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Properties

Label 5550.2.a.g.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
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] = [5550,2,Mod(1,5550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5550, 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("5550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.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: \(44.3169731218\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{21} -3.00000 q^{22} +1.00000 q^{24} +2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} -1.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} -2.00000 q^{38} +2.00000 q^{39} +9.00000 q^{41} +1.00000 q^{42} -11.0000 q^{43} +3.00000 q^{44} -1.00000 q^{48} -6.00000 q^{49} +3.00000 q^{51} -2.00000 q^{52} +9.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -2.00000 q^{57} +3.00000 q^{58} -6.00000 q^{59} -1.00000 q^{61} +1.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -8.00000 q^{67} -3.00000 q^{68} -12.0000 q^{71} -1.00000 q^{72} -8.00000 q^{73} +1.00000 q^{74} +2.00000 q^{76} +3.00000 q^{77} -2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} -9.00000 q^{82} +6.00000 q^{83} -1.00000 q^{84} +11.0000 q^{86} +3.00000 q^{87} -3.00000 q^{88} -6.00000 q^{89} -2.00000 q^{91} +1.00000 q^{93} +1.00000 q^{96} +13.0000 q^{97} +6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) −3.00000 −0.639602
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) −2.00000 −0.324443
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 − 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 1.00000 0.154303
\(43\) −11.0000 −1.67748 −0.838742 − 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) −2.00000 −0.277350
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −2.00000 −0.264906
\(58\) 3.00000 0.393919
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 1.00000 0.127000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −1.00000 −0.117851
\(73\) −8.00000 −0.936329 −0.468165 − 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 3.00000 0.341882
\(78\) −2.00000 −0.226455
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −9.00000 −0.993884
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 11.0000 1.18616
\(87\) 3.00000 0.321634
\(88\) −3.00000 −0.319801
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 13.0000 1.31995 0.659975 − 0.751288i \(-0.270567\pi\)
0.659975 + 0.751288i \(0.270567\pi\)
\(98\) 6.00000 0.606092
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −3.00000 −0.297044
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 18.0000 1.74013 0.870063 − 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −1.00000 −0.0957826 −0.0478913 − 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 1.00000 0.0944911
\(113\) −9.00000 −0.846649 −0.423324 − 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) −3.00000 −0.278543
\(117\) −2.00000 −0.184900
\(118\) 6.00000 0.552345
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 1.00000 0.0905357
\(123\) −9.00000 −0.811503
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.0000 0.968496
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) −3.00000 −0.261116
\(133\) 2.00000 0.173422
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 − 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) −6.00000 −0.501745
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 8.00000 0.662085
\(147\) 6.00000 0.494872
\(148\) −1.00000 −0.0821995
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 − 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) −2.00000 −0.162221
\(153\) −3.00000 −0.242536
\(154\) −3.00000 −0.241747
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 7.00000 0.558661 0.279330 − 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 4.00000 0.318223
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −11.0000 −0.861586 −0.430793 − 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 9.00000 0.702782
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) −11.0000 −0.838742
\(173\) −15.0000 −1.14043 −0.570214 − 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 6.00000 0.450988
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 2.00000 0.148250
\(183\) 1.00000 0.0739221
\(184\) 0 0
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) −9.00000 −0.658145
