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Properties

Label 4998.2.a.o.1.1
Level $4998$
Weight $2$
Character 4998.1
Self dual yes
Analytic conductor $39.909$
Analytic rank $0$
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] = [4998,2,Mod(1,4998)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4998, 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("4998.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4998.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: \(39.9092309302\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 4998.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +5.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -5.00000 q^{26} +1.00000 q^{27} -4.00000 q^{29} +1.00000 q^{30} -2.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +3.00000 q^{37} -8.00000 q^{38} +5.00000 q^{39} +1.00000 q^{40} +6.00000 q^{41} +13.0000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +6.00000 q^{46} +2.00000 q^{47} +1.00000 q^{48} +4.00000 q^{50} +1.00000 q^{51} +5.00000 q^{52} +5.00000 q^{53} -1.00000 q^{54} +1.00000 q^{55} +8.00000 q^{57} +4.00000 q^{58} -4.00000 q^{59} -1.00000 q^{60} -2.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} -5.00000 q^{65} +1.00000 q^{66} -5.00000 q^{67} +1.00000 q^{68} -6.00000 q^{69} +14.0000 q^{71} -1.00000 q^{72} -11.0000 q^{73} -3.00000 q^{74} -4.00000 q^{75} +8.00000 q^{76} -5.00000 q^{78} +1.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -15.0000 q^{83} -1.00000 q^{85} -13.0000 q^{86} -4.00000 q^{87} +1.00000 q^{88} -5.00000 q^{89} +1.00000 q^{90} -6.00000 q^{92} -2.00000 q^{93} -2.00000 q^{94} -8.00000 q^{95} -1.00000 q^{96} +1.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 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
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 1.00000 0.288675
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) −1.00000 −0.235702
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) −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 \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) −5.00000 −0.980581
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 1.00000 0.182574
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −8.00000 −1.29777
\(39\) 5.00000 0.800641
\(40\) 1.00000 0.158114
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 13.0000 1.98248 0.991241 − 0.132068i \(-0.0421616\pi\)
0.991241 + 0.132068i \(0.0421616\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) 6.00000 0.884652
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) 1.00000 0.140028
\(52\) 5.00000 0.693375
\(53\) 5.00000 0.686803 0.343401 − 0.939189i \(-0.388421\pi\)
0.343401 + 0.939189i \(0.388421\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 4.00000 0.525226
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −5.00000 −0.620174
\(66\) 1.00000 0.123091
\(67\) −5.00000 −0.610847 −0.305424 − 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 1.00000 0.121268
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 14.0000 1.66149 0.830747 − 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) −1.00000 −0.117851
\(73\) −11.0000 −1.28745 −0.643726 − 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) −3.00000 −0.348743
\(75\) −4.00000 −0.461880
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) −5.00000 −0.566139
\(79\) 1.00000 0.112509 0.0562544 − 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −15.0000 −1.64646 −0.823232 − 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) 0 0
\(85\) −1.00000 −0.108465
\(86\) −13.0000 −1.40183
\(87\) −4.00000 −0.428845
\(88\) 1.00000 0.106600
\(89\) −5.00000 −0.529999 −0.264999 − 0.964249i \(-0.585372\pi\)
−0.264999 + 0.964249i \(0.585372\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) −2.00000 −0.207390
\(94\) −2.00000 −0.206284
\(95\) −8.00000 −0.820783
\(96\) −1.00000 −0.102062
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −4.00000 −0.400000
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 11.0000 1.08386 0.541931 − 0.840423i \(-0.317693\pi\)
0.541931 + 0.840423i \(0.317693\pi\)
