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Properties

Label 3971.2.a.a.1.1
Level $3971$
Weight $2$
Character 3971.1
Self dual yes
Analytic conductor $31.709$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3971,2,Mod(1,3971)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3971, 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("3971.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3971 = 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3971.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: \(31.7085946427\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3971.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} -4.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} -4.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} +3.00000 q^{15} +4.00000 q^{16} +6.00000 q^{20} +4.00000 q^{21} +3.00000 q^{23} +4.00000 q^{25} +5.00000 q^{27} +8.00000 q^{28} +6.00000 q^{29} +7.00000 q^{31} -1.00000 q^{33} +12.0000 q^{35} +4.00000 q^{36} +7.00000 q^{37} +2.00000 q^{39} -10.0000 q^{43} -2.00000 q^{44} +6.00000 q^{45} -4.00000 q^{48} +9.00000 q^{49} +4.00000 q^{52} -6.00000 q^{53} -3.00000 q^{55} -3.00000 q^{59} -6.00000 q^{60} -10.0000 q^{61} +8.00000 q^{63} -8.00000 q^{64} +6.00000 q^{65} -11.0000 q^{67} -3.00000 q^{69} -15.0000 q^{71} +8.00000 q^{73} -4.00000 q^{75} -4.00000 q^{77} +16.0000 q^{79} -12.0000 q^{80} +1.00000 q^{81} -8.00000 q^{84} -6.00000 q^{87} -9.00000 q^{89} +8.00000 q^{91} -6.00000 q^{92} -7.00000 q^{93} +1.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) −2.00000 −1.00000
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 4.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 6.00000 1.34164
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 − 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 8.00000 1.51186
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 − 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) 4.00000 0.666667
\(37\) 7.00000 1.15079 0.575396 − 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) −2.00000 −0.301511
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −4.00000 −0.577350
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 − 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) −6.00000 −0.774597
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 8.00000 1.00791
\(64\) −8.00000 −1.00000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −11.0000 −1.34386 −0.671932 − 0.740613i \(-0.734535\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 − 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) −12.0000 −1.34164
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −8.00000 −0.872872
\(85\) 0 0
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 − 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) −6.00000 −0.625543
\(93\) −7.00000 −0.725866
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) −8.00000 −0.800000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) −12.0000 −1.17108
\(106\) 0 0
\(107\) 18.0000 1.74013 0.870063 − 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) −10.0000 −0.962250
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) −16.0000 −1.51186
\(113\) 9.00000 0.846649 0.423324 − 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) −9.00000 −0.839254
\(116\) −12.0000 −1.11417
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) −14.0000 −1.25724
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 − 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 0 0
\(135\) −15.0000 −1.29099
\(136\) 0 0
\(137\) 21.0000 1.79415 0.897076 − 0.441877i \(-0.145687\pi\)
0.897076 + 0.441877i \(0.145687\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 − 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) −24.0000 −2.02837
\(141\) 0 0
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) −8.00000 −0.666667
\(145\) −18.0000 −1.49482
\(146\) 0 0
\(147\) −9.00000 −0.742307
\(148\) −14.0000 −1.15079
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 − 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −21.0000 −1.68676
\(156\) −4.00000 −0.320256
\(157\) 17.0000 1.35675 0.678374 − 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 3.00000 0.233550
\(166\) 0 0
