785
\(436\) 16.0000 0.766261
\(437\) 28.0000 1.33942
\(438\) 8.00000 0.382255
\(439\) 18.0000 0.859093 0.429547 − 0.903045i \(-0.358673\pi\)
0.429547 + 0.903045i \(0.358673\pi\)
\(440\) −5.00000 −0.238366
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 30.0000 1.42534 0.712672 − 0.701498i \(-0.247485\pi\)
0.712672 + 0.701498i \(0.247485\pi\)
\(444\) −6.00000 −0.284747
\(445\) −7.00000 −0.331832
\(446\) 19.0000 0.899676
\(447\) −4.00000 −0.189194
\(448\) −1.00000 −0.0472456
\(449\) −18.0000 −0.849473 −0.424736 − 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 1.00000 0.0471405
\(451\) 10.0000 0.470882
\(452\) 0 0
\(453\) 8.00000 0.375873
\(454\) 10.0000 0.469323
\(455\) 1.00000 0.0468807
\(456\) 4.00000 0.187317
\(457\) −8.00000 −0.374224 −0.187112 − 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 7.00000 0.326377
\(461\) 2.00000 0.0931493 0.0465746 − 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 5.00000 0.232621
\(463\) −29.0000 −1.34774 −0.673872 − 0.738848i \(-0.735370\pi\)
−0.673872 + 0.738848i \(0.735370\pi\)
\(464\) −4.00000 −0.185695
\(465\) −8.00000 −0.370991
\(466\) 0 0
\(467\) −18.0000 −0.832941 −0.416470 − 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 1.00000 0.0462250
\(469\) −4.00000 −0.184703
\(470\) 7.00000 0.322886
\(471\) −17.0000 −0.783319
\(472\) 9.00000 0.414259
\(473\) 30.0000 1.37940
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 13.0000 0.595229
\(478\) −12.0000 −0.548867
\(479\) −16.0000 −0.731059 −0.365529 − 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 1.00000 0.0456435
\(481\) 6.00000 0.273576
\(482\) 7.00000 0.318841
\(483\) −7.00000 −0.318511
\(484\) 14.0000 0.636364
\(485\) 12.0000 0.544892
\(486\) −1.00000 −0.0453609
\(487\) 32.0000 1.45006 0.725029 − 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −10.0000 −0.452679
\(489\) 2.00000 0.0904431
\(490\) 6.00000 0.271052
\(491\) 3.00000 0.135388 0.0676941 − 0.997706i \(-0.478436\pi\)
0.0676941 + 0.997706i \(0.478436\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 0 0
\(494\) −4.00000 −0.179969
\(495\) −5.00000 −0.224733
\(496\) −8.00000 −0.359211
\(497\) 10.0000 0.448561
\(498\) 10.0000 0.448111
\(499\) −5.00000 −0.223831 −0.111915 − 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −15.0000 −0.670151
\(502\) −31.0000 −1.38360
\(503\) −25.0000 −1.11469 −0.557347 − 0.830279i \(-0.688181\pi\)
−0.557347 + 0.830279i \(0.688181\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 10.0000 0.444994
\(506\) −35.0000 −1.55594
\(507\) 12.0000 0.532939
\(508\) −4.00000 −0.177471
\(509\) 14.0000 0.620539 0.310270 − 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) −6.00000 −0.264135
\(517\) −35.0000 −1.53930
\(518\) −6.00000 −0.263625
\(519\) −13.0000 −0.570637
\(520\) −1.00000 −0.0438529
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −4.00000 −0.175075
\(523\) 34.0000 1.48672 0.743358 − 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) −7.00000 −0.305796
\(525\) 1.00000 0.0436436
\(526\) −31.0000 −1.35166
\(527\) 0 0
\(528\) −5.00000 −0.217597
\(529\) 26.0000 1.13043
\(530\) −13.0000 −0.564684
\(531\) 9.00000 0.390567
\(532\) 4.00000 0.173422
\(533\) 2.00000 0.0866296
\(534\) −7.00000 −0.302920
