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Properties

Label 5550.2.a.c.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5550,2,Mod(1,5550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.3169731218\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +2.00000 q^{21} +4.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} +6.00000 q^{41} -2.00000 q^{42} +4.00000 q^{43} -4.00000 q^{46} -10.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} +2.00000 q^{57} +6.00000 q^{59} +8.00000 q^{61} -8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} +8.00000 q^{73} -1.00000 q^{74} -2.00000 q^{76} -2.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +2.00000 q^{84} -4.00000 q^{86} +6.00000 q^{89} +4.00000 q^{91} +4.00000 q^{92} -8.00000 q^{93} +10.0000 q^{94} +1.00000 q^{96} -14.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.00000 0.534522
\(15\) 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
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) 2.00000 0.324443
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −2.00000 −0.308607
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −10.0000 −1.45865 −0.729325 − 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −8.00000 −1.01600
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.00000 −0.242536
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 4.00000 0.417029
\(93\) −8.00000 −0.829561
\(94\) 10.0000 1.03142
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −2.00000 −0.198030
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −12.0000 −1.14939 −0.574696 − 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) −2.00000 −0.188982
\(113\) −10.0000 −0.940721 −0.470360 − 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) −6.00000 −0.552345
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −8.00000 −0.724286
\(123\) −6.00000 −0.541002
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 6.00000 0.532414 0.266207 − 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 8.00000 0.683486 0.341743 − 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 4.00000 0.340503
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 10.0000 0.842152
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −8.00000 −0.662085
\(147\) 3.00000 0.247436
\(148\) 1.00000 0.0821995
\(149\) 14.0000 1.14692 0.573462 − 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 2.00000 0.162221
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −6.00000 −0.478852 −0.239426 − 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 16.0000 1.27289
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) −1.00000 −0.0785674
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −2.00000 −0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) −6.00000 −0.449719
\(179\) 6.00000 0.448461 0.224231 − 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 − 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) −4.00000 −0.296500
\(183\) −8.00000 −0.591377
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) −10.0000 −0.729325
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 10.0000 0.712470 0.356235 − 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 4.00000 0.278019
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 − 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −2.00000 −0.137361
\(213\) −8.00000 −0.548151
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −16.0000 −1.08615
\(218\) 12.0000 0.812743
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 1.00000 0.0671156
\(223\) −26.0000 −1.74109 −0.870544 − 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 2.00000 0.132453
\(229\) 2.00000 0.132164 0.0660819 − 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 16.0000 1.03931
\(238\) −4.00000 −0.259281
\(239\) −8.00000 −0.517477 −0.258738 − 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 − 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 11.0000 0.707107
\(243\) −1.00000 −0.0641500
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 4.00000 0.254514
\(248\) −8.00000 −0.508001
\(249\) 0 0
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 − 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) −6.00000 −0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −26.0000 −1.62184 −0.810918 − 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) 4.00000 0.249029
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) 6.00000 0.369976 0.184988 − 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) −6.00000 −0.367194
\(268\) −4.00000 −0.244339
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −2.00000 −0.121268
\(273\) −4.00000 −0.242091
\(274\) −8.00000 −0.483298
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) −14.0000 −0.841178 −0.420589 − 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) −12.0000 −0.719712
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 − 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) −10.0000 −0.595491
\(283\) −12.0000 −0.713326 −0.356663 − 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 8.00000 0.468165
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) −14.0000 −0.810998
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 10.0000 0.574485
\(304\) −2.00000 −0.114708
\(305\) 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
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 − 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) −2.00000 −0.113228
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) −26.0000 −1.46031 −0.730153 − 0.683284i \(-0.760551\pi\)
−0.730153 + 0.683284i \(0.760551\pi\)
