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Properties

Label 8771.2.a.b.1.1
Level $8771$
Weight $2$
Character 8771.1
Self dual yes
Analytic conductor $70.037$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8771,2,Mod(1,8771)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8771, 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("8771.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8771 = 7^{2} \cdot 179 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8771.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: \(70.0367876129\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 179)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8771.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.00000 q^{4} -3.00000 q^{5} -3.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} +2.00000 q^{4} -3.00000 q^{5} -3.00000 q^{9} -6.00000 q^{10} +4.00000 q^{11} +1.00000 q^{13} -4.00000 q^{16} -1.00000 q^{17} -6.00000 q^{18} +3.00000 q^{19} -6.00000 q^{20} +8.00000 q^{22} +6.00000 q^{23} +4.00000 q^{25} +2.00000 q^{26} +3.00000 q^{29} +8.00000 q^{31} -8.00000 q^{32} -2.00000 q^{34} -6.00000 q^{36} +2.00000 q^{37} +6.00000 q^{38} -12.0000 q^{41} -11.0000 q^{43} +8.00000 q^{44} +9.00000 q^{45} +12.0000 q^{46} -1.00000 q^{47} +8.00000 q^{50} +2.00000 q^{52} -12.0000 q^{55} +6.00000 q^{58} +5.00000 q^{59} -14.0000 q^{61} +16.0000 q^{62} -8.00000 q^{64} -3.00000 q^{65} -9.00000 q^{67} -2.00000 q^{68} -10.0000 q^{73} +4.00000 q^{74} +6.00000 q^{76} +10.0000 q^{79} +12.0000 q^{80} +9.00000 q^{81} -24.0000 q^{82} -17.0000 q^{83} +3.00000 q^{85} -22.0000 q^{86} +1.00000 q^{89} +18.0000 q^{90} +12.0000 q^{92} -2.00000 q^{94} -9.00000 q^{95} +14.0000 q^{97} -12.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 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
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) −6.00000 −1.89737
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −1.00000 −0.242536 −0.121268 − 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) −6.00000 −1.41421
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) −6.00000 −1.34164
\(21\) 0 0
\(22\) 8.00000 1.70561
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) −6.00000 −1.00000
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 − 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 − 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 8.00000 1.20605
\(45\) 9.00000 1.34164
\(46\) 12.0000 1.76930
\(47\) −1.00000 −0.145865 −0.0729325 − 0.997337i \(-0.523236\pi\)
−0.0729325 + 0.997337i \(0.523236\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 8.00000 1.13137
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 5.00000 0.650945 0.325472 − 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 16.0000 2.03200
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) −9.00000 −1.09952 −0.549762 − 0.835321i \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 12.0000 1.34164
\(81\) 9.00000 1.00000
\(82\) −24.0000 −2.65036
\(83\) −17.0000 −1.86599 −0.932996 − 0.359886i \(-0.882816\pi\)
−0.932996 + 0.359886i \(0.882816\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) −22.0000 −2.37232
\(87\) 0 0
\(88\) 0 0
\(89\) 1.00000 0.106000 0.0529999 − 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 18.0000 1.89737
\(91\) 0 0
\(92\) 12.0000 1.25109
\(93\) 0 0
\(94\) −2.00000 −0.206284
\(95\) −9.00000 −0.923381
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) −12.0000 −1.20605
\(100\) 8.00000 0.800000
\(101\) −9.00000 −0.895533 −0.447767 − 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 − 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) −24.0000 −2.28831
\(111\) 0 0
\(112\) 0 0
\(113\) −4.00000 −0.376288 −0.188144 − 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) −18.0000 −1.67851
\(116\) 6.00000 0.557086
\(117\) −3.00000 −0.277350
\(118\) 10.0000 0.920575
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −28.0000 −2.53500
\(123\) 0 0
\(124\) 16.0000 1.43684
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −6.00000 −0.526235
\(131\) 6.00000 0.524222 0.262111 − 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −18.0000 −1.55496
\(135\) 0 0
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 − 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) −7.00000 −0.593732 −0.296866 − 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 12.0000 1.00000
\(145\) −9.00000 −0.747409
\(146\) −20.0000 −1.65521
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) −18.0000 −1.47462 −0.737309 − 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 21.0000 1.70896 0.854478 − 0.519488i \(-0.173877\pi\)
0.854478 + 0.519488i \(0.173877\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) −24.0000 −1.92773
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 − 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 20.0000 1.59111
\(159\) 0 0
\(160\) 24.0000 1.89737
\(161\) 0 0
\(162\) 18.0000 1.41421
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) −24.0000 −1.87409
\(165\) 0 0
\(166\) −34.0000 −2.63891
\(167\) −10.0000 −0.773823 −0.386912 − 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 6.00000 0.460179
\(171\) −9.00000 −0.688247
\(172\) −22.0000 −1.67748
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −16.0000 −1.20605
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) 1.00000 0.0747435
