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Properties

Label 4998.2.a.k.1.1
Level $4998$
Weight $2$
Character 4998.1
Self dual yes
Analytic conductor $39.909$
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] = [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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 714)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +1.00000 q^{12} -1.00000 q^{13} -3.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -3.00000 q^{20} +6.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} +3.00000 q^{29} +3.00000 q^{30} -1.00000 q^{31} -1.00000 q^{32} -1.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +4.00000 q^{38} -1.00000 q^{39} +3.00000 q^{40} +3.00000 q^{41} -4.00000 q^{43} -3.00000 q^{45} -6.00000 q^{46} +3.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} +1.00000 q^{51} -1.00000 q^{52} +12.0000 q^{53} -1.00000 q^{54} -4.00000 q^{57} -3.00000 q^{58} +9.00000 q^{59} -3.00000 q^{60} -10.0000 q^{61} +1.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} +2.00000 q^{67} +1.00000 q^{68} +6.00000 q^{69} -6.00000 q^{71} -1.00000 q^{72} -16.0000 q^{73} +4.00000 q^{74} +4.00000 q^{75} -4.00000 q^{76} +1.00000 q^{78} -4.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -3.00000 q^{82} -9.00000 q^{83} -3.00000 q^{85} +4.00000 q^{86} +3.00000 q^{87} -12.0000 q^{89} +3.00000 q^{90} +6.00000 q^{92} -1.00000 q^{93} -3.00000 q^{94} +12.0000 q^{95} -1.00000 q^{96} +8.00000 q^{97} +O(q^{100})\)

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
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −3.00000 −0.670820
\(21\) 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
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 3.00000 0.547723
\(31\) −1.00000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 4.00000 0.648886
\(39\) −1.00000 −0.160128
\(40\) 3.00000 0.474342
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) −6.00000 −0.884652
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) 1.00000 0.140028
\(52\) −1.00000 −0.138675
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) −3.00000 −0.393919
\(59\) 9.00000 1.17170 0.585850 − 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) −3.00000 −0.387298
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 1.00000 0.121268
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −1.00000 −0.117851
\(73\) −16.0000 −1.87266 −0.936329 − 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 4.00000 0.464991
\(75\) 4.00000 0.461880
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) −9.00000 −0.987878 −0.493939 − 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 4.00000 0.431331
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) −1.00000 −0.103695
\(94\) −3.00000 −0.309426
\(95\) 12.0000 1.23117
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) 9.00000 0.846649 0.423324 − 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 4.00000 0.374634
\(115\) −18.0000 −1.67851
\(116\) 3.00000 0.278543
\(117\) −1.00000 −0.0924500
\(118\) −9.00000 −0.828517
\(119\) 0 0
\(120\) 3.00000 0.273861
\(121\) −11.0000 −1.00000
\(122\) 10.0000 0.905357
\(123\) 3.00000 0.270501
\(124\) −1.00000 −0.0898027
\(125\) 3.00000 0.268328
\(126\) 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
\(129\) −4.00000 −0.352180
\(130\) −3.00000 −0.263117
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) −3.00000 −0.258199
\(136\) −1.00000 −0.0857493
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −6.00000 −0.510754
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −9.00000 −0.747409
\(146\) 16.0000 1.32417
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −4.00000 −0.326599
\(151\) −10.0000 −0.813788 −0.406894 − 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 4.00000 0.324443
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) 3.00000 0.240966
\(156\) −1.00000 −0.0800641
\(157\) −13.0000 −1.03751 −0.518756 − 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 4.00000 0.318223
\(159\) 12.0000 0.951662
\(160\) 3.00000 0.237171
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −16.0000 −1.25322 −0.626608 − 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 9.00000 0.698535
