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Properties

Label 5390.2.a.r.1.1
Level $5390$
Weight $2$
Character 5390.1
Self dual yes
Analytic conductor $43.039$
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] = [5390,2,Mod(1,5390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5390, 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("5390.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.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: \(43.0393666895\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5390.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +1.00000 q^{22} -6.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -4.00000 q^{27} -2.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +2.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} -12.0000 q^{41} -4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} +6.00000 q^{46} -12.0000 q^{47} +2.00000 q^{48} -1.00000 q^{50} +12.0000 q^{51} -2.00000 q^{52} +4.00000 q^{54} -1.00000 q^{55} -4.00000 q^{57} +2.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +2.00000 q^{66} +8.00000 q^{67} +6.00000 q^{68} -12.0000 q^{69} +12.0000 q^{71} -1.00000 q^{72} -2.00000 q^{73} +4.00000 q^{74} +2.00000 q^{75} -2.00000 q^{76} +4.00000 q^{78} +14.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +12.0000 q^{82} -12.0000 q^{83} +6.00000 q^{85} +4.00000 q^{86} +1.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} -6.00000 q^{92} -16.0000 q^{93} +12.0000 q^{94} -2.00000 q^{95} -2.00000 q^{96} -8.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.00000 −0.816497
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −2.00000 −0.408248
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −2.00000 −0.365148
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 2.00000 0.324443
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) −12.0000 −1.87409 −0.937043 − 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.00000 0.149071
\(46\) 6.00000 0.884652
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 2.00000 0.288675
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 12.0000 1.68034
\(52\) −2.00000 −0.277350
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 4.00000 0.544331
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 2.00000 0.258199
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 2.00000 0.246183
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 6.00000 0.727607
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 4.00000 0.464991
\(75\) 2.00000 0.230940
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 14.0000 1.57512 0.787562 − 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) 12.0000 1.32518
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) −16.0000 −1.65912
\(94\) 12.0000 1.23771
\(95\) −2.00000 −0.205196
\(96\) −2.00000 −0.204124
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −12.0000 −1.18818
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −4.00000 −0.384900
\(109\) −16.0000 −1.53252 −0.766261 − 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 1.00000 0.0953463
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 4.00000 0.374634
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) −24.0000 −2.16401
\(124\) −8.00000 −0.718421
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.00000 −0.704361
\(130\) 2.00000 0.175412
\(131\) 18.0000 1.57267 0.786334 − 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) −4.00000 −0.344265
\(136\) −6.00000 −0.514496
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 12.0000 1.02151
\(139\) −14.0000 −1.18746 −0.593732 − 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) −12.0000 −1.00702
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −2.00000 −0.163299
\(151\) 14.0000 1.13930 0.569652 − 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 2.00000 0.162221
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) −4.00000 −0.320256
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −14.0000 −1.11378
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 11.0000 0.864242
\(163\) 8.00000 0.626608 0.313304 − 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) −12.0000 −0.937043
\(165\) −2.00000 −0.155700
\(166\) 12.0000 0.931381
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) −2.00000 −0.152944
\(172\) −4.00000 −0.304997
