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Properties

Label 1062.2.a.b.1.1
Level $1062$
Weight $2$
Character 1062.1
Self dual yes
Analytic conductor $8.480$
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] = [1062,2,Mod(1,1062)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1062, 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("1062.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1062 = 2 \cdot 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1062.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: \(8.48011269466\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 118)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1062.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.00000 q^{11} -6.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -5.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} -4.00000 q^{23} -4.00000 q^{25} +6.00000 q^{26} +3.00000 q^{28} +5.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} -3.00000 q^{35} +8.00000 q^{37} +5.00000 q^{38} +1.00000 q^{40} -7.00000 q^{41} -6.00000 q^{43} -2.00000 q^{44} +4.00000 q^{46} +2.00000 q^{47} +2.00000 q^{49} +4.00000 q^{50} -6.00000 q^{52} -9.00000 q^{53} +2.00000 q^{55} -3.00000 q^{56} -5.00000 q^{58} +1.00000 q^{59} -8.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} -2.00000 q^{67} +2.00000 q^{68} +3.00000 q^{70} -12.0000 q^{71} +4.00000 q^{73} -8.00000 q^{74} -5.00000 q^{76} -6.00000 q^{77} +5.00000 q^{79} -1.00000 q^{80} +7.00000 q^{82} -14.0000 q^{83} -2.00000 q^{85} +6.00000 q^{86} +2.00000 q^{88} -18.0000 q^{91} -4.00000 q^{92} -2.00000 q^{94} +5.00000 q^{95} +8.00000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 − 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) 3.00000 0.566947
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 5.00000 0.811107
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −7.00000 −1.09322 −0.546608 − 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) −9.00000 −1.23625 −0.618123 − 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) −5.00000 −0.656532
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 3.00000 0.358569
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −5.00000 −0.573539
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 − 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 7.00000 0.773021
\(83\) −14.0000 −1.53670 −0.768350 − 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 6.00000 0.646997
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −18.0000 −1.88691
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −2.00000 −0.206284
\(95\) 5.00000 0.512989
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −2.00000 −0.202031
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) 18.0000 1.79107 0.895533 − 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) −3.00000 −0.290021 −0.145010 − 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) −20.0000 −1.91565 −0.957826 − 0.287348i \(-0.907226\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) 3.00000 0.283473
\(113\) −14.0000 −1.31701 −0.658505 − 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 5.00000 0.464238
\(117\) 0 0
\(118\) −1.00000 −0.0920575
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 13.0000 1.15356 0.576782 − 0.816898i \(-0.304308\pi\)
0.576782 + 0.816898i \(0.304308\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −6.00000 −0.526235
\(131\) 18.0000 1.57267 0.786334 − 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) −15.0000 −1.30066
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −3.00000 −0.256307 −0.128154 − 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) −3.00000 −0.253546
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) −5.00000 −0.415227
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 − 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 5.00000 0.405554
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) −12.0000 −0.957704 −0.478852 − 0.877896i \(-0.658947\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) −5.00000 −0.397779
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −7.00000 −0.546608
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) −3.00000 −0.232147 −0.116073 − 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) −6.00000 −0.457496
\(173\) −4.00000 −0.304114 −0.152057 − 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 0 0
\(175\) −12.0000 −0.907115
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 − 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) −13.0000 −0.966282 −0.483141 − 0.875542i \(-0.660504\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(182\) 18.0000 1.33425
