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Properties

Label 6011.2.a.a.1.1
Level $6011$
Weight $2$
Character 6011.1
Self dual yes
Analytic conductor $47.998$
Analytic rank $2$
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] = [6011,2,Mod(1,6011)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6011, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6011.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6011.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: \(47.9980766550\)
Analytic rank: \(2\)
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 \(\chi\) \(=\) 6011.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} -3.00000 q^{9} +4.00000 q^{10} -2.00000 q^{11} -2.00000 q^{14} -4.00000 q^{16} -6.00000 q^{17} +6.00000 q^{18} -2.00000 q^{19} -4.00000 q^{20} +4.00000 q^{22} -6.00000 q^{23} -1.00000 q^{25} +2.00000 q^{28} -8.00000 q^{29} +2.00000 q^{31} +8.00000 q^{32} +12.0000 q^{34} -2.00000 q^{35} -6.00000 q^{36} -10.0000 q^{37} +4.00000 q^{38} +3.00000 q^{41} +1.00000 q^{43} -4.00000 q^{44} +6.00000 q^{45} +12.0000 q^{46} -6.00000 q^{47} -6.00000 q^{49} +2.00000 q^{50} -12.0000 q^{53} +4.00000 q^{55} +16.0000 q^{58} +1.00000 q^{59} +6.00000 q^{61} -4.00000 q^{62} -3.00000 q^{63} -8.00000 q^{64} +5.00000 q^{67} -12.0000 q^{68} +4.00000 q^{70} +5.00000 q^{71} +2.00000 q^{73} +20.0000 q^{74} -4.00000 q^{76} -2.00000 q^{77} -11.0000 q^{79} +8.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} -1.00000 q^{83} +12.0000 q^{85} -2.00000 q^{86} -8.00000 q^{89} -12.0000 q^{90} -12.0000 q^{92} +12.0000 q^{94} +4.00000 q^{95} -7.00000 q^{97} +12.0000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 2.00000 1.00000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 4.00000 1.26491
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 6.00000 1.41421
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −4.00000 −0.894427
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) 12.0000 2.05798
\(35\) −2.00000 −0.338062
\(36\) −6.00000 −1.00000
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) −4.00000 −0.603023
\(45\) 6.00000 0.894427
\(46\) 12.0000 1.76930
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 2.00000 0.282843
\(51\) 0 0
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 16.0000 2.10090
\(59\) 1.00000 0.130189 0.0650945 − 0.997879i \(-0.479265\pi\)
0.0650945 + 0.997879i \(0.479265\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −4.00000 −0.508001
\(63\) −3.00000 −0.377964
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) −12.0000 −1.45521
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 5.00000 0.593391 0.296695 − 0.954972i \(-0.404115\pi\)
0.296695 + 0.954972i \(0.404115\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 20.0000 2.32495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 − 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 8.00000 0.894427
\(81\) 9.00000 1.00000
\(82\) −6.00000 −0.662589
\(83\) −1.00000 −0.109764 −0.0548821 − 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) −12.0000 −1.26491
\(91\) 0 0
\(92\) −12.0000 −1.25109
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −7.00000 −0.710742 −0.355371 − 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 12.0000 1.21218
\(99\) 6.00000 0.603023
\(100\) −2.00000 −0.200000
\(101\) −3.00000 −0.298511 −0.149256 − 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) 15.0000 1.47799 0.738997 − 0.673709i \(-0.235300\pi\)
0.738997 + 0.673709i \(0.235300\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 24.0000 2.33109