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 1.00000 0.0745356
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 6.00000 0.442326
\(185\) −4.00000 −0.294086
\(186\) 16.0000 1.17318
\(187\) −6.00000 −0.438763
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) −24.0000 −1.73658 −0.868290 − 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 2.00000 0.144338
\(193\) 2.00000 0.143963 0.0719816 − 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 8.00000 0.574367
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 1.00000 0.0710669
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 16.0000 1.12855
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) −12.0000 −0.838116
\(206\) −16.0000 −1.11477
\(207\) −6.00000 −0.417029
\(208\) −2.00000 −0.138675
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 24.0000 1.64445
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) 4.00000 0.272166
\(217\) 0 0
\(218\) 16.0000 1.08366
\(219\) −4.00000 −0.270295
\(220\) −1.00000 −0.0674200
\(221\) −12.0000 −0.807207
\(222\) 8.00000 0.536925
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) −24.0000 −1.59294 −0.796468 − 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) −4.00000 −0.264906
\(229\) 10.0000 0.660819 0.330409 − 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 28.0000 1.81880
\(238\) 0 0
\(239\) 18.0000 1.16432 0.582162 − 0.813073i \(-0.302207\pi\)
0.582162 + 0.813073i \(0.302207\pi\)
\(240\) 2.00000 0.129099
\(241\) −20.0000 −1.28831 −0.644157 − 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −10.0000 −0.641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 24.0000 1.53018
\(247\) 4.00000 0.254514
\(248\) 8.00000 0.508001
\(249\) −24.0000 −1.52094
\(250\) −1.00000 −0.0632456
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) −8.00000 −0.501965
\(255\) 12.0000 0.751469
\(256\) 1.00000 0.0625000
\(257\) −24.0000 −1.49708 −0.748539 − 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 8.00000 0.498058
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) −18.0000 −1.11204
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 8.00000 0.488678
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 4.00000 0.243432
\(271\) −20.0000 −1.21491 −0.607457 − 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −1.00000 −0.0603023
\(276\) −12.0000 −0.722315
\(277\) −22.0000 −1.32185 −0.660926 − 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 14.0000 0.839664
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 24.0000 1.42918
\(283\) 16.0000 0.951101 0.475551 − 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 12.0000 0.712069
\(285\) −4.00000 −0.236940
\(286\) −2.00000 −0.118262
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) −2.00000 −0.117041
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 2.00000 0.115470
\(301\) 0 0
\(302\) −14.0000 −0.805609
\(303\) 12.0000 0.689382
\(304\) −2.00000 −0.114708
\(305\) −2.00000 −0.114520
\(306\) −6.00000 −0.342997
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 32.0000 1.82042
\(310\) 8.00000 0.454369
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 4.00000 0.226455
\(313\) −8.00000 −0.452187 −0.226093 − 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) −12.0000 −0.673987 −0.336994 − 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) −11.0000 −0.611111
\(325\) −2.00000 −0.110940
\(326\) −8.00000 −0.443079
\(327\) −32.0000 −1.76960
\(328\) 12.0000 0.662589
\(329\) 0 0
\(330\) 2.00000 0.110096
\(331\) −28.0000 −1.53902 −0.769510 − 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −12.0000 −0.658586
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 9.00000 0.489535
\(339\) −12.0000 −0.651751
\(340\) 6.00000 0.325396
\(341\) 8.00000 0.433224
\(342\) 2.00000 0.108148
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) −12.0000 −0.646058
\(346\) −18.0000 −0.967686
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 1.00000 0.0533002
\(353\) 24.0000 1.27739 0.638696 − 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 30.0000 1.58334 0.791670 − 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −15.0000 −0.789474
\(362\) 2.00000 0.105118
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 4.00000 0.209083
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −6.00000 −0.312772
\(369\) −12.0000 −0.624695
\(370\) 4.00000 0.207950