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 3.00000 0.230089
\(171\) −4.00000 −0.305888
\(172\) −4.00000 −0.304997
\(173\) −3.00000 −0.228086 −0.114043 − 0.993476i \(-0.536380\pi\)
−0.114043 + 0.993476i \(0.536380\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) 0 0
\(177\) 9.00000 0.676481
\(178\) 12.0000 0.899438
\(179\) 9.00000 0.672692 0.336346 − 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) −3.00000 −0.223607
\(181\) 20.0000 1.48659 0.743294 − 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) −6.00000 −0.442326
\(185\) 12.0000 0.882258
\(186\) 1.00000 0.0733236
\(187\) 0 0
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) −12.0000 −0.870572
\(191\) 3.00000 0.217072 0.108536 − 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 1.00000 0.0721688
\(193\) 20.0000 1.43963 0.719816 − 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) −8.00000 −0.574367
\(195\) 3.00000 0.214834
\(196\) 0 0
\(197\) −27.0000 −1.92367 −0.961835 − 0.273629i \(-0.911776\pi\)
−0.961835 + 0.273629i \(0.911776\pi\)
\(198\) 0 0
\(199\) −7.00000 −0.496217 −0.248108 − 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) −4.00000 −0.282843
\(201\) 2.00000 0.141069
\(202\) 0 0
\(203\) 0 0
\(204\) 1.00000 0.0700140
\(205\) −9.00000 −0.628587
\(206\) 4.00000 0.278693
\(207\) 6.00000 0.417029
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 − 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 12.0000 0.824163
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) 12.0000 0.818393
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) −16.0000 −1.08118
\(220\) 0 0
\(221\) −1.00000 −0.0672673
\(222\) 4.00000 0.268462
\(223\) −10.0000 −0.669650 −0.334825 − 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) −9.00000 −0.598671
\(227\) −18.0000 −1.19470 −0.597351 − 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) −4.00000 −0.264906
\(229\) −19.0000 −1.25556 −0.627778 − 0.778393i \(-0.716035\pi\)
−0.627778 + 0.778393i \(0.716035\pi\)
\(230\) 18.0000 1.18688
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 15.0000 0.982683 0.491341 − 0.870967i \(-0.336507\pi\)
0.491341 + 0.870967i \(0.336507\pi\)
\(234\) 1.00000 0.0653720
\(235\) −9.00000 −0.587095
\(236\) 9.00000 0.585850
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −27.0000 −1.74648 −0.873242 − 0.487286i \(-0.837987\pi\)
−0.873242 + 0.487286i \(0.837987\pi\)
\(240\) −3.00000 −0.193649
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 11.0000 0.707107
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) −3.00000 −0.191273
\(247\) 4.00000 0.254514
\(248\) 1.00000 0.0635001
\(249\) −9.00000 −0.570352
\(250\) −3.00000 −0.189737
\(251\) 9.00000 0.568075 0.284037 − 0.958813i \(-0.408326\pi\)
0.284037 + 0.958813i \(0.408326\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) −3.00000 −0.187867
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 3.00000 0.186052
\(261\) 3.00000 0.185695
\(262\) 12.0000 0.741362
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) −36.0000 −2.21146
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 2.00000 0.122169
\(269\) −9.00000 −0.548740 −0.274370 − 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 3.00000 0.182574
\(271\) 2.00000 0.121491 0.0607457 − 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) −22.0000 −1.32185 −0.660926 − 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −20.0000 −1.19952
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 − 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) −3.00000 −0.178647
\(283\) −25.0000 −1.48610 −0.743048 − 0.669238i \(-0.766621\pi\)
−0.743048 + 0.669238i \(0.766621\pi\)
\(284\) −6.00000 −0.356034
\(285\) 12.0000 0.710819
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) 9.00000 0.528498
\(291\) 8.00000 0.468968
\(292\) −16.0000 −0.936329
\(293\) −30.0000 −1.75262 −0.876309 − 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) −27.0000 −1.57200
\(296\) 4.00000 0.232495
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −6.00000 −0.346989
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 30.0000 1.71780
\(306\) −1.00000 −0.0571662