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 15.0000 1.08536 0.542681 − 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −2.00000 −0.143963 −0.0719816 − 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −13.0000 −0.933346
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −3.00000 −0.213201
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) −3.00000 −0.210559
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 11.0000 0.757271 0.378636 − 0.925546i \(-0.376393\pi\)
0.378636 + 0.925546i \(0.376393\pi\)
\(212\) 9.00000 0.618123
\(213\) 12.0000 0.822226
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −1.00000 −0.0678844
\(218\) 1.00000 0.0677285
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) −1.00000 −0.0671156
\(223\) −5.00000 −0.334825 −0.167412 − 0.985887i \(-0.553541\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 9.00000 0.598671
\(227\) −3.00000 −0.199117 −0.0995585 − 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) −2.00000 −0.132453
\(229\) 8.00000 0.528655 0.264327 − 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 3.00000 0.196960
\(233\) 12.0000 0.786146 0.393073 − 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 4.00000 0.259828
\(238\) 3.00000 0.194461
\(239\) −9.00000 −0.582162 −0.291081 − 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 2.00000 0.128565
\(243\) −1.00000 −0.0641500
\(244\) −1.00000 −0.0640184
\(245\) 0 0
\(246\) 9.00000 0.573819
\(247\) −4.00000 −0.254514
\(248\) 1.00000 0.0635001
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −11.0000 −0.684830
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 6.00000 0.370681
\(263\) 9.00000 0.554964 0.277482 − 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 6.00000 0.367194
\(268\) −8.00000 −0.488678
\(269\) −24.0000 −1.46331 −0.731653 − 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 − 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) −3.00000 −0.181902
\(273\) 2.00000 0.121046
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 − 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −5.00000 −0.299880
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 9.00000 0.531253
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −13.0000 −0.762073
\(292\) −8.00000 −0.468165
\(293\) −21.0000 −1.22683 −0.613417 − 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) −6.00000 −0.349927
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) −3.00000 −0.174078
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) −11.0000 −0.634029
\(302\) −2.00000 −0.115087
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 3.00000 0.170941
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 3.00000 0.170114 0.0850572 − 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) −2.00000 −0.113228
\(313\) −26.0000 −1.46961 −0.734803 − 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −7.00000 −0.395033
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 33.0000 1.85346 0.926732 − 0.375722i \(-0.122605\pi\)
0.926732 + 0.375722i \(0.122605\pi\)
\(318\) 9.00000 0.504695
\(319\) −9.00000 −0.503903
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 11.0000 0.609234
\(327\) 1.00000 0.0553001
\(328\) −9.00000 −0.496942
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 6.00000 0.329293
\(333\) −1.00000 −0.0547997
\(334\) 0 0
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −14.0000 −0.762629 −0.381314 − 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 9.00000 0.489535
\(339\) 9.00000 0.488813
\(340\) 0 0
\(341\) −3.00000 −0.162459
\(342\) −2.00000 −0.108148
\(343\) −13.0000 −0.701934
\(344\) 11.0000 0.593080
\(345\) 0 0
\(346\) 15.0000 0.806405
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 3.00000 0.160817
\(349\) −28.0000 −1.49881 −0.749403 − 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −3.00000 −0.159901
\(353\) −3.00000 −0.159674 −0.0798369 − 0.996808i \(-0.525440\pi\)
−0.0798369 + 0.996808i \(0.525440\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 3.00000 0.158777
\(358\) 12.0000 0.634220
\(359\) 6.00000 0.316668 0.158334 − 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −2.00000 −0.105118
\(363\) 2.00000 0.104973
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) −1.00000 −0.0522708
\(367\) −23.0000 −1.20059 −0.600295 − 0.799779i \(-0.704950\pi\)
−0.600295 + 0.799779i \(0.704950\pi\)