\(167\) 18.0000 1.39288 0.696441 − 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 20.0000 1.52499
\(173\) −24.0000 −1.82469 −0.912343 − 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 0 0
\(175\) −16.0000 −1.20949
\(176\) 4.00000 0.301511
\(177\) 3.00000 0.225494
\(178\) 0 0
\(179\) −15.0000 −1.12115 −0.560576 − 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) −12.0000 −0.894427
\(181\) 7.00000 0.520306 0.260153 − 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) −21.0000 −1.54395
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −20.0000 −1.45479
\(190\) 0 0
\(191\) −15.0000 −1.08536 −0.542681 − 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 8.00000 0.577350
\(193\) 4.00000 0.287926 0.143963 − 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) −6.00000 −0.429669
\(196\) −18.0000 −1.28571
\(197\) 12.0000 0.854965 0.427482 − 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 11.0000 0.775880
\(202\) 0 0
\(203\) −24.0000 −1.68447
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) −8.00000 −0.554700
\(209\) 0 0
\(210\) 0 0
\(211\) 22.0000 1.51454 0.757271 − 0.653101i \(-0.226532\pi\)
0.757271 + 0.653101i \(0.226532\pi\)
\(212\) 12.0000 0.824163
\(213\) 15.0000 1.02778
\(214\) 0 0
\(215\) 30.0000 2.04598
\(216\) 0 0
\(217\) −28.0000 −1.90076
\(218\) 0 0
\(219\) −8.00000 −0.540590
\(220\) 6.00000 0.404520
\(221\) 0 0
\(222\) 0 0
\(223\) −5.00000 −0.334825 −0.167412 − 0.985887i \(-0.553541\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) 0 0
\(225\) −8.00000 −0.533333
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −13.0000 −0.859064 −0.429532 − 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 − 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) −18.0000 −1.16432 −0.582162 − 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) 12.0000 0.774597
\(241\) −20.0000 −1.28831 −0.644157 − 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 20.0000 1.28037
\(245\) −27.0000 −1.72497
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 − 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) −16.0000 −1.00791
\(253\) 3.00000 0.188608
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) −28.0000 −1.73984
\(260\) −12.0000 −0.744208
\(261\) −12.0000 −0.742781
\(262\) 0 0
\(263\) −30.0000 −1.84988 −0.924940 − 0.380114i \(-0.875885\pi\)
−0.924940 + 0.380114i \(0.875885\pi\)
\(264\) 0 0
\(265\) 18.0000 1.10573
\(266\) 0 0
\(267\) 9.00000 0.550791
\(268\) 22.0000 1.34386
\(269\) 30.0000 1.82913 0.914566 − 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) −8.00000 −0.484182
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 6.00000 0.361158
\(277\) 26.0000 1.56219 0.781094 − 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 0 0
\(279\) −14.0000 −0.838158
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 − 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 30.0000 1.78017
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) −16.0000 −0.936329
\(293\) −18.0000 −1.05157 −0.525786 − 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) 9.00000 0.524000
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) −6.00000 −0.346989
\(300\) 8.00000 0.461880
\(301\) 40.0000 2.30556
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 30.0000 1.71780
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 8.00000 0.455842
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 − 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 0 0
\(315\) −24.0000 −1.35225
\(316\) −32.0000 −1.80014
\(317\) −15.0000 −0.842484 −0.421242 − 0.906948i \(-0.638406\pi\)
−0.421242 + 0.906948i \(0.638406\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 24.0000 1.34164
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) 0 0
\(324\) −2.00000 −0.111111
\(325\) −8.00000 −0.443760
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7.00000 0.384755 0.192377 − 0.981321i \(-0.438380\pi\)
0.192377 + 0.981321i \(0.438380\pi\)
\(332\) 0 0
\(333\) −14.0000 −0.767195
\(334\) 0 0
\(335\) 33.0000 1.80298
\(336\) 16.0000 0.872872
\(337\) 4.00000 0.217894 0.108947 − 0.994048i \(-0.465252\pi\)
0.108947 + 0.994048i \(0.465252\pi\)
\(338\) 0 0
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) 7.00000 0.379071