\(104\) −5.00000 −0.490290
\(105\) 0 0
\(106\) −5.00000 −0.485643
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) 19.0000 1.78737 0.893685 − 0.448695i \(-0.148111\pi\)
0.893685 + 0.448695i \(0.148111\pi\)
\(114\) −8.00000 −0.749269
\(115\) 6.00000 0.559503
\(116\) −4.00000 −0.371391
\(117\) 5.00000 0.462250
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −10.0000 −0.909091
\(122\) 2.00000 0.181071
\(123\) 6.00000 0.541002
\(124\) −2.00000 −0.179605
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 14.0000 1.24230 0.621150 − 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 13.0000 1.14459
\(130\) 5.00000 0.438529
\(131\) 18.0000 1.57267 0.786334 − 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) 5.00000 0.431934
\(135\) −1.00000 −0.0860663
\(136\) −1.00000 −0.0857493
\(137\) 14.0000 1.19610 0.598050 − 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) 6.00000 0.510754
\(139\) −9.00000 −0.763370 −0.381685 − 0.924292i \(-0.624656\pi\)
−0.381685 + 0.924292i \(0.624656\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) −14.0000 −1.17485
\(143\) −5.00000 −0.418121
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 11.0000 0.910366
\(147\) 0 0
\(148\) 3.00000 0.246598
\(149\) 17.0000 1.39269 0.696347 − 0.717705i \(-0.254807\pi\)
0.696347 + 0.717705i \(0.254807\pi\)
\(150\) 4.00000 0.326599
\(151\) −14.0000 −1.13930 −0.569652 − 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) −8.00000 −0.648886
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 5.00000 0.400320
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −1.00000 −0.0795557
\(159\) 5.00000 0.396526
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −24.0000 −1.87983 −0.939913 − 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 6.00000 0.468521
\(165\) 1.00000 0.0778499
\(166\) 15.0000 1.16423
\(167\) −5.00000 −0.386912 −0.193456 − 0.981109i \(-0.561970\pi\)
−0.193456 + 0.981109i \(0.561970\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 1.00000 0.0766965
\(171\) 8.00000 0.611775
\(172\) 13.0000 0.991241
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −4.00000 −0.300658
\(178\) 5.00000 0.374766
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −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
\(183\) −2.00000 −0.147844
\(184\) 6.00000 0.442326
\(185\) −3.00000 −0.220564
\(186\) 2.00000 0.146647
\(187\) −1.00000 −0.0731272
\(188\) 2.00000 0.145865
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) 15.0000 1.08536 0.542681 − 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −1.00000 −0.0717958
\(195\) −5.00000 −0.358057
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 1.00000 0.0710669
\(199\) 22.0000 1.55954 0.779769 − 0.626067i \(-0.215336\pi\)
0.779769 + 0.626067i \(0.215336\pi\)
\(200\) 4.00000 0.282843
\(201\) −5.00000 −0.352673
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) 1.00000 0.0700140
\(205\) −6.00000 −0.419058
\(206\) −11.0000 −0.766406
\(207\) −6.00000 −0.417029
\(208\) 5.00000 0.346688
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 5.00000 0.343401
\(213\) 14.0000 0.959264
\(214\) 8.00000 0.546869
\(215\) −13.0000 −0.886593
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) −11.0000 −0.743311
\(220\) 1.00000 0.0674200
\(221\) 5.00000 0.336336
\(222\) −3.00000 −0.201347
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) −19.0000 −1.26386
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 8.00000 0.529813
\(229\) −1.00000 −0.0660819 −0.0330409 − 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) 4.00000 0.262613
\(233\) −13.0000 −0.851658 −0.425829 − 0.904804i \(-0.640018\pi\)
−0.425829 + 0.904804i \(0.640018\pi\)
\(234\) −5.00000 −0.326860
\(235\) −2.00000 −0.130466
\(236\) −4.00000 −0.260378
\(237\) 1.00000 0.0649570
\(238\) 0 0
\(239\) −27.0000 −1.74648 −0.873242 − 0.487286i \(-0.837987\pi\)
−0.873242 + 0.487286i \(0.837987\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 10.0000 0.642824
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 40.0000 2.54514
\(248\) 2.00000 0.127000
