Properties

Label 1550.2.a.a.1.1
Level $1550$
Weight $2$
Character 1550.1
Self dual yes
Analytic conductor $12.377$
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] = [1550,2,Mod(1,1550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1550 = 2 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1550.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: \(12.3768123133\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 310)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -2.00000 q^{12} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{22} +4.00000 q^{23} +2.00000 q^{24} +4.00000 q^{27} -4.00000 q^{29} -1.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} +4.00000 q^{38} +6.00000 q^{41} -2.00000 q^{43} +2.00000 q^{44} -4.00000 q^{46} -2.00000 q^{48} -7.00000 q^{49} +4.00000 q^{51} -8.00000 q^{53} -4.00000 q^{54} +8.00000 q^{57} +4.00000 q^{58} +8.00000 q^{59} +1.00000 q^{62} +1.00000 q^{64} +4.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} -8.00000 q^{69} -1.00000 q^{72} -6.00000 q^{73} -8.00000 q^{74} -4.00000 q^{76} -4.00000 q^{79} -11.0000 q^{81} -6.00000 q^{82} -6.00000 q^{83} +2.00000 q^{86} +8.00000 q^{87} -2.00000 q^{88} -6.00000 q^{89} +4.00000 q^{92} +2.00000 q^{93} +2.00000 q^{96} +2.00000 q^{97} +7.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −2.00000 −0.577350
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.00000 −0.288675
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 4.00000 0.525226
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.00000 −0.242536
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −6.00000 −0.662589
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) 8.00000 0.857690
\(88\) −2.00000 −0.213201
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 7.00000 0.707107
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −4.00000 −0.396059
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 4.00000 0.384900
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) −16.0000 −1.51865
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 8.00000 0.681005
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) 14.0000 1.15470
\(148\) 8.00000 0.657596
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 4.00000 0.324443
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 4.00000 0.318223
\(159\) 16.0000 1.26888
\(160\) 0 0
\(161\) 0 0
\(162\) 11.0000 0.864242
\(163\) −24.0000 −1.87983 −0.939913 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) −2.00000 −0.152499
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) −8.00000 −0.606478
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) −16.0000 −1.20263
\(178\) 6.00000 0.449719
\(179\) 14.0000 1.04641 0.523205 0.852207i \(-0.324736\pi\)
0.523205 + 0.852207i \(0.324736\pi\)
\(180\) 0 0
\(181\) −24.0000 −1.78391 −0.891953 0.452128i \(-0.850665\pi\)
−0.891953 + 0.452128i \(0.850665\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −2.00000 −0.146647
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −2.00000 −0.144338
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) −2.00000 −0.142134
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −8.00000 −0.549442
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 0 0
\(218\) 18.0000 1.21911
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 0 0
\(222\) 16.0000 1.07385
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 8.00000 0.529813
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.00000 0.262613
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 7.00000 0.449977
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) 0 0
\(248\) 1.00000 0.0635001
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 14.0000 0.883672 0.441836 0.897096i \(-0.354327\pi\)
0.441836 + 0.897096i \(0.354327\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −10.0000 −0.623783 −0.311891 0.950118i \(-0.600963\pi\)
−0.311891 + 0.950118i \(0.600963\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) −20.0000 −1.23560
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) −4.00000 −0.244339
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) −2.00000 −0.119952
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 24.0000 1.42665 0.713326 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) −6.00000 −0.351123
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) −14.0000 −0.816497
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 8.00000 0.464207
\(298\) 14.0000 0.810998
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −4.00000 −0.230174
\(303\) −4.00000 −0.229794
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −16.0000 −0.897235