\(107\) 10.0000 0.966736 0.483368 − 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 0 0
\(109\) −15.0000 −1.43674 −0.718370 − 0.695662i \(-0.755111\pi\)
−0.718370 + 0.695662i \(0.755111\pi\)
\(110\) −8.00000 −0.762770
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) −3.00000 −0.282216 −0.141108 − 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 0 0
\(115\) 12.0000 1.11901
\(116\) −16.0000 −1.48556
\(117\) 0 0
\(118\) −2.00000 −0.184115
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −12.0000 −1.08643
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 12.0000 1.07331
\(126\) 6.00000 0.534522
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 − 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −10.0000 −0.863868
\(135\) 0 0
\(136\) 0 0
\(137\) −23.0000 −1.96502 −0.982511 − 0.186203i \(-0.940382\pi\)
−0.982511 + 0.186203i \(0.940382\pi\)
\(138\) 0 0
\(139\) 6.00000 0.508913 0.254457 − 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) −10.0000 −0.839181
\(143\) 0 0
\(144\) 12.0000 1.00000
\(145\) 16.0000 1.32873
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) −20.0000 −1.64399
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 18.0000 1.45521
\(154\) 4.00000 0.322329
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 12.0000 0.957704 0.478852 − 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 22.0000 1.75023
\(159\) 0 0
\(160\) −16.0000 −1.26491
\(161\) −6.00000 −0.472866
\(162\) −18.0000 −1.41421
\(163\) −8.00000 −0.626608 −0.313304 − 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) −9.00000 −0.696441 −0.348220 − 0.937413i \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −24.0000 −1.84072
\(171\) 6.00000 0.458831
\(172\) 2.00000 0.152499
\(173\) 16.0000 1.21646 0.608229 − 0.793762i \(-0.291880\pi\)
0.608229 + 0.793762i \(0.291880\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 8.00000 0.603023
\(177\) 0 0
\(178\) 16.0000 1.19925
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 12.0000 0.894427
\(181\) 13.0000 0.966282 0.483141 − 0.875542i \(-0.339496\pi\)
0.483141 + 0.875542i \(0.339496\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 20.0000 1.47043
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −12.0000 −0.857143
\(197\) −15.0000 −1.06871 −0.534353 − 0.845262i \(-0.679445\pi\)
−0.534353 + 0.845262i \(0.679445\pi\)
\(198\) −12.0000 −0.852803
\(199\) 5.00000 0.354441 0.177220 − 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) −30.0000 −2.09020
\(207\) 18.0000 1.25109
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 − 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −24.0000 −1.64833
\(213\) 0 0
\(214\) −20.0000 −1.36717
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 30.0000 2.03186
\(219\) 0 0
\(220\) 8.00000 0.539360
\(221\) 0 0
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 − 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 8.00000 0.534522
\(225\) 3.00000 0.200000
\(226\) 6.00000 0.399114
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −15.0000 −0.991228 −0.495614 − 0.868543i \(-0.665057\pi\)
−0.495614 + 0.868543i \(0.665057\pi\)
\(230\) −24.0000 −1.58251
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 − 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) 2.00000 0.130189
\(237\) 0 0
\(238\) 12.0000 0.777844
\(239\) 8.00000 0.517477 0.258738 − 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 − 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 14.0000 0.899954
\(243\) 0 0
\(244\) 12.0000 0.768221
\(245\) 12.0000 0.766652
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −24.0000 −1.51789