\(249\) −15.0000 −0.950586
\(250\) −9.00000 −0.569210
\(251\) 11.0000 0.694314 0.347157 − 0.937807i \(-0.387147\pi\)
0.347157 + 0.937807i \(0.387147\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) −14.0000 −0.878438
\(255\) −1.00000 −0.0626224
\(256\) 1.00000 0.0625000
\(257\) −23.0000 −1.43470 −0.717350 − 0.696713i \(-0.754645\pi\)
−0.717350 + 0.696713i \(0.754645\pi\)
\(258\) −13.0000 −0.809345
\(259\) 0 0
\(260\) −5.00000 −0.310087
\(261\) −4.00000 −0.247594
\(262\) −18.0000 −1.11204
\(263\) 11.0000 0.678289 0.339145 − 0.940734i \(-0.389862\pi\)
0.339145 + 0.940734i \(0.389862\pi\)
\(264\) 1.00000 0.0615457
\(265\) −5.00000 −0.307148
\(266\) 0 0
\(267\) −5.00000 −0.305995
\(268\) −5.00000 −0.305424
\(269\) 2.00000 0.121942 0.0609711 − 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 1.00000 0.0608581
\(271\) 29.0000 1.76162 0.880812 − 0.473466i \(-0.156997\pi\)
0.880812 + 0.473466i \(0.156997\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) −14.0000 −0.845771
\(275\) 4.00000 0.241209
\(276\) −6.00000 −0.361158
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 9.00000 0.539784
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) −2.00000 −0.119098
\(283\) −17.0000 −1.01055 −0.505273 − 0.862960i \(-0.668608\pi\)
−0.505273 + 0.862960i \(0.668608\pi\)
\(284\) 14.0000 0.830747
\(285\) −8.00000 −0.473879
\(286\) 5.00000 0.295656
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) −4.00000 −0.234888
\(291\) 1.00000 0.0586210
\(292\) −11.0000 −0.643726
\(293\) 12.0000 0.701047 0.350524 − 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) −3.00000 −0.174371
\(297\) −1.00000 −0.0580259
\(298\) −17.0000 −0.984784
\(299\) −30.0000 −1.73494
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) 14.0000 0.805609
\(303\) −10.0000 −0.574485
\(304\) 8.00000 0.458831
\(305\) 2.00000 0.114520
\(306\) −1.00000 −0.0571662
\(307\) 16.0000 0.913168 0.456584 − 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 11.0000 0.625768
\(310\) −2.00000 −0.113592
\(311\) 7.00000 0.396934 0.198467 − 0.980108i \(-0.436404\pi\)
0.198467 + 0.980108i \(0.436404\pi\)
\(312\) −5.00000 −0.283069
\(313\) 30.0000 1.69570 0.847850 − 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 1.00000 0.0562544
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −5.00000 −0.280386
\(319\) 4.00000 0.223957
\(320\) −1.00000 −0.0559017
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) −20.0000 −1.10940
\(326\) 24.0000 1.32924
\(327\) 10.0000 0.553001
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) −1.00000 −0.0550482
\(331\) −25.0000 −1.37412 −0.687062 − 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) −15.0000 −0.823232
\(333\) 3.00000 0.164399
\(334\) 5.00000 0.273588
\(335\) 5.00000 0.273179
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −12.0000 −0.652714
\(339\) 19.0000 1.03194
\(340\) −1.00000 −0.0542326
\(341\) 2.00000 0.108306
\(342\) −8.00000 −0.432590
\(343\) 0 0
\(344\) −13.0000 −0.700913
\(345\) 6.00000 0.323029
\(346\) 14.0000 0.752645
\(347\) 9.00000 0.483145 0.241573 − 0.970383i \(-0.422337\pi\)
0.241573 + 0.970383i \(0.422337\pi\)
\(348\) −4.00000 −0.214423
\(349\) −33.0000 −1.76645 −0.883225 − 0.468950i \(-0.844632\pi\)
−0.883225 + 0.468950i \(0.844632\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 1.00000 0.0533002
\(353\) 27.0000 1.43706 0.718532 − 0.695493i \(-0.244814\pi\)
0.718532 + 0.695493i \(0.244814\pi\)
\(354\) 4.00000 0.212598
\(355\) −14.0000 −0.743043
\(356\) −5.00000 −0.264999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 4.00000 0.211112 0.105556 − 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 1.00000 0.0527046
\(361\) 45.0000 2.36842
\(362\) 2.00000 0.105118
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 11.0000 0.575766
\(366\) 2.00000 0.104542
\(367\) −32.0000 −1.67039 −0.835193 − 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) −6.00000 −0.312772
\(369\) 6.00000 0.312348
\(370\) 3.00000 0.155963
\(371\) 0 0
\(372\) −2.00000 −0.103695
\(373\) 34.0000 1.76045 0.880227 − 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) 1.00000 0.0517088