\(368\) 0 0
\(369\) 9.00000 0.468521
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) 1.00000 0.0518476
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 1.00000 0.0514344
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) −15.0000 −0.767467
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) −11.0000 −0.559161
\(388\) 13.0000 0.659975
\(389\) −33.0000 −1.67317 −0.836583 − 0.547840i \(-0.815450\pi\)
−0.836583 + 0.547840i \(0.815450\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 6.00000 0.303046
\(393\) 6.00000 0.302660
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) −14.0000 −0.702640 −0.351320 − 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 4.00000 0.200502
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −8.00000 −0.399004
\(403\) 2.00000 0.0996271
\(404\) 0 0
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) −3.00000 −0.148704
\(408\) −3.00000 −0.148522
\(409\) −22.0000 −1.08783 −0.543915 − 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 4.00000 0.197066
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −5.00000 −0.244851
\(418\) −6.00000 −0.293470
\(419\) 36.0000 1.75872 0.879358 − 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −11.0000 −0.535472
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) −1.00000 −0.0483934
\(428\) 18.0000 0.870063
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) 3.00000 0.144505 0.0722525 − 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −20.0000 −0.961139 −0.480569 − 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) 1.00000 0.0480015
\(435\) 0 0
\(436\) −1.00000 −0.0478913
\(437\) 0 0
\(438\) −8.00000 −0.382255
\(439\) 17.0000 0.811366 0.405683 − 0.914014i \(-0.367034\pi\)
0.405683 + 0.914014i \(0.367034\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −6.00000 −0.285391
\(443\) −6.00000 −0.285069 −0.142534 − 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 1.00000 0.0474579
\(445\) 0 0
\(446\) 5.00000 0.236757
\(447\) 6.00000 0.283790
\(448\) 1.00000 0.0472456
\(449\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 27.0000 1.27138
\(452\) −9.00000 −0.423324
\(453\) −2.00000 −0.0939682
\(454\) 3.00000 0.140797
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) −35.0000 −1.63723 −0.818615 − 0.574342i \(-0.805258\pi\)
−0.818615 + 0.574342i \(0.805258\pi\)
\(458\) −8.00000 −0.373815
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) 39.0000 1.81641 0.908206 − 0.418524i \(-0.137453\pi\)
0.908206 + 0.418524i \(0.137453\pi\)
\(462\) 3.00000 0.139573
\(463\) −26.0000 −1.20832 −0.604161 − 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) 15.0000 0.694117 0.347059 − 0.937843i \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −7.00000 −0.322543
\(472\) 6.00000 0.276172
\(473\) −33.0000 −1.51734
\(474\) −4.00000 −0.183726
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) 9.00000 0.412082
\(478\) 9.00000 0.411650
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −2.00000 −0.0906287 −0.0453143 − 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 1.00000 0.0452679
\(489\) 11.0000 0.497437
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −9.00000 −0.405751
\(493\) 9.00000 0.405340
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) −12.0000 −0.538274
\(498\) 6.00000 0.268866
\(499\) 32.0000 1.43252 0.716258 − 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) −8.00000 −0.354943
\(509\) −36.0000 −1.59567 −0.797836 − 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −1.00000 −0.0441942
\(513\) −2.00000 −0.0883022
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 11.0000 0.484248
\(517\) 0 0
\(518\) 1.00000 0.0439375
\(519\) 15.0000 0.658427
\(520\) 0 0
\(521\) 27.0000 1.18289 0.591446 − 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) 3.00000 0.131306
\(523\) 40.0000 1.74908 0.874539 − 0.484955i \(-0.161164\pi\)
0.874539 + 0.484955i \(0.161164\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) 3.00000 0.130682
\(528\) −3.00000 −0.130558
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 2.00000 0.0867110
\(533\) −18.0000 −0.779667
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 12.0000 0.517838
\(538\) 24.0000 1.03471
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 − 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 4.00000 0.171815
\(543\) −2.00000 −0.0858282