\(535\) 12.0000 0.518805
\(536\) 4.00000 0.172774
\(537\) −21.0000 −0.906217
\(538\) −6.00000 −0.258678
\(539\) −30.0000 −1.29219
\(540\) 1.00000 0.0430331
\(541\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) −18.0000 −0.773166
\(543\) 22.0000 0.944110
\(544\) 0 0
\(545\) −16.0000 −0.685365
\(546\) 1.00000 0.0427960
\(547\) 38.0000 1.62476 0.812381 − 0.583127i \(-0.198171\pi\)
0.812381 + 0.583127i \(0.198171\pi\)
\(548\) −12.0000 −0.512615
\(549\) −10.0000 −0.426790
\(550\) 5.00000 0.213201
\(551\) 16.0000 0.681623
\(552\) 7.00000 0.297940
\(553\) 0 0
\(554\) −1.00000 −0.0424859
\(555\) 6.00000 0.254686
\(556\) 4.00000 0.169638
\(557\) −17.0000 −0.720313 −0.360157 − 0.932892i \(-0.617277\pi\)
−0.360157 + 0.932892i \(0.617277\pi\)
\(558\) −8.00000 −0.338667
\(559\) 6.00000 0.253773
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −17.0000 −0.717102
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 7.00000 0.294753
\(565\) 0 0
\(566\) 10.0000 0.420331
\(567\) −1.00000 −0.0419961
\(568\) −10.0000 −0.419591
\(569\) 6.00000 0.251533 0.125767 − 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) −4.00000 −0.167542
\(571\) 1.00000 0.0418487 0.0209243 − 0.999781i \(-0.493339\pi\)
0.0209243 + 0.999781i \(0.493339\pi\)
\(572\) 5.00000 0.209061
\(573\) −4.00000 −0.167102
\(574\) −2.00000 −0.0834784
\(575\) −7.00000 −0.291920
\(576\) 1.00000 0.0416667
\(577\) −42.0000 −1.74848 −0.874241 − 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) 10.0000 0.414870
\(582\) 12.0000 0.497416
\(583\) 65.0000 2.69202
\(584\) −8.00000 −0.331042
\(585\) −1.00000 −0.0413449
\(586\) −29.0000 −1.19798
\(587\) 26.0000 1.07313 0.536567 − 0.843857i \(-0.319721\pi\)
0.536567 + 0.843857i \(0.319721\pi\)
\(588\) 6.00000 0.247436
\(589\) 32.0000 1.31854
\(590\) −9.00000 −0.370524
\(591\) −7.00000 −0.287942
\(592\) 6.00000 0.246598
\(593\) 24.0000 0.985562 0.492781 − 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) 20.0000 0.818546
\(598\) −7.00000 −0.286251
\(599\) −18.0000 −0.735460 −0.367730 − 0.929933i \(-0.619865\pi\)
−0.367730 + 0.929933i \(0.619865\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 21.0000 0.856608 0.428304 − 0.903635i \(-0.359111\pi\)
0.428304 + 0.903635i \(0.359111\pi\)
\(602\) −6.00000 −0.244542
\(603\) 4.00000 0.162893
\(604\) −8.00000 −0.325515
\(605\) −14.0000 −0.569181
\(606\) 10.0000 0.406222
\(607\) 25.0000 1.01472 0.507359 − 0.861735i \(-0.330622\pi\)
0.507359 + 0.861735i \(0.330622\pi\)
\(608\) −4.00000 −0.162221
\(609\) −4.00000 −0.162088
\(610\) 10.0000 0.404888
\(611\) −7.00000 −0.283190
\(612\) 0 0
\(613\) −29.0000 −1.17130 −0.585649 − 0.810564i \(-0.699160\pi\)
−0.585649 + 0.810564i \(0.699160\pi\)
\(614\) −10.0000 −0.403567
\(615\) 2.00000 0.0806478
\(616\) −5.00000 −0.201456
\(617\) −12.0000 −0.483102 −0.241551 − 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) 11.0000 0.442127 0.221064 − 0.975259i \(-0.429047\pi\)
0.221064 + 0.975259i \(0.429047\pi\)
\(620\) 8.00000 0.321288
\(621\) 7.00000 0.280900
\(622\) −28.0000 −1.12270
\(623\) −7.00000 −0.280449
\(624\) −1.00000 −0.0400320
\(625\) 1.00000 0.0400000
\(626\) −8.00000 −0.319744
\(627\) 20.0000 0.798723
\(628\) 17.0000 0.678374
\(629\) 0 0