\(307\) 26.0000 1.48390 0.741949 − 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) −3.00000 −0.170389
\(311\) −30.0000 −1.70114 −0.850572 − 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 1.00000 0.0566139
\(313\) 26.0000 1.46961 0.734803 − 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 13.0000 0.733632
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 33.0000 1.85346 0.926732 − 0.375722i \(-0.122605\pi\)
0.926732 + 0.375722i \(0.122605\pi\)
\(318\) −12.0000 −0.672927
\(319\) 0 0
\(320\) −3.00000 −0.167705
\(321\) 0 0
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 16.0000 0.886158
\(327\) 2.00000 0.110600
\(328\) −3.00000 −0.165647
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −9.00000 −0.493939
\(333\) −4.00000 −0.219199
\(334\) 12.0000 0.656611
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 − 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 12.0000 0.652714
\(339\) 9.00000 0.488813
\(340\) −3.00000 −0.162698
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) −18.0000 −0.969087
\(346\) 3.00000 0.161281
\(347\) −18.0000 −0.966291 −0.483145 − 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 3.00000 0.160817
\(349\) −19.0000 −1.01705 −0.508523 − 0.861048i \(-0.669808\pi\)
−0.508523 + 0.861048i \(0.669808\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) −9.00000 −0.478345
\(355\) 18.0000 0.955341
\(356\) −12.0000 −0.635999
\(357\) 0 0
\(358\) −9.00000 −0.475665
\(359\) −21.0000 −1.10834 −0.554169 − 0.832404i \(-0.686964\pi\)
−0.554169 + 0.832404i \(0.686964\pi\)
\(360\) 3.00000 0.158114
\(361\) −3.00000 −0.157895
\(362\) −20.0000 −1.05118
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 48.0000 2.51243
\(366\) 10.0000 0.522708
\(367\) 32.0000 1.67039 0.835193 − 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 6.00000 0.312772
\(369\) 3.00000 0.156174
\(370\) −12.0000 −0.623850
\(371\) 0 0
\(372\) −1.00000 −0.0518476
\(373\) −7.00000 −0.362446 −0.181223 − 0.983442i \(-0.558006\pi\)
−0.181223 + 0.983442i \(0.558006\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) −3.00000 −0.154713
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 12.0000 0.615587
\(381\) 8.00000 0.409852
\(382\) −3.00000 −0.153493
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −20.0000 −1.01797
\(387\) −4.00000 −0.203331
\(388\) 8.00000 0.406138
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) −3.00000 −0.151911
\(391\) 6.00000 0.303433
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 27.0000 1.36024
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 − 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 7.00000 0.350878
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 30.0000 1.49813 0.749064 − 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 1.00000 0.0498135
\(404\) 0 0
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) 0 0
\(408\) −1.00000 −0.0495074
\(409\) 26.0000 1.28562 0.642809 − 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 9.00000 0.444478
\(411\) −6.00000 −0.295958
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) 27.0000 1.32538
\(416\) 1.00000 0.0490290
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 − 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 − 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 13.0000 0.632830
\(423\) 3.00000 0.145865
\(424\) −12.0000 −0.582772
\(425\) 4.00000 0.194029
\(426\) 6.00000 0.290701
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) −12.0000 −0.578691
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) 11.0000 0.528626 0.264313 − 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 2.00000 0.0957826
\(437\) −24.0000 −1.14808
\(438\) 16.0000 0.764510
\(439\) 17.0000 0.811366 0.405683 − 0.914014i \(-0.367034\pi\)
0.405683 + 0.914014i \(0.367034\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.00000 0.0475651
\(443\) −9.00000 −0.427603 −0.213801 − 0.976877i \(-0.568585\pi\)
−0.213801 + 0.976877i \(0.568585\pi\)
\(444\) −4.00000 −0.189832
\(445\) 36.0000 1.70656
\(446\) 10.0000 0.473514
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 − 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) −4.00000 −0.188562
\(451\) 0 0
