↑

Properties

Label 5550.2.a.bn.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
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 \(\chi\) \(=\) 5550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +6.00000 q^{11} +1.00000 q^{12} +5.00000 q^{13} +1.00000 q^{16} +7.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} +6.00000 q^{22} -8.00000 q^{23} +1.00000 q^{24} +5.00000 q^{26} +1.00000 q^{27} +3.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} +7.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} -2.00000 q^{38} +5.00000 q^{39} -10.0000 q^{41} -4.00000 q^{43} +6.00000 q^{44} -8.00000 q^{46} +3.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +7.00000 q^{51} +5.00000 q^{52} -5.00000 q^{53} +1.00000 q^{54} -2.00000 q^{57} +11.0000 q^{59} +6.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} -13.0000 q^{67} +7.00000 q^{68} -8.00000 q^{69} +1.00000 q^{71} +1.00000 q^{72} +7.00000 q^{73} +1.00000 q^{74} -2.00000 q^{76} +5.00000 q^{78} +7.00000 q^{79} +1.00000 q^{81} -10.0000 q^{82} -4.00000 q^{83} -4.00000 q^{86} +6.00000 q^{88} -14.0000 q^{89} -8.00000 q^{92} +3.00000 q^{93} +3.00000 q^{94} +1.00000 q^{96} -4.00000 q^{97} -7.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 1.00000 0.288675
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 − 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 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
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 5.00000 0.980581
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.00000 1.04447
\(34\) 7.00000 1.20049
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) −2.00000 −0.324443
\(39\) 5.00000 0.800641
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 7.00000 0.980196
\(52\) 5.00000 0.693375
\(53\) −5.00000 −0.686803 −0.343401 − 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 11.0000 1.43208 0.716039 − 0.698060i \(-0.245953\pi\)
0.716039 + 0.698060i \(0.245953\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 3.00000 0.381000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) −13.0000 −1.58820 −0.794101 − 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) 7.00000 0.848875
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 1.00000 0.118678 0.0593391 − 0.998238i \(-0.481101\pi\)
0.0593391 + 0.998238i \(0.481101\pi\)
\(72\) 1.00000 0.117851
\(73\) 7.00000 0.819288 0.409644 − 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) 5.00000 0.566139
\(79\) 7.00000 0.787562 0.393781 − 0.919204i \(-0.371167\pi\)
0.393781 + 0.919204i \(0.371167\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 3.00000 0.311086
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −4.00000 −0.406138 −0.203069 − 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) −7.00000 −0.707107
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −1.00000 −0.0995037 −0.0497519 − 0.998762i \(-0.515843\pi\)
−0.0497519 + 0.998762i \(0.515843\pi\)
\(102\) 7.00000 0.693103
\(103\) −11.0000 −1.08386 −0.541931 − 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(104\) 5.00000 0.490290
\(105\) 0 0
\(106\) −5.00000 −0.485643
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) 1.00000 0.0957826 0.0478913 − 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 0 0
\(113\) −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
\(117\) 5.00000 0.462250
\(118\) 11.0000 1.01263
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 6.00000 0.543214
\(123\) −10.0000 −0.901670
\(124\) 3.00000 0.269408
\(125\) 0 0
\(126\) 0 0
\(127\) 18.0000 1.59724 0.798621 − 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 9.00000 0.786334 0.393167 − 0.919467i \(-0.371379\pi\)
