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Properties

Label 161.2.a.a.1.1
Level $161$
Weight $2$
Character 161.1
Self dual yes
Analytic conductor $1.286$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [161,2,Mod(1,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, 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("161.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 161.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: \(1.28559147254\)
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 \(\chi\) \(=\) 161.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} -2.00000 q^{10} +4.00000 q^{11} +6.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} -4.00000 q^{22} -1.00000 q^{23} -1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{28} -2.00000 q^{29} -4.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} +2.00000 q^{35} +3.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} +6.00000 q^{40} -6.00000 q^{41} +12.0000 q^{43} -4.00000 q^{44} -6.00000 q^{45} +1.00000 q^{46} -12.0000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} -10.0000 q^{53} +8.00000 q^{55} +3.00000 q^{56} +2.00000 q^{58} +2.00000 q^{61} +4.00000 q^{62} -3.00000 q^{63} +7.00000 q^{64} +12.0000 q^{65} +12.0000 q^{67} +2.00000 q^{68} -2.00000 q^{70} +8.00000 q^{71} -9.00000 q^{72} -14.0000 q^{73} +2.00000 q^{74} -4.00000 q^{76} +4.00000 q^{77} +8.00000 q^{79} -2.00000 q^{80} +9.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} -4.00000 q^{85} -12.0000 q^{86} +12.0000 q^{88} +6.00000 q^{89} +6.00000 q^{90} +6.00000 q^{91} +1.00000 q^{92} +12.0000 q^{94} +8.00000 q^{95} -10.0000 q^{97} -1.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

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 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1.00000 −0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 3.00000 1.06066
\(9\) −3.00000 −1.00000
\(10\) −2.00000 −0.632456
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 3.00000 0.707107
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 2.00000 0.338062
\(36\) 3.00000 0.500000
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) 6.00000 0.948683
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −4.00000 −0.603023
\(45\) −6.00000 −0.894427
\(46\) 1.00000 0.147442
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 4.00000 0.508001
\(63\) −3.00000 −0.377964
\(64\) 7.00000 0.875000
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −9.00000 −1.06066
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −2.00000 −0.223607
\(81\) 9.00000 1.00000
\(82\) 6.00000 0.662589
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −12.0000 −1.29399
\(87\) 0 0
\(88\) 12.0000 1.27920
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 6.00000 0.632456
\(91\) 6.00000 0.628971
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −1.00000 −0.101015
\(99\) −12.0000 −1.20605
\(100\) 1.00000 0.100000
\(101\) 14.0000 1.39305 0.696526 − 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 18.0000 1.76505
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −8.00000 −0.762770
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −14.0000 −1.31701 −0.658505 − 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 2.00000 0.185695
\(117\) −18.0000 −1.66410
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) −12.0000 −1.07331
\(126\) 3.00000 0.267261
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) −12.0000 −1.05247
\(131\) −8.00000 −0.698963 −0.349482 − 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −14.0000 −1.19610 −0.598050 − 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 − 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 24.0000 2.00698
\(144\) 3.00000 0.250000
\(145\) −4.00000 −0.332182
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 12.0000 0.973329
\(153\) 6.00000 0.485071
\(154\) −4.00000 −0.322329
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 − 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −10.0000 −0.790569
\(161\) −1.00000 −0.0788110
\(162\) −9.00000 −0.707107
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 4.00000 0.306786
\(171\) −12.0000 −0.917663
\(172\) −12.0000 −0.914991
\(173\) −2.00000 −0.152057 −0.0760286 − 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 6.00000 0.447214
\(181\) −14.0000 −1.04061 −0.520306 − 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) −3.00000 −0.221163
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 − 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 12.0000 0.852803
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −14.0000 −0.985037
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 8.00000 0.557386