\(452\) 9.00000 0.423324
\(453\) −10.0000 −0.469841
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 29.0000 1.35656 0.678281 − 0.734802i \(-0.262725\pi\)
0.678281 + 0.734802i \(0.262725\pi\)
\(458\) 19.0000 0.887812
\(459\) 1.00000 0.0466760
\(460\) −18.0000 −0.839254
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 − 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 3.00000 0.139272
\(465\) 3.00000 0.139122
\(466\) −15.0000 −0.694862
\(467\) 15.0000 0.694117 0.347059 − 0.937843i \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 9.00000 0.415139
\(471\) −13.0000 −0.599008
\(472\) −9.00000 −0.414259
\(473\) 0 0
\(474\) 4.00000 0.183726
\(475\) −16.0000 −0.734130
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 27.0000 1.23495
\(479\) 18.0000 0.822441 0.411220 − 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 3.00000 0.136931
\(481\) 4.00000 0.182384
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −24.0000 −1.08978
\(486\) −1.00000 −0.0453609
\(487\) 11.0000 0.498458 0.249229 − 0.968445i \(-0.419823\pi\)
0.249229 + 0.968445i \(0.419823\pi\)
\(488\) 10.0000 0.452679
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 − 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 3.00000 0.135250
\(493\) 3.00000 0.135113
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 0 0
\(498\) 9.00000 0.403300
\(499\) −31.0000 −1.38775 −0.693875 − 0.720095i \(-0.744098\pi\)
−0.693875 + 0.720095i \(0.744098\pi\)
\(500\) 3.00000 0.134164
\(501\) −12.0000 −0.536120
\(502\) −9.00000 −0.401690
\(503\) −30.0000 −1.33763 −0.668817 − 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 8.00000 0.354943
\(509\) −12.0000 −0.531891 −0.265945 − 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(510\) 3.00000 0.132842
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) 18.0000 0.793946
\(515\) 12.0000 0.528783
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) −3.00000 −0.131685
\(520\) −3.00000 −0.131559
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −3.00000 −0.131306
\(523\) −22.0000 −0.961993 −0.480996 − 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −1.00000 −0.0435607
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 36.0000 1.56374
\(531\) 9.00000 0.390567
\(532\) 0 0
\(533\) −3.00000 −0.129944
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) 9.00000 0.388379
\(538\) 9.00000 0.388018
\(539\) 0 0
\(540\) −3.00000 −0.129099
\(541\) −40.0000 −1.71973 −0.859867 − 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) −2.00000 −0.0859074
\(543\) 20.0000 0.858282
\(544\) −1.00000 −0.0428746
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −1.00000 −0.0427569 −0.0213785 − 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) −6.00000 −0.256307
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) −6.00000 −0.255377
\(553\) 0 0
\(554\) 22.0000 0.934690
\(555\) 12.0000 0.509372
\(556\) 20.0000 0.848189
\(557\) −42.0000 −1.77960 −0.889799 − 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 1.00000 0.0423334
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) −24.0000 −1.01238
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 3.00000 0.126323
\(565\) −27.0000 −1.13590
\(566\) 25.0000 1.05083
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 36.0000 1.50920 0.754599 − 0.656186i \(-0.227831\pi\)
0.754599 + 0.656186i \(0.227831\pi\)
\(570\) −12.0000 −0.502625
\(571\) −40.0000 −1.67395 −0.836974 − 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 3.00000 0.125327
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 1.00000 0.0416667
\(577\) −19.0000 −0.790980 −0.395490 − 0.918470i \(-0.629425\pi\)
−0.395490 + 0.918470i \(0.629425\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 20.0000 0.831172
\(580\) −9.00000 −0.373705
\(581\) 0 0
\(582\) −8.00000 −0.331611
\(583\) 0 0
\(584\) 16.0000 0.662085
\(585\) 3.00000 0.124035
\(586\) 30.0000 1.23929
\(587\) −36.0000 −1.48588 −0.742940 − 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 27.0000 1.11157
\(591\) −27.0000 −1.11063
\(592\) −4.00000 −0.164399
\(593\) −12.0000 −0.492781 −0.246390 − 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −7.00000 −0.286491
\(598\) 6.00000 0.245358