0.393167 + 0.919467i \(0.371379\pi\)
\(132\) 6.00000 0.522233
\(133\) 0 0
\(134\) −13.0000 −1.12303
\(135\) 0 0
\(136\) 7.00000 0.600245
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −8.00000 −0.681005
\(139\) −15.0000 −1.27228 −0.636142 − 0.771572i \(-0.719471\pi\)
−0.636142 + 0.771572i \(0.719471\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 1.00000 0.0839181
\(143\) 30.0000 2.50873
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 7.00000 0.579324
\(147\) −7.00000 −0.577350
\(148\) 1.00000 0.0821995
\(149\) −15.0000 −1.22885 −0.614424 − 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 − 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −2.00000 −0.162221
\(153\) 7.00000 0.565916
\(154\) 0 0
\(155\) 0 0
\(156\) 5.00000 0.400320
\(157\) −16.0000 −1.27694 −0.638470 − 0.769647i \(-0.720432\pi\)
−0.638470 + 0.769647i \(0.720432\pi\)
\(158\) 7.00000 0.556890
\(159\) −5.00000 −0.396526
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −6.00000 −0.469956 −0.234978 − 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) −2.00000 −0.154765 −0.0773823 − 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) −4.00000 −0.304997
\(173\) 1.00000 0.0760286 0.0380143 − 0.999277i \(-0.487897\pi\)
0.0380143 + 0.999277i \(0.487897\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) 11.0000 0.826811
\(178\) −14.0000 −1.04934
\(179\) −1.00000 −0.0747435 −0.0373718 − 0.999301i \(-0.511899\pi\)
−0.0373718 + 0.999301i \(0.511899\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) 3.00000 0.219971
\(187\) 42.0000 3.07134
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) 0 0
\(191\) −22.0000 −1.59186 −0.795932 − 0.605386i \(-0.793019\pi\)
−0.795932 + 0.605386i \(0.793019\pi\)
\(192\) 1.00000 0.0721688
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 6.00000 0.426401
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −13.0000 −0.916949
\(202\) −1.00000 −0.0703598
\(203\) 0 0
\(204\) 7.00000 0.490098
\(205\) 0 0
\(206\) −11.0000 −0.766406
\(207\) −8.00000 −0.556038
\(208\) 5.00000 0.346688
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 − 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) −5.00000 −0.343401
\(213\) 1.00000 0.0685189
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 1.00000 0.0677285
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) 35.0000 2.35435
\(222\) 1.00000 0.0671156
\(223\) 2.00000 0.133930 0.0669650 − 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −10.0000 −0.665190
\(227\) 21.0000 1.39382 0.696909 − 0.717159i \(-0.254558\pi\)
0.696909 + 0.717159i \(0.254558\pi\)
\(228\) −2.00000 −0.132453
\(229\) −4.00000 −0.264327 −0.132164 − 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.0000 0.917170 0.458585 − 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 5.00000 0.326860
\(235\) 0 0
\(236\) 11.0000 0.716039
\(237\) 7.00000 0.454699
\(238\) 0 0
\(239\) 26.0000 1.68180 0.840900 − 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 25.0000 1.60706
\(243\) 1.00000 0.0641500
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) −10.0000 −0.636285
\(248\) 3.00000 0.190500
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 − 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) −48.0000 −3.01773
\(254\) 18.0000 1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 19.0000 1.18519 0.592594 − 0.805502i \(-0.298104\pi\)
0.592594 + 0.805502i \(0.298104\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 9.00000 0.556022
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) −13.0000 −0.794101
\(269\) 31.0000 1.89010 0.945052 − 0.326921i \(-0.106011\pi\)
0.945052 + 0.326921i \(0.106011\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 7.00000 0.424437