\(251\) −5.00000 −0.315597 −0.157799 − 0.987471i \(-0.550440\pi\)
−0.157799 + 0.987471i \(0.550440\pi\)
\(252\) −6.00000 −0.377964
\(253\) 12.0000 0.754434
\(254\) −32.0000 −2.00786
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 13.0000 0.810918 0.405459 − 0.914113i \(-0.367112\pi\)
0.405459 + 0.914113i \(0.367112\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) 0 0
\(261\) 24.0000 1.48556
\(262\) 30.0000 1.85341
\(263\) −28.0000 −1.72655 −0.863277 − 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) 24.0000 1.47431
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) −2.00000 −0.121942 −0.0609711 − 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) −22.0000 −1.33640 −0.668202 − 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) 24.0000 1.45521
\(273\) 0 0
\(274\) 46.0000 2.77896
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) −7.00000 −0.420589 −0.210295 − 0.977638i \(-0.567442\pi\)
−0.210295 + 0.977638i \(0.567442\pi\)
\(278\) −12.0000 −0.719712
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 − 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) 0 0
\(287\) 3.00000 0.177084
\(288\) −24.0000 −1.41421
\(289\) 19.0000 1.11765
\(290\) −32.0000 −1.87910
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) −22.0000 −1.28525 −0.642627 − 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) 0 0
\(295\) −2.00000 −0.116445
\(296\) 0 0
\(297\) 0 0
\(298\) 20.0000 1.15857
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) −12.0000 −0.687118
\(306\) −36.0000 −2.05798
\(307\) 17.0000 0.970241 0.485121 − 0.874447i \(-0.338776\pi\)
0.485121 + 0.874447i \(0.338776\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) 32.0000 1.81455 0.907277 − 0.420534i \(-0.138157\pi\)
0.907277 + 0.420534i \(0.138157\pi\)
\(312\) 0 0
\(313\) −17.0000 −0.960897 −0.480448 − 0.877023i \(-0.659526\pi\)
−0.480448 + 0.877023i \(0.659526\pi\)
\(314\) −24.0000 −1.35440
\(315\) 6.00000 0.338062
\(316\) −22.0000 −1.23760
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 16.0000 0.895828
\(320\) 16.0000 0.894427
\(321\) 0 0
\(322\) 12.0000 0.668734
\(323\) 12.0000 0.667698
\(324\) 18.0000 1.00000
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) 0 0
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 − 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −2.00000 −0.109764
\(333\) 30.0000 1.64399
\(334\) 18.0000 0.984916
\(335\) −10.0000 −0.546358
\(336\) 0 0
\(337\) 3.00000 0.163420 0.0817102 − 0.996656i \(-0.473962\pi\)
0.0817102 + 0.996656i \(0.473962\pi\)
\(338\) 26.0000 1.41421
\(339\) 0 0
\(340\) 24.0000 1.30158
\(341\) −4.00000 −0.216612
\(342\) −12.0000 −0.648886
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) −32.0000 −1.72033
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) −16.0000 −0.852803
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) −10.0000 −0.530745
\(356\) −16.0000 −0.847998
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) 27.0000 1.42501 0.712503 − 0.701669i \(-0.247562\pi\)
0.712503 + 0.701669i \(0.247562\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −26.0000 −1.36653
\(363\) 0 0
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 − 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 24.0000 1.25109
\(369\) −9.00000 −0.468521
\(370\) −40.0000 −2.07950
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 − 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) −14.0000 −0.715367 −0.357683 − 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 20.0000 1.01797
\(387\) −3.00000 −0.152499