\(371\) 0 0
\(372\) −16.0000 −0.829561
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 6.00000 0.310253
\(375\) 2.00000 0.103280
\(376\) 12.0000 0.618853
\(377\) 0 0
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 − 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) −2.00000 −0.102598
\(381\) 16.0000 0.819705
\(382\) 24.0000 1.22795
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) −8.00000 −0.406138
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 4.00000 0.202548
\(391\) −36.0000 −1.82060
\(392\) 0 0
\(393\) 36.0000 1.81596
\(394\) −6.00000 −0.302276
\(395\) 14.0000 0.704416
\(396\) −1.00000 −0.0502519
\(397\) −38.0000 −1.90717 −0.953583 − 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 30.0000 1.49813 0.749064 − 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −16.0000 −0.798007
\(403\) 16.0000 0.797017
\(404\) 6.00000 0.298511
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) −12.0000 −0.594089
\(409\) 4.00000 0.197787 0.0988936 − 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 12.0000 0.592638
\(411\) −12.0000 −0.591916
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) −12.0000 −0.589057
\(416\) 2.00000 0.0980581
\(417\) −28.0000 −1.37117
\(418\) −2.00000 −0.0978232
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.00000 0.194717
\(423\) −12.0000 −0.583460
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) −24.0000 −1.16280
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 4.00000 0.193122
\(430\) 4.00000 0.192897
\(431\) 30.0000 1.44505 0.722525 − 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) −4.00000 −0.192450
\(433\) 4.00000 0.192228 0.0961139 − 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 12.0000 0.574038
\(438\) 4.00000 0.191127
\(439\) 28.0000 1.33637 0.668184 − 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) −8.00000 −0.379663
\(445\) −6.00000 −0.284427
\(446\) 8.00000 0.378811
\(447\) 0 0
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 − 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 12.0000 0.565058
\(452\) −6.00000 −0.282216
\(453\) 28.0000 1.31555
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −10.0000 −0.467269
\(459\) −24.0000 −1.12022
\(460\) −6.00000 −0.279751
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 − 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 0 0
\(465\) −16.0000 −0.741982
\(466\) −6.00000 −0.277945
\(467\) 6.00000 0.277647 0.138823 − 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 12.0000 0.553519
\(471\) −28.0000 −1.29017
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) −28.0000 −1.28608
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) 0 0
\(478\) −18.0000 −0.823301
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 8.00000 0.364769
\(482\) 20.0000 0.910975
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −8.00000 −0.363261
\(486\) 10.0000 0.453609
\(487\) 2.00000 0.0906287 0.0453143 − 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 2.00000 0.0905357
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 − 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) −24.0000 −1.08200
\(493\) 0 0
\(494\) −4.00000 −0.179969
\(495\) −1.00000 −0.0449467
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 24.0000 1.07547
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) −6.00000 −0.266733
\(507\) −18.0000 −0.799408
\(508\) 8.00000 0.354943
\(509\) −18.0000 −0.797836 −0.398918 − 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) −12.0000 −0.531369
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 8.00000 0.353209
\(514\) 24.0000 1.05859
\(515\) 16.0000 0.705044
\(516\) −8.00000 −0.352180
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 36.0000 1.58022
\(520\) 2.00000 0.0877058
\(521\) −30.0000 −1.31432 −0.657162 − 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) −44.0000 −1.92399 −0.961993 − 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −48.0000 −2.09091
\(528\) −2.00000 −0.0870388
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) 12.0000 0.519291
\(535\) 12.0000 0.518805
\(536\) −8.00000 −0.345547
\(537\) −24.0000 −1.03568
\(538\) −18.0000 −0.776035
\(539\) 0 0
\(540\) −4.00000 −0.172133
\(541\) 20.0000 0.859867 0.429934 − 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 20.0000 0.859074
\(543\) −4.00000 −0.171656
\(544\) −6.00000 −0.257248