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) −13.0000 −0.781094 −0.390547 − 0.920583i \(-0.627714\pi\)
−0.390547 + 0.920583i \(0.627714\pi\)
\(278\) −15.0000 −0.899640
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 − 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 3.00000 0.178647
\(283\) −14.0000 −0.832214 −0.416107 − 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 1.00000 0.0593391
\(285\) 0 0
\(286\) 30.0000 1.77394
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) 7.00000 0.409644
\(293\) 3.00000 0.175262 0.0876309 − 0.996153i \(-0.472070\pi\)
0.0876309 + 0.996153i \(0.472070\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 6.00000 0.348155
\(298\) −15.0000 −0.868927
\(299\) −40.0000 −2.31326
\(300\) 0 0
\(301\) 0 0
\(302\) 10.0000 0.575435
\(303\) −1.00000 −0.0574485
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 7.00000 0.400163
\(307\) −21.0000 −1.19853 −0.599267 − 0.800549i \(-0.704541\pi\)
−0.599267 + 0.800549i \(0.704541\pi\)
\(308\) 0 0
\(309\) −11.0000 −0.625768
\(310\) 0 0
\(311\) 32.0000 1.81455 0.907277 − 0.420534i \(-0.138157\pi\)
0.907277 + 0.420534i \(0.138157\pi\)
\(312\) 5.00000 0.283069
\(313\) −4.00000 −0.226093 −0.113047 − 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) −16.0000 −0.902932
\(315\) 0 0
\(316\) 7.00000 0.393781
\(317\) 19.0000 1.06715 0.533573 − 0.845754i \(-0.320849\pi\)
0.533573 + 0.845754i \(0.320849\pi\)
\(318\) −5.00000 −0.280386
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −14.0000 −0.778981
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −6.00000 −0.332309
\(327\) 1.00000 0.0553001
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 − 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) −4.00000 −0.219529
\(333\) 1.00000 0.0547997
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 − 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 12.0000 0.652714
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) 18.0000 0.974755
\(342\) −2.00000 −0.108148
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 1.00000 0.0537603
\(347\) −33.0000 −1.77153 −0.885766 − 0.464131i \(-0.846367\pi\)
−0.885766 + 0.464131i \(0.846367\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 − 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 6.00000 0.319801
\(353\) −29.0000 −1.54351 −0.771757 − 0.635917i \(-0.780622\pi\)
−0.771757 + 0.635917i \(0.780622\pi\)
\(354\) 11.0000 0.584643
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) −1.00000 −0.0528516
\(359\) −27.0000 −1.42501 −0.712503 − 0.701669i \(-0.752438\pi\)
−0.712503 + 0.701669i \(0.752438\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −2.00000 −0.105118
\(363\) 25.0000 1.31216
\(364\) 0 0
\(365\) 0 0
\(366\) 6.00000 0.313625
\(367\) −12.0000 −0.626395 −0.313197 − 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) −8.00000 −0.417029
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 3.00000 0.155543
\(373\) −24.0000 −1.24267 −0.621336 − 0.783544i \(-0.713410\pi\)
−0.621336 + 0.783544i \(0.713410\pi\)
\(374\) 42.0000 2.17177
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) 0 0
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 − 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) −22.0000 −1.12562
\(383\) −6.00000 −0.306586 −0.153293 − 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −4.00000 −0.203331
\(388\) −4.00000 −0.203069
\(389\) −36.0000 −1.82527 −0.912636 − 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) −56.0000 −2.83204
\(392\) −7.00000 −0.353553
\(393\) 9.00000 0.453990
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) −12.0000 −0.602263 −0.301131 − 0.953583i \(-0.597364\pi\)