\(388\) −14.0000 −0.710742
\(389\) 9.00000 0.456318 0.228159 − 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) 0 0
\(393\) 0 0
\(394\) 30.0000 1.51138
\(395\) 22.0000 1.10694
\(396\) 12.0000 0.603023
\(397\) −17.0000 −0.853206 −0.426603 − 0.904439i \(-0.640290\pi\)
−0.426603 + 0.904439i \(0.640290\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −2.00000 −0.0998752 −0.0499376 − 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) −18.0000 −0.894427
\(406\) 16.0000 0.794067
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 − 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 12.0000 0.592638
\(411\) 0 0
\(412\) 30.0000 1.47799
\(413\) 1.00000 0.0492068
\(414\) −36.0000 −1.76930
\(415\) 2.00000 0.0981761
\(416\) 0 0
\(417\) 0 0
\(418\) −8.00000 −0.391293
\(419\) −26.0000 −1.27018 −0.635092 − 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 − 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 32.0000 1.55774
\(423\) 18.0000 0.875190
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 20.0000 0.966736
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −12.0000 −0.578020 −0.289010 − 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −30.0000 −1.43674
\(437\) 12.0000 0.574038
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 − 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 − 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 16.0000 0.758473
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) −8.00000 −0.377964
\(449\) −21.0000 −0.991051 −0.495526 − 0.868593i \(-0.665025\pi\)
−0.495526 + 0.868593i \(0.665025\pi\)
\(450\) −6.00000 −0.282843
\(451\) −6.00000 −0.282529
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) 0 0
\(457\) −34.0000 −1.59045 −0.795226 − 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) 30.0000 1.40181
\(459\) 0 0
\(460\) 24.0000 1.11901
\(461\) −34.0000 −1.58354 −0.791769 − 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 0 0
\(463\) 27.0000 1.25480 0.627398 − 0.778699i \(-0.284120\pi\)
0.627398 + 0.778699i \(0.284120\pi\)
\(464\) 32.0000 1.48556
\(465\) 0 0
\(466\) 48.0000 2.22356
\(467\) 26.0000 1.20314 0.601568 − 0.798821i \(-0.294543\pi\)
0.601568 + 0.798821i \(0.294543\pi\)
\(468\) 0 0
\(469\) 5.00000 0.230879
\(470\) −24.0000 −1.10704
\(471\) 0 0
\(472\) 0 0
\(473\) −2.00000 −0.0919601
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) −12.0000 −0.550019
\(477\) 36.0000 1.64833
\(478\) −16.0000 −0.731823
\(479\) −16.0000 −0.731059 −0.365529 − 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −44.0000 −2.00415
\(483\) 0 0
\(484\) −14.0000 −0.636364
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 − 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −24.0000 −1.08421
\(491\) 27.0000 1.21849 0.609246 − 0.792981i \(-0.291472\pi\)
0.609246 + 0.792981i \(0.291472\pi\)
\(492\) 0 0
\(493\) 48.0000 2.16181
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) −8.00000 −0.359211
\(497\) 5.00000 0.224281
\(498\) 0 0
\(499\) −29.0000 −1.29822 −0.649109 − 0.760695i \(-0.724858\pi\)
−0.649109 + 0.760695i \(0.724858\pi\)
\(500\) 24.0000 1.07331
\(501\) 0 0
\(502\) 10.0000 0.446322
\(503\) 25.0000 1.11469 0.557347 − 0.830279i \(-0.311819\pi\)
0.557347 + 0.830279i \(0.311819\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) −24.0000 −1.06693
\(507\) 0 0
\(508\) 32.0000 1.41977
\(509\) 11.0000 0.487566 0.243783 − 0.969830i \(-0.421611\pi\)
0.243783 + 0.969830i \(0.421611\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) −26.0000 −1.14681
\(515\) −30.0000 −1.32196
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 20.0000 0.878750