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) −16.0000 −0.893033
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) 24.0000 1.32924
\(327\) 36.0000 1.99080
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −30.0000 −1.64895 −0.824475 0.565899i \(-0.808529\pi\)
−0.824475 + 0.565899i \(0.808529\pi\)
\(332\) −6.00000 −0.329293
\(333\) 8.00000 0.438397
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 13.0000 0.707107
\(339\) 28.0000 1.52075
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) 22.0000 1.18102 0.590511 0.807030i \(-0.298926\pi\)
0.590511 + 0.807030i \(0.298926\pi\)
\(348\) 8.00000 0.428845
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 16.0000 0.850390
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −14.0000 −0.739923
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 24.0000 1.26141
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 4.00000 0.208514
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 2.00000 0.103695
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 32.0000 1.63941
\(382\) 8.00000 0.409316
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) −2.00000 −0.101666
\(388\) 2.00000 0.101535
\(389\) −36.0000 −1.82527 −0.912636 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 7.00000 0.353553
\(393\) −40.0000 −2.01773
\(394\) 8.00000 0.403034
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) 38.0000 1.89763 0.948815 0.315833i \(-0.102284\pi\)
0.948815 + 0.315833i \(0.102284\pi\)
\(402\) −8.00000 −0.399004
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) 16.0000 0.793091
\(408\) −4.00000 −0.198030
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 36.0000 1.77575
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 8.00000 0.391293
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 16.0000 0.778868
\(423\) 0 0
\(424\) 8.00000 0.388514
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 4.00000 0.192450
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) −16.0000 −0.765384
\(438\) −12.0000 −0.573382
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) −16.0000 −0.759326
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 28.0000 1.32435
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) −14.0000 −0.658505
\(453\) −8.00000 −0.375873
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) −8.00000 −0.374634
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −4.00000 −0.186908
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) −8.00000 −0.372597 −0.186299 0.982493i \(-0.559649\pi\)
−0.186299 + 0.982493i \(0.559649\pi\)
\(462\) 0 0
\(463\) −36.0000 −1.67306 −0.836531 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 12.0000 0.552931
\(472\) −8.00000 −0.368230
\(473\) −4.00000 −0.183920
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 0 0
\(477\) −8.00000 −0.366295
\(478\) −12.0000 −0.548867
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 0 0
\(489\) 48.0000 2.17064
\(490\) 0 0
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) −12.0000 −0.541002
\(493\) 8.00000 0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) −14.0000 −0.624851
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −8.00000 −0.355643
\(507\) 26.0000 1.15470
\(508\) −16.0000 −0.709885
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −16.0000 −0.706417
\(514\) 10.0000 0.441081
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) 4.00000 0.175075
\(523\) −38.0000 −1.66162 −0.830812 0.556553i \(-0.812124\pi\)
−0.830812 + 0.556553i \(0.812124\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 2.00000 0.0871214
\(528\) −4.00000 −0.174078
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) −28.0000 −1.20829
\(538\) 0 0
\(539\) −14.0000 −0.603023
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 20.0000 0.859074
\(543\) 48.0000 2.05988
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) 32.0000 1.36822 0.684111 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) −18.0000 −0.768922
\(549\) 0 0
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) 8.00000 0.340503
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 1.00000 0.0423334
\(559\) 0 0
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −10.0000 −0.421825
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −24.0000 −1.00880
\(567\) 0 0
\(568\) 0 0
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −26.0000 −1.08807 −0.544033 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) 13.0000 0.540729
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 0 0
\(582\) 4.00000 0.165805
\(583\) −16.0000 −0.662652