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 2.00000 0.145865
\(189\) 0 0
\(190\) −5.00000 −0.362738
\(191\) 18.0000 1.30243 0.651217 − 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) 19.0000 1.36765 0.683825 − 0.729646i \(-0.260315\pi\)
0.683825 + 0.729646i \(0.260315\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −5.00000 −0.354441 −0.177220 − 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) 4.00000 0.282843
\(201\) 0 0
\(202\) −18.0000 −1.26648
\(203\) 15.0000 1.05279
\(204\) 0 0
\(205\) 7.00000 0.488901
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) 10.0000 0.691714
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 − 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −9.00000 −0.618123
\(213\) 0 0
\(214\) 3.00000 0.205076
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 20.0000 1.35457
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 − 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −5.00000 −0.328266
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −2.00000 −0.130466
\(236\) 1.00000 0.0650945
\(237\) 0 0
\(238\) −6.00000 −0.388922
\(239\) 25.0000 1.61712 0.808558 − 0.588417i \(-0.200249\pi\)
0.808558 + 0.588417i \(0.200249\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 − 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 30.0000 1.90885
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) −9.00000 −0.569210
\(251\) 23.0000 1.45175 0.725874 − 0.687828i \(-0.241436\pi\)
0.725874 + 0.687828i \(0.241436\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) −13.0000 −0.815693
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 27.0000 1.68421 0.842107 − 0.539311i \(-0.181315\pi\)
0.842107 + 0.539311i \(0.181315\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) −18.0000 −1.11204
\(263\) 21.0000 1.29492 0.647458 − 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 15.0000 0.919709
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) 10.0000 0.609711 0.304855 − 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 17.0000 1.03268 0.516338 − 0.856385i \(-0.327295\pi\)
0.516338 + 0.856385i \(0.327295\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 3.00000 0.181237
\(275\) 8.00000 0.482418
\(276\) 0 0
\(277\) −27.0000 −1.62227 −0.811136 − 0.584857i \(-0.801151\pi\)
−0.811136 + 0.584857i \(0.801151\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) 13.0000 0.775515 0.387757 − 0.921761i \(-0.373250\pi\)
0.387757 + 0.921761i \(0.373250\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 − 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) −21.0000 −1.23959
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 5.00000 0.293610
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) −19.0000 −1.10999 −0.554996 − 0.831853i \(-0.687280\pi\)
−0.554996 + 0.831853i \(0.687280\pi\)
\(294\) 0 0
\(295\) −1.00000 −0.0582223
\(296\) −8.00000 −0.464991
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −18.0000 −1.03750
\(302\) −2.00000 −0.115087
\(303\) 0 0
\(304\) −5.00000 −0.286770
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 13.0000 0.741949 0.370975 − 0.928643i \(-0.379024\pi\)
0.370975 + 0.928643i \(0.379024\pi\)
\(308\) −6.00000 −0.341882
\(309\) 0 0
\(310\) 2.00000 0.113592
\(311\) −7.00000 −0.396934 −0.198467 − 0.980108i \(-0.563596\pi\)
−0.198467 + 0.980108i \(0.563596\pi\)
\(312\) 0 0
\(313\) 24.0000 1.35656 0.678280 − 0.734803i \(-0.262726\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(314\) 12.0000 0.677199
\(315\) 0 0
\(316\) 5.00000 0.281272
\(317\) 2.00000 0.112331 0.0561656 − 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) −10.0000 −0.559893
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 12.0000 0.668734
\(323\) −10.0000 −0.556415
\(324\) 0 0
\(325\) 24.0000 1.33128
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 7.00000 0.386510
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 27.0000 1.48405 0.742027 − 0.670370i \(-0.233865\pi\)
0.742027 + 0.670370i \(0.233865\pi\)
\(332\) −14.0000 −0.768350
\(333\) 0 0
\(334\) 3.00000 0.164153
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 − 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) 4.00000 0.215041
\(347\) 22.0000 1.18102 0.590511 − 0.807030i \(-0.298926\pi\)
0.590511 + 0.807030i \(0.298926\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 − 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 12.0000 0.641427
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) −4.00000 −0.212899 −0.106449 − 0.994318i \(-0.533948\pi\)
