Properties

Label 310.2.a.b.1.1
Level $310$
Weight $2$
Character 310.1
Self dual yes
Analytic conductor $2.475$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [310,2,Mod(1,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, 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("310.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 310.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: \(2.47536246266\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 310.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +2.00000 q^{12} -2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} -4.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} -4.00000 q^{27} -4.00000 q^{29} -2.00000 q^{30} -1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} -4.00000 q^{38} -1.00000 q^{40} +6.00000 q^{41} +2.00000 q^{43} +2.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} +2.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} +8.00000 q^{53} -4.00000 q^{54} -2.00000 q^{55} -8.00000 q^{57} -4.00000 q^{58} +8.00000 q^{59} -2.00000 q^{60} -1.00000 q^{62} +1.00000 q^{64} +4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} +1.00000 q^{72} +6.00000 q^{73} -8.00000 q^{74} +2.00000 q^{75} -4.00000 q^{76} -4.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +6.00000 q^{83} -2.00000 q^{85} +2.00000 q^{86} -8.00000 q^{87} +2.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} -4.00000 q^{92} -2.00000 q^{93} +4.00000 q^{95} +2.00000 q^{96} -2.00000 q^{97} -7.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 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
\(10\) −1.00000 −0.316228
\(11\) 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
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 2.00000 0.408248
\(25\) 1.00000 0.200000
\(26\) 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 \(0.621119\pi\)
\(30\) −2.00000 −0.365148
\(31\) −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 \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 2.00000 0.301511
\(45\) −1.00000 −0.149071
\(46\) −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
\(50\) 1.00000 0.141421
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) −4.00000 −0.544331
\(55\) −2.00000 −0.269680
\(56\) 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 \(0.325650\pi\)
\(60\) −2.00000 −0.258199
\(61\) 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 \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 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 \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −8.00000 −0.929981
\(75\) 2.00000 0.230940
\(76\) −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 \(0.572244\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 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 \(0.603011\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 2.00000 0.204124
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −7.00000 −0.707107
\(99\) 2.00000 0.201008
\(100\) 1.00000 0.100000
\(101\) 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 \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −4.00000 −0.384900
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) −2.00000 −0.190693
\(111\) −16.0000 −1.51865
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) −8.00000 −0.749269
\(115\) 4.00000 0.373002
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) 8.00000 0.736460
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 12.0000 1.08200
\(124\) −1.00000 −0.0898027
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 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
\(135\) 4.00000 0.344265
\(136\) 2.00000 0.171499
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) −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
\(145\) 4.00000 0.332182
\(146\) 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 \(0.694400\pi\)
\(150\) 2.00000 0.163299
\(151\) 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
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) −4.00000 −0.318223
\(159\) 16.0000 1.26888
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −11.0000 −0.864242
\(163\) 24.0000 1.87983 0.939913 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) 6.00000 0.468521
\(165\) −4.00000 −0.311400
\(166\) 6.00000 0.465690
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −2.00000 −0.153393
\(171\) −4.00000 −0.305888
\(172\) 2.00000 0.152499
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −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 \(0.324736\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −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
\(185\) 8.00000 0.588172
\(186\) −2.00000 −0.146647
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) −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 \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 2.00000 0.142134
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.00000 0.564276
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) −6.00000 −0.419058
\(206\) 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
\(215\) −2.00000 −0.136399
\(216\) −4.00000 −0.272166
\(217\) 0 0
\(218\) −18.0000 −1.21911
\(219\) 12.0000 0.810885
\(220\) −2.00000 −0.134840