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 2.00000 0.0825488 0.0412744 0.999148i \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) 14.0000 0.577350
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 16.0000 0.658152
\(592\) 8.00000 0.328798
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) −8.00000 −0.328244
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) −32.0000 −1.30967
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) 4.00000 0.162489
\(607\) 24.0000 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) 44.0000 1.77714 0.888572 0.458738i \(-0.151698\pi\)
0.888572 + 0.458738i \(0.151698\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) −16.0000 −0.643614
\(619\) −34.0000 −1.36658 −0.683288 0.730149i \(-0.739451\pi\)
−0.683288 + 0.730149i \(0.739451\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) −8.00000 −0.320771
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) 16.0000 0.638978
\(628\) −6.00000 −0.239426
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 4.00000 0.159111
\(633\) 32.0000 1.27189
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) 16.0000 0.634441
\(637\) 0 0
\(638\) 8.00000 0.316723
\(639\) 0 0
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 16.0000 0.631470
\(643\) 2.00000 0.0788723 0.0394362 0.999222i \(-0.487444\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 11.0000 0.432121
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) −24.0000 −0.939913
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) −36.0000 −1.40771
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 30.0000 1.16598
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) −16.0000 −0.619522
\(668\) −12.0000 −0.464294
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 48.0000 1.84479 0.922395 0.386248i \(-0.126229\pi\)
0.922395 + 0.386248i \(0.126229\pi\)
\(678\) −28.0000 −1.07533
\(679\) 0 0
\(680\) 0 0
\(681\) 16.0000 0.613121
\(682\) 2.00000 0.0765840
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) −8.00000 −0.305219
\(688\) −2.00000 −0.0762493
\(689\) 0 0
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) −22.0000 −0.835109
\(695\) 0 0
\(696\) −8.00000 −0.303239
\(697\) −12.0000 −0.454532
\(698\) −10.0000 −0.378506
\(699\) −52.0000 −1.96682
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 0 0
\(708\) −16.0000 −0.601317
\(709\) −16.0000 −0.600893 −0.300446 0.953799i \(-0.597136\pi\)
−0.300446 + 0.953799i \(0.597136\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 6.00000 0.224860
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) 14.0000 0.523205
\(717\) −24.0000 −0.896296
\(718\) 8.00000 0.298557
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 4.00000 0.148762
\(724\) −24.0000 −0.891953
\(725\) 0 0
\(726\) −14.0000 −0.519589
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −8.00000 −0.294684
\(738\) −6.00000 −0.220863
\(739\) 26.0000 0.956425 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 0 0
\(746\) −26.0000 −0.951928
\(747\) −6.00000 −0.219529
\(748\) −4.00000 −0.146254
\(749\) 0 0
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 0 0
\(753\) −28.0000 −1.02038
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −12.0000 −0.436147 −0.218074 0.975932i \(-0.569977\pi\)
−0.218074 + 0.975932i \(0.569977\pi\)
\(758\) −16.0000 −0.581146
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) −32.0000 −1.15924
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 0 0
\(768\) −2.00000 −0.0721688
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 20.0000 0.720282
\(772\) −2.00000 −0.0719816
\(773\) 28.0000 1.00709 0.503545 0.863969i \(-0.332029\pi\)
0.503545 + 0.863969i \(0.332029\pi\)
\(774\) 2.00000 0.0718885
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 36.0000 1.29066
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) −16.0000 −0.571793
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 40.0000 1.42675
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) −8.00000 −0.284988
\(789\) −48.0000 −1.70885
\(790\) 0 0
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) 0 0
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −24.0000 −0.850124 −0.425062 0.905164i \(-0.639748\pi\)
−0.425062 + 0.905164i \(0.639748\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) −38.0000 −1.34183
\(803\) −12.0000 −0.423471
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −2.00000 −0.0703598
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 24.0000 0.842754 0.421377 0.906886i \(-0.361547\pi\)
0.421377 + 0.906886i \(0.361547\pi\)
\(812\) 0 0
\(813\) 40.0000 1.40286
