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Properties

Label 4018.2.a.g.1.1
Level $4018$
Weight $2$
Character 4018.1
Self dual yes
Analytic conductor $32.084$
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] = [4018,2,Mod(1,4018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4018, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4018.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: \(32.0838915322\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 574)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4018.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} +2.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +6.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +2.00000 q^{26} -5.00000 q^{27} -9.00000 q^{29} +1.00000 q^{30} -3.00000 q^{31} -1.00000 q^{32} -5.00000 q^{34} -2.00000 q^{36} -8.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +1.00000 q^{41} +7.00000 q^{43} +2.00000 q^{45} -6.00000 q^{46} +10.0000 q^{47} +1.00000 q^{48} +4.00000 q^{50} +5.00000 q^{51} -2.00000 q^{52} -13.0000 q^{53} +5.00000 q^{54} +4.00000 q^{57} +9.00000 q^{58} +14.0000 q^{59} -1.00000 q^{60} +5.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +4.00000 q^{67} +5.00000 q^{68} +6.00000 q^{69} -15.0000 q^{71} +2.00000 q^{72} +6.00000 q^{73} +8.00000 q^{74} -4.00000 q^{75} +4.00000 q^{76} +2.00000 q^{78} -3.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -1.00000 q^{82} -16.0000 q^{83} -5.00000 q^{85} -7.00000 q^{86} -9.00000 q^{87} -13.0000 q^{89} -2.00000 q^{90} +6.00000 q^{92} -3.00000 q^{93} -10.0000 q^{94} -4.00000 q^{95} -1.00000 q^{96} -17.0000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) 1.00000 0.316228
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 2.00000 0.471405
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) 2.00000 0.392232
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 1.00000 0.182574
\(31\) −3.00000 −0.538816 −0.269408 − 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −5.00000 −0.857493
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) −4.00000 −0.648886
\(39\) −2.00000 −0.320256
\(40\) 1.00000 0.158114
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) 7.00000 1.06749 0.533745 − 0.845645i \(-0.320784\pi\)
0.533745 + 0.845645i \(0.320784\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) −6.00000 −0.884652
\(47\) 10.0000 1.45865 0.729325 − 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) 5.00000 0.700140
\(52\) −2.00000 −0.277350
\(53\) −13.0000 −1.78569 −0.892844 − 0.450367i \(-0.851293\pi\)
−0.892844 + 0.450367i \(0.851293\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 9.00000 1.18176
\(59\) 14.0000 1.82264 0.911322 − 0.411693i \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) −1.00000 −0.129099
\(61\) 5.00000 0.640184 0.320092 − 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 3.00000 0.381000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 5.00000 0.606339
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 − 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 2.00000 0.235702
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 8.00000 0.929981
\(75\) −4.00000 −0.461880
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) −3.00000 −0.337526 −0.168763 − 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −1.00000 −0.110432
\(83\) −16.0000 −1.75623 −0.878114 − 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) −5.00000 −0.542326
\(86\) −7.00000 −0.754829
\(87\) −9.00000 −0.964901
\(88\) 0 0
\(89\) −13.0000 −1.37800 −0.688999 − 0.724763i \(-0.741949\pi\)
−0.688999 + 0.724763i \(0.741949\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) −3.00000 −0.311086
\(94\) −10.0000 −1.03142
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) −17.0000 −1.72609 −0.863044 − 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) −18.0000 −1.79107 −0.895533 − 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) −5.00000 −0.495074