−0.106449 + 0.994318i \(0.533948\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) 15.0000 0.791670 0.395835 − 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 13.0000 0.683265
\(363\) 0 0
\(364\) −18.0000 −0.943456
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) −2.00000 −0.104399 −0.0521996 − 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) −27.0000 −1.40177
\(372\) 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
\(375\) 0 0
\(376\) −2.00000 −0.103142
\(377\) −30.0000 −1.54508
\(378\) 0 0
\(379\) 5.00000 0.256833 0.128416 − 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) 5.00000 0.256495
\(381\) 0 0
\(382\) −18.0000 −0.920960
\(383\) 16.0000 0.817562 0.408781 − 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 6.00000 0.305788
\(386\) −19.0000 −0.967075
\(387\) 0 0
\(388\) 8.00000 0.406138
\(389\) 10.0000 0.507020 0.253510 − 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) −2.00000 −0.101015
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) −5.00000 −0.251577
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 5.00000 0.250627
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −12.0000 −0.599251 −0.299626 − 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) −12.0000 −0.597763
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) −15.0000 −0.744438
\(407\) −16.0000 −0.793091
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 − 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) −7.00000 −0.345705
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) 3.00000 0.147620
\(414\) 0 0
\(415\) 14.0000 0.687233
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) −10.0000 −0.489116
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 − 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 9.00000 0.437079
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) −24.0000 −1.16144
\(428\) −3.00000 −0.145010
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) 18.0000 0.867029 0.433515 − 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 19.0000 0.913082 0.456541 − 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) −20.0000 −0.957826
\(437\) 20.0000 0.956730
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −24.0000 −1.14027 −0.570137 − 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −24.0000 −1.13643
\(447\) 0 0
\(448\) 3.00000 0.141737
\(449\) −5.00000 −0.235965 −0.117982 − 0.993016i \(-0.537643\pi\)
−0.117982 + 0.993016i \(0.537643\pi\)
\(450\) 0 0
\(451\) 14.0000 0.659234
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) 8.00000 0.375459
\(455\) 18.0000 0.843853
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 10.0000 0.467269
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −36.0000 −1.67306 −0.836531 − 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −28.0000 −1.29569 −0.647843 − 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) −6.00000 −0.277054
\(470\) 2.00000 0.0922531
\(471\) 0 0
\(472\) −1.00000 −0.0460287
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) 20.0000 0.917663
\(476\) 6.00000 0.275010
\(477\) 0 0
\(478\) −25.0000 −1.14347
\(479\) −40.0000 −1.82765 −0.913823 − 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) 0 0
\(481\) −48.0000 −2.18861
\(482\) −17.0000 −0.774329
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) −17.0000 −0.770344 −0.385172 − 0.922845i \(-0.625858\pi\)
−0.385172 + 0.922845i \(0.625858\pi\)
\(488\) 8.00000 0.362143
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) −37.0000 −1.66979 −0.834893 − 0.550412i \(-0.814471\pi\)
−0.834893 + 0.550412i \(0.814471\pi\)
\(492\) 0 0
\(493\) 10.0000 0.450377
\(494\) −30.0000 −1.34976
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −36.0000 −1.61482
\(498\) 0 0
\(499\) 15.0000 0.671492 0.335746 − 0.941953i \(-0.391012\pi\)
0.335746 + 0.941953i \(0.391012\pi\)
\(500\) 9.00000 0.402492
\(501\) 0 0
\(502\) −23.0000 −1.02654
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) 13.0000 0.576782
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −27.0000 −1.19092
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) −4.00000 −0.175920
\(518\) −24.0000 −1.05450
\(519\) 0 0
\(520\) −6.00000 −0.263117
\(521\) −22.0000 −0.963837 −0.481919 − 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) −11.0000 −0.480996 −0.240498 − 0.970650i \(-0.577311\pi\)
−0.240498 + 0.970650i \(0.577311\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) −21.0000 −0.915644
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −9.00000 −0.390935