−0.301131 + 0.953583i \(0.597364\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) 9.00000 0.449439 0.224719 − 0.974424i \(-0.427853\pi\)
0.224719 + 0.974424i \(0.427853\pi\)
\(402\) −13.0000 −0.648381
\(403\) 15.0000 0.747203
\(404\) −1.00000 −0.0497519
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 7.00000 0.346552
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) −11.0000 −0.541931
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) 5.00000 0.245145
\(417\) −15.0000 −0.734553
\(418\) −12.0000 −0.586939
\(419\) −38.0000 −1.85642 −0.928211 − 0.372055i \(-0.878653\pi\)
−0.928211 + 0.372055i \(0.878653\pi\)
\(420\) 0 0
\(421\) 17.0000 0.828529 0.414265 − 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) 13.0000 0.632830
\(423\) 3.00000 0.145865
\(424\) −5.00000 −0.242821
\(425\) 0 0
\(426\) 1.00000 0.0484502
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 30.0000 1.44841
\(430\) 0 0
\(431\) 32.0000 1.54139 0.770693 − 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 1.00000 0.0481125
\(433\) −3.00000 −0.144171 −0.0720854 − 0.997398i \(-0.522965\pi\)
−0.0720854 + 0.997398i \(0.522965\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.00000 0.0478913
\(437\) 16.0000 0.765384
\(438\) 7.00000 0.334473
\(439\) 24.0000 1.14546 0.572729 − 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 35.0000 1.66478
\(443\) −20.0000 −0.950229 −0.475114 − 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 1.00000 0.0474579
\(445\) 0 0
\(446\) 2.00000 0.0947027
\(447\) −15.0000 −0.709476
\(448\) 0 0
\(449\) −13.0000 −0.613508 −0.306754 − 0.951789i \(-0.599243\pi\)
−0.306754 + 0.951789i \(0.599243\pi\)
\(450\) 0 0
\(451\) −60.0000 −2.82529
\(452\) −10.0000 −0.470360
\(453\) 10.0000 0.469841
\(454\) 21.0000 0.985579
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −28.0000 −1.30978 −0.654892 − 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) −4.00000 −0.186908
\(459\) 7.00000 0.326732
\(460\) 0 0
\(461\) 32.0000 1.49039 0.745194 − 0.666847i \(-0.232357\pi\)
0.745194 + 0.666847i \(0.232357\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 − 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) −39.0000 −1.80470 −0.902352 − 0.430999i \(-0.858161\pi\)
−0.902352 + 0.430999i \(0.858161\pi\)
\(468\) 5.00000 0.231125
\(469\) 0 0
\(470\) 0 0
\(471\) −16.0000 −0.737241
\(472\) 11.0000 0.506316
\(473\) −24.0000 −1.10352
\(474\) 7.00000 0.321521
\(475\) 0 0
\(476\) 0 0
\(477\) −5.00000 −0.228934
\(478\) 26.0000 1.18921
\(479\) 4.00000 0.182765 0.0913823 − 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) 5.00000 0.227980
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 23.0000 1.04223 0.521115 − 0.853487i \(-0.325516\pi\)
0.521115 + 0.853487i \(0.325516\pi\)
\(488\) 6.00000 0.271607
\(489\) −6.00000 −0.271329
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) −10.0000 −0.450835
\(493\) 0 0
\(494\) −10.0000 −0.449921
\(495\) 0 0
\(496\) 3.00000 0.134704
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) −26.0000 −1.16392 −0.581960 − 0.813217i \(-0.697714\pi\)
−0.581960 + 0.813217i \(0.697714\pi\)
\(500\) 0 0
\(501\) −2.00000 −0.0893534
\(502\) 20.0000 0.892644
\(503\) 20.0000 0.891756 0.445878 − 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −48.0000 −2.13386
\(507\) 12.0000 0.532939
\(508\) 18.0000 0.798621
\(509\) 6.00000 0.265945 0.132973 − 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −2.00000 −0.0883022
\(514\) 19.0000 0.838054
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 18.0000 0.791639
\(518\) 0 0
\(519\) 1.00000 0.0438951
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 − 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 − 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 9.00000 0.393167