\(103\) −7.00000 −0.689730 −0.344865 − 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 13.0000 1.26267
\(107\) 3.00000 0.290021 0.145010 − 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) −5.00000 −0.481125
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 1.00000 0.0940721 0.0470360 − 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) −4.00000 −0.374634
\(115\) −6.00000 −0.559503
\(116\) −9.00000 −0.835629
\(117\) 4.00000 0.369800
\(118\) −14.0000 −1.28880
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −11.0000 −1.00000
\(122\) −5.00000 −0.452679
\(123\) 1.00000 0.0901670
\(124\) −3.00000 −0.269408
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 18.0000 1.59724 0.798621 − 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.00000 0.616316
\(130\) −2.00000 −0.175412
\(131\) 18.0000 1.57267 0.786334 − 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 5.00000 0.430331
\(136\) −5.00000 −0.428746
\(137\) −18.0000 −1.53784 −0.768922 − 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) −6.00000 −0.510754
\(139\) −14.0000 −1.18746 −0.593732 − 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 10.0000 0.842152
\(142\) 15.0000 1.25877
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) 9.00000 0.747409
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) 7.00000 0.573462 0.286731 − 0.958011i \(-0.407431\pi\)
0.286731 + 0.958011i \(0.407431\pi\)
\(150\) 4.00000 0.326599
\(151\) −5.00000 −0.406894 −0.203447 − 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) −4.00000 −0.324443
\(153\) −10.0000 −0.808452
\(154\) 0 0
\(155\) 3.00000 0.240966
\(156\) −2.00000 −0.160128
\(157\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 3.00000 0.238667
\(159\) −13.0000 −1.03097
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 1.00000 0.0780869
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 5.00000 0.383482
\(171\) −8.00000 −0.611775
\(172\) 7.00000 0.533745
\(173\) −9.00000 −0.684257 −0.342129 − 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) 9.00000 0.682288
\(175\) 0 0
\(176\) 0 0
\(177\) 14.0000 1.05230
\(178\) 13.0000 0.974391
\(179\) −24.0000 −1.79384 −0.896922 − 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 2.00000 0.149071
\(181\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 5.00000 0.369611
\(184\) −6.00000 −0.442326
\(185\) 8.00000 0.588172
\(186\) 3.00000 0.219971
\(187\) 0 0
\(188\) 10.0000 0.729325
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) −15.0000 −1.08536 −0.542681 − 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 1.00000 0.0721688
\(193\) 16.0000 1.15171 0.575853 − 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 17.0000 1.22053
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 4.00000 0.282843
\(201\) 4.00000 0.282138
\(202\) 18.0000 1.26648
\(203\) 0 0
\(204\) 5.00000 0.350070
\(205\) −1.00000 −0.0698430
\(206\) 7.00000 0.487713
\(207\) −12.0000 −0.834058
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −13.0000 −0.892844
\(213\) −15.0000 −1.02778
\(214\) −3.00000 −0.205076
\(215\) −7.00000 −0.477396
\(216\) 5.00000 0.340207
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) −10.0000 −0.672673
\(222\) 8.00000 0.536925
\(223\) −3.00000 −0.200895 −0.100447 − 0.994942i \(-0.532027\pi\)
−0.100447 + 0.994942i \(0.532027\pi\)
\(224\) 0 0
\(225\) 8.00000 0.533333
\(226\) −1.00000 −0.0665190
\(227\) −7.00000 −0.464606 −0.232303 − 0.972643i \(-0.574626\pi\)
−0.232303 + 0.972643i \(0.574626\pi\)
\(228\) 4.00000 0.264906
\(229\) −8.00000 −0.528655 −0.264327 − 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 9.00000 0.590879
\(233\) −12.0000 −0.786146 −0.393073 − 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) −4.00000 −0.261488
\(235\) −10.0000 −0.652328
\(236\) 14.0000 0.911322
\(237\) −3.00000 −0.194871
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 20.0000 1.28831 0.644157 − 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 11.0000 0.707107
\(243\) 16.0000 1.02640
\(244\) 5.00000 0.320092
\(245\) 0 0
\(246\) −1.00000 −0.0637577