\(531\) 0 0
\(532\) −15.0000 −0.650332
\(533\) 42.0000 1.81922
\(534\) 0 0
\(535\) 3.00000 0.129701
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) −10.0000 −0.431131
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 − 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) −17.0000 −0.730213
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −3.00000 −0.128154
\(549\) 0 0
\(550\) −8.00000 −0.341121
\(551\) −25.0000 −1.06504
\(552\) 0 0
\(553\) 15.0000 0.637865
\(554\) 27.0000 1.14712
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) −33.0000 −1.39825 −0.699127 − 0.714997i \(-0.746428\pi\)
−0.699127 + 0.714997i \(0.746428\pi\)
\(558\) 0 0
\(559\) 36.0000 1.52264
\(560\) −3.00000 −0.126773
\(561\) 0 0
\(562\) −13.0000 −0.548372
\(563\) −4.00000 −0.168580 −0.0842900 − 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 14.0000 0.588984
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 22.0000 0.920671 0.460336 − 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) 21.0000 0.876523
\(575\) 16.0000 0.667246
\(576\) 0 0
\(577\) −7.00000 −0.291414 −0.145707 − 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) −5.00000 −0.207614
\(581\) −42.0000 −1.74245
\(582\) 0 0
\(583\) 18.0000 0.745484
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 19.0000 0.784883
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −10.0000 −0.412043
\(590\) 1.00000 0.0411693
\(591\) 0 0
\(592\) 8.00000 0.328798
\(593\) 11.0000 0.451716 0.225858 − 0.974160i \(-0.427481\pi\)
0.225858 + 0.974160i \(0.427481\pi\)
\(594\) 0 0
\(595\) −6.00000 −0.245976
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) −24.0000 −0.981433
\(599\) 15.0000 0.612883 0.306442 − 0.951889i \(-0.400862\pi\)
0.306442 + 0.951889i \(0.400862\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 − 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 18.0000 0.733625
\(603\) 0 0
\(604\) 2.00000 0.0813788
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) 3.00000 0.121766 0.0608831 − 0.998145i \(-0.480608\pi\)
0.0608831 + 0.998145i \(0.480608\pi\)
\(608\) 5.00000 0.202777
\(609\) 0 0
\(610\) −8.00000 −0.323911
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 − 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) −13.0000 −0.524637
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) 7.00000 0.281809 0.140905 − 0.990023i \(-0.454999\pi\)
0.140905 + 0.990023i \(0.454999\pi\)
\(618\) 0 0
\(619\) 5.00000 0.200967 0.100483 − 0.994939i \(-0.467961\pi\)
0.100483 + 0.994939i \(0.467961\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 0 0
\(622\) 7.00000 0.280674
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −24.0000 −0.959233
\(627\) 0 0
\(628\) −12.0000 −0.478852
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 − 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) −5.00000 −0.198889
\(633\) 0 0
\(634\) −2.00000 −0.0794301
\(635\) −13.0000 −0.515889
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 10.0000 0.395904
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −42.0000 −1.65890 −0.829450 − 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 19.0000 0.749287 0.374643 − 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) 10.0000 0.393445
\(647\) −23.0000 −0.904223 −0.452112 − 0.891961i \(-0.649329\pi\)
−0.452112 + 0.891961i \(0.649329\pi\)
\(648\) 0 0
\(649\) −2.00000 −0.0785069
\(650\) −24.0000 −0.941357
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −9.00000 −0.352197 −0.176099 − 0.984373i \(-0.556348\pi\)
−0.176099 + 0.984373i \(0.556348\pi\)
\(654\) 0 0
\(655\) −18.0000 −0.703318
\(656\) −7.00000 −0.273304
\(657\) 0 0
\(658\) −6.00000 −0.233904
\(659\) 40.0000 1.55818 0.779089 − 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) 17.0000 0.661223 0.330612 − 0.943767i \(-0.392745\pi\)
0.330612 + 0.943767i \(0.392745\pi\)
\(662\) −27.0000 −1.04938
\(663\) 0 0
\(664\) 14.0000 0.543305
\(665\) 15.0000 0.581675
\(666\) 0 0
\(667\) −20.0000 −0.774403
\(668\) −3.00000 −0.116073
\(669\) 0 0
\(670\) −2.00000 −0.0772667
\(671\) 16.0000 0.617673
\(672\) 0 0
\(673\) −36.0000 −1.38770 −0.693849 − 0.720121i \(-0.744086\pi\)
−0.693849 + 0.720121i \(0.744086\pi\)
\(674\) 32.0000 1.23259
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 24.0000 0.921035
\(680\) 2.00000 0.0766965
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) 26.0000 0.994862 0.497431 − 0.867503i \(-0.334277\pi\)
0.497431 + 0.867503i \(0.334277\pi\)
\(684\) 0 0
\(685\) 3.00000 0.114624
\(686\) 15.0000 0.572703
\(687\) 0 0
\(688\) −6.00000 −0.228748