\(630\) 1.00000 0.0398410
\(631\) 20.0000 0.796187 0.398094 − 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 0 0
\(633\) 23.0000 0.914168
\(634\) 22.0000 0.873732
\(635\) 4.00000 0.158735
\(636\) −13.0000 −0.515484
\(637\) −6.00000 −0.237729
\(638\) −20.0000 −0.791808
\(639\) −10.0000 −0.395594
\(640\) −1.00000 −0.0395285
\(641\) 21.0000 0.829450 0.414725 − 0.909947i \(-0.363878\pi\)
0.414725 + 0.909947i \(0.363878\pi\)
\(642\) 12.0000 0.473602
\(643\) −20.0000 −0.788723 −0.394362 − 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 7.00000 0.275839
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) −25.0000 −0.982851 −0.491426 − 0.870919i \(-0.663524\pi\)
−0.491426 + 0.870919i \(0.663524\pi\)
\(648\) 1.00000 0.0392837
\(649\) 45.0000 1.76640
\(650\) 1.00000 0.0392232
\(651\) −8.00000 −0.313545
\(652\) −2.00000 −0.0783260
\(653\) −21.0000 −0.821794 −0.410897 − 0.911682i \(-0.634784\pi\)
−0.410897 + 0.911682i \(0.634784\pi\)
\(654\) −16.0000 −0.625650
\(655\) 7.00000 0.273513
\(656\) 2.00000 0.0780869
\(657\) −8.00000 −0.312110
\(658\) 7.00000 0.272888
\(659\) 20.0000 0.779089 0.389545 − 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 5.00000 0.194625
\(661\) −4.00000 −0.155582 −0.0777910 − 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) 25.0000 0.971653
\(663\) 0 0
\(664\) −10.0000 −0.388075
\(665\) −4.00000 −0.155113
\(666\) 6.00000 0.232495
\(667\) 28.0000 1.08416
\(668\) 15.0000 0.580367
\(669\) −19.0000 −0.734582
\(670\) −4.00000 −0.154533
\(671\) −50.0000 −1.93023
\(672\) 1.00000 0.0385758
\(673\) −50.0000 −1.92736 −0.963679 − 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) 8.00000 0.308148
\(675\) −1.00000 −0.0384900
\(676\) −12.0000 −0.461538
\(677\) 27.0000 1.03769 0.518847 − 0.854867i \(-0.326361\pi\)
0.518847 + 0.854867i \(0.326361\pi\)
\(678\) 0 0
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) −10.0000 −0.383201
\(682\) −40.0000 −1.53168
\(683\) −24.0000 −0.918334 −0.459167 − 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) −4.00000 −0.152944
\(685\) 12.0000 0.458496
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) 6.00000 0.228748
\(689\) 13.0000 0.495261
\(690\) −7.00000 −0.266485
\(691\) −11.0000 −0.418460 −0.209230 − 0.977866i \(-0.567096\pi\)
−0.209230 + 0.977866i \(0.567096\pi\)
\(692\) 13.0000 0.494186
\(693\) −5.00000 −0.189934
\(694\) 2.00000 0.0759190
\(695\) −4.00000 −0.151729
\(696\) 4.00000 0.151620
\(697\) 0 0
\(698\) −8.00000 −0.302804
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −42.0000 −1.58632 −0.793159 − 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −24.0000 −0.905177
\(704\) 5.00000 0.188445
\(705\) −7.00000 −0.263635
\(706\) −28.0000 −1.05379
\(707\) 10.0000 0.376089
\(708\) −9.00000 −0.338241
\(709\) 52.0000 1.95290 0.976450 − 0.215742i \(-0.0692169\pi\)
0.976450 + 0.215742i \(0.0692169\pi\)
\(710\) 10.0000 0.375293
\(711\) 0 0
\(712\) 7.00000 0.262336
\(713\) 56.0000 2.09722
\(714\) 0 0
\(715\) −5.00000 −0.186989
\(716\) 21.0000 0.784807
\(717\) 12.0000 0.448148
\(718\) −6.00000 −0.223918
\(719\) −50.0000 −1.86469 −0.932343 − 0.361576i \(-0.882239\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −7.00000 −0.260333