\(525\) 0 0
\(526\) 0 0
\(527\) 21.0000 0.914774
\(528\) 6.00000 0.261116
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 11.0000 0.477359
\(532\) 0 0
\(533\) −50.0000 −2.16574
\(534\) −14.0000 −0.605839
\(535\) 0 0
\(536\) −13.0000 −0.561514
\(537\) −1.00000 −0.0431532
\(538\) 31.0000 1.33650
\(539\) −42.0000 −1.80907
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 16.0000 0.687259
\(543\) −2.00000 −0.0858282
\(544\) 7.00000 0.300123
\(545\) 0 0
\(546\) 0 0
\(547\) −24.0000 −1.02617 −0.513083 − 0.858339i \(-0.671497\pi\)
−0.513083 + 0.858339i \(0.671497\pi\)
\(548\) −6.00000 −0.256307
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 0 0
\(552\) −8.00000 −0.340503
\(553\) 0 0
\(554\) −13.0000 −0.552317
\(555\) 0 0
\(556\) −15.0000 −0.636142
\(557\) 26.0000 1.10166 0.550828 − 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) 3.00000 0.127000
\(559\) −20.0000 −0.845910
\(560\) 0 0
\(561\) 42.0000 1.77324
\(562\) −15.0000 −0.632737
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 3.00000 0.126323
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) 1.00000 0.0419591
\(569\) 39.0000 1.63497 0.817483 − 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(570\) 0 0
\(571\) −15.0000 −0.627730 −0.313865 − 0.949468i \(-0.601624\pi\)
−0.313865 + 0.949468i \(0.601624\pi\)
\(572\) 30.0000 1.25436
\(573\) −22.0000 −0.919063
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −24.0000 −0.999133 −0.499567 − 0.866276i \(-0.666507\pi\)
−0.499567 + 0.866276i \(0.666507\pi\)
\(578\) 32.0000 1.33102
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) 0 0
\(582\) −4.00000 −0.165805
\(583\) −30.0000 −1.24247
\(584\) 7.00000 0.289662
\(585\) 0 0
\(586\) 3.00000 0.123929
\(587\) 4.00000 0.165098 0.0825488 − 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) −7.00000 −0.288675
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 1.00000 0.0410997
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) −15.0000 −0.614424
\(597\) 16.0000 0.654836
\(598\) −40.0000 −1.63572
\(599\) 7.00000 0.286012 0.143006 − 0.989722i \(-0.454323\pi\)
0.143006 + 0.989722i \(0.454323\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) −13.0000 −0.529401
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) −1.00000 −0.0406222
\(607\) −11.0000 −0.446476 −0.223238 − 0.974764i \(-0.571663\pi\)
−0.223238 + 0.974764i \(0.571663\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 0 0
\(611\) 15.0000 0.606835
\(612\) 7.00000 0.282958
\(613\) −14.0000 −0.565455 −0.282727 − 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −21.0000 −0.847491
\(615\) 0 0
\(616\) 0 0
\(617\) −38.0000 −1.52982 −0.764911 − 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) −11.0000 −0.442485
\(619\) −23.0000 −0.924448 −0.462224 − 0.886763i \(-0.652948\pi\)
−0.462224 + 0.886763i \(0.652948\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 32.0000 1.28308
\(623\) 0 0
\(624\) 5.00000 0.200160
\(625\) 0 0
\(626\) −4.00000 −0.159872
\(627\) −12.0000 −0.479234
\(628\) −16.0000 −0.638470
\(629\) 7.00000 0.279108
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 − 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 7.00000 0.278445
\(633\) 13.0000 0.516704
\(634\) 19.0000 0.754586
\(635\) 0 0
\(636\) −5.00000 −0.198263
\(637\) −35.0000 −1.38675
\(638\) 0 0
\(639\) 1.00000 0.0395594
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 − 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 12.0000 0.473602
\(643\) 14.0000 0.552106 0.276053 − 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −14.0000 −0.550823