\(599\) −9.00000 −0.367730 −0.183865 − 0.982952i \(-0.558861\pi\)
−0.183865 + 0.982952i \(0.558861\pi\)
\(600\) −4.00000 −0.163299
\(601\) −4.00000 −0.163163 −0.0815817 − 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) −10.0000 −0.406894
\(605\) 33.0000 1.34164
\(606\) 0 0
\(607\) −25.0000 −1.01472 −0.507359 − 0.861735i \(-0.669378\pi\)
−0.507359 + 0.861735i \(0.669378\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −30.0000 −1.21466
\(611\) −3.00000 −0.121367
\(612\) 1.00000 0.0404226
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −26.0000 −1.04927
\(615\) −9.00000 −0.362915
\(616\) 0 0
\(617\) 15.0000 0.603877 0.301939 − 0.953327i \(-0.402366\pi\)
0.301939 + 0.953327i \(0.402366\pi\)
\(618\) 4.00000 0.160904
\(619\) 20.0000 0.803868 0.401934 − 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 3.00000 0.120483
\(621\) 6.00000 0.240772
\(622\) 30.0000 1.20289
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) −29.0000 −1.16000
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) −13.0000 −0.518756
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 − 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 4.00000 0.159111
\(633\) −13.0000 −0.516704
\(634\) −33.0000 −1.31060
\(635\) −24.0000 −0.952411
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 3.00000 0.118585
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 17.0000 0.670415 0.335207 − 0.942144i \(-0.391194\pi\)
0.335207 + 0.942144i \(0.391194\pi\)
\(644\) 0 0
\(645\) 12.0000 0.472500
\(646\) 4.00000 0.157378
\(647\) 21.0000 0.825595 0.412798 − 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) 6.00000 0.234798 0.117399 − 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 36.0000 1.40664
\(656\) 3.00000 0.117130
\(657\) −16.0000 −0.624219
\(658\) 0 0
\(659\) 33.0000 1.28550 0.642749 − 0.766077i \(-0.277794\pi\)
0.642749 + 0.766077i \(0.277794\pi\)
\(660\) 0 0
\(661\) −7.00000 −0.272268 −0.136134 − 0.990690i \(-0.543468\pi\)
−0.136134 + 0.990690i \(0.543468\pi\)
\(662\) −20.0000 −0.777322
\(663\) −1.00000 −0.0388368
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 18.0000 0.696963
\(668\) −12.0000 −0.464294
\(669\) −10.0000 −0.386622
\(670\) 6.00000 0.231800
\(671\) 0 0
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 − 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −8.00000 −0.308148
\(675\) 4.00000 0.153960
\(676\) −12.0000 −0.461538
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −9.00000 −0.345643
\(679\) 0 0
\(680\) 3.00000 0.115045
\(681\) −18.0000 −0.689761
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −4.00000 −0.152944
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) −19.0000 −0.724895
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 18.0000 0.685248
\(691\) 5.00000 0.190209 0.0951045 − 0.995467i \(-0.469681\pi\)
0.0951045 + 0.995467i \(0.469681\pi\)
\(692\) −3.00000 −0.114043
\(693\) 0 0
\(694\) 18.0000 0.683271
\(695\) −60.0000 −2.27593
\(696\) −3.00000 −0.113715
\(697\) 3.00000 0.113633
\(698\) 19.0000 0.719161
\(699\) 15.0000 0.567352
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 − 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 1.00000 0.0377426
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) −9.00000 −0.338960
\(706\) 0 0
\(707\) 0 0
\(708\) 9.00000 0.338241
\(709\) 26.0000 0.976450 0.488225 − 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) −18.0000 −0.675528
\(711\) −4.00000 −0.150012
\(712\) 12.0000 0.449719
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 9.00000 0.336346
\(717\) −27.0000 −1.00833
\(718\) 21.0000 0.783713
\(719\) −42.0000 −1.56634 −0.783168 − 0.621810i \(-0.786397\pi\)
−0.783168 + 0.621810i \(0.786397\pi\)
\(720\) −3.00000 −0.111803
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −10.0000 −0.371904
\(724\) 20.0000 0.743294
\(725\) 12.0000 0.445669
\(726\) 11.0000 0.408248
\(727\) 8.00000 0.296704 0.148352 − 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −48.0000 −1.77656
\(731\) −4.00000 −0.147945
\(732\) −10.0000 −0.369611
\(733\) −46.0000 −1.69905 −0.849524 − 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 0 0
\(738\) −3.00000 −0.110432