\(814\) −16.0000 −0.560800
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 8.00000 0.279885
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) 0 0
\(821\) 20.0000 0.698005 0.349002 0.937122i \(-0.386521\pi\)
0.349002 + 0.937122i \(0.386521\pi\)
\(822\) −36.0000 −1.25564
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) 14.0000 0.486828 0.243414 0.969923i \(-0.421733\pi\)
0.243414 + 0.969923i \(0.421733\pi\)
\(828\) 4.00000 0.139010
\(829\) 32.0000 1.11141 0.555703 0.831381i \(-0.312449\pi\)
0.555703 + 0.831381i \(0.312449\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 14.0000 0.485071
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) −4.00000 −0.138260
\(838\) 24.0000 0.829066
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 22.0000 0.758170
\(843\) −20.0000 −0.688837
\(844\) −16.0000 −0.550743
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −8.00000 −0.274721
\(849\) −48.0000 −1.64736
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) 42.0000 1.43302 0.716511 0.697576i \(-0.245738\pi\)
0.716511 + 0.697576i \(0.245738\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) 26.0000 0.883006
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 0 0
\(872\) 18.0000 0.609557
\(873\) 2.00000 0.0676897
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) −40.0000 −1.34993
\(879\) −36.0000 −1.21425
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 7.00000 0.235702
\(883\) 26.0000 0.874970 0.437485 0.899226i \(-0.355869\pi\)
0.437485 + 0.899226i \(0.355869\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −28.0000 −0.940678
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 16.0000 0.536925
\(889\) 0 0
\(890\) 0 0
\(891\) −22.0000 −0.737028
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) −28.0000 −0.936460
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) 16.0000 0.533037
\(902\) −12.0000 −0.399556
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) 32.0000 1.06254 0.531271 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) −8.00000 −0.265489
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 8.00000 0.264906
\(913\) −12.0000 −0.397142
\(914\) 2.00000 0.0661541
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) 0 0
\(918\) 8.00000 0.264039
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) −32.0000 −1.05444
\(922\) 8.00000 0.263466
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 36.0000 1.18303
\(927\) −8.00000 −0.262754
\(928\) 4.00000 0.131306
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) 26.0000 0.851658
\(933\) −16.0000 −0.523816
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 0 0
\(939\) −44.0000 −1.43589
\(940\) 0 0
\(941\) −60.0000 −1.95594 −0.977972 0.208736i \(-0.933065\pi\)
−0.977972 + 0.208736i \(0.933065\pi\)
\(942\) −12.0000 −0.390981
\(943\) 24.0000 0.781548
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 14.0000 0.454939 0.227469 0.973785i \(-0.426955\pi\)
0.227469 + 0.973785i \(0.426955\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 8.00000 0.259010
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 16.0000 0.517207
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 8.00000 0.257796
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 7.00000 0.224989
\(969\) −16.0000 −0.513994
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 10.0000 0.320750
\(973\) 0 0
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) 0 0
\(977\) −34.0000 −1.08776 −0.543878 0.839164i \(-0.683045\pi\)
−0.543878 + 0.839164i \(0.683045\pi\)
\(978\) −48.0000 −1.53487
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) −2.00000 −0.0638226
\(983\) −8.00000 −0.255160 −0.127580 0.991828i \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) 12.0000 0.382546
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) 0 0
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 1.00000 0.0317500
\(993\) 60.0000 1.90404
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) −10.0000 −0.316544
\(999\) 32.0000 1.01244
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 1550.2.a.a.1.1 1
5.2 odd 4 1550.2.b.e.249.1 2
5.3 odd 4 1550.2.b.e.249.2 2
5.4 even 2 310.2.a.b.1.1 1
15.14 odd 2 2790.2.a.h.1.1 1
20.19 odd 2 2480.2.a.c.1.1 1
40.19 odd 2 9920.2.a.bg.1.1 1
40.29 even 2 9920.2.a.d.1.1 1
155.154 odd 2 9610.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
310.2.a.b.1.1 1 5.4 even 2
1550.2.a.a.1.1 1 1.1 even 1 trivial
1550.2.b.e.249.1 2 5.2 odd 4
1550.2.b.e.249.2 2 5.3 odd 4
2480.2.a.c.1.1 1 20.19 odd 2
2790.2.a.h.1.1 1 15.14 odd 2
9610.2.a.a.1.1 1 155.154 odd 2
9920.2.a.d.1.1 1 40.29 even 2
9920.2.a.bg.1.1 1 40.19 odd 2