\(247\) −8.00000 −0.509028
\(248\) 3.00000 0.190500
\(249\) −16.0000 −1.01396
\(250\) −9.00000 −0.569210
\(251\) 8.00000 0.504956 0.252478 − 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −18.0000 −1.12942
\(255\) −5.00000 −0.313112
\(256\) 1.00000 0.0625000
\(257\) −3.00000 −0.187135 −0.0935674 − 0.995613i \(-0.529827\pi\)
−0.0935674 + 0.995613i \(0.529827\pi\)
\(258\) −7.00000 −0.435801
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) 18.0000 1.11417
\(262\) −18.0000 −1.11204
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 13.0000 0.798584
\(266\) 0 0
\(267\) −13.0000 −0.795587
\(268\) 4.00000 0.244339
\(269\) −10.0000 −0.609711 −0.304855 − 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) −5.00000 −0.304290
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 5.00000 0.303170
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) 20.0000 1.20168 0.600842 − 0.799368i \(-0.294832\pi\)
0.600842 + 0.799368i \(0.294832\pi\)
\(278\) 14.0000 0.839664
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 − 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) −10.0000 −0.595491
\(283\) 12.0000 0.713326 0.356663 − 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) −15.0000 −0.890086
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) 0 0
\(288\) 2.00000 0.117851
\(289\) 8.00000 0.470588
\(290\) −9.00000 −0.528498
\(291\) −17.0000 −0.996558
\(292\) 6.00000 0.351123
\(293\) −24.0000 −1.40209 −0.701047 − 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 0 0
\(295\) −14.0000 −0.815112
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) −7.00000 −0.405499
\(299\) −12.0000 −0.693978
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) 5.00000 0.287718
\(303\) −18.0000 −1.03407
\(304\) 4.00000 0.229416
\(305\) −5.00000 −0.286299
\(306\) 10.0000 0.571662
\(307\) 16.0000 0.913168 0.456584 − 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) −3.00000 −0.170389
\(311\) −18.0000 −1.02069 −0.510343 − 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 2.00000 0.113228
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −3.00000 −0.168763
\(317\) −22.0000 −1.23564 −0.617822 − 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 13.0000 0.729004
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 3.00000 0.167444
\(322\) 0 0
\(323\) 20.0000 1.11283
\(324\) 1.00000 0.0555556
\(325\) 8.00000 0.443760
\(326\) −12.0000 −0.664619
\(327\) −2.00000 −0.110600
\(328\) −1.00000 −0.0552158
\(329\) 0 0
\(330\) 0 0
\(331\) 26.0000 1.42909 0.714545 − 0.699590i \(-0.246634\pi\)
0.714545 + 0.699590i \(0.246634\pi\)
\(332\) −16.0000 −0.878114
\(333\) 16.0000 0.876795
\(334\) 12.0000 0.656611
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −35.0000 −1.90657 −0.953286 − 0.302070i \(-0.902322\pi\)
−0.953286 + 0.302070i \(0.902322\pi\)
\(338\) 9.00000 0.489535
\(339\) 1.00000 0.0543125
\(340\) −5.00000 −0.271163
\(341\) 0 0
\(342\) 8.00000 0.432590
\(343\) 0 0
\(344\) −7.00000 −0.377415
\(345\) −6.00000 −0.323029
\(346\) 9.00000 0.483843
\(347\) 24.0000 1.28839 0.644194 − 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) −9.00000 −0.482451
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 10.0000 0.533761
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 − 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −14.0000 −0.744092
\(355\) 15.0000 0.796117
\(356\) −13.0000 −0.688999
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) −5.00000 −0.261354
\(367\) 19.0000 0.991792 0.495896 − 0.868382i \(-0.334840\pi\)
0.495896 + 0.868382i \(0.334840\pi\)
\(368\) 6.00000 0.312772
\(369\) −2.00000 −0.104116
\(370\) −8.00000 −0.415900
\(371\) 0 0
\(372\) −3.00000 −0.155543
\(373\) −16.0000 −0.828449 −0.414224 − 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 0 0
\(375\) 9.00000 0.464758
\(376\) −10.0000 −0.515711
\(377\) 18.0000 0.927047
\(378\) 0 0
\(379\) −15.0000 −0.770498 −0.385249 − 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) −4.00000 −0.205196
\(381\) 18.0000 0.922168
\(382\) 15.0000 0.767467