\(318\) −2.00000 −0.112154
\(319\) 0 0
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 8.00000 0.445823
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) 12.0000 0.663602
\(328\) −6.00000 −0.331295
\(329\) 20.0000 1.10264
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 − 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 0 0
\(333\) 1.00000 0.0547997
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 8.00000 0.435788 0.217894 − 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 9.00000 0.489535
\(339\) 10.0000 0.543125
\(340\) 0 0
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) 20.0000 1.07990
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) 28.0000 1.50312 0.751559 − 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 − 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 − 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) −4.00000 −0.211702
\(358\) −6.00000 −0.317110
\(359\) −16.0000 −0.844448 −0.422224 − 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 6.00000 0.315353
\(363\) 11.0000 0.577350
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 8.00000 0.418167
\(367\) 22.0000 1.14839 0.574195 − 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 4.00000 0.208514
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) −8.00000 −0.414781
\(373\) 2.00000 0.103556 0.0517780 − 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 10.0000 0.515711
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) −8.00000 −0.409316
\(383\) 4.00000 0.204390 0.102195 − 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 4.00000 0.203331
\(388\) −14.0000 −0.710742
\(389\) −16.0000 −0.811232 −0.405616 − 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 3.00000 0.151523
\(393\) 6.00000 0.302660
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 − 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 16.0000 0.802008
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −4.00000 −0.199502
\(403\) −16.0000 −0.797017
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −2.00000 −0.0990148
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −8.00000 −0.394611
\(412\) 8.00000 0.394132
\(413\) −12.0000 −0.590481
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −20.0000 −0.977064 −0.488532 − 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 − 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 8.00000 0.389434
\(423\) −10.0000 −0.486217
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) −16.0000 −0.774294
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 − 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −16.0000 −0.768911 −0.384455 − 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −12.0000 −0.574696
\(437\) −8.00000 −0.382692
\(438\) 8.00000 0.382255
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −4.00000 −0.190261
\(443\) 12.0000 0.570137 0.285069 − 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 0 0
\(446\) 26.0000 1.23114
\(447\) −14.0000 −0.662177
\(448\) −2.00000 −0.0944911
\(449\) −2.00000 −0.0943858 −0.0471929 − 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −10.0000 −0.470360
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −14.0000 −0.654892 −0.327446 − 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) −2.00000 −0.0934539
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 − 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) −6.00000 −0.276172
\(473\) 0 0
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) −2.00000 −0.0915737
\(478\) 8.00000 0.365911
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 22.0000 1.00207
\(483\) 8.00000 0.364013
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) −8.00000 −0.362143
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 8.00000 0.361035 0.180517 − 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) −6.00000 −0.270501
\(493\) 0 0
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −6.00000 −0.268597 −0.134298 − 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −30.0000 −1.33897
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 6.00000 0.266207
\(509\) −10.0000 −0.443242 −0.221621 − 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) 26.0000 1.14681
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 − 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) 36.0000 1.57417 0.787085 − 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −6.00000 −0.261612
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 4.00000 0.173422
\(533\) −12.0000 −0.519778
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) −6.00000 −0.258919
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 − 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 0 0
\(543\) 6.00000 0.257485
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) −12.0000 −0.513083 −0.256541 − 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 8.00000 0.341743
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 0 0
\(552\) 4.00000 0.170251
\(553\) 32.0000 1.36078
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −8.00000 −0.338667
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 10.0000 0.421076
\(565\) 0 0
\(566\) 12.0000 0.504398
\(567\) −2.00000 −0.0839921
\(568\) −8.00000 −0.335673
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 − 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −10.0000 −0.416305 −0.208153 − 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 13.0000 0.540729
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) −14.0000 −0.580319
\(583\) 0 0
\(584\) −8.00000 −0.331042
