Properties

Label 9610.2.a.a.1.1
Level $9610$
Weight $2$
Character 9610.1
Self dual yes
Analytic conductor $76.736$
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] = [9610,2,Mod(1,9610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9610, 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("9610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9610 = 2 \cdot 5 \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9610.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: \(76.7362363425\)
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 \(\chi\) \(=\) 9610.1

$q$-expansion

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

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 \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 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 \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −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 \(-0.577979\pi\)
−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 \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 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
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0
\(32\) 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
\(40\) −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 \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) −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 \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 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 \(0.5\pi\)
\(62\) 0 0
\(63\) 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 \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 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 \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 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 \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 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
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 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
\(135\) −4.00000 −0.344265
\(136\) −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 \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 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 \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −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 \(-0.423041\pi\)
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 \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 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 \(-0.675264\pi\)
−0.523205 + 0.852207i \(0.675264\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 24.0000 1.78391 0.891953 0.452128i \(-0.149335\pi\)
0.891953 + 0.452128i \(0.149335\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 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 \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) −2.00000 −0.142134
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 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 \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 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 \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) −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 \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 2.00000 0.129099
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −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
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) −1.00000 −0.0632456
\(251\) −14.0000 −0.883672 −0.441836 0.897096i \(-0.645673\pi\)
−0.441836 + 0.897096i \(0.645673\pi\)
\(252\) 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 \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 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 \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −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
\(279\) 0 0
\(280\) 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
\(310\) 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 \(-0.482113\pi\)
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 \(-0.191471\pi\)
0.824475 + 0.565899i \(0.191471\pi\)
\(332\) −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 \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −13.0000 −0.707107
\(339\) −28.0000 −1.52075
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) −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 \(-0.298926\pi\)
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 \(0.567708\pi\)
\(360\) −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 \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 4.00000 0.208514
\(369\) 6.00000 0.312348
\(370\) −8.00000 −0.415900
\(371\) 0 0
\(372\) 0 0
\(373\) −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 \(-0.634046\pi\)
−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 \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) 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 \(-0.897716\pi\)
−0.948815 + 0.315833i \(0.897716\pi\)
\(402\) −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 \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −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 \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 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 \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 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 \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −4.00000 −0.186908
\(459\) −8.00000 −0.373408
\(460\) −4.00000 −0.186501
\(461\) 8.00000 0.372597 0.186299 0.982493i \(-0.440351\pi\)
0.186299 + 0.982493i \(0.440351\pi\)
\(462\) 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 \(-0.440740\pi\)
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 \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 0 0
\(489\) −48.0000 −2.17064
\(490\) 7.00000 0.316228
\(491\) −2.00000 −0.0902587 −0.0451294 0.998981i \(-0.514370\pi\)
−0.0451294 + 0.998981i \(0.514370\pi\)
\(492\) −12.0000 −0.541002
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) −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 \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) −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 \(-0.812124\pi\)
−0.830812 + 0.556553i \(0.812124\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) 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 \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) 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 \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) −8.00000 −0.335083
\(571\) 26.0000 1.08807 0.544033 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) 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 \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) 14.0000 0.577350
\(589\) 0 0
\(590\) −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 \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 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 \(-0.151698\pi\)
0.888572 + 0.458738i \(0.151698\pi\)
\(614\) −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 \(-0.260549\pi\)
0.683288 + 0.730149i \(0.260549\pi\)
\(620\) 0 0
\(621\) 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 \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) 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 \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 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
\(645\) −4.00000 −0.157500
\(646\) 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 \(-0.537456\pi\)
−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 \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −14.0000 −0.539260
\(675\) 4.00000 0.153960
\(676\) −13.0000 −0.500000
\(677\) 48.0000 1.84479 0.922395 0.386248i \(-0.126229\pi\)
0.922395 + 0.386248i \(0.126229\pi\)
\(678\) −28.0000 −1.07533
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) −16.0000 −0.613121
\(682\) 0 0
\(683\) −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 \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 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 \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 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 \(-0.569977\pi\)
−0.218074 + 0.975932i \(0.569977\pi\)
\(758\) 16.0000 0.581146
\(759\) 16.0000 0.580763
\(760\) 4.00000 0.145095
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 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 \(-0.332029\pi\)
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
\(785\) −6.00000 −0.214149
\(786\) −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
\(790\) −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 \(-0.639748\pi\)
−0.425062 + 0.905164i \(0.639748\pi\)
\(798\) 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 \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 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 \(-0.613479\pi\)
−0.349002 + 0.937122i \(0.613479\pi\)
\(822\) 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
\(825\) 4.00000 0.139262
\(826\) 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 \(-0.687551\pi\)
−0.555703 + 0.831381i \(0.687551\pi\)
\(830\) 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
\(837\) 0 0
\(838\) −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 \(-0.754262\pi\)
−0.716511 + 0.697576i \(0.754262\pi\)
\(860\) 2.00000 0.0681994
\(861\) 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
\(865\) 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 \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) −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
\(885\) 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
\(899\) 0 0
\(900\) 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 \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 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 \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) −26.0000 −0.851658
\(933\) −16.0000 −0.523816
\(934\) 8.00000 0.261768
\(935\) −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 \(-0.0669349\pi\)
0.977972 + 0.208736i \(0.0669349\pi\)
\(942\) −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
\(950\) −4.00000 −0.129777
\(951\) −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
\(955\) 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
\(961\) 0 0
\(962\) 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 \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −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 \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) −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 \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) −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 9610.2.a.a.1.1 1
31.30 odd 2 310.2.a.b.1.1 1
93.92 even 2 2790.2.a.h.1.1 1
124.123 even 2 2480.2.a.c.1.1 1
155.92 even 4 1550.2.b.e.249.2 2
155.123 even 4 1550.2.b.e.249.1 2
155.154 odd 2 1550.2.a.a.1.1 1
248.61 odd 2 9920.2.a.d.1.1 1
248.123 even 2 9920.2.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
310.2.a.b.1.1 1 31.30 odd 2
1550.2.a.a.1.1 1 155.154 odd 2
1550.2.b.e.249.1 2 155.123 even 4
1550.2.b.e.249.2 2 155.92 even 4
2480.2.a.c.1.1 1 124.123 even 2
2790.2.a.h.1.1 1 93.92 even 2
9610.2.a.a.1.1 1 1.1 even 1 trivial
9920.2.a.d.1.1 1 248.61 odd 2
9920.2.a.bg.1.1 1 248.123 even 2