\(221\) 0 0
\(222\) −16.0000 −1.07385
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 14.0000 0.931266
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) −8.00000 −0.529813
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 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 \(0.373129\pi\)
\(240\) −2.00000 −0.129099
\(241\) −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
\(245\) 7.00000 0.447214
\(246\) 12.0000 0.765092
\(247\) 0 0
\(248\) −1.00000 −0.0635001
\(249\) 12.0000 0.760469
\(250\) −1.00000 −0.0632456
\(251\) 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
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) 10.0000 0.623783 0.311891 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) 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 \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 4.00000 0.246183
\(265\) −8.00000 −0.491436
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 4.00000 0.244339
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 4.00000 0.243432
\(271\) −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
\(275\) 2.00000 0.120605
\(276\) −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 \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) 0 0
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 4.00000 0.234888
\(291\) −4.00000 −0.234484
\(292\) 6.00000 0.351123
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) −14.0000 −0.816497
\(295\) −8.00000 −0.465778
\(296\) −8.00000 −0.464991
\(297\) −8.00000 −0.464207
\(298\) −14.0000 −0.810998
\(299\) 0 0
\(300\) 2.00000 0.115470
\(301\) 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 \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 1.00000 0.0567962
\(311\) 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 \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 16.0000 0.897235
\(319\) −8.00000 −0.447914
\(320\) −1.00000 −0.0559017
\(321\) −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
\(330\) −4.00000 −0.220193
\(331\) −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
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −13.0000 −0.707107
\(339\) 28.0000 1.52075
\(340\) −2.00000 −0.108465
\(341\) −2.00000 −0.108306
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) 8.00000 0.430706
\(346\) −2.00000 −0.107521
\(347\) −22.0000 −1.18102 −0.590511 0.807030i \(-0.701074\pi\)
−0.590511 + 0.807030i \(0.701074\pi\)
\(348\) −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 \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 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 \(0.567708\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) −24.0000 −1.26141
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) −4.00000 −0.208514
\(369\) 6.00000 0.312348
\(370\) 8.00000 0.415900
\(371\) 0 0
\(372\) −2.00000 −0.103695
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 4.00000 0.206835
\(375\) −2.00000 −0.103280
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 4.00000 0.205196
\(381\) 32.0000 1.63941
\(382\) −8.00000 −0.409316
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 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
\(395\) 4.00000 0.201262
\(396\) 2.00000 0.100504
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 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
\(405\) 11.0000 0.546594
\(406\) 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 \(0.452607\pi\)
\(410\) −6.00000 −0.296319
\(411\) 36.0000 1.77575
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) −6.00000 −0.294528
\(416\) 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
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) −2.00000 −0.0964486
\(431\) −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 \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) 8.00000 0.383571
\(436\) −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 \(0.0963387\pi\)
\(440\) −2.00000 −0.0953463
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) −28.0000 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) −16.0000 −0.759326
\(445\) 6.00000 0.284427
\(446\) −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 \(0.454782\pi\)
\(450\) 1.00000 0.0471405
\(451\) 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 \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 4.00000 0.186908
\(459\) −8.00000 −0.373408
\(460\) 4.00000 0.186501
\(461\) −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 \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) −4.00000 −0.185695
\(465\) 2.00000 0.0927478
\(466\) −26.0000 −1.20443
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 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
\(475\) −4.00000 −0.183533
\(476\) 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 \(0.315277\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 0 0
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 2.00000 0.0908153
\(486\) −10.0000 −0.453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 0 0
\(489\) 48.0000 2.17064
\(490\) 7.00000 0.316228
\(491\) 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
\(495\) −2.00000 −0.0898933