\(519\) 0 0
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 − 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) −48.0000 −2.10090
\(523\) 14.0000 0.612177 0.306089 − 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) −30.0000 −1.31056
\(525\) 0 0
\(526\) 56.0000 2.44172
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −48.0000 −2.08499
\(531\) −3.00000 −0.130189
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 0 0
\(535\) −20.0000 −0.864675
\(536\) 0 0
\(537\) 0 0
\(538\) 4.00000 0.172452
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) 7.00000 0.300954 0.150477 − 0.988614i \(-0.451919\pi\)
0.150477 + 0.988614i \(0.451919\pi\)
\(542\) 44.0000 1.88996
\(543\) 0 0
\(544\) −48.0000 −2.05798
\(545\) 30.0000 1.28506
\(546\) 0 0
\(547\) 23.0000 0.983409 0.491704 − 0.870762i \(-0.336374\pi\)
0.491704 + 0.870762i \(0.336374\pi\)
\(548\) −46.0000 −1.96502
\(549\) −18.0000 −0.768221
\(550\) −4.00000 −0.170561
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) −11.0000 −0.467768
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −5.00000 −0.211857 −0.105928 − 0.994374i \(-0.533781\pi\)
−0.105928 + 0.994374i \(0.533781\pi\)
\(558\) 12.0000 0.508001
\(559\) 0 0
\(560\) 8.00000 0.338062
\(561\) 0 0
\(562\) −4.00000 −0.168730
\(563\) −18.0000 −0.758610 −0.379305 − 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 8.00000 0.336265
\(567\) 9.00000 0.377964
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 − 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) 23.0000 0.962520 0.481260 − 0.876578i \(-0.340179\pi\)
0.481260 + 0.876578i \(0.340179\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 6.00000 0.250217
\(576\) 24.0000 1.00000
\(577\) 18.0000 0.749350 0.374675 − 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) −38.0000 −1.58059
\(579\) 0 0
\(580\) 32.0000 1.32873
\(581\) −1.00000 −0.0414870
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 0 0
\(585\) 0 0
\(586\) 44.0000 1.81762
\(587\) −6.00000 −0.247647 −0.123823 − 0.992304i \(-0.539516\pi\)
−0.123823 + 0.992304i \(0.539516\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) 40.0000 1.64399
\(593\) 14.0000 0.574911 0.287456 − 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) −20.0000 −0.819232
\(597\) 0 0
\(598\) 0 0
\(599\) 5.00000 0.204294 0.102147 − 0.994769i \(-0.467429\pi\)
0.102147 + 0.994769i \(0.467429\pi\)
\(600\) 0 0
\(601\) 39.0000 1.59084 0.795422 − 0.606057i \(-0.207249\pi\)
0.795422 + 0.606057i \(0.207249\pi\)
\(602\) −2.00000 −0.0815139
\(603\) −15.0000 −0.610847
\(604\) 0 0
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 − 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −16.0000 −0.648886
\(609\) 0 0
\(610\) 24.0000 0.971732
\(611\) 0 0
\(612\) 36.0000 1.45521
\(613\) −1.00000 −0.0403896 −0.0201948 − 0.999796i \(-0.506429\pi\)
−0.0201948 + 0.999796i \(0.506429\pi\)
\(614\) −34.0000 −1.37213
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 − 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) 14.0000 0.562708 0.281354 − 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) −64.0000 −2.56617
\(623\) −8.00000 −0.320513
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 34.0000 1.35891
\(627\) 0 0
\(628\) 24.0000 0.957704
\(629\) 60.0000 2.39236
\(630\) −12.0000 −0.478091
\(631\) −14.0000 −0.557331 −0.278666 − 0.960388i \(-0.589892\pi\)
−0.278666 + 0.960388i \(0.589892\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 12.0000 0.476581
\(635\) −32.0000 −1.26988
\(636\) 0 0
\(637\) 0 0
\(638\) −32.0000 −1.26689