\(739\) −40.0000 −1.47142 −0.735712 − 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 12.0000 0.441129
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 1.00000 0.0366618
\(745\) 18.0000 0.659469
\(746\) 7.00000 0.256288
\(747\) −9.00000 −0.329293
\(748\) 0 0
\(749\) 0 0
\(750\) −3.00000 −0.109545
\(751\) −13.0000 −0.474377 −0.237188 − 0.971464i \(-0.576226\pi\)
−0.237188 + 0.971464i \(0.576226\pi\)
\(752\) 3.00000 0.109399
\(753\) 9.00000 0.327978
\(754\) 3.00000 0.109254
\(755\) 30.0000 1.09181
\(756\) 0 0
\(757\) −7.00000 −0.254419 −0.127210 − 0.991876i \(-0.540602\pi\)
−0.127210 + 0.991876i \(0.540602\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) −12.0000 −0.435286
\(761\) 18.0000 0.652499 0.326250 − 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) 3.00000 0.108536
\(765\) −3.00000 −0.108465
\(766\) 0 0
\(767\) −9.00000 −0.324971
\(768\) 1.00000 0.0360844
\(769\) 2.00000 0.0721218 0.0360609 − 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 20.0000 0.719816
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 4.00000 0.143777
\(775\) −4.00000 −0.143684
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 3.00000 0.107417
\(781\) 0 0
\(782\) −6.00000 −0.214560
\(783\) 3.00000 0.107211
\(784\) 0 0
\(785\) 39.0000 1.39197
\(786\) 12.0000 0.428026
\(787\) −49.0000 −1.74666 −0.873331 − 0.487128i \(-0.838045\pi\)
−0.873331 + 0.487128i \(0.838045\pi\)
\(788\) −27.0000 −0.961835
\(789\) −12.0000 −0.427211
\(790\) −12.0000 −0.426941
\(791\) 0 0
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 34.0000 1.20661
\(795\) −36.0000 −1.27679
\(796\) −7.00000 −0.248108
\(797\) 24.0000 0.850124 0.425062 − 0.905164i \(-0.360252\pi\)
0.425062 + 0.905164i \(0.360252\pi\)
\(798\) 0 0
\(799\) 3.00000 0.106132
\(800\) −4.00000 −0.141421
\(801\) −12.0000 −0.423999
\(802\) −30.0000 −1.05934
\(803\) 0 0
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) −1.00000 −0.0352235
\(807\) −9.00000 −0.316815
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 − 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 3.00000 0.105409
\(811\) 47.0000 1.65039 0.825197 − 0.564846i \(-0.191064\pi\)
0.825197 + 0.564846i \(0.191064\pi\)
\(812\) 0 0
\(813\) 2.00000 0.0701431
\(814\) 0 0
\(815\) 48.0000 1.68137
\(816\) 1.00000 0.0350070
\(817\) 16.0000 0.559769
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) 42.0000 1.46581 0.732905 − 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 6.00000 0.209274
\(823\) −31.0000 −1.08059 −0.540296 − 0.841475i \(-0.681688\pi\)
−0.540296 + 0.841475i \(0.681688\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 30.0000 1.04320 0.521601 − 0.853189i \(-0.325335\pi\)
0.521601 + 0.853189i \(0.325335\pi\)
\(828\) 6.00000 0.208514
\(829\) −34.0000 −1.18087 −0.590434 − 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) −27.0000 −0.937184
\(831\) −22.0000 −0.763172
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −20.0000 −0.692543
\(835\) 36.0000 1.24583
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 30.0000 1.03633
\(839\) 30.0000 1.03572 0.517858 − 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 1.00000 0.0344623
\(843\) 24.0000 0.826604
\(844\) −13.0000 −0.447478
\(845\) 36.0000 1.23844
\(846\) −3.00000 −0.103142
\(847\) 0 0
\(848\) 12.0000 0.412082
\(849\) −25.0000 −0.857998
\(850\) −4.00000 −0.137199
\(851\) −24.0000 −0.822709
\(852\) −6.00000 −0.205557
\(853\) 2.00000 0.0684787 0.0342393 − 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) 0 0
\(855\) 12.0000 0.410391
\(856\) 0 0
\(857\) 45.0000 1.53717 0.768585 − 0.639747i \(-0.220961\pi\)
0.768585 + 0.639747i \(0.220961\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 − 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) 0 0
\(863\) −48.0000 −1.63394 −0.816970 − 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 9.00000 0.306009
\(866\) −11.0000 −0.373795
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 0 0
\(870\) 9.00000 0.305129
\(871\) −2.00000 −0.0677674
\(872\) −2.00000 −0.0677285
\(873\) 8.00000 0.270759
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) −16.0000 −0.540590
\(877\) 32.0000 1.08056 0.540282 − 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −17.0000 −0.573722