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 4.00000 0.165098 0.0825488 − 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 3.00000 0.123718
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 1.00000 0.0410997
\(593\) 16.0000 0.657041 0.328521 − 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) 16.0000 0.654836
\(598\) 8.00000 0.327144
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −42.0000 −1.71322 −0.856608 − 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 8.00000 0.326056
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) −44.0000 −1.78590 −0.892952 − 0.450151i \(-0.851370\pi\)
−0.892952 + 0.450151i \(0.851370\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 0 0
\(611\) 20.0000 0.809113
\(612\) −2.00000 −0.0808452
\(613\) −2.00000 −0.0807792 −0.0403896 − 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) −4.00000 −0.161034 −0.0805170 − 0.996753i \(-0.525657\pi\)
−0.0805170 + 0.996753i \(0.525657\pi\)
\(618\) 8.00000 0.321807
\(619\) 40.0000 1.60774 0.803868 − 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 4.00000 0.160385
\(623\) −12.0000 −0.480770
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) −12.0000 −0.477712 −0.238856 − 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 16.0000 0.636446
\(633\) 8.00000 0.317971
\(634\) 26.0000 1.03259
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 6.00000 0.237729
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −34.0000 −1.34292 −0.671460 − 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 4.00000 0.157867
\(643\) −28.0000 −1.10421 −0.552106 − 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) −20.0000 −0.783260
\(653\) −14.0000 −0.547862 −0.273931 − 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) −12.0000 −0.469237
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 8.00000 0.312110
\(658\) −20.0000 −0.779681
\(659\) −20.0000 −0.779089 −0.389545 − 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 − 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) −10.0000 −0.388661
\(663\) −4.00000 −0.155347
\(664\) 0 0
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) 0 0
\(668\) 0 0
\(669\) 26.0000 1.00522
\(670\) 0 0
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) −28.0000 −1.07932 −0.539660 − 0.841883i \(-0.681447\pi\)
−0.539660 + 0.841883i \(0.681447\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −10.0000 −0.384048
\(679\) 28.0000 1.07454
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) −2.00000 −0.0763048
\(688\) 4.00000 0.152499
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 − 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) −28.0000 −1.06287
\(695\) 0 0
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) 0 0
\(701\) −4.00000 −0.151078 −0.0755390 − 0.997143i \(-0.524068\pi\)
−0.0755390 + 0.997143i \(0.524068\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) 20.0000 0.752177
\(708\) −6.00000 −0.225494
\(709\) −32.0000 −1.20179 −0.600893 − 0.799330i \(-0.705188\pi\)
−0.600893 + 0.799330i \(0.705188\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) −6.00000 −0.224860
\(713\) 32.0000 1.19841
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) 8.00000 0.298765
\(718\) 16.0000 0.597115
\(719\) −16.0000 −0.596699 −0.298350 − 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 15.0000 0.558242
\(723\) 22.0000 0.818189
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) 52.0000 1.92857 0.964287 − 0.264861i \(-0.0853260\pi\)
0.964287 + 0.264861i \(0.0853260\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) −8.00000 −0.295689
\(733\) 22.0000 0.812589 0.406294 − 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) −22.0000 −0.812035
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) −52.0000 −1.91285 −0.956425 − 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) −4.00000 −0.146845
\(743\) −38.0000 −1.39408 −0.697042 − 0.717030i \(-0.745501\pi\)
−0.697042 + 0.717030i \(0.745501\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) −2.00000 −0.0732252
\(747\) 0 0
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 − 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) −10.0000 −0.364662
\(753\) −30.0000 −1.09326
\(754\) 0 0
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 − 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 6.00000 0.217357
\(763\) 24.0000 0.868858
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) −12.0000 −0.433295
\(768\) −1.00000 −0.0360844
\(769\) −38.0000 −1.37032 −0.685158 − 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(770\) 0 0
\(771\) 26.0000 0.936367
\(772\) 14.0000 0.503871
\(773\) −42.0000 −1.51064 −0.755318 − 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 2.00000 0.0717496
\(778\) 16.0000 0.573628
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −6.00000 −0.214013
\(787\) 36.0000 1.28326 0.641631 − 0.767014i \(-0.278258\pi\)
0.641631 + 0.767014i \(0.278258\pi\)
\(788\) 10.0000 0.356235
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) 20.0000 0.711118
\(792\) 0 0
\(793\) −16.0000 −0.568177
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 42.0000 1.48772 0.743858 − 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 4.00000 0.141598
\(799\) 20.0000 0.707549
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) −6.00000 −0.211210
\(808\) 10.0000 0.351799
\(809\) −54.0000 −1.89854 −0.949269 − 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 − 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) −8.00000 −0.279885
\(818\) −14.0000 −0.489499