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) 66.0000 2.59073
\(650\) 0 0
\(651\) 0 0
\(652\) −6.00000 −0.234978
\(653\) 16.0000 0.626128 0.313064 − 0.949732i \(-0.398644\pi\)
0.313064 + 0.949732i \(0.398644\pi\)
\(654\) 1.00000 0.0391031
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) 7.00000 0.273096
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −9.00000 −0.350059 −0.175030 − 0.984563i \(-0.556002\pi\)
−0.175030 + 0.984563i \(0.556002\pi\)
\(662\) 2.00000 0.0777322
\(663\) 35.0000 1.35929
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) −2.00000 −0.0773823
\(669\) 2.00000 0.0773245
\(670\) 0 0
\(671\) 36.0000 1.38976
\(672\) 0 0
\(673\) −18.0000 −0.693849 −0.346925 − 0.937893i \(-0.612774\pi\)
−0.346925 + 0.937893i \(0.612774\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) 13.0000 0.499631 0.249815 − 0.968294i \(-0.419630\pi\)
0.249815 + 0.968294i \(0.419630\pi\)
\(678\) −10.0000 −0.384048
\(679\) 0 0
\(680\) 0 0
\(681\) 21.0000 0.804722
\(682\) 18.0000 0.689256
\(683\) −37.0000 −1.41577 −0.707883 − 0.706330i \(-0.750350\pi\)
−0.707883 + 0.706330i \(0.750350\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) 0 0
\(687\) −4.00000 −0.152610
\(688\) −4.00000 −0.152499
\(689\) −25.0000 −0.952424
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 − 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 1.00000 0.0380143
\(693\) 0 0
\(694\) −33.0000 −1.25266
\(695\) 0 0
\(696\) 0 0
\(697\) −70.0000 −2.65144
\(698\) 2.00000 0.0757011
\(699\) 14.0000 0.529529
\(700\) 0 0
\(701\) −20.0000 −0.755390 −0.377695 − 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) 5.00000 0.188713
\(703\) −2.00000 −0.0754314
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) −29.0000 −1.09143
\(707\) 0 0
\(708\) 11.0000 0.413405
\(709\) 38.0000 1.42712 0.713560 − 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) 7.00000 0.262521
\(712\) −14.0000 −0.524672
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) −1.00000 −0.0373718
\(717\) 26.0000 0.970988
\(718\) −27.0000 −1.00763
\(719\) 29.0000 1.08152 0.540759 − 0.841178i \(-0.318137\pi\)
0.540759 + 0.841178i \(0.318137\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −15.0000 −0.558242
\(723\) 10.0000 0.371904
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 25.0000 0.927837
\(727\) −36.0000 −1.33517 −0.667583 − 0.744535i \(-0.732671\pi\)
−0.667583 + 0.744535i \(0.732671\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −28.0000 −1.03562
\(732\) 6.00000 0.221766
\(733\) 46.0000 1.69905 0.849524 − 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) −12.0000 −0.442928
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) −78.0000 −2.87317
\(738\) −10.0000 −0.368105
\(739\) 37.0000 1.36107 0.680534 − 0.732717i \(-0.261748\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) −10.0000 −0.367359
\(742\) 0 0
\(743\) 37.0000 1.35740 0.678699 − 0.734416i \(-0.262544\pi\)
0.678699 + 0.734416i \(0.262544\pi\)
\(744\) 3.00000 0.109985
\(745\) 0 0
\(746\) −24.0000 −0.878702
\(747\) −4.00000 −0.146352
\(748\) 42.0000 1.53567
\(749\) 0 0
\(750\) 0 0
\(751\) 38.0000 1.38664 0.693320 − 0.720630i \(-0.256147\pi\)
0.693320 + 0.720630i \(0.256147\pi\)
\(752\) 3.00000 0.109399
\(753\) 20.0000 0.728841
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −17.0000 −0.617876 −0.308938 − 0.951082i \(-0.599973\pi\)
−0.308938 + 0.951082i \(0.599973\pi\)
\(758\) −12.0000 −0.435860
\(759\) −48.0000 −1.74229
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 − 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 18.0000 0.652071
\(763\) 0 0
\(764\) −22.0000 −0.795932
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) 55.0000 1.98593
\(768\) 1.00000 0.0360844