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) −11.0000 −0.470326 −0.235163 − 0.971956i \(-0.575562\pi\)
−0.235163 + 0.971956i \(0.575562\pi\)
\(548\) 18.0000 0.768922
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) 5.00000 0.212047
\(557\) −6.00000 −0.254228 −0.127114 − 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 1.00000 0.0423334
\(559\) 22.0000 0.930501
\(560\) 0 0
\(561\) 9.00000 0.379980
\(562\) −6.00000 −0.253095
\(563\) −39.0000 −1.64365 −0.821827 − 0.569737i \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 1.00000 0.0419961
\(568\) 12.0000 0.503509
\(569\) −18.0000 −0.754599 −0.377300 − 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −13.0000 −0.544033 −0.272017 − 0.962293i \(-0.587691\pi\)
−0.272017 + 0.962293i \(0.587691\pi\)
\(572\) −6.00000 −0.250873
\(573\) −15.0000 −0.626634
\(574\) −9.00000 −0.375653
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 − 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 8.00000 0.332756
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 13.0000 0.538867
\(583\) 27.0000 1.11823
\(584\) 8.00000 0.331042
\(585\) 0 0
\(586\) 21.0000 0.867502
\(587\) −33.0000 −1.36206 −0.681028 − 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 6.00000 0.247436
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) −1.00000 −0.0410997
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 4.00000 0.163709
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 − 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −31.0000 −1.26452 −0.632258 − 0.774758i \(-0.717872\pi\)
−0.632258 + 0.774758i \(0.717872\pi\)
\(602\) 11.0000 0.448327
\(603\) −8.00000 −0.325785
\(604\) 2.00000 0.0813788
\(605\) 0 0
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 − 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 3.00000 0.121566
\(610\) 0 0
\(611\) 0 0
\(612\) −3.00000 −0.121268
\(613\) 25.0000 1.00974 0.504870 − 0.863195i \(-0.331540\pi\)
0.504870 + 0.863195i \(0.331540\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 4.00000 0.160904
\(619\) 5.00000 0.200967 0.100483 − 0.994939i \(-0.467961\pi\)
0.100483 + 0.994939i \(0.467961\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −3.00000 −0.120289
\(623\) −6.00000 −0.240385
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 26.0000 1.03917
\(627\) −6.00000 −0.239617
\(628\) 7.00000 0.279330
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) 47.0000 1.87104 0.935520 − 0.353273i \(-0.114931\pi\)
0.935520 + 0.353273i \(0.114931\pi\)
\(632\) 4.00000 0.159111
\(633\) −11.0000 −0.437211
\(634\) −33.0000 −1.31060
\(635\) 0 0
\(636\) −9.00000 −0.356873
\(637\) 12.0000 0.475457
\(638\) 9.00000 0.356313
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 9.00000 0.355479 0.177739 − 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) 18.0000 0.710403
\(643\) −23.0000 −0.907031 −0.453516 − 0.891248i \(-0.649830\pi\)
−0.453516 + 0.891248i \(0.649830\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) 12.0000 0.471769 0.235884 − 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −18.0000 −0.706562
\(650\) 0 0
\(651\) 1.00000 0.0391931
\(652\) −11.0000 −0.430793
\(653\) −24.0000 −0.939193 −0.469596 − 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) −1.00000 −0.0391031
\(655\) 0 0
\(656\) 9.00000 0.351391
\(657\) −8.00000 −0.312110
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 17.0000 0.661223 0.330612 − 0.943767i \(-0.392745\pi\)
0.330612 + 0.943767i \(0.392745\pi\)
\(662\) −20.0000 −0.777322
\(663\) −6.00000 −0.233021
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) 0 0
\(669\) 5.00000 0.193311
\(670\) 0 0
\(671\) −3.00000 −0.115814
\(672\) 1.00000 0.0385758
\(673\) 28.0000 1.07932 0.539660 − 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −9.00000 −0.345643
\(679\) 13.0000 0.498894
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 3.00000 0.114876
\(683\) −27.0000 −1.03313 −0.516563 − 0.856249i \(-0.672789\pi\)
−0.516563 + 0.856249i \(0.672789\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) −8.00000 −0.305219
\(688\) −11.0000 −0.419371
\(689\) −18.0000 −0.685745
\(690\) 0 0
\(691\) −25.0000 −0.951045 −0.475522 − 0.879704i \(-0.657741\pi\)
−0.475522 + 0.879704i \(0.657741\pi\)
\(692\) −15.0000 −0.570214
\(693\) 3.00000 0.113961
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) −27.0000 −1.02270