\(724\) −22.0000 −0.817624
\(725\) −4.00000 −0.148556
\(726\) −14.0000 −0.519589
\(727\) 17.0000 0.630495 0.315248 − 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 1.00000 0.0370370
\(730\) 8.00000 0.296093
\(731\) 0 0
\(732\) 10.0000 0.369611
\(733\) 30.0000 1.10808 0.554038 − 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) −11.0000 −0.406017
\(735\) −6.00000 −0.221313
\(736\) −7.00000 −0.258023
\(737\) 20.0000 0.736709
\(738\) 2.00000 0.0736210
\(739\) 51.0000 1.87607 0.938033 − 0.346547i \(-0.112646\pi\)
0.938033 + 0.346547i \(0.112646\pi\)
\(740\) −6.00000 −0.220564
\(741\) 4.00000 0.146944
\(742\) −13.0000 −0.477245
\(743\) 13.0000 0.476924 0.238462 − 0.971152i \(-0.423357\pi\)
0.238462 + 0.971152i \(0.423357\pi\)
\(744\) 8.00000 0.293294
\(745\) −4.00000 −0.146549
\(746\) −7.00000 −0.256288
\(747\) −10.0000 −0.365881
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 1.00000 0.0365148
\(751\) 48.0000 1.75154 0.875772 − 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) −7.00000 −0.255264
\(753\) 31.0000 1.12970
\(754\) −4.00000 −0.145671
\(755\) 8.00000 0.291150
\(756\) 1.00000 0.0363696
\(757\) 35.0000 1.27210 0.636048 − 0.771649i \(-0.280568\pi\)
0.636048 + 0.771649i \(0.280568\pi\)
\(758\) −28.0000 −1.01701
\(759\) 35.0000 1.27042
\(760\) 4.00000 0.145095
\(761\) −9.00000 −0.326250 −0.163125 − 0.986605i \(-0.552157\pi\)
−0.163125 + 0.986605i \(0.552157\pi\)
\(762\) 4.00000 0.144905
\(763\) −16.0000 −0.579239
\(764\) 4.00000 0.144715
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 9.00000 0.324971
\(768\) −1.00000 −0.0360844
\(769\) 35.0000 1.26213 0.631066 − 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(770\) 5.00000 0.180187
\(771\) −6.00000 −0.216085
\(772\) 0 0
\(773\) 30.0000 1.07903 0.539513 − 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 6.00000 0.215666
\(775\) −8.00000 −0.287368
\(776\) −12.0000 −0.430775
\(777\) 6.00000 0.215249
\(778\) −36.0000 −1.29066
\(779\) −8.00000 −0.286630
\(780\) 1.00000 0.0358057
\(781\) −50.0000 −1.78914
\(782\) 0 0
\(783\) 4.00000 0.142948
\(784\) −6.00000 −0.214286
\(785\) −17.0000 −0.606756
\(786\) 7.00000 0.249682
\(787\) −48.0000 −1.71102 −0.855508 − 0.517790i \(-0.826755\pi\)
−0.855508 + 0.517790i \(0.826755\pi\)
\(788\) 7.00000 0.249365
\(789\) 31.0000 1.10363
\(790\) 0 0
\(791\) 0 0
\(792\) 5.00000 0.177667
\(793\) −10.0000 −0.355110
\(794\) −9.00000 −0.319398
\(795\) 13.0000 0.461062
\(796\) −20.0000 −0.708881
\(797\) 37.0000 1.31061 0.655304 − 0.755366i \(-0.272541\pi\)
0.655304 + 0.755366i \(0.272541\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 7.00000 0.247333
\(802\) 27.0000 0.953403
\(803\) −40.0000 −1.41157
\(804\) −4.00000 −0.141069
\(805\) −7.00000 −0.246718
\(806\) −8.00000 −0.281788
\(807\) 6.00000 0.211210
\(808\) −10.0000 −0.351799
\(809\) 42.0000 1.47664 0.738321 − 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 29.0000 1.01833 0.509164 − 0.860670i \(-0.329955\pi\)
0.509164 + 0.860670i \(0.329955\pi\)
\(812\) 4.00000 0.140372
\(813\) 18.0000 0.631288
\(814\) 30.0000 1.05150
\(815\) 2.00000 0.0700569
\(816\) 0 0
\(817\) −24.0000 −0.839654
\(818\) −11.0000 −0.384606
\(819\) −1.00000 −0.0349428
\(820\) −2.00000 −0.0698430