\(496\) −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 \(0.428144\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 24.0000 1.07224
\(502\) 14.0000 0.624851
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) −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 \(0.321482\pi\)
\(510\) −4.00000 −0.177123
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 16.0000 0.706417
\(514\) 10.0000 0.441081
\(515\) −8.00000 −0.352522
\(516\) 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 \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) 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
\(530\) −8.00000 −0.347498
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) −12.0000 −0.519291
\(535\) 8.00000 0.345870
\(536\) 4.00000 0.172774
\(537\) 28.0000 1.20829
\(538\) 0 0
\(539\) −14.0000 −0.603023
\(540\) 4.00000 0.172133
\(541\) 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
\(545\) 18.0000 0.771035
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) 2.00000 0.0852803
\(551\) 16.0000 0.681623
\(552\) −8.00000 −0.340503
\(553\) 0 0
\(554\) 0 0
\(555\) 16.0000 0.679162
\(556\) 2.00000 0.0848189
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) −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 \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) −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 \(0.842703\pi\)
\(570\) 8.00000 0.335083
\(571\) −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
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) −13.0000 −0.540729
\(579\) 4.00000 0.166234
\(580\) 4.00000 0.166091
\(581\) 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 \(-0.513142\pi\)
−0.0412744 + 0.999148i \(0.513142\pi\)
\(588\) −14.0000 −0.577350
\(589\) 4.00000 0.164817
\(590\) −8.00000 −0.329355
\(591\) 16.0000 0.658152
\(592\) −8.00000 −0.328798
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) −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 \(0.605994\pi\)
\(600\) 2.00000 0.0816497
\(601\) −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
\(605\) 7.00000 0.284590
\(606\) 4.00000 0.162489
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) −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 \(-0.848302\pi\)
−0.888572 + 0.458738i \(0.848302\pi\)
\(614\) −16.0000 −0.645707
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 16.0000 0.643614
\(619\) −34.0000 −1.36658 −0.683288 0.730149i \(-0.739451\pi\)
−0.683288 + 0.730149i \(0.739451\pi\)
\(620\) 1.00000 0.0401610
\(621\) 16.0000 0.642058
\(622\) 8.00000 0.320771
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −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
\(635\) −16.0000 −0.634941
\(636\) 16.0000 0.634441
\(637\) 0 0
\(638\) −8.00000 −0.316723
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 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 \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) −8.00000 −0.314756
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −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 \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) −36.0000 −1.40771
\(655\) −20.0000 −0.781465
\(656\) 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 \(0.747343\pi\)
\(660\) −4.00000 −0.155700
\(661\) −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
\(670\) −4.00000 −0.154533
\(671\) 0 0
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 14.0000 0.539260
\(675\) −4.00000 −0.153960
\(676\) −13.0000 −0.500000
\(677\) −48.0000 −1.84479 −0.922395 0.386248i \(-0.873771\pi\)
−0.922395 + 0.386248i \(0.873771\pi\)
\(678\) 28.0000 1.07533
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) 16.0000 0.613121
\(682\) −2.00000 −0.0765840
\(683\) −48.0000 −1.83667 −0.918334 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) −4.00000 −0.152944
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) 8.00000 0.305219
\(688\) 2.00000 0.0762493
\(689\) 0 0
\(690\) 8.00000 0.304555
\(691\) 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
\(695\) −2.00000 −0.0758643
\(696\) −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 \(0.5\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −4.00000 −0.148762
\(724\) −24.0000 −0.891953
\(725\) −4.00000 −0.148556
\(726\) −14.0000 −0.519589
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) −6.00000 −0.222070
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 4.00000 0.147643
\(735\) 14.0000 0.516398
\(736\) −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 \(0.341285\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 14.0000 0.512920
\(746\) −26.0000 −0.951928
\(747\) 6.00000 0.219529
\(748\) 4.00000 0.146254
\(749\) 0 0
\(750\) −2.00000 −0.0730297
\(751\) 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
\(755\) −4.00000 −0.145575
\(756\) 0 0
\(757\) 12.0000 0.436147 0.218074 0.975932i \(-0.430023\pi\)
0.218074 + 0.975932i \(0.430023\pi\)
\(758\) 16.0000 0.581146
\(759\) −16.0000 −0.580763
\(760\) 4.00000 0.145095
\(761\) 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
\(765\) −2.00000 −0.0723102
\(766\) 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 \(-0.667971\pi\)
−0.503545 + 0.863969i \(0.667971\pi\)
\(774\) 2.00000 0.0718885
\(775\) −1.00000 −0.0359211