\(383\) 8.00000 0.408781 0.204390 − 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −16.0000 −0.814379
\(387\) −14.0000 −0.711660
\(388\) −17.0000 −0.863044
\(389\) 20.0000 1.01404 0.507020 − 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) −2.00000 −0.101274
\(391\) 30.0000 1.51717
\(392\) 0 0
\(393\) 18.0000 0.907980
\(394\) 0 0
\(395\) 3.00000 0.150946
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 − 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −5.00000 −0.249688 −0.124844 − 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(402\) −4.00000 −0.199502
\(403\) 6.00000 0.298881
\(404\) −18.0000 −0.895533
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) −5.00000 −0.247537
\(409\) −32.0000 −1.58230 −0.791149 − 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 1.00000 0.0493865
\(411\) −18.0000 −0.887875
\(412\) −7.00000 −0.344865
\(413\) 0 0
\(414\) 12.0000 0.589768
\(415\) 16.0000 0.785409
\(416\) 2.00000 0.0980581
\(417\) −14.0000 −0.685583
\(418\) 0 0
\(419\) 14.0000 0.683945 0.341972 − 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) −7.00000 −0.341159 −0.170580 − 0.985344i \(-0.554564\pi\)
−0.170580 + 0.985344i \(0.554564\pi\)
\(422\) −20.0000 −0.973585
\(423\) −20.0000 −0.972433
\(424\) 13.0000 0.631336
\(425\) −20.0000 −0.970143
\(426\) 15.0000 0.726752
\(427\) 0 0
\(428\) 3.00000 0.145010
\(429\) 0 0
\(430\) 7.00000 0.337570
\(431\) 8.00000 0.385346 0.192673 − 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) −5.00000 −0.240563
\(433\) −6.00000 −0.288342 −0.144171 − 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 0 0
\(435\) 9.00000 0.431517
\(436\) −2.00000 −0.0957826
\(437\) 24.0000 1.14808
\(438\) −6.00000 −0.286691
\(439\) 28.0000 1.33637 0.668184 − 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 10.0000 0.475651
\(443\) 39.0000 1.85295 0.926473 − 0.376361i \(-0.122825\pi\)
0.926473 + 0.376361i \(0.122825\pi\)
\(444\) −8.00000 −0.379663
\(445\) 13.0000 0.616259
\(446\) 3.00000 0.142054
\(447\) 7.00000 0.331089
\(448\) 0 0
\(449\) −33.0000 −1.55737 −0.778683 − 0.627417i \(-0.784112\pi\)
−0.778683 + 0.627417i \(0.784112\pi\)
\(450\) −8.00000 −0.377124
\(451\) 0 0
\(452\) 1.00000 0.0470360
\(453\) −5.00000 −0.234920
\(454\) 7.00000 0.328526
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −18.0000 −0.842004 −0.421002 − 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 8.00000 0.373815
\(459\) −25.0000 −1.16690
\(460\) −6.00000 −0.279751
\(461\) −21.0000 −0.978068 −0.489034 − 0.872265i \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −9.00000 −0.417815
\(465\) 3.00000 0.139122
\(466\) 12.0000 0.555889
\(467\) −32.0000 −1.48078 −0.740392 − 0.672176i \(-0.765360\pi\)
−0.740392 + 0.672176i \(0.765360\pi\)
\(468\) 4.00000 0.184900
\(469\) 0 0
\(470\) 10.0000 0.461266
\(471\) 0 0
\(472\) −14.0000 −0.644402
\(473\) 0 0
\(474\) 3.00000 0.137795
\(475\) −16.0000 −0.734130
\(476\) 0 0
\(477\) 26.0000 1.19046
\(478\) 0 0
\(479\) 26.0000 1.18797 0.593985 − 0.804476i \(-0.297554\pi\)
0.593985 + 0.804476i \(0.297554\pi\)
\(480\) 1.00000 0.0456435
\(481\) 16.0000 0.729537
\(482\) −20.0000 −0.910975
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 17.0000 0.771930
\(486\) −16.0000 −0.725775
\(487\) 26.0000 1.17817 0.589086 − 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) −5.00000 −0.226339
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 5.00000 0.225647 0.112823 − 0.993615i \(-0.464011\pi\)
0.112823 + 0.993615i \(0.464011\pi\)
\(492\) 1.00000 0.0450835
\(493\) −45.0000 −2.02670
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) 0 0
\(498\) 16.0000 0.716977
\(499\) −2.00000 −0.0895323 −0.0447661 − 0.998997i \(-0.514254\pi\)
−0.0447661 + 0.998997i \(0.514254\pi\)
\(500\) 9.00000 0.402492
\(501\) −12.0000 −0.536120
\(502\) −8.00000 −0.357057
\(503\) 10.0000 0.445878 0.222939 − 0.974832i \(-0.428435\pi\)
0.222939 + 0.974832i \(0.428435\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 18.0000 0.798621