\(545\) −16.0000 −0.685365
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −6.00000 −0.256307
\(549\) −2.00000 −0.0853579
\(550\) 1.00000 0.0426401
\(551\) 0 0
\(552\) 12.0000 0.510754
\(553\) 0 0
\(554\) 22.0000 0.934690
\(555\) −8.00000 −0.339581
\(556\) −14.0000 −0.593732
\(557\) 6.00000 0.254228 0.127114 − 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 8.00000 0.338667
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) −6.00000 −0.253095
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −24.0000 −1.01058
\(565\) −6.00000 −0.252422
\(566\) −16.0000 −0.672530
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 4.00000 0.167542
\(571\) 32.0000 1.33916 0.669579 − 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 2.00000 0.0836242
\(573\) −48.0000 −2.00523
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) −8.00000 −0.333044 −0.166522 − 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) −19.0000 −0.790296
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 0 0
\(582\) 16.0000 0.663221
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) −2.00000 −0.0826898
\(586\) −6.00000 −0.247858
\(587\) 30.0000 1.23823 0.619116 − 0.785299i \(-0.287491\pi\)
0.619116 + 0.785299i \(0.287491\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) −4.00000 −0.164399
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 0 0
\(597\) 32.0000 1.30967
\(598\) −12.0000 −0.490716
\(599\) −12.0000 −0.490307 −0.245153 − 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) −2.00000 −0.0816497
\(601\) −32.0000 −1.30531 −0.652654 − 0.757656i \(-0.726344\pi\)
−0.652654 + 0.757656i \(0.726344\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 14.0000 0.569652
\(605\) 1.00000 0.0406558
\(606\) −12.0000 −0.487467
\(607\) 40.0000 1.62355 0.811775 − 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) 24.0000 0.970936
\(612\) 6.00000 0.242536
\(613\) 14.0000 0.565455 0.282727 − 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −4.00000 −0.161427
\(615\) −24.0000 −0.967773
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −32.0000 −1.28723
\(619\) 16.0000 0.643094 0.321547 − 0.946894i \(-0.395797\pi\)
0.321547 + 0.946894i \(0.395797\pi\)
\(620\) −8.00000 −0.321288
\(621\) 24.0000 0.963087
\(622\) 24.0000 0.962312
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 8.00000 0.319744
\(627\) 4.00000 0.159745
\(628\) −14.0000 −0.558661
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 − 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −14.0000 −0.556890
\(633\) −8.00000 −0.317971
\(634\) 12.0000 0.476581
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −24.0000 −0.947204
\(643\) 34.0000 1.34083 0.670415 − 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 12.0000 0.472134
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 11.0000 0.432121
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 24.0000 0.939193 0.469596 − 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) 32.0000 1.25130
\(655\) 18.0000 0.703318
\(656\) −12.0000 −0.468521
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 22.0000 0.855701 0.427850 − 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 28.0000 1.08825
\(663\) −24.0000 −0.932083
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 0 0
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) −8.00000 −0.309067
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 − 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −14.0000 −0.539260
\(675\) −4.00000 −0.153960
\(676\) −9.00000 −0.346154
\(677\) 42.0000 1.61419 0.807096 − 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 12.0000 0.460857
\(679\) 0 0
\(680\) −6.00000 −0.230089
\(681\) −48.0000 −1.83936
\(682\) −8.00000 −0.306336
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 20.0000 0.763048
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 12.0000 0.456832
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −14.0000 −0.531050
\(696\) 0 0
\(697\) −72.0000 −2.72719
\(698\) 14.0000 0.529908
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 − 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) −8.00000 −0.301941
\(703\) 8.00000 0.301726
\(704\) −1.00000 −0.0376889
\(705\) −24.0000 −0.903892
\(706\) −24.0000 −0.903252
\(707\) 0 0
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 − 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) −12.0000 −0.450352