\(821\) −48.0000 −1.67521 −0.837606 − 0.546275i \(-0.816045\pi\)
−0.837606 + 0.546275i \(0.816045\pi\)
\(822\) 12.0000 0.418548
\(823\) 32.0000 1.11545 0.557725 − 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 0 0
\(825\) −5.00000 −0.174078
\(826\) −9.00000 −0.313150
\(827\) 2.00000 0.0695468 0.0347734 − 0.999395i \(-0.488929\pi\)
0.0347734 + 0.999395i \(0.488929\pi\)
\(828\) −7.00000 −0.243267
\(829\) 20.0000 0.694629 0.347314 − 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 10.0000 0.347105
\(831\) 1.00000 0.0346896
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) −4.00000 −0.138509
\(835\) −15.0000 −0.519096
\(836\) −20.0000 −0.691714
\(837\) 8.00000 0.276520
\(838\) 15.0000 0.518166
\(839\) 42.0000 1.45000 0.725001 − 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −13.0000 −0.448276
\(842\) −12.0000 −0.413547
\(843\) 17.0000 0.585511
\(844\) −23.0000 −0.791693
\(845\) 12.0000 0.412813
\(846\) −7.00000 −0.240665
\(847\) −14.0000 −0.481046
\(848\) 13.0000 0.446422
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) −42.0000 −1.43974
\(852\) 10.0000 0.342594
\(853\) −37.0000 −1.26686 −0.633428 − 0.773802i \(-0.718353\pi\)
−0.633428 + 0.773802i \(0.718353\pi\)
\(854\) 10.0000 0.342193
\(855\) 4.00000 0.136797
\(856\) −12.0000 −0.410152
\(857\) 32.0000 1.09310 0.546550 − 0.837427i \(-0.315941\pi\)
0.546550 + 0.837427i \(0.315941\pi\)
\(858\) −5.00000 −0.170697
\(859\) 4.00000 0.136478 0.0682391 − 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −6.00000 −0.204598
\(861\) 2.00000 0.0681598
\(862\) −40.0000 −1.36241
\(863\) −43.0000 −1.46374 −0.731869 − 0.681446i \(-0.761351\pi\)
−0.731869 + 0.681446i \(0.761351\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −13.0000 −0.442013
\(866\) 16.0000 0.543702
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) 0 0
\(870\) −4.00000 −0.135613
\(871\) 4.00000 0.135535
\(872\) 16.0000 0.541828
\(873\) −12.0000 −0.406138
\(874\) 28.0000 0.947114
\(875\) 1.00000 0.0338062
\(876\) 8.00000 0.270295
\(877\) 41.0000 1.38447 0.692236 − 0.721671i \(-0.256626\pi\)
0.692236 + 0.721671i \(0.256626\pi\)
\(878\) 18.0000 0.607471
\(879\) 29.0000 0.978146
\(880\) −5.00000 −0.168550
\(881\) 3.00000 0.101073 0.0505363 − 0.998722i \(-0.483907\pi\)
0.0505363 + 0.998722i \(0.483907\pi\)
\(882\) −6.00000 −0.202031
\(883\) 50.0000 1.68263 0.841317 − 0.540542i \(-0.181781\pi\)
0.841317 + 0.540542i \(0.181781\pi\)
\(884\) 0 0
\(885\) 9.00000 0.302532
\(886\) 30.0000 1.00787
\(887\) 24.0000 0.805841 0.402921 − 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) −6.00000 −0.201347
\(889\) 4.00000 0.134156
\(890\) −7.00000 −0.234641
\(891\) 5.00000 0.167506
\(892\) 19.0000 0.636167
\(893\) 28.0000 0.936984
\(894\) −4.00000 −0.133780
\(895\) −21.0000 −0.701953
\(896\) −1.00000 −0.0334077
\(897\) 7.00000 0.233723
\(898\) −18.0000 −0.600668
\(899\) 32.0000 1.06726
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 10.0000 0.332964
\(903\) 6.00000 0.199667
\(904\) 0 0
\(905\) 22.0000 0.731305
\(906\) 8.00000 0.265782
\(907\) 28.0000 0.929725 0.464862 − 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 10.0000 0.331862
\(909\) −10.0000 −0.331679
\(910\) 1.00000 0.0331497