\(698\) 28.0000 1.05982
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 − 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −2.00000 −0.0754314
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 3.00000 0.112906
\(707\) 0 0
\(708\) 6.00000 0.225494
\(709\) −13.0000 −0.488225 −0.244113 − 0.969747i \(-0.578497\pi\)
−0.244113 + 0.969747i \(0.578497\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 9.00000 0.336111
\(718\) −6.00000 −0.223918
\(719\) 30.0000 1.11881 0.559406 − 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 15.0000 0.558242
\(723\) 10.0000 0.371904
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) −26.0000 −0.964287 −0.482143 − 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 33.0000 1.22055
\(732\) 1.00000 0.0369611
\(733\) 1.00000 0.0369358 0.0184679 − 0.999829i \(-0.494121\pi\)
0.0184679 + 0.999829i \(0.494121\pi\)
\(734\) 23.0000 0.848945
\(735\) 0 0
\(736\) 0 0
\(737\) −24.0000 −0.884051
\(738\) −9.00000 −0.331295
\(739\) −7.00000 −0.257499 −0.128750 − 0.991677i \(-0.541096\pi\)
−0.128750 + 0.991677i \(0.541096\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) −9.00000 −0.330400
\(743\) −39.0000 −1.43077 −0.715386 − 0.698730i \(-0.753749\pi\)
−0.715386 + 0.698730i \(0.753749\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) 6.00000 0.219529
\(748\) −9.00000 −0.329073
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 − 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −6.00000 −0.218507
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) 10.0000 0.363456 0.181728 − 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) −15.0000 −0.543750 −0.271875 − 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) −8.00000 −0.289809
\(763\) −1.00000 −0.0362024
\(764\) 15.0000 0.542681
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) −1.00000 −0.0360844
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) −2.00000 −0.0719816
\(773\) 51.0000 1.83434 0.917171 − 0.398493i \(-0.130467\pi\)
0.917171 + 0.398493i \(0.130467\pi\)
\(774\) 11.0000 0.395387
\(775\) 0 0
\(776\) −13.0000 −0.466673
\(777\) 1.00000 0.0358748
\(778\) 33.0000 1.18311
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) 3.00000 0.107211
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) −6.00000 −0.214013
\(787\) −8.00000 −0.285169 −0.142585 − 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 6.00000 0.213741
\(789\) −9.00000 −0.320408
\(790\) 0 0
\(791\) −9.00000 −0.320003
\(792\) −3.00000 −0.106600
\(793\) 2.00000 0.0710221
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 2.00000 0.0707992
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 30.0000 1.05934
\(803\) −24.0000 −0.846942
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −2.00000 −0.0704470
\(807\) 24.0000 0.844840
\(808\) 0 0
\(809\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −3.00000 −0.105279
\(813\) 4.00000 0.140286
\(814\) 3.00000 0.105150
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) −22.0000 −0.769683
\(818\) 22.0000 0.769212
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 − 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 18.0000 0.627822
\(823\) 16.0000 0.557725 0.278862 − 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 9.00000 0.312961 0.156480 − 0.987681i \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(828\) 0 0
\(829\) −13.0000 −0.451509 −0.225754 − 0.974184i \(-0.572485\pi\)
−0.225754 + 0.974184i \(0.572485\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) −2.00000 −0.0693375
\(833\) 18.0000 0.623663
\(834\) 5.00000 0.173136
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) 1.00000 0.0345651
\(838\) −36.0000 −1.24360
\(839\) 18.0000 0.621429 0.310715 − 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 10.0000 0.344623
\(843\) −6.00000 −0.206651
\(844\) 11.0000 0.378636
\(845\) 0 0
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 9.00000 0.309061
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 0 0
\(852\) 12.0000 0.411113
\(853\) 46.0000 1.57501 0.787505 − 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 1.00000 0.0342193
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) −45.0000 −1.53717 −0.768585 − 0.639747i \(-0.779039\pi\)
−0.768585 + 0.639747i \(0.779039\pi\)
\(858\) −6.00000 −0.204837