\(207\) 3.00000 0.208514
\(208\) −6.00000 −0.416025
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 24.0000 1.63679
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) −8.00000 −0.539360
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 28.0000 1.87502 0.937509 − 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) −5.00000 −0.334077
\(225\) 3.00000 0.200000
\(226\) 14.0000 0.931266
\(227\) 4.00000 0.265489 0.132745 − 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 18.0000 1.17670
\(235\) −24.0000 −1.56559
\(236\) 0 0
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) −8.00000 −0.517477 −0.258738 − 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 − 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 24.0000 1.52708
\(248\) −12.0000 −0.762001
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 3.00000 0.188982
\(253\) −4.00000 −0.251478
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) −12.0000 −0.744208
\(261\) 6.00000 0.371391
\(262\) 8.00000 0.494242
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) −20.0000 −1.22859
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) 30.0000 1.82913 0.914566 − 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 14.0000 0.845771
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 16.0000 0.959616
\(279\) 12.0000 0.718421
\(280\) 6.00000 0.358569
\(281\) −14.0000 −0.835170 −0.417585 − 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 − 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) −6.00000 −0.354169
\(288\) 15.0000 0.883883
\(289\) −13.0000 −0.764706
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) −30.0000 −1.75262 −0.876309 − 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 4.00000 0.229039
\(306\) −6.00000 −0.342997
\(307\) −16.0000 −0.913168 −0.456584 − 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) 4.00000 0.226819 0.113410 − 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −18.0000 −1.01580
\(315\) −6.00000 −0.338062
\(316\) −8.00000 −0.450035
\(317\) 30.0000 1.68497 0.842484 − 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 14.0000 0.782624
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) −8.00000 −0.445132
\(324\) −9.00000 −0.500000
\(325\) −6.00000 −0.332820
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) −18.0000 −0.993884
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) 12.0000 0.656611
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 − 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) −16.0000 −0.866449
\(342\) 12.0000 0.648886
\(343\) 1.00000 0.0539949
\(344\) 36.0000 1.94099
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 − 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) −20.0000 −1.06600
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 16.0000 0.849192
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −32.0000 −1.68890 −0.844448 − 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) −18.0000 −0.948683
\(361\) −3.00000 −0.157895
\(362\) 14.0000 0.735824
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) −28.0000 −1.46559
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 1.00000 0.0521286
\(369\) 18.0000 0.937043
\(370\) 4.00000 0.207950
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 − 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) −36.0000 −1.85656
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 − 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) −2.00000 −0.101797
\(387\) −36.0000 −1.82998
\(388\) 10.0000 0.507673
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 16.0000 0.805047
\(396\) 12.0000 0.603023
\(397\) 22.0000 1.10415 0.552074 − 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 10.0000 0.499376 0.249688 − 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) −24.0000 −1.19553
\(404\) −14.0000 −0.696526
\(405\) 18.0000 0.894427
\(406\) 2.00000 0.0992583
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 − 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 12.0000 0.592638
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −3.00000 −0.147442
\(415\) −8.00000 −0.392705
\(416\) −30.0000 −1.47087
\(417\) 0 0
\(418\) −16.0000 −0.782586
\(419\) 20.0000 0.977064 0.488532 − 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 − 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 4.00000 0.194717
\(423\) 36.0000 1.75038
\(424\) −30.0000 −1.45693
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −24.0000 −1.15738
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −10.0000 −0.480569 −0.240285 − 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 − 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 24.0000 1.14416