\(639\) −15.0000 −0.593391
\(640\) 0 0
\(641\) −15.0000 −0.592464 −0.296232 − 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 0 0
\(643\) −5.00000 −0.197181 −0.0985904 − 0.995128i \(-0.531433\pi\)
−0.0985904 + 0.995128i \(0.531433\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) −16.0000 −0.629025 −0.314512 − 0.949253i \(-0.601841\pi\)
−0.314512 + 0.949253i \(0.601841\pi\)
\(648\) 0 0
\(649\) −2.00000 −0.0785069
\(650\) 0 0
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −27.0000 −1.05659 −0.528296 − 0.849060i \(-0.677169\pi\)
−0.528296 + 0.849060i \(0.677169\pi\)
\(654\) 0 0
\(655\) 30.0000 1.17220
\(656\) −12.0000 −0.468521
\(657\) −6.00000 −0.234082
\(658\) 12.0000 0.467809
\(659\) 18.0000 0.701180 0.350590 − 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) 0 0
\(661\) 28.0000 1.08907 0.544537 − 0.838737i \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) −16.0000 −0.621858
\(663\) 0 0
\(664\) 0 0
\(665\) 4.00000 0.155113
\(666\) −60.0000 −2.32495
\(667\) 48.0000 1.85857
\(668\) −18.0000 −0.696441
\(669\) 0 0
\(670\) 20.0000 0.772667
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 − 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) −26.0000 −1.00000
\(677\) −14.0000 −0.538064 −0.269032 − 0.963131i \(-0.586704\pi\)
−0.269032 + 0.963131i \(0.586704\pi\)
\(678\) 0 0
\(679\) −7.00000 −0.268635
\(680\) 0 0
\(681\) 0 0
\(682\) 8.00000 0.306336
\(683\) −19.0000 −0.727015 −0.363507 − 0.931591i \(-0.618421\pi\)
−0.363507 + 0.931591i \(0.618421\pi\)
\(684\) 12.0000 0.458831
\(685\) 46.0000 1.75757
\(686\) 26.0000 0.992685
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 32.0000 1.21646
\(693\) 6.00000 0.227921
\(694\) −24.0000 −0.911028
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 28.0000 1.05982
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) −21.0000 −0.793159 −0.396580 − 0.918000i \(-0.629803\pi\)
−0.396580 + 0.918000i \(0.629803\pi\)
\(702\) 0 0
\(703\) 20.0000 0.754314
\(704\) 16.0000 0.603023
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) −3.00000 −0.112827
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 − 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 20.0000 0.750587
\(711\) 33.0000 1.23760
\(712\) 0 0
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) −54.0000 −2.01526
\(719\) −12.0000 −0.447524 −0.223762 − 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) −24.0000 −0.894427
\(721\) 15.0000 0.558629
\(722\) 30.0000 1.11648
\(723\) 0 0
\(724\) 26.0000 0.966282
\(725\) 8.00000 0.297113
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 − 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 8.00000 0.296093
\(731\) −6.00000 −0.221918
\(732\) 0 0
\(733\) −9.00000 −0.332423 −0.166211 − 0.986090i \(-0.553153\pi\)
−0.166211 + 0.986090i \(0.553153\pi\)
\(734\) −40.0000 −1.47643
\(735\) 0 0
\(736\) −48.0000 −1.76930
\(737\) −10.0000 −0.368355
\(738\) 18.0000 0.662589
\(739\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 40.0000 1.47043
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) 16.0000 0.586983 0.293492 − 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 20.0000 0.732743
\(746\) 32.0000 1.17160
\(747\) 3.00000 0.109764
\(748\) 24.0000 0.877527
\(749\) 10.0000 0.365392
\(750\) 0 0
\(751\) −35.0000 −1.27717 −0.638584 − 0.769552i \(-0.720480\pi\)
−0.638584 + 0.769552i \(0.720480\pi\)
\(752\) 24.0000 0.875190
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 − 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −40.0000 −1.45287
\(759\) 0 0