\(911\) −40.0000 −1.32526 −0.662630 − 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 4.00000 0.132453
\(913\) −50.0000 −1.65476
\(914\) −8.00000 −0.264616
\(915\) −10.0000 −0.330590
\(916\) 0 0
\(917\) 7.00000 0.231160
\(918\) 0 0
\(919\) 28.0000 0.923635 0.461817 − 0.886975i \(-0.347198\pi\)
0.461817 + 0.886975i \(0.347198\pi\)
\(920\) 7.00000 0.230783
\(921\) 10.0000 0.329511
\(922\) 2.00000 0.0658665
\(923\) −10.0000 −0.329154
\(924\) 5.00000 0.164488
\(925\) 6.00000 0.197279
\(926\) −29.0000 −0.952999
\(927\) 0 0
\(928\) −4.00000 −0.131306
\(929\) 30.0000 0.984268 0.492134 − 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) −8.00000 −0.262330
\(931\) 24.0000 0.786568
\(932\) 0 0
\(933\) 28.0000 0.916679
\(934\) −18.0000 −0.588978
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 20.0000 0.653372 0.326686 − 0.945133i \(-0.394068\pi\)
0.326686 + 0.945133i \(0.394068\pi\)
\(938\) −4.00000 −0.130605
\(939\) 8.00000 0.261070
\(940\) 7.00000 0.228315
\(941\) 58.0000 1.89075 0.945373 − 0.325991i \(-0.105698\pi\)
0.945373 + 0.325991i \(0.105698\pi\)
\(942\) −17.0000 −0.553890
\(943\) −14.0000 −0.455903
\(944\) 9.00000 0.292925
\(945\) −1.00000 −0.0325300
\(946\) 30.0000 0.975384
\(947\) −22.0000 −0.714904 −0.357452 − 0.933932i \(-0.616354\pi\)
−0.357452 + 0.933932i \(0.616354\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) −4.00000 −0.129777
\(951\) −22.0000 −0.713399
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 − 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 13.0000 0.420891
\(955\) −4.00000 −0.129437
\(956\) −12.0000 −0.388108
\(957\) 20.0000 0.646508
\(958\) −16.0000 −0.516937
\(959\) 12.0000 0.387500
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) 6.00000 0.193448
\(963\) −12.0000 −0.386695
\(964\) 7.00000 0.225455
\(965\) 0 0
\(966\) −7.00000 −0.225221
\(967\) 32.0000 1.02905 0.514525 − 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 14.0000 0.449977
\(969\) 0 0
\(970\) 12.0000 0.385297
\(971\) −20.0000 −0.641831 −0.320915 − 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −4.00000 −0.128234
\(974\) 32.0000 1.02535
\(975\) −1.00000 −0.0320256
\(976\) −10.0000 −0.320092
\(977\) −6.00000 −0.191957 −0.0959785 − 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 2.00000 0.0639529
\(979\) 35.0000 1.11860
\(980\) 6.00000 0.191663
\(981\) 16.0000 0.510841
\(982\) 3.00000 0.0957338
\(983\) 32.0000 1.02064 0.510321 − 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −7.00000 −0.223039
\(986\) 0 0
\(987\) −7.00000 −0.222812
\(988\) −4.00000 −0.127257
\(989\) −42.0000 −1.33552
\(990\) −5.00000 −0.158910
\(991\) 2.00000 0.0635321 0.0317660 − 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) −8.00000 −0.254000
\(993\) −25.0000 −0.793351
\(994\) 10.0000 0.317181
\(995\) 20.0000 0.634043
\(996\) 10.0000 0.316862
\(997\) −23.0000 −0.728417 −0.364209 − 0.931317i \(-0.618661\pi\)
−0.364209 + 0.931317i \(0.618661\pi\)
\(998\) −5.00000 −0.158272
\(999\) −6.00000 −0.189832
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 8670.2.a.p.1.1 ✓ 1
17.16 even 2 8670.2.a.z.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8670.2.a.p.1.1 ✓ 1 1.1 even 1 trivial
8670.2.a.z.1.1 yes 1 17.16 even 2