\(441\) −3.00000 −0.142857
\(442\) 12.0000 0.570782
\(443\) 12.0000 0.570137 0.285069 − 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) −28.0000 −1.32584
\(447\) 0 0
\(448\) 7.00000 0.330719
\(449\) 18.0000 0.849473 0.424736 − 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) −3.00000 −0.141421
\(451\) −24.0000 −1.13012
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) 42.0000 1.96468 0.982339 − 0.187112i \(-0.0599128\pi\)
0.982339 + 0.187112i \(0.0599128\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) 2.00000 0.0932505
\(461\) 30.0000 1.39724 0.698620 − 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 18.0000 0.832050
\(469\) 12.0000 0.554109
\(470\) 24.0000 1.10704
\(471\) 0 0
\(472\) 0 0
\(473\) 48.0000 2.20704
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 2.00000 0.0916698
\(477\) 30.0000 1.37361
\(478\) 8.00000 0.365911
\(479\) −16.0000 −0.731059 −0.365529 − 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −14.0000 −0.637683
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) −20.0000 −0.908153
\(486\) 0 0
\(487\) 40.0000 1.81257 0.906287 − 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) −20.0000 −0.902587 −0.451294 − 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) 4.00000 0.180151
\(494\) −24.0000 −1.07981
\(495\) −24.0000 −1.07872
\(496\) 4.00000 0.179605
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 12.0000 0.537194 0.268597 − 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) −8.00000 −0.356702 −0.178351 − 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) −9.00000 −0.400892
\(505\) 28.0000 1.24598
\(506\) 4.00000 0.177822
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 30.0000 1.32973 0.664863 − 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) −48.0000 −2.11104
\(518\) 2.00000 0.0878750
\(519\) 0 0
\(520\) 36.0000 1.57870
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −6.00000 −0.262613
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 20.0000 0.868744
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) −36.0000 −1.55933
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) 36.0000 1.55496
\(537\) 0 0
\(538\) −30.0000 −1.29339
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) 10.0000 0.428746
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 14.0000 0.598050
\(549\) −6.00000 −0.256074
\(550\) 4.00000 0.170561
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −12.0000 −0.508001
\(559\) 72.0000 3.04528
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 14.0000 0.590554
\(563\) 20.0000 0.842900 0.421450 − 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 0 0
\(565\) −28.0000 −1.17797
\(566\) −12.0000 −0.504398
\(567\) 9.00000 0.377964
\(568\) 24.0000 1.00702
\(569\) 2.00000 0.0838444 0.0419222 − 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 − 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −24.0000 −1.00349
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 1.00000 0.0417029
\(576\) −21.0000 −0.875000
\(577\) −38.0000 −1.58196 −0.790980 − 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) −40.0000 −1.65663
\(584\) −42.0000 −1.73797
\(585\) −36.0000 −1.48842
\(586\) 30.0000 1.23929
\(587\) −32.0000 −1.32078 −0.660391 − 0.750922i \(-0.729609\pi\)
−0.660391 + 0.750922i \(0.729609\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) −14.0000 −0.574911 −0.287456 − 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 6.00000 0.245358
\(599\) 16.0000 0.653742 0.326871 − 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) 34.0000 1.38689 0.693444 − 0.720510i \(-0.256092\pi\)
0.693444 + 0.720510i \(0.256092\pi\)
\(602\) −12.0000 −0.489083
\(603\) −36.0000 −1.46603
\(604\) −16.0000 −0.651031
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) 4.00000 0.162355 0.0811775 − 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) −20.0000 −0.811107
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) −72.0000 −2.91281
\(612\) −6.00000 −0.242536
\(613\) 14.0000 0.565455 0.282727 − 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 12.0000 0.483494
\(617\) 2.00000 0.0805170 0.0402585 − 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) −44.0000 −1.76851 −0.884255 − 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) −4.00000 −0.160385
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) 4.00000 0.159490
\(630\) 6.00000 0.239046
\(631\) 32.0000 1.27390 0.636950 − 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 24.0000 0.954669
\(633\) 0 0
\(634\) −30.0000 −1.19145
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) 6.00000 0.237729
\(638\) 8.00000 0.316723
\(639\) −24.0000 −0.949425
\(640\) 6.00000 0.237171