\(509\) 42.0000 1.86162 0.930809 − 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 5.00000 0.221404
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −20.0000 −0.883022
\(514\) 3.00000 0.132324
\(515\) 7.00000 0.308457
\(516\) 7.00000 0.308158
\(517\) 0 0
\(518\) 0 0
\(519\) −9.00000 −0.395056
\(520\) −2.00000 −0.0877058
\(521\) −14.0000 −0.613351 −0.306676 − 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) −18.0000 −0.787839
\(523\) −6.00000 −0.262362 −0.131181 − 0.991358i \(-0.541877\pi\)
−0.131181 + 0.991358i \(0.541877\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −15.0000 −0.653410
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −13.0000 −0.564684
\(531\) −28.0000 −1.21510
\(532\) 0 0
\(533\) −2.00000 −0.0866296
\(534\) 13.0000 0.562565
\(535\) −3.00000 −0.129701
\(536\) −4.00000 −0.172774
\(537\) −24.0000 −1.03568
\(538\) 10.0000 0.431131
\(539\) 0 0
\(540\) 5.00000 0.215166
\(541\) 12.0000 0.515920 0.257960 − 0.966156i \(-0.416950\pi\)
0.257960 + 0.966156i \(0.416950\pi\)
\(542\) 16.0000 0.687259
\(543\) 0 0
\(544\) −5.00000 −0.214373
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −36.0000 −1.53925 −0.769624 − 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) −18.0000 −0.768922
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −36.0000 −1.53365
\(552\) −6.00000 −0.255377
\(553\) 0 0
\(554\) −20.0000 −0.849719
\(555\) 8.00000 0.339581
\(556\) −14.0000 −0.593732
\(557\) 13.0000 0.550828 0.275414 − 0.961326i \(-0.411185\pi\)
0.275414 + 0.961326i \(0.411185\pi\)
\(558\) −6.00000 −0.254000
\(559\) −14.0000 −0.592137
\(560\) 0 0
\(561\) 0 0
\(562\) 22.0000 0.928014
\(563\) −20.0000 −0.842900 −0.421450 − 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 10.0000 0.421076
\(565\) −1.00000 −0.0420703
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) 15.0000 0.629386
\(569\) 27.0000 1.13190 0.565949 − 0.824440i \(-0.308510\pi\)
0.565949 + 0.824440i \(0.308510\pi\)
\(570\) 4.00000 0.167542
\(571\) −26.0000 −1.08807 −0.544033 − 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) 0 0
\(573\) −15.0000 −0.626634
\(574\) 0 0
\(575\) −24.0000 −1.00087
\(576\) −2.00000 −0.0833333
\(577\) 42.0000 1.74848 0.874241 − 0.485491i \(-0.161359\pi\)
0.874241 + 0.485491i \(0.161359\pi\)
\(578\) −8.00000 −0.332756
\(579\) 16.0000 0.664937
\(580\) 9.00000 0.373705
\(581\) 0 0
\(582\) 17.0000 0.704673
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) −4.00000 −0.165380
\(586\) 24.0000 0.991431
\(587\) 15.0000 0.619116 0.309558 − 0.950881i \(-0.399819\pi\)
0.309558 + 0.950881i \(0.399819\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 14.0000 0.576371
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) −15.0000 −0.615976 −0.307988 − 0.951390i \(-0.599656\pi\)
−0.307988 + 0.951390i \(0.599656\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.00000 0.286731
\(597\) −4.00000 −0.163709
\(598\) 12.0000 0.490716
\(599\) 6.00000 0.245153 0.122577 − 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 4.00000 0.163299
\(601\) 37.0000 1.50926 0.754631 − 0.656150i \(-0.227816\pi\)
0.754631 + 0.656150i \(0.227816\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) −5.00000 −0.203447
\(605\) 11.0000 0.447214
\(606\) 18.0000 0.731200
\(607\) −3.00000 −0.121766 −0.0608831 − 0.998145i \(-0.519392\pi\)
−0.0608831 + 0.998145i \(0.519392\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 5.00000 0.202444
\(611\) −20.0000 −0.809113
\(612\) −10.0000 −0.404226
\(613\) 28.0000 1.13091 0.565455 − 0.824779i \(-0.308701\pi\)
0.565455 + 0.824779i \(0.308701\pi\)
\(614\) −16.0000 −0.645707
\(615\) −1.00000 −0.0403239
\(616\) 0 0
\(617\) −26.0000 −1.04672 −0.523360 − 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 7.00000 0.281581
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 3.00000 0.120483
\(621\) −30.0000 −1.20386
\(622\) 18.0000 0.721734
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 11.0000 0.440000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) 0 0