\(859\) 14.0000 0.477674 0.238837 − 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) −3.00000 −0.102180
\(863\) −9.00000 −0.306364 −0.153182 − 0.988198i \(-0.548952\pi\)
−0.153182 + 0.988198i \(0.548952\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 20.0000 0.679628
\(867\) 8.00000 0.271694
\(868\) −1.00000 −0.0339422
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 1.00000 0.0338643
\(873\) 13.0000 0.439983
\(874\) 0 0
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) −41.0000 −1.38447 −0.692236 − 0.721671i \(-0.743374\pi\)
−0.692236 + 0.721671i \(0.743374\pi\)
\(878\) −17.0000 −0.573722
\(879\) 21.0000 0.708312
\(880\) 0 0
\(881\) 9.00000 0.303218 0.151609 − 0.988441i \(-0.451555\pi\)
0.151609 + 0.988441i \(0.451555\pi\)
\(882\) 6.00000 0.202031
\(883\) 43.0000 1.44707 0.723533 − 0.690290i \(-0.242517\pi\)
0.723533 + 0.690290i \(0.242517\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) −3.00000 −0.100730 −0.0503651 − 0.998731i \(-0.516038\pi\)
−0.0503651 + 0.998731i \(0.516038\pi\)
\(888\) −1.00000 −0.0335578
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) −5.00000 −0.167412
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 0 0
\(899\) 3.00000 0.100056
\(900\) 0 0
\(901\) −27.0000 −0.899500
\(902\) −27.0000 −0.899002
\(903\) 11.0000 0.366057
\(904\) 9.00000 0.299336
\(905\) 0 0
\(906\) 2.00000 0.0664455
\(907\) 52.0000 1.72663 0.863316 − 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) −3.00000 −0.0995585
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 18.0000 0.595713
\(914\) 35.0000 1.15770
\(915\) 0 0
\(916\) 8.00000 0.264327
\(917\) −6.00000 −0.198137
\(918\) −3.00000 −0.0990148
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) −39.0000 −1.28440
\(923\) 24.0000 0.789970
\(924\) −3.00000 −0.0986928
\(925\) 0 0
\(926\) 26.0000 0.854413
\(927\) 4.00000 0.131377
\(928\) 3.00000 0.0984798
\(929\) 15.0000 0.492134 0.246067 − 0.969253i \(-0.420862\pi\)
0.246067 + 0.969253i \(0.420862\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 12.0000 0.393073
\(933\) −3.00000 −0.0982156
\(934\) −15.0000 −0.490815
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 16.0000 0.522697 0.261349 − 0.965244i \(-0.415833\pi\)
0.261349 + 0.965244i \(0.415833\pi\)
\(938\) 8.00000 0.261209
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 7.00000 0.228072
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 33.0000 1.07292
\(947\) −27.0000 −0.877382 −0.438691 − 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 4.00000 0.129914
\(949\) 16.0000 0.519382
\(950\) 0 0
\(951\) −33.0000 −1.07010
\(952\) 3.00000 0.0972306
\(953\) −24.0000 −0.777436 −0.388718 − 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) −9.00000 −0.291386
\(955\) 0 0
\(956\) −9.00000 −0.291081
\(957\) 9.00000 0.290929
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) −2.00000 −0.0644826
\(963\) 18.0000 0.580042
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 − 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 2.00000 0.0642824
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) −51.0000 −1.63667 −0.818334 − 0.574743i \(-0.805102\pi\)
−0.818334 + 0.574743i \(0.805102\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 5.00000 0.160293
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) −15.0000 −0.479893 −0.239946 − 0.970786i \(-0.577130\pi\)
−0.239946 + 0.970786i \(0.577130\pi\)
\(978\) −11.0000 −0.351741
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) −1.00000 −0.0319275
\(982\) 12.0000 0.382935
\(983\) 39.0000 1.24391 0.621953 − 0.783054i \(-0.286339\pi\)
0.621953 + 0.783054i \(0.286339\pi\)
\(984\) 9.00000 0.286910
\(985\) 0 0
\(986\) −9.00000 −0.286618
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) 29.0000 0.921215 0.460608 − 0.887604i \(-0.347632\pi\)
0.460608 + 0.887604i \(0.347632\pi\)
\(992\) 1.00000 0.0317500
\(993\) −20.0000 −0.634681
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) 10.0000 0.316703 0.158352 − 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) −32.0000 −1.01294
\(999\) 1.00000 0.0316386
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 5550.2.a.g.1.1 1
5.4 even 2 1110.2.a.n.1.1 ✓ 1
15.14 odd 2 3330.2.a.h.1.1 1
20.19 odd 2 8880.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.n.1.1 ✓ 1 5.4 even 2
3330.2.a.h.1.1 1 15.14 odd 2
5550.2.a.g.1.1 1 1.1 even 1 trivial
8880.2.a.g.1.1 1 20.19 odd 2