\(641\) −6.00000 −0.236986 −0.118493 − 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 0 0
\(643\) −12.0000 −0.473234 −0.236617 − 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) 20.0000 0.786281 0.393141 − 0.919478i \(-0.371389\pi\)
0.393141 + 0.919478i \(0.371389\pi\)
\(648\) 27.0000 1.06066
\(649\) 0 0
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 30.0000 1.17399 0.586995 − 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 0 0
\(655\) −16.0000 −0.625172
\(656\) 6.00000 0.234261
\(657\) 42.0000 1.63858
\(658\) 12.0000 0.467809
\(659\) 20.0000 0.779089 0.389545 − 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 − 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 8.00000 0.310227
\(666\) −6.00000 −0.232495
\(667\) 2.00000 0.0774403
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) −24.0000 −0.927201
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 18.0000 0.693849 0.346925 − 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) −23.0000 −0.884615
\(677\) 42.0000 1.61419 0.807096 − 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) −12.0000 −0.460179
\(681\) 0 0
\(682\) 16.0000 0.612672
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 12.0000 0.458831
\(685\) −28.0000 −1.06983
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −12.0000 −0.457496
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) −16.0000 −0.608669 −0.304334 − 0.952565i \(-0.598434\pi\)
−0.304334 + 0.952565i \(0.598434\pi\)
\(692\) 2.00000 0.0760286
\(693\) −12.0000 −0.455842
\(694\) 12.0000 0.455514
\(695\) −32.0000 −1.21383
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) −14.0000 −0.529908
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 6.00000 0.226617 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 28.0000 1.05529
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 14.0000 0.526524
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 − 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) −16.0000 −0.600469
\(711\) −24.0000 −0.900070
\(712\) 18.0000 0.674579
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 48.0000 1.79510
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 32.0000 1.19423
\(719\) 12.0000 0.447524 0.223762 − 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 6.00000 0.223607
\(721\) −8.00000 −0.297936
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) 14.0000 0.520306
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 18.0000 0.667124
\(729\) −27.0000 −1.00000
\(730\) 28.0000 1.03633
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 10.0000 0.369358 0.184679 − 0.982799i \(-0.440875\pi\)
0.184679 + 0.982799i \(0.440875\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 5.00000 0.184302
\(737\) 48.0000 1.76810
\(738\) −18.0000 −0.662589
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) 10.0000 0.367112
\(743\) −32.0000 −1.17397 −0.586983 − 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) −6.00000 −0.219676
\(747\) 12.0000 0.439057
\(748\) 8.00000 0.292509
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 32.0000 1.16460
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 − 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 28.0000 1.01701
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) 18.0000 0.652499 0.326250 − 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) 0 0
\(765\) 12.0000 0.433861
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) −8.00000 −0.288300
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) −6.00000 −0.215805 −0.107903 − 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 36.0000 1.29399
\(775\) 4.00000 0.143684
\(776\) −30.0000 −1.07694
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) −2.00000 −0.0715199
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 36.0000 1.28490
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 − 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −16.0000 −0.569254
\(791\) −14.0000 −0.497783
\(792\) −36.0000 −1.27920
\(793\) 12.0000 0.426132
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 10.0000 0.354218 0.177109 − 0.984191i \(-0.443325\pi\)
0.177109 + 0.984191i \(0.443325\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 5.00000 0.176777
\(801\) −18.0000 −0.635999
\(802\) −10.0000 −0.353112
\(803\) −56.0000 −1.97620
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) 24.0000 0.845364
\(807\) 0 0
\(808\) 42.0000 1.47755
\(809\) −54.0000 −1.89854 −0.949269 − 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) −18.0000 −0.632456
\(811\) 32.0000 1.12367 0.561836 − 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) 48.0000 1.67931
\(818\) 30.0000 1.04893
\(819\) −18.0000 −0.628971