\(375\) 9.00000 0.464758
\(376\) −2.00000 −0.103142
\(377\) −20.0000 −1.03005
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 − 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) −8.00000 −0.410391
\(381\) 14.0000 0.717242
\(382\) −15.0000 −0.767467
\(383\) 6.00000 0.306586 0.153293 − 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 13.0000 0.660827
\(388\) 1.00000 0.0507673
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 5.00000 0.253185
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 18.0000 0.907980
\(394\) −18.0000 −0.906827
\(395\) −1.00000 −0.0503155
\(396\) −1.00000 −0.0502519
\(397\) 8.00000 0.401508 0.200754 − 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) −22.0000 −1.10276
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 5.00000 0.249377
\(403\) −10.0000 −0.498135
\(404\) −10.0000 −0.497519
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −3.00000 −0.148704
\(408\) −1.00000 −0.0495074
\(409\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 6.00000 0.296319
\(411\) 14.0000 0.690569
\(412\) 11.0000 0.541931
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 15.0000 0.736321
\(416\) −5.00000 −0.245145
\(417\) −9.00000 −0.440732
\(418\) 8.00000 0.391293
\(419\) −16.0000 −0.781651 −0.390826 − 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 4.00000 0.194717
\(423\) 2.00000 0.0972433
\(424\) −5.00000 −0.242821
\(425\) −4.00000 −0.194029
\(426\) −14.0000 −0.678302
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) −5.00000 −0.241402
\(430\) 13.0000 0.626916
\(431\) 10.0000 0.481683 0.240842 − 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) 10.0000 0.478913
\(437\) −48.0000 −2.29615
\(438\) 11.0000 0.525600
\(439\) 18.0000 0.859093 0.429547 − 0.903045i \(-0.358673\pi\)
0.429547 + 0.903045i \(0.358673\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) −5.00000 −0.237826
\(443\) 30.0000 1.42534 0.712672 − 0.701498i \(-0.247485\pi\)
0.712672 + 0.701498i \(0.247485\pi\)
\(444\) 3.00000 0.142374
\(445\) 5.00000 0.237023
\(446\) −16.0000 −0.757622
\(447\) 17.0000 0.804072
\(448\) 0 0
\(449\) −33.0000 −1.55737 −0.778683 − 0.627417i \(-0.784112\pi\)
−0.778683 + 0.627417i \(0.784112\pi\)
\(450\) 4.00000 0.188562
\(451\) −6.00000 −0.282529
\(452\) 19.0000 0.893685
\(453\) −14.0000 −0.657777
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) −8.00000 −0.374634
\(457\) 11.0000 0.514558 0.257279 − 0.966337i \(-0.417174\pi\)
0.257279 + 0.966337i \(0.417174\pi\)
\(458\) 1.00000 0.0467269
\(459\) 1.00000 0.0466760
\(460\) 6.00000 0.279751
\(461\) 30.0000 1.39724 0.698620 − 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 − 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −4.00000 −0.185695
\(465\) 2.00000 0.0927478
\(466\) 13.0000 0.602213
\(467\) −32.0000 −1.48078 −0.740392 − 0.672176i \(-0.765360\pi\)
−0.740392 + 0.672176i \(0.765360\pi\)
\(468\) 5.00000 0.231125
\(469\) 0 0
\(470\) 2.00000 0.0922531
\(471\) 14.0000 0.645086
\(472\) 4.00000 0.184115
\(473\) −13.0000 −0.597741
\(474\) −1.00000 −0.0459315
\(475\) −32.0000 −1.46826
\(476\) 0 0
\(477\) 5.00000 0.228934
\(478\) 27.0000 1.23495
\(479\) −15.0000 −0.685367 −0.342684 − 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) 1.00000 0.0456435
\(481\) 15.0000 0.683941
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) −1.00000 −0.0454077
\(486\) −1.00000 −0.0453609
\(487\) 7.00000 0.317200 0.158600 − 0.987343i \(-0.449302\pi\)
0.158600 + 0.987343i \(0.449302\pi\)
\(488\) 2.00000 0.0905357
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000 0.270501
\(493\) −4.00000 −0.180151
\(494\) −40.0000 −1.79969
\(495\) 1.00000 0.0449467
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) 15.0000 0.672166
\(499\) 6.00000 0.268597 0.134298 − 0.990941i \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) 9.00000 0.402492
\(501\) −5.00000 −0.223384
\(502\) −11.0000 −0.490954
\(503\) 28.0000 1.24846 0.624229 − 0.781241i \(-0.285413\pi\)
0.624229 + 0.781241i \(0.285413\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) −6.00000 −0.266733