\(711\) 14.0000 0.525041
\(712\) 6.00000 0.224860
\(713\) 48.0000 1.79761
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) −12.0000 −0.448461
\(717\) 36.0000 1.34444
\(718\) −30.0000 −1.11959
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 15.0000 0.558242
\(723\) −40.0000 −1.48762
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) −20.0000 −0.741759 −0.370879 − 0.928681i \(-0.620944\pi\)
−0.370879 + 0.928681i \(0.620944\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 2.00000 0.0740233
\(731\) −24.0000 −0.887672
\(732\) −4.00000 −0.147844
\(733\) −14.0000 −0.517102 −0.258551 − 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) −8.00000 −0.294684
\(738\) 12.0000 0.441726
\(739\) −4.00000 −0.147142 −0.0735712 − 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) −4.00000 −0.147043
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 16.0000 0.586588
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) −12.0000 −0.439057
\(748\) −6.00000 −0.219382
\(749\) 0 0
\(750\) −2.00000 −0.0730297
\(751\) −4.00000 −0.145962 −0.0729810 − 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 0 0
\(755\) 14.0000 0.509512
\(756\) 0 0
\(757\) −16.0000 −0.581530 −0.290765 − 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 28.0000 1.01701
\(759\) 12.0000 0.435572
\(760\) 2.00000 0.0725476
\(761\) 12.0000 0.435000 0.217500 − 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) −16.0000 −0.579619
\(763\) 0 0
\(764\) −24.0000 −0.868290
\(765\) 6.00000 0.216930
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) 2.00000 0.0721688
\(769\) 40.0000 1.44244 0.721218 − 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) −48.0000 −1.72868
\(772\) 2.00000 0.0719816
\(773\) −42.0000 −1.51064 −0.755318 − 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 4.00000 0.143777
\(775\) −8.00000 −0.287368
\(776\) 8.00000 0.287183
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 24.0000 0.859889
\(780\) −4.00000 −0.143223
\(781\) −12.0000 −0.429394
\(782\) 36.0000 1.28736
\(783\) 0 0
\(784\) 0 0
\(785\) −14.0000 −0.499681
\(786\) −36.0000 −1.28408
\(787\) 16.0000 0.570338 0.285169 − 0.958477i \(-0.407950\pi\)
0.285169 + 0.958477i \(0.407950\pi\)
\(788\) 6.00000 0.213741
\(789\) 48.0000 1.70885
\(790\) −14.0000 −0.498098
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 4.00000 0.142044
\(794\) 38.0000 1.34857
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) 42.0000 1.48772 0.743858 − 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) −30.0000 −1.05934
\(803\) 2.00000 0.0705785
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 36.0000 1.26726
\(808\) −6.00000 −0.211079
\(809\) 30.0000 1.05474 0.527372 − 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 11.0000 0.386501
\(811\) −38.0000 −1.33436 −0.667180 − 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 0 0
\(813\) −40.0000 −1.40286
\(814\) −4.00000 −0.140200
\(815\) 8.00000 0.280228
\(816\) 12.0000 0.420084
\(817\) 8.00000 0.279885
\(818\) −4.00000 −0.139857
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) 12.0000 0.418803 0.209401 − 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 12.0000 0.418548
\(823\) 50.0000 1.74289 0.871445 − 0.490493i \(-0.163183\pi\)
0.871445 + 0.490493i \(0.163183\pi\)
\(824\) −16.0000 −0.557386
\(825\) −2.00000 −0.0696311
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −6.00000 −0.208514
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 12.0000 0.416526
\(831\) −44.0000 −1.52634
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 28.0000 0.969561
\(835\) 0 0
\(836\) 2.00000 0.0691714
\(837\) 32.0000 1.10608
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 10.0000 0.344623
\(843\) 12.0000 0.413302
\(844\) −4.00000 −0.137686
\(845\) −9.00000 −0.309609
\(846\) 12.0000 0.412568
\(847\) 0 0
\(848\) 0 0
\(849\) 32.0000 1.09824
\(850\) −6.00000 −0.205798
\(851\) 24.0000 0.822709
\(852\) 24.0000 0.822226
\(853\) 10.0000 0.342393 0.171197 − 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) −12.0000 −0.410152
\(857\) −42.0000 −1.43469 −0.717346 − 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) −4.00000 −0.136558
\(859\) 16.0000 0.545913 0.272956 − 0.962026i \(-0.411998\pi\)
0.272956 + 0.962026i \(0.411998\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −30.0000 −1.02180