\(776\) −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
\(785\) −6.00000 −0.214149
\(786\) 40.0000 1.42675
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 8.00000 0.284988
\(789\) −48.0000 −1.70885
\(790\) 4.00000 0.142314
\(791\) 0 0
\(792\) 2.00000 0.0710669
\(793\) 0 0
\(794\) −14.0000 −0.496841
\(795\) −16.0000 −0.567462
\(796\) 16.0000 0.567105
\(797\) 24.0000 0.850124 0.425062 0.905164i \(-0.360252\pi\)
0.425062 + 0.905164i \(0.360252\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −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 \(0.556248\pi\)
\(810\) 11.0000 0.386501
\(811\) 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
\(815\) −24.0000 −0.840683
\(816\) 4.00000 0.140028
\(817\) −8.00000 −0.279885
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 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 \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 8.00000 0.278693
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −14.0000 −0.486828 −0.243414 0.969923i \(-0.578267\pi\)
−0.243414 + 0.969923i \(0.578267\pi\)
\(828\) −4.00000 −0.139010
\(829\) 32.0000 1.11141 0.555703 0.831381i \(-0.312449\pi\)
0.555703 + 0.831381i \(0.312449\pi\)
\(830\) −6.00000 −0.208263
\(831\) 0 0
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 4.00000 0.138509
\(835\) −12.0000 −0.415277
\(836\) −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
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) 0 0
\(848\) 8.00000 0.274721
\(849\) −48.0000 −1.64736
\(850\) 2.00000 0.0685994
\(851\) 32.0000 1.09695
\(852\) 0 0
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) −8.00000 −0.273434
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 42.0000 1.43302 0.716511 0.697576i \(-0.245738\pi\)
0.716511 + 0.697576i \(0.245738\pi\)
\(860\) −2.00000 −0.0681994
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −4.00000 −0.136083
\(865\) 2.00000 0.0680020
\(866\) 26.0000 0.883516
\(867\) −26.0000 −0.883006
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 8.00000 0.271225
\(871\) 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 \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 40.0000 1.34993
\(879\) −36.0000 −1.21425
\(880\) −2.00000 −0.0674200
\(881\) −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 \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(884\) 0 0
\(885\) −16.0000 −0.537834
\(886\) −28.0000 −0.940678
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) −16.0000 −0.536925
\(889\) 0 0
\(890\) 6.00000 0.201120
\(891\) −22.0000 −0.737028
\(892\) −8.00000 −0.267860
\(893\) 0 0
\(894\) −28.0000 −0.936460
\(895\) −14.0000 −0.467968
\(896\) 0 0
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) 4.00000 0.133407
\(900\) 1.00000 0.0333333
\(901\) 16.0000 0.533037
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 24.0000 0.797787
\(906\) 8.00000 0.265782
\(907\) −32.0000 −1.06254 −0.531271 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) 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 \(0.270667\pi\)
\(920\) 4.00000 0.131876
\(921\) −32.0000 −1.05444
\(922\) −8.00000 −0.263466
\(923\) 0 0
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) 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 \(0.336217\pi\)
\(930\) 2.00000 0.0655826
\(931\) 28.0000 0.917663
\(932\) −26.0000 −0.851658
\(933\) 16.0000 0.523816
\(934\) 8.00000 0.261768
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 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 \(-0.573045\pi\)
−0.227469 + 0.973785i \(0.573045\pi\)
\(948\) −8.00000 −0.259828
\(949\) 0 0
\(950\) −4.00000 −0.129777
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 8.00000 0.259010
\(955\) 8.00000 0.258874
\(956\) 12.0000 0.388108
\(957\) −16.0000 −0.517207
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −8.00000 −0.257796
\(964\) −2.00000 −0.0644157
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −7.00000 −0.224989
\(969\) −16.0000 −0.513994
\(970\) 2.00000 0.0642161
\(971\) 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 \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) 48.0000 1.53487
\(979\) −12.0000 −0.383522
\(980\) 7.00000 0.223607
\(981\) −18.0000 −0.574696
\(982\) 2.00000 0.0638226
\(983\) 8.00000 0.255160 0.127580 0.991828i \(-0.459279\pi\)
0.127580 + 0.991828i \(0.459279\pi\)
\(984\) 12.0000 0.382546
\(985\) −8.00000 −0.254901
\(986\) −8.00000 −0.254772
\(987\) 0 0
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) −2.00000 −0.0635642
\(991\) −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
\(995\) −16.0000 −0.507234
\(996\) 12.0000 0.380235
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 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 310.2.a.b.1.1 1
3.2 odd 2 2790.2.a.h.1.1 1
4.3 odd 2 2480.2.a.c.1.1 1
5.2 odd 4 1550.2.b.e.249.2 2
5.3 odd 4 1550.2.b.e.249.1 2
5.4 even 2 1550.2.a.a.1.1 1
8.3 odd 2 9920.2.a.bg.1.1 1
8.5 even 2 9920.2.a.d.1.1 1
31.30 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 1.1 even 1 trivial
1550.2.a.a.1.1 1 5.4 even 2
1550.2.b.e.249.1 2 5.3 odd 4
1550.2.b.e.249.2 2 5.2 odd 4
2480.2.a.c.1.1 1 4.3 odd 2
2790.2.a.h.1.1 1 3.2 odd 2
9610.2.a.a.1.1 1 31.30 odd 2
9920.2.a.d.1.1 1 8.5 even 2
9920.2.a.bg.1.1 1 8.3 odd 2