\(507\) 12.0000 0.532939
\(508\) 14.0000 0.621150
\(509\) −20.0000 −0.886484 −0.443242 − 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 1.00000 0.0442807
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 8.00000 0.353209
\(514\) 23.0000 1.01449
\(515\) −11.0000 −0.484718
\(516\) 13.0000 0.572293
\(517\) −2.00000 −0.0879599
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 5.00000 0.219265
\(521\) −2.00000 −0.0876216 −0.0438108 − 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) 4.00000 0.175075
\(523\) −16.0000 −0.699631 −0.349816 − 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) −11.0000 −0.479623
\(527\) −2.00000 −0.0871214
\(528\) −1.00000 −0.0435194
\(529\) 13.0000 0.565217
\(530\) 5.00000 0.217186
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 30.0000 1.29944
\(534\) 5.00000 0.216371
\(535\) 8.00000 0.345870
\(536\) 5.00000 0.215967
\(537\) 12.0000 0.517838
\(538\) −2.00000 −0.0862261
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −29.0000 −1.24681 −0.623404 − 0.781900i \(-0.714251\pi\)
−0.623404 + 0.781900i \(0.714251\pi\)
\(542\) −29.0000 −1.24566
\(543\) −2.00000 −0.0858282
\(544\) −1.00000 −0.0428746
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 − 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) 14.0000 0.598050
\(549\) −2.00000 −0.0853579
\(550\) −4.00000 −0.170561
\(551\) −32.0000 −1.36325
\(552\) 6.00000 0.255377
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) −3.00000 −0.127343
\(556\) −9.00000 −0.381685
\(557\) 30.0000 1.27114 0.635570 − 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 2.00000 0.0846668
\(559\) 65.0000 2.74921
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) −18.0000 −0.759284
\(563\) −11.0000 −0.463595 −0.231797 − 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(564\) 2.00000 0.0842152
\(565\) −19.0000 −0.799336
\(566\) 17.0000 0.714563
\(567\) 0 0
\(568\) −14.0000 −0.587427
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 8.00000 0.335083
\(571\) −6.00000 −0.251092 −0.125546 − 0.992088i \(-0.540068\pi\)
−0.125546 + 0.992088i \(0.540068\pi\)
\(572\) −5.00000 −0.209061
\(573\) 15.0000 0.626634
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 14.0000 0.581820
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) −1.00000 −0.0414513
\(583\) −5.00000 −0.207079
\(584\) 11.0000 0.455183
\(585\) −5.00000 −0.206725
\(586\) −12.0000 −0.495715
\(587\) 5.00000 0.206372 0.103186 − 0.994662i \(-0.467096\pi\)
0.103186 + 0.994662i \(0.467096\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) −4.00000 −0.164677
\(591\) 18.0000 0.740421
\(592\) 3.00000 0.123299
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 17.0000 0.696347
\(597\) 22.0000 0.900400
\(598\) 30.0000 1.22679
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 4.00000 0.163299
\(601\) −34.0000 −1.38689 −0.693444 − 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) −5.00000 −0.203616
\(604\) −14.0000 −0.569652
\(605\) 10.0000 0.406558
\(606\) 10.0000 0.406222
\(607\) −18.0000 −0.730597 −0.365299 − 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 10.0000 0.404557
\(612\) 1.00000 0.0404226
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −16.0000 −0.645707
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 7.00000 0.281809 0.140905 − 0.990023i \(-0.454999\pi\)
0.140905 + 0.990023i \(0.454999\pi\)
\(618\) −11.0000 −0.442485
\(619\) −35.0000 −1.40677 −0.703384 − 0.710810i \(-0.748329\pi\)
−0.703384 + 0.710810i \(0.748329\pi\)
\(620\) 2.00000 0.0803219
\(621\) −6.00000 −0.240772
\(622\) −7.00000 −0.280674
\(623\) 0 0
\(624\) 5.00000 0.200160
\(625\) 11.0000 0.440000
\(626\) −30.0000 −1.19904
\(627\) −8.00000 −0.319489
\(628\) 14.0000 0.558661
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) 42.0000 1.67199 0.835997 − 0.548734i \(-0.184890\pi\)
0.835997 + 0.548734i \(0.184890\pi\)
\(632\) −1.00000 −0.0397779
\(633\) −4.00000 −0.158986
\(634\) 6.00000 0.238290
\(635\) −14.0000 −0.555573
\(636\) 5.00000 0.198263
\(637\) 0 0
\(638\) −4.00000 −0.158362
\(639\) 14.0000 0.553831