\(863\) 18.0000 0.612727 0.306364 − 0.951915i \(-0.400888\pi\)
0.306364 + 0.951915i \(0.400888\pi\)
\(864\) 4.00000 0.136083
\(865\) 18.0000 0.612018
\(866\) −4.00000 −0.135926
\(867\) 38.0000 1.29055
\(868\) 0 0
\(869\) −14.0000 −0.474917
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 16.0000 0.541828
\(873\) −8.00000 −0.270759
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) −10.0000 −0.337676 −0.168838 − 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) −28.0000 −0.944954
\(879\) 12.0000 0.404750
\(880\) −1.00000 −0.0337100
\(881\) 30.0000 1.01073 0.505363 − 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) −48.0000 −1.61168 −0.805841 − 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 8.00000 0.268462
\(889\) 0 0
\(890\) 6.00000 0.201120
\(891\) 11.0000 0.368514
\(892\) −8.00000 −0.267860
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 24.0000 0.801337
\(898\) 18.0000 0.600668
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) −12.0000 −0.399556
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −2.00000 −0.0664822
\(906\) −28.0000 −0.930238
\(907\) 20.0000 0.664089 0.332045 − 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) −24.0000 −0.796468
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 − 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) −4.00000 −0.132453
\(913\) 12.0000 0.397142
\(914\) 10.0000 0.330771
\(915\) −4.00000 −0.132236
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 24.0000 0.792118
\(919\) −34.0000 −1.12156 −0.560778 − 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 6.00000 0.197814
\(921\) 8.00000 0.263609
\(922\) −18.0000 −0.592798
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 22.0000 0.722965
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) −54.0000 −1.77168 −0.885841 − 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 16.0000 0.524661
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) −48.0000 −1.57145
\(934\) −6.00000 −0.196326
\(935\) −6.00000 −0.196221
\(936\) 2.00000 0.0653720
\(937\) −38.0000 −1.24141 −0.620703 − 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) −16.0000 −0.522140
\(940\) −12.0000 −0.391397
\(941\) −18.0000 −0.586783 −0.293392 − 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 28.0000 0.912289
\(943\) 72.0000 2.34464
\(944\) 0 0
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 28.0000 0.909398
\(949\) 4.00000 0.129845
\(950\) 2.00000 0.0648886
\(951\) −24.0000 −0.778253
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 − 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) 18.0000 0.582162
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) 33.0000 1.06452
\(962\) −8.00000 −0.257930
\(963\) 12.0000 0.386695
\(964\) −20.0000 −0.644157
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −24.0000 −0.770991
\(970\) 8.00000 0.256865
\(971\) 48.0000 1.54039 0.770197 − 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) −10.0000 −0.320750
\(973\) 0 0
\(974\) −2.00000 −0.0640841
\(975\) −4.00000 −0.128103
\(976\) −2.00000 −0.0640184
\(977\) 42.0000 1.34370 0.671850 − 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) −16.0000 −0.511624
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 36.0000 1.14881
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 24.0000 0.765092
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 24.0000 0.763156
\(990\) 1.00000 0.0317821
\(991\) 32.0000 1.01651 0.508257 − 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 8.00000 0.254000
\(993\) −56.0000 −1.77711
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) −24.0000 −0.760469
\(997\) −38.0000 −1.20347 −0.601736 − 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) 4.00000 0.126618
\(999\) 16.0000 0.506218
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 5390.2.a.r.1.1 1
7.6 odd 2 770.2.a.a.1.1 ✓ 1
21.20 even 2 6930.2.a.bm.1.1 1
28.27 even 2 6160.2.a.k.1.1 1
35.13 even 4 3850.2.c.o.1849.2 2
35.27 even 4 3850.2.c.o.1849.1 2
35.34 odd 2 3850.2.a.ba.1.1 1
77.76 even 2 8470.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.a.1.1 ✓ 1 7.6 odd 2
3850.2.a.ba.1.1 1 35.34 odd 2
3850.2.c.o.1849.1 2 35.27 even 4
3850.2.c.o.1849.2 2 35.13 even 4
5390.2.a.r.1.1 1 1.1 even 1 trivial
6160.2.a.k.1.1 1 28.27 even 2
6930.2.a.bm.1.1 1 21.20 even 2
8470.2.a.r.1.1 1 77.76 even 2