\(820\) 12.0000 0.419058
\(821\) −42.0000 −1.46581 −0.732905 − 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 − 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −3.00000 −0.104257
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 8.00000 0.277684
\(831\) 0 0
\(832\) 42.0000 1.45609
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) −20.0000 −0.690889
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 46.0000 1.58245
\(846\) −36.0000 −1.23771
\(847\) 5.00000 0.171802
\(848\) 10.0000 0.343401
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) −50.0000 −1.71197 −0.855984 − 0.517003i \(-0.827048\pi\)
−0.855984 + 0.517003i \(0.827048\pi\)
\(854\) −2.00000 −0.0684386
\(855\) −24.0000 −0.820783
\(856\) −12.0000 −0.410152
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 16.0000 0.545913 0.272956 − 0.962026i \(-0.411998\pi\)
0.272956 + 0.962026i \(0.411998\pi\)
\(860\) −24.0000 −0.818393
\(861\) 0 0
\(862\) 0 0
\(863\) 48.0000 1.63394 0.816970 − 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) −4.00000 −0.136004
\(866\) 10.0000 0.339814
\(867\) 0 0
\(868\) 4.00000 0.135769
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 72.0000 2.43963
\(872\) −30.0000 −1.01593
\(873\) 30.0000 1.01535
\(874\) 4.00000 0.135302
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 − 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −4.00000 −0.134993
\(879\) 0 0
\(880\) −8.00000 −0.269680
\(881\) 54.0000 1.81931 0.909653 − 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) 3.00000 0.101015
\(883\) −20.0000 −0.673054 −0.336527 − 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −12.0000 −0.402921 −0.201460 − 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) −12.0000 −0.402241
\(891\) 36.0000 1.20605
\(892\) −28.0000 −0.937509
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 8.00000 0.266815
\(900\) −3.00000 −0.100000
\(901\) 20.0000 0.666297
\(902\) 24.0000 0.799113
\(903\) 0 0
\(904\) −42.0000 −1.39690
\(905\) −28.0000 −0.930751
\(906\) 0 0
\(907\) −4.00000 −0.132818 −0.0664089 − 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) −4.00000 −0.132745
\(909\) −42.0000 −1.39305
\(910\) −12.0000 −0.397796
\(911\) 24.0000 0.795155 0.397578 − 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) −42.0000 −1.38924
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) −8.00000 −0.264183
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) −6.00000 −0.197814
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 24.0000 0.788689
\(927\) 24.0000 0.788263
\(928\) 10.0000 0.328266
\(929\) 34.0000 1.11550 0.557752 − 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) −16.0000 −0.523256
\(936\) −54.0000 −1.76505
\(937\) −34.0000 −1.11073 −0.555366 − 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) −12.0000 −0.391814
\(939\) 0 0
\(940\) 24.0000 0.782794
\(941\) 42.0000 1.36916 0.684580 − 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) −84.0000 −2.72676
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) −6.00000 −0.194461
\(953\) −6.00000 −0.194359 −0.0971795 − 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −30.0000 −0.971286
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) 16.0000 0.516937
\(959\) −14.0000 −0.452084
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 12.0000 0.386896
\(963\) 12.0000 0.386695
\(964\) −14.0000 −0.450910
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 − 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 15.0000 0.482118
\(969\) 0 0
\(970\) 20.0000 0.642161
\(971\) 4.00000 0.128366 0.0641831 − 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) 0 0
\(973\) −16.0000 −0.512936
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −6.00000 −0.191957 −0.0959785 − 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) −2.00000 −0.0638877
\(981\) 30.0000 0.957826
\(982\) 20.0000 0.638226
\(983\) −48.0000 −1.53096 −0.765481 − 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) −12.0000 −0.381578
\(990\) 24.0000 0.762770
\(991\) 32.0000 1.01651 0.508257 − 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 20.0000 0.635001
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) −42.0000 −1.33015 −0.665077 − 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) −12.0000 −0.379853
\(999\) 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 161.2.a.a.1.1 ✓ 1
3.2 odd 2 1449.2.a.d.1.1 1
4.3 odd 2 2576.2.a.j.1.1 1
5.4 even 2 4025.2.a.e.1.1 1
7.6 odd 2 1127.2.a.a.1.1 1
23.22 odd 2 3703.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
1127.2.a.a.1.1 1 7.6 odd 2
1449.2.a.d.1.1 1 3.2 odd 2
2576.2.a.j.1.1 1 4.3 odd 2
3703.2.a.a.1.1 1 23.22 odd 2
4025.2.a.e.1.1 1 5.4 even 2