\(640\) 1.00000 0.0395285
\(641\) −41.0000 −1.61940 −0.809701 − 0.586842i \(-0.800371\pi\)
−0.809701 + 0.586842i \(0.800371\pi\)
\(642\) 8.00000 0.315735
\(643\) 27.0000 1.06478 0.532388 − 0.846500i \(-0.321295\pi\)
0.532388 + 0.846500i \(0.321295\pi\)
\(644\) 0 0
\(645\) −13.0000 −0.511875
\(646\) −8.00000 −0.314756
\(647\) 6.00000 0.235884 0.117942 − 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 4.00000 0.157014
\(650\) 20.0000 0.784465
\(651\) 0 0
\(652\) −24.0000 −0.939913
\(653\) 14.0000 0.547862 0.273931 − 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −10.0000 −0.391031
\(655\) −18.0000 −0.703318
\(656\) 6.00000 0.234261
\(657\) −11.0000 −0.429151
\(658\) 0 0
\(659\) 16.0000 0.623272 0.311636 − 0.950202i \(-0.399123\pi\)
0.311636 + 0.950202i \(0.399123\pi\)
\(660\) 1.00000 0.0389249
\(661\) 38.0000 1.47803 0.739014 − 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 25.0000 0.971653
\(663\) 5.00000 0.194184
\(664\) 15.0000 0.582113
\(665\) 0 0
\(666\) −3.00000 −0.116248
\(667\) 24.0000 0.929284
\(668\) −5.00000 −0.193456
\(669\) 16.0000 0.618596
\(670\) −5.00000 −0.193167
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 − 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −14.0000 −0.539260
\(675\) −4.00000 −0.153960
\(676\) 12.0000 0.461538
\(677\) −9.00000 −0.345898 −0.172949 − 0.984931i \(-0.555330\pi\)
−0.172949 + 0.984931i \(0.555330\pi\)
\(678\) −19.0000 −0.729691
\(679\) 0 0
\(680\) 1.00000 0.0383482
\(681\) −8.00000 −0.306561
\(682\) −2.00000 −0.0765840
\(683\) −20.0000 −0.765279 −0.382639 − 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 8.00000 0.305888
\(685\) −14.0000 −0.534913
\(686\) 0 0
\(687\) −1.00000 −0.0381524
\(688\) 13.0000 0.495620
\(689\) 25.0000 0.952424
\(690\) −6.00000 −0.228416
\(691\) 37.0000 1.40755 0.703773 − 0.710425i \(-0.251497\pi\)
0.703773 + 0.710425i \(0.251497\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) −9.00000 −0.341635
\(695\) 9.00000 0.341389
\(696\) 4.00000 0.151620
\(697\) 6.00000 0.227266
\(698\) 33.0000 1.24907
\(699\) −13.0000 −0.491705
\(700\) 0 0
\(701\) 7.00000 0.264386 0.132193 − 0.991224i \(-0.457798\pi\)
0.132193 + 0.991224i \(0.457798\pi\)
\(702\) −5.00000 −0.188713
\(703\) 24.0000 0.905177
\(704\) −1.00000 −0.0376889
\(705\) −2.00000 −0.0753244
\(706\) −27.0000 −1.01616
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) 19.0000 0.713560 0.356780 − 0.934188i \(-0.383875\pi\)
0.356780 + 0.934188i \(0.383875\pi\)
\(710\) 14.0000 0.525411
\(711\) 1.00000 0.0375029
\(712\) 5.00000 0.187383
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 5.00000 0.186989
\(716\) 12.0000 0.448461
\(717\) −27.0000 −1.00833
\(718\) −4.00000 −0.149279
\(719\) 36.0000 1.34257 0.671287 − 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) −45.0000 −1.67473
\(723\) 10.0000 0.371904
\(724\) −2.00000 −0.0743294
\(725\) 16.0000 0.594225
\(726\) 10.0000 0.371135
\(727\) −1.00000 −0.0370879 −0.0185440 − 0.999828i \(-0.505903\pi\)
−0.0185440 + 0.999828i \(0.505903\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −11.0000 −0.407128
\(731\) 13.0000 0.480822
\(732\) −2.00000 −0.0739221
\(733\) −14.0000 −0.517102 −0.258551 − 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 5.00000 0.184177
\(738\) −6.00000 −0.220863
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) −3.00000 −0.110282
\(741\) 40.0000 1.46944
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 − 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 2.00000 0.0733236
\(745\) −17.0000 −0.622832
\(746\) −34.0000 −1.24483
\(747\) −15.0000 −0.548821
\(748\) −1.00000 −0.0365636
\(749\) 0 0
\(750\) −9.00000 −0.328634
\(751\) −24.0000 −0.875772 −0.437886 − 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 2.00000 0.0729325
\(753\) 11.0000 0.400862
\(754\) 20.0000 0.728357
\(755\) 14.0000 0.509512
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 − 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) −12.0000 −0.435860
\(759\) 6.00000 0.217786
\(760\) 8.00000 0.290191
\(761\) 5.00000 0.181250 0.0906249 − 0.995885i \(-0.471114\pi\)