\(879\) −30.0000 −1.01187
\(880\) 0 0
\(881\) 27.0000 0.909653 0.454827 − 0.890580i \(-0.349701\pi\)
0.454827 + 0.890580i \(0.349701\pi\)
\(882\) 0 0
\(883\) −10.0000 −0.336527 −0.168263 − 0.985742i \(-0.553816\pi\)
−0.168263 + 0.985742i \(0.553816\pi\)
\(884\) −1.00000 −0.0336336
\(885\) −27.0000 −0.907595
\(886\) 9.00000 0.302361
\(887\) 24.0000 0.805841 0.402921 − 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 4.00000 0.134231
\(889\) 0 0
\(890\) −36.0000 −1.20672
\(891\) 0 0
\(892\) −10.0000 −0.334825
\(893\) −12.0000 −0.401565
\(894\) 6.00000 0.200670
\(895\) −27.0000 −0.902510
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) −30.0000 −1.00111
\(899\) −3.00000 −0.100056
\(900\) 4.00000 0.133333
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) −9.00000 −0.299336
\(905\) −60.0000 −1.99447
\(906\) 10.0000 0.332228
\(907\) −25.0000 −0.830111 −0.415056 − 0.909796i \(-0.636238\pi\)
−0.415056 + 0.909796i \(0.636238\pi\)
\(908\) −18.0000 −0.597351
\(909\) 0 0
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 − 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) −4.00000 −0.132453
\(913\) 0 0
\(914\) −29.0000 −0.959235
\(915\) 30.0000 0.991769
\(916\) −19.0000 −0.627778
\(917\) 0 0
\(918\) −1.00000 −0.0330049
\(919\) 44.0000 1.45143 0.725713 − 0.687998i \(-0.241510\pi\)
0.725713 + 0.687998i \(0.241510\pi\)
\(920\) 18.0000 0.593442
\(921\) 26.0000 0.856729
\(922\) 18.0000 0.592798
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) −16.0000 −0.526077
\(926\) 4.00000 0.131448
\(927\) −4.00000 −0.131377
\(928\) −3.00000 −0.0984798
\(929\) −15.0000 −0.492134 −0.246067 − 0.969253i \(-0.579138\pi\)
−0.246067 + 0.969253i \(0.579138\pi\)
\(930\) −3.00000 −0.0983739
\(931\) 0 0
\(932\) 15.0000 0.491341
\(933\) −30.0000 −0.982156
\(934\) −15.0000 −0.490815
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) −22.0000 −0.718709 −0.359354 − 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) −9.00000 −0.293548
\(941\) −6.00000 −0.195594 −0.0977972 − 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 13.0000 0.423563
\(943\) 18.0000 0.586161
\(944\) 9.00000 0.292925
\(945\) 0 0
\(946\) 0 0
\(947\) −24.0000 −0.779895 −0.389948 − 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) −4.00000 −0.129914
\(949\) 16.0000 0.519382
\(950\) 16.0000 0.519109
\(951\) 33.0000 1.07010
\(952\) 0 0
\(953\) −48.0000 −1.55487 −0.777436 − 0.628962i \(-0.783480\pi\)
−0.777436 + 0.628962i \(0.783480\pi\)
\(954\) −12.0000 −0.388514
\(955\) −9.00000 −0.291233
\(956\) −27.0000 −0.873242
\(957\) 0 0
\(958\) −18.0000 −0.581554
\(959\) 0 0
\(960\) −3.00000 −0.0968246
\(961\) −30.0000 −0.967742
\(962\) −4.00000 −0.128965
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −60.0000 −1.93147
\(966\) 0 0
\(967\) 26.0000 0.836104 0.418052 − 0.908423i \(-0.362713\pi\)
0.418052 + 0.908423i \(0.362713\pi\)
\(968\) 11.0000 0.353553
\(969\) −4.00000 −0.128499
\(970\) 24.0000 0.770594
\(971\) 36.0000 1.15529 0.577647 − 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −11.0000 −0.352463
\(975\) −4.00000 −0.128103
\(976\) −10.0000 −0.320092
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 16.0000 0.511624
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) −36.0000 −1.14881
\(983\) 24.0000 0.765481 0.382741 − 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −3.00000 −0.0956365
\(985\) 81.0000 2.58087
\(986\) −3.00000 −0.0955395
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −13.0000 −0.412959 −0.206479 − 0.978451i \(-0.566201\pi\)
−0.206479 + 0.978451i \(0.566201\pi\)
\(992\) 1.00000 0.0317500
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) 21.0000 0.665745
\(996\) −9.00000 −0.285176
\(997\) 20.0000 0.633406 0.316703 − 0.948525i \(-0.397424\pi\)
0.316703 + 0.948525i \(0.397424\pi\)
\(998\) 31.0000 0.981288
\(999\) −4.00000 −0.126554
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.k.1.1 1
7.2 even 3 714.2.i.h.613.1 yes 2
7.4 even 3 714.2.i.h.205.1 ✓ 2
7.6 odd 2 4998.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
714.2.i.h.205.1 ✓ 2 7.4 even 3
714.2.i.h.613.1 yes 2 7.2 even 3
4998.2.a.j.1.1 1 7.6 odd 2
4998.2.a.k.1.1 1 1.1 even 1 trivial