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 9.00000 0.484544
\(346\) 0 0
\(347\) −18.0000 −0.966291 −0.483145 − 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 12.0000 0.643268
\(349\) −28.0000 −1.49881 −0.749403 − 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) −10.0000 −0.533761
\(352\) 0 0
\(353\) −21.0000 −1.11772 −0.558859 − 0.829263i \(-0.688761\pi\)
−0.558859 + 0.829263i \(0.688761\pi\)
\(354\) 0 0
\(355\) 45.0000 2.38835
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 − 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) −16.0000 −0.838628
\(365\) −24.0000 −1.25622
\(366\) 0 0
\(367\) 23.0000 1.20059 0.600295 − 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) 12.0000 0.625543
\(369\) 0 0
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 14.0000 0.725866
\(373\) −14.0000 −0.724893 −0.362446 − 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 − 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 0 0
\(383\) −33.0000 −1.68622 −0.843111 − 0.537740i \(-0.819278\pi\)
−0.843111 + 0.537740i \(0.819278\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) 0 0
\(387\) 20.0000 1.01666
\(388\) −2.00000 −0.101535
\(389\) −3.00000 −0.152106 −0.0760530 − 0.997104i \(-0.524232\pi\)
−0.0760530 + 0.997104i \(0.524232\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) 0 0
\(395\) −48.0000 −2.41514
\(396\) 4.00000 0.201008
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 16.0000 0.800000
\(401\) 30.0000 1.49813 0.749064 − 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) −14.0000 −0.697390
\(404\) 0 0
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) 7.00000 0.346977
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 − 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 0 0
\(411\) −21.0000 −1.03585
\(412\) −32.0000 −1.57653
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 24.0000 1.17108
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 40.0000 1.93574
\(428\) −36.0000 −1.74013
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 − 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 20.0000 0.962250
\(433\) 13.0000 0.624740 0.312370 − 0.949960i \(-0.398877\pi\)
0.312370 + 0.949960i \(0.398877\pi\)
\(434\) 0 0
\(435\) 18.0000 0.863034
\(436\) 4.00000 0.191565
\(437\) 0 0
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 − 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) 0 0
\(443\) −27.0000 −1.28281 −0.641404 − 0.767203i \(-0.721648\pi\)
−0.641404 + 0.767203i \(0.721648\pi\)
\(444\) 14.0000 0.664411
\(445\) 27.0000 1.27992
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 32.0000 1.51186
\(449\) 27.0000 1.27421 0.637104 − 0.770778i \(-0.280132\pi\)
0.637104 + 0.770778i \(0.280132\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −18.0000 −0.846649
\(453\) −10.0000 −0.469841
\(454\) 0 0
\(455\) −24.0000 −1.12514
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 18.0000 0.839254
\(461\) 12.0000 0.558896 0.279448 − 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) 29.0000 1.34774 0.673872 − 0.738848i \(-0.264630\pi\)
0.673872 + 0.738848i \(0.264630\pi\)
\(464\) 24.0000 1.11417
\(465\) 21.0000 0.973852
\(466\) 0 0
\(467\) −3.00000 −0.138823 −0.0694117 − 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) −8.00000 −0.369800
\(469\) 44.0000 2.03173
\(470\) 0 0
\(471\) −17.0000 −0.783319
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) −6.00000 −0.274147 −0.137073 − 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) −14.0000 −0.638345
\(482\) 0 0
\(483\) 12.0000 0.546019
\(484\) −2.00000 −0.0909091
\(485\) −3.00000 −0.136223
\(486\) 0 0
\(487\) 7.00000 0.317200 0.158600 − 0.987343i \(-0.449302\pi\)
0.158600 + 0.987343i \(0.449302\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 28.0000 1.25724
\(497\) 60.0000 2.69137
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 − 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) −6.00000 −0.268328
\(501\) −18.0000 −0.804181
\(502\) 0 0
\(503\) 18.0000 0.802580 0.401290 − 0.915951i \(-0.368562\pi\)
0.401290 + 0.915951i \(0.368562\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 4.00000 0.177471
\(509\) −21.0000 −0.930809 −0.465404 − 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 0 0