0.0906249 + 0.995885i \(0.471114\pi\)
\(762\) −14.0000 −0.507166
\(763\) 0 0
\(764\) 15.0000 0.542681
\(765\) −1.00000 −0.0361551
\(766\) −6.00000 −0.216789
\(767\) −20.0000 −0.722158
\(768\) 1.00000 0.0360844
\(769\) −16.0000 −0.576975 −0.288487 − 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) −23.0000 −0.828325
\(772\) 14.0000 0.503871
\(773\) −36.0000 −1.29483 −0.647415 − 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) −13.0000 −0.467275
\(775\) 8.00000 0.287368
\(776\) −1.00000 −0.0358979
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 48.0000 1.71978
\(780\) −5.00000 −0.179029
\(781\) −14.0000 −0.500959
\(782\) 6.00000 0.214560
\(783\) −4.00000 −0.142948
\(784\) 0 0
\(785\) −14.0000 −0.499681
\(786\) −18.0000 −0.642039
\(787\) 1.00000 0.0356462 0.0178231 − 0.999841i \(-0.494326\pi\)
0.0178231 + 0.999841i \(0.494326\pi\)
\(788\) 18.0000 0.641223
\(789\) 11.0000 0.391610
\(790\) 1.00000 0.0355784
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) −10.0000 −0.355110
\(794\) −8.00000 −0.283909
\(795\) −5.00000 −0.177332
\(796\) 22.0000 0.779769
\(797\) −6.00000 −0.212531 −0.106265 − 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 2.00000 0.0707549
\(800\) 4.00000 0.141421
\(801\) −5.00000 −0.176666
\(802\) 6.00000 0.211867
\(803\) 11.0000 0.388182
\(804\) −5.00000 −0.176336
\(805\) 0 0
\(806\) 10.0000 0.352235
\(807\) 2.00000 0.0704033
\(808\) 10.0000 0.351799
\(809\) −18.0000 −0.632846 −0.316423 − 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 1.00000 0.0351364
\(811\) −37.0000 −1.29925 −0.649623 − 0.760257i \(-0.725073\pi\)
−0.649623 + 0.760257i \(0.725073\pi\)
\(812\) 0 0
\(813\) 29.0000 1.01707
\(814\) 3.00000 0.105150
\(815\) 24.0000 0.840683
\(816\) 1.00000 0.0350070
\(817\) 104.000 3.63850
\(818\) 0 0
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 44.0000 1.53561 0.767805 − 0.640683i \(-0.221349\pi\)
0.767805 + 0.640683i \(0.221349\pi\)
\(822\) −14.0000 −0.488306
\(823\) −47.0000 −1.63832 −0.819159 − 0.573567i \(-0.805559\pi\)
−0.819159 + 0.573567i \(0.805559\pi\)
\(824\) −11.0000 −0.383203
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −9.00000 −0.312961 −0.156480 − 0.987681i \(-0.550015\pi\)
−0.156480 + 0.987681i \(0.550015\pi\)
\(828\) −6.00000 −0.208514
\(829\) 1.00000 0.0347314 0.0173657 − 0.999849i \(-0.494472\pi\)
0.0173657 + 0.999849i \(0.494472\pi\)
\(830\) −15.0000 −0.520658
\(831\) −2.00000 −0.0693792
\(832\) 5.00000 0.173344
\(833\) 0 0
\(834\) 9.00000 0.311645
\(835\) 5.00000 0.173032
\(836\) −8.00000 −0.276686
\(837\) −2.00000 −0.0691301
\(838\) 16.0000 0.552711
\(839\) 21.0000 0.725001 0.362500 − 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 18.0000 0.619953
\(844\) −4.00000 −0.137686
\(845\) −12.0000 −0.412813
\(846\) −2.00000 −0.0687614
\(847\) 0 0
\(848\) 5.00000 0.171701
\(849\) −17.0000 −0.583438
\(850\) 4.00000 0.137199
\(851\) −18.0000 −0.617032
\(852\) 14.0000 0.479632
\(853\) −4.00000 −0.136957 −0.0684787 − 0.997653i \(-0.521815\pi\)
−0.0684787 + 0.997653i \(0.521815\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 8.00000 0.273434
\(857\) −12.0000 −0.409912 −0.204956 − 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 5.00000 0.170697
\(859\) 6.00000 0.204717 0.102359 − 0.994748i \(-0.467361\pi\)
0.102359 + 0.994748i \(0.467361\pi\)
\(860\) −13.0000 −0.443296
\(861\) 0 0
\(862\) −10.0000 −0.340601
\(863\) 3.00000 0.102121 0.0510606 − 0.998696i \(-0.483740\pi\)
0.0510606 + 0.998696i \(0.483740\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 14.0000 0.476014
\(866\) −14.0000 −0.475739
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −1.00000 −0.0339227
\(870\) −4.00000 −0.135613
\(871\) −25.0000 −0.847093
\(872\) −10.0000 −0.338643
\(873\) 1.00000 0.0338449
\(874\) 48.0000 1.62362
\(875\) 0 0
\(876\) −11.0000 −0.371656
\(877\) 33.0000 1.11433 0.557165 − 0.830402i \(-0.311889\pi\)
0.557165 + 0.830402i \(0.311889\pi\)
\(878\) −18.0000 −0.607471
\(879\) 12.0000 0.404750
\(880\) 1.00000 0.0337100