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 − 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) 3.00000 0.119334
\(633\) 20.0000 0.794929
\(634\) 22.0000 0.873732
\(635\) −18.0000 −0.714308
\(636\) −13.0000 −0.515484
\(637\) 0 0
\(638\) 0 0
\(639\) 30.0000 1.18678
\(640\) 1.00000 0.0395285
\(641\) 38.0000 1.50091 0.750455 − 0.660922i \(-0.229834\pi\)
0.750455 + 0.660922i \(0.229834\pi\)
\(642\) −3.00000 −0.118401
\(643\) −19.0000 −0.749287 −0.374643 − 0.927169i \(-0.622235\pi\)
−0.374643 + 0.927169i \(0.622235\pi\)
\(644\) 0 0
\(645\) −7.00000 −0.275625
\(646\) −20.0000 −0.786889
\(647\) −39.0000 −1.53325 −0.766624 − 0.642096i \(-0.778065\pi\)
−0.766624 + 0.642096i \(0.778065\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −8.00000 −0.313786
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) −9.00000 −0.352197 −0.176099 − 0.984373i \(-0.556348\pi\)
−0.176099 + 0.984373i \(0.556348\pi\)
\(654\) 2.00000 0.0782062
\(655\) −18.0000 −0.703318
\(656\) 1.00000 0.0390434
\(657\) −12.0000 −0.468165
\(658\) 0 0
\(659\) −18.0000 −0.701180 −0.350590 − 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) 0 0
\(661\) 50.0000 1.94477 0.972387 − 0.233373i \(-0.0749763\pi\)
0.972387 + 0.233373i \(0.0749763\pi\)
\(662\) −26.0000 −1.01052
\(663\) −10.0000 −0.388368
\(664\) 16.0000 0.620920
\(665\) 0 0
\(666\) −16.0000 −0.619987
\(667\) −54.0000 −2.09089
\(668\) −12.0000 −0.464294
\(669\) −3.00000 −0.115987
\(670\) 4.00000 0.154533
\(671\) 0 0
\(672\) 0 0
\(673\) −32.0000 −1.23351 −0.616755 − 0.787155i \(-0.711553\pi\)
−0.616755 + 0.787155i \(0.711553\pi\)
\(674\) 35.0000 1.34815
\(675\) 20.0000 0.769800
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −1.00000 −0.0384048
\(679\) 0 0
\(680\) 5.00000 0.191741
\(681\) −7.00000 −0.268241
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −8.00000 −0.305888
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) −8.00000 −0.305219
\(688\) 7.00000 0.266872
\(689\) 26.0000 0.990521
\(690\) 6.00000 0.228416
\(691\) 13.0000 0.494543 0.247272 − 0.968946i \(-0.420466\pi\)
0.247272 + 0.968946i \(0.420466\pi\)
\(692\) −9.00000 −0.342129
\(693\) 0 0
\(694\) −24.0000 −0.911028
\(695\) 14.0000 0.531050
\(696\) 9.00000 0.341144
\(697\) 5.00000 0.189389
\(698\) 2.00000 0.0757011
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 − 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) −10.0000 −0.377426
\(703\) −32.0000 −1.20690
\(704\) 0 0
\(705\) −10.0000 −0.376622
\(706\) −2.00000 −0.0752710
\(707\) 0 0
\(708\) 14.0000 0.526152
\(709\) −1.00000 −0.0375558 −0.0187779 − 0.999824i \(-0.505978\pi\)
−0.0187779 + 0.999824i \(0.505978\pi\)
\(710\) −15.0000 −0.562940
\(711\) 6.00000 0.225018
\(712\) 13.0000 0.487196
\(713\) −18.0000 −0.674105
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) 0 0
\(719\) 30.0000 1.11881 0.559406 − 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 20.0000 0.743808
\(724\) 0 0
\(725\) 36.0000 1.33701
\(726\) 11.0000 0.408248
\(727\) 32.0000 1.18681 0.593407 − 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 6.00000 0.222070
\(731\) 35.0000 1.29452
\(732\) 5.00000 0.184805
\(733\) −19.0000 −0.701781 −0.350891 − 0.936416i \(-0.614121\pi\)
−0.350891 + 0.936416i \(0.614121\pi\)
\(734\) −19.0000 −0.701303
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 0 0
\(738\) 2.00000 0.0736210
\(739\) 19.0000 0.698926 0.349463 − 0.936950i \(-0.386364\pi\)
0.349463 + 0.936950i \(0.386364\pi\)
\(740\) 8.00000 0.294086
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) 2.00000 0.0733729 0.0366864 − 0.999327i \(-0.488320\pi\)
0.0366864 + 0.999327i \(0.488320\pi\)
\(744\) 3.00000 0.109985
\(745\) −7.00000 −0.256460
\(746\) 16.0000 0.585802
\(747\) 32.0000 1.17082
\(748\) 0 0
\(749\) 0 0
\(750\) −9.00000 −0.328634
\(751\) −32.0000 −1.16770 −0.583848 − 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 10.0000 0.364662
\(753\) 8.00000 0.291536
\(754\) −18.0000 −0.655521
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) 15.0000 0.545184 0.272592 − 0.962130i \(-0.412119\pi\)