\(511\) −32.0000 −1.41560
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −48.0000 −2.11513
\(516\) −20.0000 −0.880451
\(517\) 0 0
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 − 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 12.0000 0.524222
\(525\) 16.0000 0.698297
\(526\) 0 0
\(527\) 0 0
\(528\) −4.00000 −0.174078
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −54.0000 −2.33462
\(536\) 0 0
\(537\) 15.0000 0.647298
\(538\) 0 0
\(539\) 9.00000 0.387657
\(540\) 30.0000 1.29099
\(541\) −10.0000 −0.429934 −0.214967 − 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) −7.00000 −0.300399
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 − 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) −42.0000 −1.79415
\(549\) 20.0000 0.853579
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −64.0000 −2.72156
\(554\) 0 0
\(555\) 21.0000 0.891400
\(556\) −16.0000 −0.678551
\(557\) −30.0000 −1.27114 −0.635570 − 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 20.0000 0.845910
\(560\) 48.0000 2.02837
\(561\) 0 0
\(562\) 0 0
\(563\) 6.00000 0.252870 0.126435 − 0.991975i \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) 0 0
\(565\) −27.0000 −1.13590
\(566\) 0 0
\(567\) −4.00000 −0.167984
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −46.0000 −1.92504 −0.962520 − 0.271211i \(-0.912576\pi\)
−0.962520 + 0.271211i \(0.912576\pi\)
\(572\) 4.00000 0.167248
\(573\) 15.0000 0.626634
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 16.0000 0.666667
\(577\) −7.00000 −0.291414 −0.145707 − 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 36.0000 1.49482
\(581\) 0 0
\(582\) 0 0
\(583\) −6.00000 −0.248495
\(584\) 0 0
\(585\) −12.0000 −0.496139
\(586\) 0 0
\(587\) 24.0000 0.990586 0.495293 − 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 18.0000 0.742307
\(589\) 0 0
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 28.0000 1.15079
\(593\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −20.0000 −0.815817 −0.407909 − 0.913023i \(-0.633742\pi\)
−0.407909 + 0.913023i \(0.633742\pi\)
\(602\) 0 0
\(603\) 22.0000 0.895909
\(604\) −20.0000 −0.813788
\(605\) −3.00000 −0.121967
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 − 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 − 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −13.0000 −0.522514 −0.261257 − 0.965269i \(-0.584137\pi\)
−0.261257 + 0.965269i \(0.584137\pi\)
\(620\) 42.0000 1.68676
\(621\) 15.0000 0.601929
\(622\) 0 0
\(623\) 36.0000 1.44231
\(624\) 8.00000 0.320256
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) −34.0000 −1.35675
\(629\) 0 0
\(630\) 0 0
\(631\) −1.00000 −0.0398094 −0.0199047 − 0.999802i \(-0.506336\pi\)
−0.0199047 + 0.999802i \(0.506336\pi\)
\(632\) 0 0
\(633\) −22.0000 −0.874421
\(634\) 0 0
\(635\) 6.00000 0.238103
\(636\) −12.0000 −0.475831
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) 30.0000 1.18678
\(640\) 0 0
\(641\) 15.0000 0.592464 0.296232 − 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 0 0
\(643\) −7.00000 −0.276053 −0.138027 − 0.990429i \(-0.544076\pi\)
−0.138027 + 0.990429i \(0.544076\pi\)
\(644\) 24.0000 0.945732
\(645\) −30.0000 −1.18125
\(646\) 0 0
\(647\) 21.0000 0.825595 0.412798 − 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) 0 0
\(649\) −3.00000 −0.117760
\(650\) 0 0
\(651\) 28.0000 1.09741
\(652\) 8.00000 0.313304
\(653\) 27.0000 1.05659 0.528296 − 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) 0 0
\(657\) −16.0000 −0.624219
\(658\) 0 0
\(659\) 18.0000 0.701180 0.350590 − 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) −6.00000 −0.233550
\(661\) 13.0000 0.505641 0.252821 − 0.967513i \(-0.418642\pi\)
0.252821 + 0.967513i \(0.418642\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) −36.0000 −1.39288
\(669\) 5.00000 0.193311
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 − 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 20.0000 0.769800
\(676\) 18.0000 0.692308
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −63.0000 −2.40711
\(686\) 0 0
\(687\) 13.0000 0.495981
\(688\) −40.0000 −1.52499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 − 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) 48.0000 1.82469