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 − 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) −15.0000 −0.543036
\(764\) 16.0000 0.578860
\(765\) −36.0000 −1.30158
\(766\) 28.0000 1.01168
\(767\) 0 0
\(768\) 0 0
\(769\) 9.00000 0.324548 0.162274 − 0.986746i \(-0.448117\pi\)
0.162274 + 0.986746i \(0.448117\pi\)
\(770\) −8.00000 −0.288300
\(771\) 0 0
\(772\) −20.0000 −0.719816
\(773\) 39.0000 1.40273 0.701366 − 0.712801i \(-0.252574\pi\)
0.701366 + 0.712801i \(0.252574\pi\)
\(774\) 6.00000 0.215666
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) −10.0000 −0.357828
\(782\) −72.0000 −2.57471
\(783\) 0 0
\(784\) 24.0000 0.857143
\(785\) −24.0000 −0.856597
\(786\) 0 0
\(787\) 45.0000 1.60408 0.802038 − 0.597272i \(-0.203749\pi\)
0.802038 + 0.597272i \(0.203749\pi\)
\(788\) −30.0000 −1.06871
\(789\) 0 0
\(790\) −44.0000 −1.56545
\(791\) −3.00000 −0.106668
\(792\) 0 0
\(793\) 0 0
\(794\) 34.0000 1.20661
\(795\) 0 0
\(796\) 10.0000 0.354441
\(797\) −12.0000 −0.425062 −0.212531 − 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) −8.00000 −0.282843
\(801\) 24.0000 0.847998
\(802\) 4.00000 0.141245
\(803\) −4.00000 −0.141157
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 − 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 36.0000 1.26491
\(811\) −43.0000 −1.50993 −0.754967 − 0.655763i \(-0.772347\pi\)
−0.754967 + 0.655763i \(0.772347\pi\)
\(812\) −16.0000 −0.561490
\(813\) 0 0
\(814\) −40.0000 −1.40200
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) −2.00000 −0.0699711
\(818\) −20.0000 −0.699284
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) 34.0000 1.18661 0.593304 − 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(822\) 0 0
\(823\) 13.0000 0.453152 0.226576 − 0.973994i \(-0.427247\pi\)
0.226576 + 0.973994i \(0.427247\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −2.00000 −0.0695889
\(827\) −22.0000 −0.765015 −0.382507 − 0.923952i \(-0.624939\pi\)
−0.382507 + 0.923952i \(0.624939\pi\)
\(828\) 36.0000 1.25109
\(829\) −5.00000 −0.173657 −0.0868286 − 0.996223i \(-0.527673\pi\)
−0.0868286 + 0.996223i \(0.527673\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) 0 0
\(833\) 36.0000 1.24733
\(834\) 0 0
\(835\) 18.0000 0.622916
\(836\) 8.00000 0.276686
\(837\) 0 0
\(838\) 52.0000 1.79631
\(839\) 30.0000 1.03572 0.517858 − 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 4.00000 0.137849
\(843\) 0 0
\(844\) −32.0000 −1.10149
\(845\) 26.0000 0.894427
\(846\) −36.0000 −1.23771
\(847\) −7.00000 −0.240523
\(848\) 48.0000 1.64833
\(849\) 0 0
\(850\) −12.0000 −0.411597
\(851\) 60.0000 2.05677
\(852\) 0 0
\(853\) −9.00000 −0.308154 −0.154077 − 0.988059i \(-0.549240\pi\)
−0.154077 + 0.988059i \(0.549240\pi\)
\(854\) −12.0000 −0.410632
\(855\) −12.0000 −0.410391
\(856\) 0 0
\(857\) −44.0000 −1.50301 −0.751506 − 0.659727i \(-0.770672\pi\)
−0.751506 + 0.659727i \(0.770672\pi\)
\(858\) 0 0
\(859\) 56.0000 1.91070 0.955348 − 0.295484i \(-0.0954809\pi\)
0.955348 + 0.295484i \(0.0954809\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −38.0000 −1.29354 −0.646768 − 0.762687i \(-0.723880\pi\)
−0.646768 + 0.762687i \(0.723880\pi\)
\(864\) 0 0
\(865\) −32.0000 −1.08803
\(866\) 4.00000 0.135926
\(867\) 0 0
\(868\) 4.00000 0.135769
\(869\) 22.0000 0.746299
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 21.0000 0.710742
\(874\) −24.0000 −0.811812
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) −10.0000 −0.337676 −0.168838 − 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) −16.0000 −0.539360
\(881\) 14.0000 0.471672 0.235836 − 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) −36.0000 −1.21218