\(881\) 2.00000 0.0673817 0.0336909 − 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) −44.0000 −1.48072 −0.740359 − 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 5.00000 0.168168
\(885\) 4.00000 0.134459
\(886\) −30.0000 −1.00787
\(887\) −39.0000 −1.30949 −0.654746 − 0.755849i \(-0.727224\pi\)
−0.654746 + 0.755849i \(0.727224\pi\)
\(888\) −3.00000 −0.100673
\(889\) 0 0
\(890\) −5.00000 −0.167600
\(891\) −1.00000 −0.0335013
\(892\) 16.0000 0.535720
\(893\) 16.0000 0.535420
\(894\) −17.0000 −0.568565
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) −30.0000 −1.00167
\(898\) 33.0000 1.10122
\(899\) 8.00000 0.266815
\(900\) −4.00000 −0.133333
\(901\) 5.00000 0.166574
\(902\) 6.00000 0.199778
\(903\) 0 0
\(904\) −19.0000 −0.631931
\(905\) 2.00000 0.0664822
\(906\) 14.0000 0.465119
\(907\) 40.0000 1.32818 0.664089 − 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) −8.00000 −0.265489
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 − 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 8.00000 0.264906
\(913\) 15.0000 0.496428
\(914\) −11.0000 −0.363848
\(915\) 2.00000 0.0661180
\(916\) −1.00000 −0.0330409
\(917\) 0 0
\(918\) −1.00000 −0.0330049
\(919\) 12.0000 0.395843 0.197922 − 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) −6.00000 −0.197814
\(921\) 16.0000 0.527218
\(922\) −30.0000 −0.987997
\(923\) 70.0000 2.30408
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) −8.00000 −0.262896
\(927\) 11.0000 0.361287
\(928\) 4.00000 0.131306
\(929\) −60.0000 −1.96854 −0.984268 − 0.176682i \(-0.943464\pi\)
−0.984268 + 0.176682i \(0.943464\pi\)
\(930\) −2.00000 −0.0655826
\(931\) 0 0
\(932\) −13.0000 −0.425829
\(933\) 7.00000 0.229170
\(934\) 32.0000 1.04707
\(935\) 1.00000 0.0327035
\(936\) −5.00000 −0.163430
\(937\) −12.0000 −0.392023 −0.196011 − 0.980602i \(-0.562799\pi\)
−0.196011 + 0.980602i \(0.562799\pi\)
\(938\) 0 0
\(939\) 30.0000 0.979013
\(940\) −2.00000 −0.0652328
\(941\) −15.0000 −0.488986 −0.244493 − 0.969651i \(-0.578622\pi\)
−0.244493 + 0.969651i \(0.578622\pi\)
\(942\) −14.0000 −0.456145
\(943\) −36.0000 −1.17232
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 13.0000 0.422666
\(947\) 32.0000 1.03986 0.519930 − 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 1.00000 0.0324785
\(949\) −55.0000 −1.78538
\(950\) 32.0000 1.03822
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) −5.00000 −0.161881
\(955\) −15.0000 −0.485389
\(956\) −27.0000 −0.873242
\(957\) 4.00000 0.129302
\(958\) 15.0000 0.484628
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) −27.0000 −0.870968
\(962\) −15.0000 −0.483619
\(963\) −8.00000 −0.257796
\(964\) 10.0000 0.322078
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) −50.0000 −1.60789 −0.803946 − 0.594703i \(-0.797270\pi\)
−0.803946 + 0.594703i \(0.797270\pi\)
\(968\) 10.0000 0.321412
\(969\) 8.00000 0.256997
\(970\) 1.00000 0.0321081
\(971\) 41.0000 1.31575 0.657876 − 0.753126i \(-0.271455\pi\)
0.657876 + 0.753126i \(0.271455\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −7.00000 −0.224294
\(975\) −20.0000 −0.640513
\(976\) −2.00000 −0.0640184
\(977\) 38.0000 1.21573 0.607864 − 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) 24.0000 0.767435
\(979\) 5.00000 0.159801
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) −16.0000 −0.510321 −0.255160 − 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) −6.00000 −0.191273
\(985\) −18.0000 −0.573528
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 40.0000 1.27257
\(989\) −78.0000 −2.48026
\(990\) −1.00000 −0.0317821
\(991\) −43.0000 −1.36594 −0.682970 − 0.730446i \(-0.739312\pi\)
−0.682970 + 0.730446i \(0.739312\pi\)
\(992\) 2.00000 0.0635001
\(993\) −25.0000 −0.793351
\(994\) 0 0
\(995\) −22.0000 −0.697447
\(996\) −15.0000 −0.475293
\(997\) −46.0000 −1.45683 −0.728417 − 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) −6.00000 −0.189927
\(999\) 3.00000 0.0949158
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 4998.2.a.o.1.1 yes 1
7.6 odd 2 4998.2.a.f.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4998.2.a.f.1.1 ✓ 1 7.6 odd 2
4998.2.a.o.1.1 yes 1 1.1 even 1 trivial