\(693\) 8.00000 0.303895
\(694\) 0 0
\(695\) −24.0000 −0.910372
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 32.0000 1.20949
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −8.00000 −0.301511
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) −6.00000 −0.225494
\(709\) −25.0000 −0.938895 −0.469447 − 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 0 0
\(711\) −32.0000 −1.20009
\(712\) 0 0
\(713\) 21.0000 0.786456
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) 30.0000 1.12115
\(717\) 18.0000 0.672222
\(718\) 0 0
\(719\) −9.00000 −0.335643 −0.167822 − 0.985817i \(-0.553673\pi\)
−0.167822 + 0.985817i \(0.553673\pi\)
\(720\) 24.0000 0.894427
\(721\) −64.0000 −2.38348
\(722\) 0 0
\(723\) 20.0000 0.743808
\(724\) −14.0000 −0.520306
\(725\) 24.0000 0.891338
\(726\) 0 0
\(727\) 35.0000 1.29808 0.649039 − 0.760755i \(-0.275171\pi\)
0.649039 + 0.760755i \(0.275171\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) −20.0000 −0.739221
\(733\) −22.0000 −0.812589 −0.406294 − 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) 27.0000 0.995910
\(736\) 0 0
\(737\) −11.0000 −0.405190
\(738\) 0 0
\(739\) 44.0000 1.61857 0.809283 − 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) 42.0000 1.54395
\(741\) 0 0
\(742\) 0 0
\(743\) 30.0000 1.10059 0.550297 − 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −72.0000 −2.63082
\(750\) 0 0
\(751\) 1.00000 0.0364905 0.0182453 − 0.999834i \(-0.494192\pi\)
0.0182453 + 0.999834i \(0.494192\pi\)
\(752\) 0 0
\(753\) −21.0000 −0.765283
\(754\) 0 0
\(755\) −30.0000 −1.09181
\(756\) 40.0000 1.45479
\(757\) 14.0000 0.508839 0.254419 − 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(758\) 0 0
\(759\) −3.00000 −0.108893
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) 30.0000 1.08536
\(765\) 0 0
\(766\) 0 0
\(767\) 6.00000 0.216647
\(768\) −16.0000 −0.577350
\(769\) −28.0000 −1.00971 −0.504853 − 0.863205i \(-0.668453\pi\)
−0.504853 + 0.863205i \(0.668453\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) −8.00000 −0.287926
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) 28.0000 1.00579
\(776\) 0 0
\(777\) 28.0000 1.00449
\(778\) 0 0
\(779\) 0 0
\(780\) 12.0000 0.429669
\(781\) −15.0000 −0.536742
\(782\) 0 0
\(783\) 30.0000 1.07211
\(784\) 36.0000 1.28571
\(785\) −51.0000 −1.82027
\(786\) 0 0
\(787\) −20.0000 −0.712923 −0.356462 − 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) −24.0000 −0.854965
\(789\) 30.0000 1.06803
\(790\) 0 0
\(791\) −36.0000 −1.28001
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) 0 0
\(795\) −18.0000 −0.638394
\(796\) 32.0000 1.13421
\(797\) −51.0000 −1.80651 −0.903256 − 0.429101i \(-0.858830\pi\)
−0.903256 + 0.429101i \(0.858830\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) 8.00000 0.282314
\(804\) −22.0000 −0.775880
\(805\) 36.0000 1.26883
\(806\) 0 0
\(807\) −30.0000 −1.05605
\(808\) 0 0
\(809\) −24.0000 −0.843795 −0.421898 − 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 − 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 48.0000 1.68447
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −16.0000 −0.559085
\(820\) 0 0
\(821\) 36.0000 1.25641 0.628204 − 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) 0 0
\(823\) −13.0000 −0.453152 −0.226576 − 0.973994i \(-0.572753\pi\)
−0.226576 + 0.973994i \(0.572753\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 42.0000 1.46048 0.730242 − 0.683189i \(-0.239408\pi\)
0.730242 + 0.683189i \(0.239408\pi\)
\(828\) 12.0000 0.417029
\(829\) −11.0000 −0.382046 −0.191023 − 0.981586i \(-0.561180\pi\)
−0.191023 + 0.981586i \(0.561180\pi\)
\(830\) 0 0
\(831\) −26.0000 −0.901930
\(832\) 16.0000 0.554700
\(833\) 0 0
\(834\) 0 0
\(835\) −54.0000 −1.86875
\(836\) 0 0
\(837\) 35.0000 1.20978
\(838\) 0 0
\(839\) −33.0000 −1.13929 −0.569643 − 0.821892i \(-0.692919\pi\)
−0.569643 + 0.821892i \(0.692919\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −30.0000 −1.03325
\(844\) −44.0000 −1.51454
\(845\) 27.0000 0.928828
\(846\) 0 0
\(847\) −4.00000 −0.137442
\(848\) −24.0000 −0.824163
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 21.0000 0.719871
\(852\) −30.0000 −1.02778