\(883\) −28.0000 −0.942275 −0.471138 − 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) −7.00000 −0.235037 −0.117518 − 0.993071i \(-0.537494\pi\)
−0.117518 + 0.993071i \(0.537494\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) −32.0000 −1.07264
\(891\) −18.0000 −0.603023
\(892\) 8.00000 0.267860
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) 0 0
\(898\) 42.0000 1.40156
\(899\) −16.0000 −0.533630
\(900\) 6.00000 0.200000
\(901\) 72.0000 2.39867
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) 0 0
\(905\) −26.0000 −0.864269
\(906\) 0 0
\(907\) −22.0000 −0.730498 −0.365249 − 0.930910i \(-0.619016\pi\)
−0.365249 + 0.930910i \(0.619016\pi\)
\(908\) 24.0000 0.796468
\(909\) 9.00000 0.298511
\(910\) 0 0
\(911\) 39.0000 1.29213 0.646064 − 0.763283i \(-0.276414\pi\)
0.646064 + 0.763283i \(0.276414\pi\)
\(912\) 0 0
\(913\) 2.00000 0.0661903
\(914\) 68.0000 2.24924
\(915\) 0 0
\(916\) −30.0000 −0.991228
\(917\) −15.0000 −0.495344
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 − 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 68.0000 2.23946
\(923\) 0 0
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) −54.0000 −1.77455
\(927\) −45.0000 −1.47799
\(928\) −64.0000 −2.10090
\(929\) −1.00000 −0.0328089 −0.0164045 − 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) −48.0000 −1.57229
\(933\) 0 0
\(934\) −52.0000 −1.70149
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) 21.0000 0.686040 0.343020 − 0.939328i \(-0.388550\pi\)
0.343020 + 0.939328i \(0.388550\pi\)
\(938\) −10.0000 −0.326512
\(939\) 0 0
\(940\) 24.0000 0.782794
\(941\) −13.0000 −0.423788 −0.211894 − 0.977293i \(-0.567963\pi\)
−0.211894 + 0.977293i \(0.567963\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 42.0000 1.36482 0.682408 − 0.730971i \(-0.260933\pi\)
0.682408 + 0.730971i \(0.260933\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 0 0
\(953\) −26.0000 −0.842223 −0.421111 − 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) −72.0000 −2.33109
\(955\) −16.0000 −0.517748
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) 32.0000 1.03387
\(959\) −23.0000 −0.742709
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −30.0000 −0.966736
\(964\) 44.0000 1.41714
\(965\) 20.0000 0.643823
\(966\) 0 0
\(967\) 19.0000 0.610999 0.305499 − 0.952192i \(-0.401177\pi\)
0.305499 + 0.952192i \(0.401177\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −28.0000 −0.899026
\(971\) 24.0000 0.770197 0.385098 − 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 0 0
\(973\) 6.00000 0.192351
\(974\) 4.00000 0.128168
\(975\) 0 0
\(976\) −24.0000 −0.768221
\(977\) −24.0000 −0.767828 −0.383914 − 0.923369i \(-0.625424\pi\)
−0.383914 + 0.923369i \(0.625424\pi\)
\(978\) 0 0
\(979\) 16.0000 0.511362
\(980\) 24.0000 0.766652
\(981\) 45.0000 1.43674
\(982\) −54.0000 −1.72321
\(983\) −41.0000 −1.30770 −0.653848 − 0.756626i \(-0.726847\pi\)
−0.653848 + 0.756626i \(0.726847\pi\)
\(984\) 0 0
\(985\) 30.0000 0.955879
\(986\) −96.0000 −3.05726
\(987\) 0 0
\(988\) 0 0
\(989\) −6.00000 −0.190789
\(990\) 24.0000 0.762770
\(991\) 6.00000 0.190596 0.0952981 − 0.995449i \(-0.469620\pi\)
0.0952981 + 0.995449i \(0.469620\pi\)
\(992\) 16.0000 0.508001
\(993\) 0 0
\(994\) −10.0000 −0.317181
\(995\) −10.0000 −0.317021
\(996\) 0 0
\(997\) −2.00000 −0.0633406 −0.0316703 − 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) 58.0000 1.83596
\(999\) 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 6011.2.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6011.2.a.a.1.1 ✓ 1 1.1 even 1 trivial