0.272592 + 0.962130i \(0.412119\pi\)
\(758\) 15.0000 0.544825
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) 14.0000 0.507500 0.253750 − 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) −18.0000 −0.652071
\(763\) 0 0
\(764\) −15.0000 −0.542681
\(765\) 10.0000 0.361551
\(766\) −8.00000 −0.289052
\(767\) −28.0000 −1.01102
\(768\) 1.00000 0.0360844
\(769\) 42.0000 1.51456 0.757279 − 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 0 0
\(771\) −3.00000 −0.108042
\(772\) 16.0000 0.575853
\(773\) 2.00000 0.0719350 0.0359675 − 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) 14.0000 0.503220
\(775\) 12.0000 0.431053
\(776\) 17.0000 0.610264
\(777\) 0 0
\(778\) −20.0000 −0.717035
\(779\) 4.00000 0.143315
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) −30.0000 −1.07280
\(783\) 45.0000 1.60817
\(784\) 0 0
\(785\) 0 0
\(786\) −18.0000 −0.642039
\(787\) 40.0000 1.42585 0.712923 − 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) −3.00000 −0.106735
\(791\) 0 0
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) −2.00000 −0.0709773
\(795\) 13.0000 0.461062
\(796\) −4.00000 −0.141776
\(797\) 17.0000 0.602171 0.301085 − 0.953597i \(-0.402651\pi\)
0.301085 + 0.953597i \(0.402651\pi\)
\(798\) 0 0
\(799\) 50.0000 1.76887
\(800\) 4.00000 0.141421
\(801\) 26.0000 0.918665
\(802\) 5.00000 0.176556
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −6.00000 −0.211341
\(807\) −10.0000 −0.352017
\(808\) 18.0000 0.633238
\(809\) 32.0000 1.12506 0.562530 − 0.826777i \(-0.309828\pi\)
0.562530 + 0.826777i \(0.309828\pi\)
\(810\) 1.00000 0.0351364
\(811\) 6.00000 0.210688 0.105344 − 0.994436i \(-0.466406\pi\)
0.105344 + 0.994436i \(0.466406\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) −12.0000 −0.420342
\(816\) 5.00000 0.175035
\(817\) 28.0000 0.979596
\(818\) 32.0000 1.11885
\(819\) 0 0
\(820\) −1.00000 −0.0349215
\(821\) 52.0000 1.81481 0.907406 − 0.420255i \(-0.138059\pi\)
0.907406 + 0.420255i \(0.138059\pi\)
\(822\) 18.0000 0.627822
\(823\) 5.00000 0.174289 0.0871445 − 0.996196i \(-0.472226\pi\)
0.0871445 + 0.996196i \(0.472226\pi\)
\(824\) 7.00000 0.243857
\(825\) 0 0
\(826\) 0 0
\(827\) 6.00000 0.208640 0.104320 − 0.994544i \(-0.466733\pi\)
0.104320 + 0.994544i \(0.466733\pi\)
\(828\) −12.0000 −0.417029
\(829\) −37.0000 −1.28506 −0.642532 − 0.766259i \(-0.722116\pi\)
−0.642532 + 0.766259i \(0.722116\pi\)
\(830\) −16.0000 −0.555368
\(831\) 20.0000 0.693792
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 14.0000 0.484780
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) 15.0000 0.518476
\(838\) −14.0000 −0.483622
\(839\) −20.0000 −0.690477 −0.345238 − 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 7.00000 0.241236
\(843\) −22.0000 −0.757720
\(844\) 20.0000 0.688428
\(845\) 9.00000 0.309609
\(846\) 20.0000 0.687614
\(847\) 0 0
\(848\) −13.0000 −0.446422
\(849\) 12.0000 0.411839
\(850\) 20.0000 0.685994
\(851\) −48.0000 −1.64542
\(852\) −15.0000 −0.513892
\(853\) 7.00000 0.239675 0.119838 − 0.992793i \(-0.461763\pi\)
0.119838 + 0.992793i \(0.461763\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) −3.00000 −0.102538
\(857\) 30.0000 1.02478 0.512390 − 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) −22.0000 −0.750630 −0.375315 − 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) −7.00000 −0.238698
\(861\) 0 0
\(862\) −8.00000 −0.272481
\(863\) −34.0000 −1.15737 −0.578687 − 0.815550i \(-0.696435\pi\)
−0.578687 + 0.815550i \(0.696435\pi\)
\(864\) 5.00000 0.170103
\(865\) 9.00000 0.306009
\(866\) 6.00000 0.203888
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 0 0
\(870\) −9.00000 −0.305129
\(871\) −8.00000 −0.271070
\(872\) 2.00000 0.0677285
\(873\) 34.0000 1.15073
\(874\) −24.0000 −0.811812
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 12.0000 0.405211 0.202606 − 0.979260i \(-0.435059\pi\)
0.202606 + 0.979260i \(0.435059\pi\)
\(878\) −28.0000 −0.944954
\(879\) −24.0000 −0.809500
\(880\) 0 0