\(689\) 54.0000 2.05724
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 − 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) −4.00000 −0.152057
\(693\) 0 0
\(694\) −22.0000 −0.835109
\(695\) −20.0000 −0.758643
\(696\) 0 0
\(697\) −14.0000 −0.530288
\(698\) −10.0000 −0.378506
\(699\) 0 0
\(700\) −12.0000 −0.453557
\(701\) −12.0000 −0.453234 −0.226617 − 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 0 0
\(703\) −40.0000 −1.50863
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 4.00000 0.150542
\(707\) 54.0000 2.03088
\(708\) 0 0
\(709\) −5.00000 −0.187779 −0.0938895 − 0.995583i \(-0.529930\pi\)
−0.0938895 + 0.995583i \(0.529930\pi\)
\(710\) −12.0000 −0.450352
\(711\) 0 0
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) −20.0000 −0.747435
\(717\) 0 0
\(718\) −15.0000 −0.559795
\(719\) 20.0000 0.745874 0.372937 − 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) −6.00000 −0.223297
\(723\) 0 0
\(724\) −13.0000 −0.483141
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 − 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 18.0000 0.667124
\(729\) 0 0
\(730\) 4.00000 0.148047
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 − 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 2.00000 0.0738213
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) 27.0000 0.991201
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 26.0000 0.951928
\(747\) 0 0
\(748\) −4.00000 −0.146254
\(749\) −9.00000 −0.328853
\(750\) 0 0
\(751\) −18.0000 −0.656829 −0.328415 − 0.944534i \(-0.606514\pi\)
−0.328415 + 0.944534i \(0.606514\pi\)
\(752\) 2.00000 0.0729325
\(753\) 0 0
\(754\) 30.0000 1.09254
\(755\) −2.00000 −0.0727875
\(756\) 0 0
\(757\) 23.0000 0.835949 0.417975 − 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) −5.00000 −0.181608
\(759\) 0 0
\(760\) −5.00000 −0.181369
\(761\) 43.0000 1.55875 0.779374 − 0.626559i \(-0.215537\pi\)
0.779374 + 0.626559i \(0.215537\pi\)
\(762\) 0 0
\(763\) −60.0000 −2.17215
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) −6.00000 −0.216647
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 − 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) −6.00000 −0.216225
\(771\) 0 0
\(772\) 19.0000 0.683825
\(773\) −24.0000 −0.863220 −0.431610 − 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) −10.0000 −0.358517
\(779\) 35.0000 1.25401
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) −12.0000 −0.427754 −0.213877 − 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) 5.00000 0.177892
\(791\) −42.0000 −1.49335
\(792\) 0 0
\(793\) 48.0000 1.70453
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −5.00000 −0.177220
\(797\) −8.00000 −0.283375 −0.141687 − 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) 12.0000 0.423735
\(803\) −8.00000 −0.282314
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) 12.0000 0.422682
\(807\) 0 0
\(808\) −18.0000 −0.633238
\(809\) −30.0000 −1.05474 −0.527372 − 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 42.0000 1.47482 0.737410 − 0.675446i \(-0.236049\pi\)
0.737410 + 0.675446i \(0.236049\pi\)
\(812\) 15.0000 0.526397
\(813\) 0 0
\(814\) 16.0000 0.560800
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 30.0000 1.04957
\(818\) 20.0000 0.699284
\(819\) 0 0
\(820\) 7.00000 0.244451
\(821\) 8.00000 0.279202 0.139601 − 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 − 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) −3.00000 −0.104383
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −5.00000 −0.173657 −0.0868286 − 0.996223i \(-0.527673\pi\)
−0.0868286 + 0.996223i \(0.527673\pi\)
\(830\) −14.0000 −0.485947
\(831\) 0 0
\(832\) −6.00000 −0.208013
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 3.00000 0.103819
\(836\) 10.0000 0.345857
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 8.00000 0.275698
\(843\) 0 0
\(844\) −8.00000 −0.275371
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) −21.0000 −0.721569
\(848\) −9.00000 −0.309061
\(849\) 0 0
\(850\) 8.00000 0.274398
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) −21.0000 −0.719026 −0.359513 − 0.933140i \(-0.617057\pi\)
−0.359513 + 0.933140i \(0.617057\pi\)
\(854\) 24.0000 0.821263
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) 52.0000 1.77629 0.888143 − 0.459567i \(-0.151995\pi\)
0.888143 + 0.459567i \(0.151995\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 6.00000 0.204598
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) −4.00000 −0.136162 −0.0680808 − 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 0 0
\(865\) 4.00000 0.136004