\(853\) −22.0000 −0.753266 −0.376633 − 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −30.0000 −1.02478 −0.512390 − 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 0 0
\(859\) 5.00000 0.170598 0.0852989 − 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) −60.0000 −2.04598
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 72.0000 2.44807
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) 56.0000 1.90076
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 22.0000 0.745442
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) 16.0000 0.540590
\(877\) 4.00000 0.135070 0.0675352 − 0.997717i \(-0.478487\pi\)
0.0675352 + 0.997717i \(0.478487\pi\)
\(878\) 0 0
\(879\) 18.0000 0.607125
\(880\) −12.0000 −0.404520
\(881\) 9.00000 0.303218 0.151609 − 0.988441i \(-0.451555\pi\)
0.151609 + 0.988441i \(0.451555\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 − 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) −9.00000 −0.302532
\(886\) 0 0
\(887\) −54.0000 −1.81314 −0.906571 − 0.422053i \(-0.861310\pi\)
−0.906571 + 0.422053i \(0.861310\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 10.0000 0.334825
\(893\) 0 0
\(894\) 0 0
\(895\) 45.0000 1.50418
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) 0 0
\(899\) 42.0000 1.40078
\(900\) 16.0000 0.533333
\(901\) 0 0
\(902\) 0 0
\(903\) −40.0000 −1.33112
\(904\) 0 0
\(905\) −21.0000 −0.698064
\(906\) 0 0
\(907\) −8.00000 −0.265636 −0.132818 − 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) −24.0000 −0.796468
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −30.0000 −0.991769
\(916\) 26.0000 0.859064
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) −4.00000 −0.131948 −0.0659739 − 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) 0 0
\(923\) 30.0000 0.987462
\(924\) −8.00000 −0.263181
\(925\) 28.0000 0.920634
\(926\) 0 0
\(927\) −32.0000 −1.05102
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 − 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −36.0000 −1.17922
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 − 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 1.00000 0.0326338
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 − 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 60.0000 1.95180
\(946\) 0 0
\(947\) −3.00000 −0.0974869 −0.0487435 − 0.998811i \(-0.515522\pi\)
−0.0487435 + 0.998811i \(0.515522\pi\)
\(948\) 32.0000 1.03931
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) 15.0000 0.486408
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 − 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 45.0000 1.45617
\(956\) 36.0000 1.16432
\(957\) −6.00000 −0.193952
\(958\) 0 0
\(959\) −84.0000 −2.71250
\(960\) −24.0000 −0.774597
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) −36.0000 −1.16008
\(964\) 40.0000 1.28831
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 − 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 15.0000 0.481373 0.240686 − 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 32.0000 1.02640
\(973\) −32.0000 −1.02587
\(974\) 0 0
\(975\) 8.00000 0.256205
\(976\) −40.0000 −1.28037
\(977\) 9.00000 0.287936 0.143968 − 0.989582i \(-0.454014\pi\)
0.143968 + 0.989582i \(0.454014\pi\)
\(978\) 0 0
\(979\) −9.00000 −0.287641
\(980\) 54.0000 1.72497
\(981\) 4.00000 0.127710
\(982\) 0 0
\(983\) 15.0000 0.478426 0.239213 − 0.970967i \(-0.423111\pi\)
0.239213 + 0.970967i \(0.423111\pi\)
\(984\) 0 0
\(985\) −36.0000 −1.14706
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −30.0000 −0.953945
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 − 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) −7.00000 −0.222138
\(994\) 0 0
\(995\) 48.0000 1.52170
\(996\) 0 0
\(997\) −40.0000 −1.26681 −0.633406 − 0.773819i \(-0.718344\pi\)
−0.633406 + 0.773819i \(0.718344\pi\)
\(998\) 0 0
\(999\) 35.0000 1.10735
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 3971.2.a.a.1.1 1
19.18 odd 2 209.2.a.a.1.1 ✓ 1
57.56 even 2 1881.2.a.c.1.1 1
76.75 even 2 3344.2.a.d.1.1 1
95.94 odd 2 5225.2.a.b.1.1 1
209.208 even 2 2299.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.a.1.1 ✓ 1 19.18 odd 2
1881.2.a.c.1.1 1 57.56 even 2
2299.2.a.c.1.1 1 209.208 even 2
3344.2.a.d.1.1 1 76.75 even 2
3971.2.a.a.1.1 1 1.1 even 1 trivial
5225.2.a.b.1.1 1 95.94 odd 2