\(769\) 28.0000 1.00971 0.504853 − 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(770\) 0 0
\(771\) 19.0000 0.684268
\(772\) −10.0000 −0.359908
\(773\) −34.0000 −1.22290 −0.611448 − 0.791285i \(-0.709412\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −4.00000 −0.143592
\(777\) 0 0
\(778\) −36.0000 −1.29066
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) −56.0000 −2.00256
\(783\) 0 0
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 9.00000 0.321019
\(787\) 11.0000 0.392108 0.196054 − 0.980593i \(-0.437187\pi\)
0.196054 + 0.980593i \(0.437187\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 6.00000 0.213201
\(793\) 30.0000 1.06533
\(794\) −12.0000 −0.425864
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) 54.0000 1.91278 0.956389 − 0.292096i \(-0.0943526\pi\)
0.956389 + 0.292096i \(0.0943526\pi\)
\(798\) 0 0
\(799\) 21.0000 0.742927
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 9.00000 0.317801
\(803\) 42.0000 1.48215
\(804\) −13.0000 −0.458475
\(805\) 0 0
\(806\) 15.0000 0.528352
\(807\) 31.0000 1.09125
\(808\) −1.00000 −0.0351799
\(809\) 34.0000 1.19538 0.597688 − 0.801729i \(-0.296086\pi\)
0.597688 + 0.801729i \(0.296086\pi\)
\(810\) 0 0
\(811\) −17.0000 −0.596951 −0.298475 − 0.954417i \(-0.596478\pi\)
−0.298475 + 0.954417i \(0.596478\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 6.00000 0.210300
\(815\) 0 0
\(816\) 7.00000 0.245049
\(817\) 8.00000 0.279885
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) −35.0000 −1.22151 −0.610754 − 0.791820i \(-0.709134\pi\)
−0.610754 + 0.791820i \(0.709134\pi\)
\(822\) −6.00000 −0.209274
\(823\) 6.00000 0.209147 0.104573 − 0.994517i \(-0.466652\pi\)
0.104573 + 0.994517i \(0.466652\pi\)
\(824\) −11.0000 −0.383203
\(825\) 0 0
\(826\) 0 0
\(827\) 19.0000 0.660695 0.330347 − 0.943859i \(-0.392834\pi\)
0.330347 + 0.943859i \(0.392834\pi\)
\(828\) −8.00000 −0.278019
\(829\) −50.0000 −1.73657 −0.868286 − 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 0 0
\(831\) −13.0000 −0.450965
\(832\) 5.00000 0.173344
\(833\) −49.0000 −1.69775
\(834\) −15.0000 −0.519408
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 3.00000 0.103695
\(838\) −38.0000 −1.31269
\(839\) 15.0000 0.517858 0.258929 − 0.965896i \(-0.416631\pi\)
0.258929 + 0.965896i \(0.416631\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 17.0000 0.585859
\(843\) −15.0000 −0.516627
\(844\) 13.0000 0.447478
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) 0 0
\(848\) −5.00000 −0.171701
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 1.00000 0.0342594
\(853\) −50.0000 −1.71197 −0.855984 − 0.517003i \(-0.827048\pi\)
−0.855984 + 0.517003i \(0.827048\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 9.00000 0.307434 0.153717 − 0.988115i \(-0.450876\pi\)
0.153717 + 0.988115i \(0.450876\pi\)
\(858\) 30.0000 1.02418
\(859\) −36.0000 −1.22830 −0.614152 − 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 32.0000 1.08992
\(863\) 48.0000 1.63394 0.816970 − 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −3.00000 −0.101944
\(867\) 32.0000 1.08678
\(868\) 0 0
\(869\) 42.0000 1.42475
\(870\) 0 0
\(871\) −65.0000 −2.20244
\(872\) 1.00000 0.0338643
\(873\) −4.00000 −0.135379
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) 7.00000 0.236508
\(877\) 42.0000 1.41824 0.709120 − 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) 24.0000 0.809961
\(879\) 3.00000 0.101187
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 − 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) −7.00000 −0.235702
\(883\) 28.0000 0.942275 0.471138 − 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 35.0000 1.17718