\(866\) −19.0000 −0.645646
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) −10.0000 −0.339227
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 20.0000 0.677285
\(873\) 0 0
\(874\) −20.0000 −0.676510
\(875\) 27.0000 0.912767
\(876\) 0 0
\(877\) −37.0000 −1.24940 −0.624701 − 0.780864i \(-0.714779\pi\)
−0.624701 + 0.780864i \(0.714779\pi\)
\(878\) −20.0000 −0.674967
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) −2.00000 −0.0673817 −0.0336909 − 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 29.0000 0.975928 0.487964 − 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) −48.0000 −1.61168 −0.805841 − 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 39.0000 1.30802
\(890\) 0 0
\(891\) 0 0
\(892\) 24.0000 0.803579
\(893\) −10.0000 −0.334637
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) 5.00000 0.166852
\(899\) 10.0000 0.333519
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) −14.0000 −0.466149
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 13.0000 0.432135
\(906\) 0 0
\(907\) −37.0000 −1.22856 −0.614282 − 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) −8.00000 −0.265489
\(909\) 0 0
\(910\) −18.0000 −0.596694
\(911\) 13.0000 0.430709 0.215355 − 0.976536i \(-0.430909\pi\)
0.215355 + 0.976536i \(0.430909\pi\)
\(912\) 0 0
\(913\) 28.0000 0.926665
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 54.0000 1.78324
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 − 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) −4.00000 −0.131876
\(921\) 0 0
\(922\) −18.0000 −0.592798
\(923\) 72.0000 2.36991
\(924\) 0 0
\(925\) −32.0000 −1.05215
\(926\) 36.0000 1.18303
\(927\) 0 0
\(928\) −5.00000 −0.164133
\(929\) −50.0000 −1.64045 −0.820223 − 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) −10.0000 −0.327737
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 28.0000 0.916188
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 − 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 6.00000 0.195907
\(939\) 0 0
\(940\) −2.00000 −0.0652328
\(941\) 8.00000 0.260793 0.130396 − 0.991462i \(-0.458375\pi\)
0.130396 + 0.991462i \(0.458375\pi\)
\(942\) 0 0
\(943\) 28.0000 0.911805
\(944\) 1.00000 0.0325472
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) −43.0000 −1.39731 −0.698656 − 0.715458i \(-0.746218\pi\)
−0.698656 + 0.715458i \(0.746218\pi\)
\(948\) 0 0
\(949\) −24.0000 −0.779073
\(950\) −20.0000 −0.648886
\(951\) 0 0
\(952\) −6.00000 −0.194461
\(953\) −34.0000 −1.10137 −0.550684 − 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) −18.0000 −0.582466
\(956\) 25.0000 0.808558
\(957\) 0 0
\(958\) 40.0000 1.29234
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 48.0000 1.54758
\(963\) 0 0
\(964\) 17.0000 0.547533
\(965\) −19.0000 −0.611632
\(966\) 0 0
\(967\) −42.0000 −1.35063 −0.675314 − 0.737530i \(-0.735992\pi\)
−0.675314 + 0.737530i \(0.735992\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 8.00000 0.256865
\(971\) 3.00000 0.0962746 0.0481373 − 0.998841i \(-0.484672\pi\)
0.0481373 + 0.998841i \(0.484672\pi\)
\(972\) 0 0
\(973\) 60.0000 1.92351
\(974\) 17.0000 0.544715
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) 42.0000 1.34370 0.671850 − 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) 37.0000 1.18072
\(983\) 6.00000 0.191370 0.0956851 − 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) −10.0000 −0.318465
\(987\) 0 0
\(988\) 30.0000 0.954427
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −18.0000 −0.571789 −0.285894 − 0.958261i \(-0.592291\pi\)
−0.285894 + 0.958261i \(0.592291\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) 36.0000 1.14185
\(995\) 5.00000 0.158511
\(996\) 0 0
\(997\) 23.0000 0.728417 0.364209 − 0.931317i \(-0.381339\pi\)
0.364209 + 0.931317i \(0.381339\pi\)
\(998\) −15.0000 −0.474817
\(999\) 0 0
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 1062.2.a.b.1.1 1
3.2 odd 2 118.2.a.c.1.1 ✓ 1
4.3 odd 2 8496.2.a.h.1.1 1
12.11 even 2 944.2.a.j.1.1 1
15.2 even 4 2950.2.c.d.1299.2 2
15.8 even 4 2950.2.c.d.1299.1 2
15.14 odd 2 2950.2.a.h.1.1 1
21.20 even 2 5782.2.a.j.1.1 1
24.5 odd 2 3776.2.a.o.1.1 1
24.11 even 2 3776.2.a.h.1.1 1
177.176 even 2 6962.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
118.2.a.c.1.1 ✓ 1 3.2 odd 2
944.2.a.j.1.1 1 12.11 even 2
1062.2.a.b.1.1 1 1.1 even 1 trivial
2950.2.a.h.1.1 1 15.14 odd 2
2950.2.c.d.1299.1 2 15.8 even 4
2950.2.c.d.1299.2 2 15.2 even 4
3776.2.a.h.1.1 1 24.11 even 2
3776.2.a.o.1.1 1 24.5 odd 2
5782.2.a.j.1.1 1 21.20 even 2
6962.2.a.b.1.1 1 177.176 even 2
8496.2.a.h.1.1 1 4.3 odd 2