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 54.0000 1.88461 0.942306 − 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) 8.00000 0.279032
\(823\) −2.00000 −0.0697156 −0.0348578 − 0.999392i \(-0.511098\pi\)
−0.0348578 + 0.999392i \(0.511098\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) −44.0000 −1.53003 −0.765015 − 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 4.00000 0.139010
\(829\) 4.00000 0.138926 0.0694629 − 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) 14.0000 0.485655
\(832\) −2.00000 −0.0693375
\(833\) 6.00000 0.207888
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 20.0000 0.690889
\(839\) 48.0000 1.65714 0.828572 − 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 20.0000 0.689246
\(843\) 10.0000 0.344418
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 10.0000 0.343807
\(847\) 22.0000 0.755929
\(848\) −2.00000 −0.0686803
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) −8.00000 −0.274075
\(853\) −14.0000 −0.479351 −0.239675 − 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 16.0000 0.547509
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 34.0000 1.16142 0.580709 − 0.814111i \(-0.302775\pi\)
0.580709 + 0.814111i \(0.302775\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 − 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) −16.0000 −0.544962
\(863\) −46.0000 −1.56586 −0.782929 − 0.622111i \(-0.786275\pi\)
−0.782929 + 0.622111i \(0.786275\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) 13.0000 0.441503
\(868\) −16.0000 −0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 12.0000 0.406371
\(873\) −14.0000 −0.473828
\(874\) 8.00000 0.270604
\(875\) 0 0
\(876\) −8.00000 −0.270295
\(877\) 2.00000 0.0675352 0.0337676 − 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 8.00000 0.269987
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 − 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 3.00000 0.101015
\(883\) 36.0000 1.21150 0.605748 − 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 30.0000 1.00730 0.503651 − 0.863907i \(-0.331990\pi\)
0.503651 + 0.863907i \(0.331990\pi\)
\(888\) 1.00000 0.0335578
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) 0 0
\(892\) −26.0000 −0.870544
\(893\) 20.0000 0.669274
\(894\) 14.0000 0.468230
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 8.00000 0.267112
\(898\) 2.00000 0.0667409
\(899\) 0 0
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 10.0000 0.332595
\(905\) 0 0
\(906\) 0 0
\(907\) 44.0000 1.46100 0.730498 − 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −12.0000 −0.398234
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 − 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 2.00000 0.0662266
\(913\) 0 0
\(914\) 14.0000 0.463079
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) 12.0000 0.396275
\(918\) −2.00000 −0.0660098
\(919\) −20.0000 −0.659739 −0.329870 − 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) 0 0
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 14.0000 0.459325 0.229663 − 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) 4.00000 0.130954
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 12.0000 0.392023 0.196011 − 0.980602i \(-0.437201\pi\)
0.196011 + 0.980602i \(0.437201\pi\)
\(938\) −8.00000 −0.261209
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 26.0000 0.847576 0.423788 − 0.905761i \(-0.360700\pi\)
0.423788 + 0.905761i \(0.360700\pi\)
\(942\) −6.00000 −0.195491
\(943\) 24.0000 0.781548
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) 4.00000 0.129983 0.0649913 − 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 16.0000 0.519656
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) 26.0000 0.843108
\(952\) −4.00000 −0.129641
\(953\) 16.0000 0.518291 0.259145 − 0.965838i \(-0.416559\pi\)
0.259145 + 0.965838i \(0.416559\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) −16.0000 −0.516667
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 2.00000 0.0644826
\(963\) 4.00000 0.128898
\(964\) −22.0000 −0.708572
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) −4.00000 −0.128631 −0.0643157 − 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 11.0000 0.353553
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) −32.0000 −1.02693 −0.513464 − 0.858111i \(-0.671638\pi\)
−0.513464 + 0.858111i \(0.671638\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −24.0000 −0.769405
\(974\) 0 0
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) −20.0000 −0.639529
\(979\) 0 0
\(980\) 0 0
\(981\) −12.0000 −0.383131
\(982\) −8.00000 −0.255290
\(983\) 10.0000 0.318950 0.159475 − 0.987202i \(-0.449020\pi\)
0.159475 + 0.987202i \(0.449020\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 0 0
\(987\) −20.0000 −0.636607
\(988\) 4.00000 0.127257
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 44.0000 1.39771 0.698853 − 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) −8.00000 −0.254000
\(993\) −10.0000 −0.317340
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 − 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 6.00000 0.189927
\(999\) −1.00000 −0.0316386
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5550.2.a.c.1.1 1
5.2 odd 4 1110.2.d.a.889.1 ✓ 2
5.3 odd 4 1110.2.d.a.889.2 yes 2
5.4 even 2 5550.2.a.bp.1.1 1
15.2 even 4 3330.2.d.g.1999.2 2
15.8 even 4 3330.2.d.g.1999.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.d.a.889.1 ✓ 2 5.2 odd 4
1110.2.d.a.889.2 yes 2 5.3 odd 4
3330.2.d.g.1999.1 2 15.8 even 4
3330.2.d.g.1999.2 2 15.2 even 4
5550.2.a.c.1.1 1 1.1 even 1 trivial
5550.2.a.bp.1.1 1 5.4 even 2