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) −53.0000 −1.77957 −0.889783 − 0.456384i \(-0.849144\pi\)
−0.889783 + 0.456384i \(0.849144\pi\)
\(888\) 1.00000 0.0335578
\(889\) 0 0
\(890\) 0 0
\(891\) 6.00000 0.201008
\(892\) 2.00000 0.0669650
\(893\) −6.00000 −0.200782
\(894\) −15.0000 −0.501675
\(895\) 0 0
\(896\) 0 0
\(897\) −40.0000 −1.33556
\(898\) −13.0000 −0.433816
\(899\) 0 0
\(900\) 0 0
\(901\) −35.0000 −1.16602
\(902\) −60.0000 −1.99778
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) 0 0
\(906\) 10.0000 0.332228
\(907\) −28.0000 −0.929725 −0.464862 − 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 21.0000 0.696909
\(909\) −1.00000 −0.0331679
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 − 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −2.00000 −0.0662266
\(913\) −24.0000 −0.794284
\(914\) −28.0000 −0.926158
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) 0 0
\(918\) 7.00000 0.231034
\(919\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −21.0000 −0.691974
\(922\) 32.0000 1.05386
\(923\) 5.00000 0.164577
\(924\) 0 0
\(925\) 0 0
\(926\) −40.0000 −1.31448
\(927\) −11.0000 −0.361287
\(928\) 0 0
\(929\) −20.0000 −0.656179 −0.328089 − 0.944647i \(-0.606405\pi\)
−0.328089 + 0.944647i \(0.606405\pi\)
\(930\) 0 0
\(931\) 14.0000 0.458831
\(932\) 14.0000 0.458585
\(933\) 32.0000 1.04763
\(934\) −39.0000 −1.27612
\(935\) 0 0
\(936\) 5.00000 0.163430
\(937\) 35.0000 1.14340 0.571700 − 0.820463i \(-0.306284\pi\)
0.571700 + 0.820463i \(0.306284\pi\)
\(938\) 0 0
\(939\) −4.00000 −0.130535
\(940\) 0 0
\(941\) −45.0000 −1.46696 −0.733479 − 0.679712i \(-0.762105\pi\)
−0.733479 + 0.679712i \(0.762105\pi\)
\(942\) −16.0000 −0.521308
\(943\) 80.0000 2.60516
\(944\) 11.0000 0.358020
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) −35.0000 −1.13735 −0.568674 − 0.822563i \(-0.692543\pi\)
−0.568674 + 0.822563i \(0.692543\pi\)
\(948\) 7.00000 0.227349
\(949\) 35.0000 1.13615
\(950\) 0 0
\(951\) 19.0000 0.616117
\(952\) 0 0
\(953\) 12.0000 0.388718 0.194359 − 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) −5.00000 −0.161881
\(955\) 0 0
\(956\) 26.0000 0.840900
\(957\) 0 0
\(958\) 4.00000 0.129234
\(959\) 0 0
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 5.00000 0.161206
\(963\) 12.0000 0.386695
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −31.0000 −0.996893 −0.498446 − 0.866921i \(-0.666096\pi\)
−0.498446 + 0.866921i \(0.666096\pi\)
\(968\) 25.0000 0.803530
\(969\) −14.0000 −0.449745
\(970\) 0 0
\(971\) 50.0000 1.60458 0.802288 − 0.596937i \(-0.203616\pi\)
0.802288 + 0.596937i \(0.203616\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 23.0000 0.736968
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) −57.0000 −1.82359 −0.911796 − 0.410644i \(-0.865304\pi\)
−0.911796 + 0.410644i \(0.865304\pi\)
\(978\) −6.00000 −0.191859
\(979\) −84.0000 −2.68465
\(980\) 0 0
\(981\) 1.00000 0.0319275
\(982\) 30.0000 0.957338
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −10.0000 −0.318788
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −10.0000 −0.318142
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 25.0000 0.794151 0.397076 − 0.917786i \(-0.370025\pi\)
0.397076 + 0.917786i \(0.370025\pi\)
\(992\) 3.00000 0.0952501
\(993\) 2.00000 0.0634681
\(994\) 0 0
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) −45.0000 −1.42516 −0.712582 − 0.701589i \(-0.752474\pi\)
−0.712582 + 0.701589i \(0.752474\pi\)
\(998\) −26.0000 −0.823016
\(999\) 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.bn.1.1 yes 1
5.4 even 2 5550.2.a.f.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5550.2.a.f.1.1 ✓ 1 5.4 even 2
5550.2.a.bn.1.1 yes 1 1.1 even 1 trivial