\(881\) −40.0000 −1.34763 −0.673817 − 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 0 0
\(883\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) −10.0000 −0.336336
\(885\) −14.0000 −0.470605
\(886\) −39.0000 −1.31023
\(887\) 10.0000 0.335767 0.167884 − 0.985807i \(-0.446307\pi\)
0.167884 + 0.985807i \(0.446307\pi\)
\(888\) 8.00000 0.268462
\(889\) 0 0
\(890\) −13.0000 −0.435761
\(891\) 0 0
\(892\) −3.00000 −0.100447
\(893\) 40.0000 1.33855
\(894\) −7.00000 −0.234115
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) −12.0000 −0.400668
\(898\) 33.0000 1.10122
\(899\) 27.0000 0.900500
\(900\) 8.00000 0.266667
\(901\) −65.0000 −2.16546
\(902\) 0 0
\(903\) 0 0
\(904\) −1.00000 −0.0332595
\(905\) 0 0
\(906\) 5.00000 0.166114
\(907\) 35.0000 1.16216 0.581078 − 0.813848i \(-0.302631\pi\)
0.581078 + 0.813848i \(0.302631\pi\)
\(908\) −7.00000 −0.232303
\(909\) 36.0000 1.19404
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 − 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) 18.0000 0.595387
\(915\) −5.00000 −0.165295
\(916\) −8.00000 −0.264327
\(917\) 0 0
\(918\) 25.0000 0.825123
\(919\) −9.00000 −0.296883 −0.148441 − 0.988921i \(-0.547426\pi\)
−0.148441 + 0.988921i \(0.547426\pi\)
\(920\) 6.00000 0.197814
\(921\) 16.0000 0.527218
\(922\) 21.0000 0.691598
\(923\) 30.0000 0.987462
\(924\) 0 0
\(925\) 32.0000 1.05215
\(926\) 24.0000 0.788689
\(927\) 14.0000 0.459820
\(928\) 9.00000 0.295439
\(929\) −34.0000 −1.11550 −0.557752 − 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) −3.00000 −0.0983739
\(931\) 0 0
\(932\) −12.0000 −0.393073
\(933\) −18.0000 −0.589294
\(934\) 32.0000 1.04707
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) −9.00000 −0.294017 −0.147009 − 0.989135i \(-0.546964\pi\)
−0.147009 + 0.989135i \(0.546964\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) −10.0000 −0.326164
\(941\) −10.0000 −0.325991 −0.162995 − 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) 14.0000 0.455661
\(945\) 0 0
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −3.00000 −0.0974355
\(949\) −12.0000 −0.389536
\(950\) 16.0000 0.519109
\(951\) −22.0000 −0.713399
\(952\) 0 0
\(953\) −26.0000 −0.842223 −0.421111 − 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) −26.0000 −0.841781
\(955\) 15.0000 0.485389
\(956\) 0 0
\(957\) 0 0
\(958\) −26.0000 −0.840022
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) −22.0000 −0.709677
\(962\) −16.0000 −0.515861
\(963\) −6.00000 −0.193347
\(964\) 20.0000 0.644157
\(965\) −16.0000 −0.515058
\(966\) 0 0
\(967\) −27.0000 −0.868261 −0.434131 − 0.900850i \(-0.642944\pi\)
−0.434131 + 0.900850i \(0.642944\pi\)
\(968\) 11.0000 0.353553
\(969\) 20.0000 0.642493
\(970\) −17.0000 −0.545837
\(971\) 21.0000 0.673922 0.336961 − 0.941519i \(-0.390601\pi\)
0.336961 + 0.941519i \(0.390601\pi\)
\(972\) 16.0000 0.513200
\(973\) 0 0
\(974\) −26.0000 −0.833094
\(975\) 8.00000 0.256205
\(976\) 5.00000 0.160046
\(977\) −6.00000 −0.191957 −0.0959785 − 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) −12.0000 −0.383718
\(979\) 0 0
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) −5.00000 −0.159556
\(983\) 21.0000 0.669796 0.334898 − 0.942254i \(-0.391298\pi\)
0.334898 + 0.942254i \(0.391298\pi\)
\(984\) −1.00000 −0.0318788
\(985\) 0 0
\(986\) 45.0000 1.43309
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 42.0000 1.33552
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 − 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 3.00000 0.0952501
\(993\) 26.0000 0.825085
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) −16.0000 −0.506979
\(997\) 14.0000 0.443384 0.221692 − 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 2.00000 0.0633089
\(999\) 40.0000 1.26554
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 4018.2.a.g.1.1 1
7.6 odd 2 574.2.a.b.1.1 ✓ 1
21.20 even 2 5166.2.a.bc.1.1 1
28.27 even 2 4592.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
574.2.a.b.1.1 ✓ 1 7.6 odd 2
4018.2.a.g.1.1 1 1.1 even 1 trivial
4592.2.a.j.1.1 1 28.27 even 2
5166.2.a.bc.1.1 1 21.20 even 2