\(180\) 18.0000 1.34164
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) −2.00000 −0.145865
\(189\) 0 0
\(190\) −18.0000 −1.30586
\(191\) −13.0000 −0.940647 −0.470323 − 0.882494i \(-0.655863\pi\)
−0.470323 + 0.882494i \(0.655863\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 − 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 28.0000 2.01028
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −24.0000 −1.70561
\(199\) −17.0000 −1.20510 −0.602549 − 0.798082i \(-0.705848\pi\)
−0.602549 + 0.798082i \(0.705848\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −18.0000 −1.26648
\(203\) 0 0
\(204\) 0 0
\(205\) 36.0000 2.51435
\(206\) 12.0000 0.836080
\(207\) −18.0000 −1.25109
\(208\) −4.00000 −0.277350
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 − 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) 33.0000 2.25058
\(216\) 0 0
\(217\) 0 0
\(218\) −28.0000 −1.89640
\(219\) 0 0
\(220\) −24.0000 −1.61808
\(221\) −1.00000 −0.0672673
\(222\) 0 0
\(223\) 26.0000 1.74109 0.870544 − 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) 0 0
\(225\) −12.0000 −0.800000
\(226\) −8.00000 −0.532152
\(227\) −21.0000 −1.39382 −0.696909 − 0.717159i \(-0.745442\pi\)
−0.696909 + 0.717159i \(0.745442\pi\)
\(228\) 0 0
\(229\) −8.00000 −0.528655 −0.264327 − 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) −36.0000 −2.37377
\(231\) 0 0
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 − 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) −6.00000 −0.392232
\(235\) 3.00000 0.195698
\(236\) 10.0000 0.650945
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 − 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 10.0000 0.642824
\(243\) 0 0
\(244\) −28.0000 −1.79252
\(245\) 0 0
\(246\) 0 0
\(247\) 3.00000 0.190885
\(248\) 0 0
\(249\) 0 0
\(250\) 6.00000 0.379473
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) −16.0000 −1.00393
\(255\) 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
\(259\) 0 0
\(260\) −6.00000 −0.372104
\(261\) −9.00000 −0.557086
\(262\) 12.0000 0.741362
\(263\) −4.00000 −0.246651 −0.123325 − 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −18.0000 −1.09952
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −10.0000 −0.607457 −0.303728 − 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −20.0000 −1.20824
\(275\) 16.0000 0.964836
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 − 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) −14.0000 −0.839664
\(279\) −24.0000 −1.43684
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 2.00000 0.118888 0.0594438 − 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) 24.0000 1.41421
\(289\) −16.0000 −0.941176
\(290\) −18.0000 −1.05700
\(291\) 0 0
\(292\) −20.0000 −1.17041
\(293\) 8.00000 0.467365 0.233682 − 0.972313i \(-0.424922\pi\)
0.233682 + 0.972313i \(0.424922\pi\)
\(294\) 0 0
\(295\) −15.0000 −0.873334
\(296\) 0 0
\(297\) 0 0
\(298\) −36.0000 −2.08542
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) 42.0000 2.41683
\(303\) 0 0
\(304\) −12.0000 −0.688247
\(305\) 42.0000 2.40491
\(306\) 6.00000 0.342997
\(307\) −6.00000 −0.342438 −0.171219 − 0.985233i \(-0.554771\pi\)
−0.171219 + 0.985233i \(0.554771\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −48.0000 −2.72622
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 − 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −8.00000 −0.451466
\(315\) 0 0
\(316\) 20.0000 1.12509
\(317\) 29.0000 1.62880 0.814401 − 0.580302i \(-0.197066\pi\)
0.814401 + 0.580302i \(0.197066\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 24.0000 1.34164
\(321\) 0 0
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 18.0000 1.00000
\(325\) 4.00000 0.221880
\(326\) −40.0000 −2.21540
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −34.0000 −1.86599
\(333\) −6.00000 −0.328798
\(334\) −20.0000 −1.09435
\(335\) 27.0000 1.47517
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 − 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) −24.0000 −1.30543
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) 32.0000 1.73290
\(342\) −18.0000 −0.973329
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 28.0000 1.50529
\(347\) 15.0000 0.805242 0.402621 − 0.915367i \(-0.368099\pi\)
0.402621 + 0.915367i \(0.368099\pi\)
\(348\) 0 0
\(349\) 18.0000 0.963518 0.481759 − 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −32.0000 −1.70561
\(353\) 24.0000 1.27739 0.638696 − 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 2.00000 0.105703
\(359\) 32.0000 1.68890 0.844448 − 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) −4.00000 −0.210235
\(363\) 0 0
\(364\) 0 0
\(365\) 30.0000 1.57027
\(366\) 0 0
\(367\) 11.0000 0.574195 0.287098 − 0.957901i \(-0.407310\pi\)
0.287098 + 0.957901i \(0.407310\pi\)
\(368\) −24.0000 −1.25109
\(369\) 36.0000 1.87409
\(370\) −12.0000 −0.623850
\(371\) 0 0
\(372\) 0 0
\(373\) 37.0000 1.91579 0.957894 − 0.287123i \(-0.0926989\pi\)
0.957894 + 0.287123i \(0.0926989\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) −18.0000 −0.924598 −0.462299 − 0.886724i \(-0.652975\pi\)
−0.462299 + 0.886724i \(0.652975\pi\)
\(380\) −18.0000 −0.923381
\(381\) 0 0
\(382\) −26.0000 −1.33028
\(383\) −21.0000 −1.07305 −0.536525 − 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −12.0000 −0.610784
\(387\) 33.0000 1.67748
\(388\) 28.0000 1.42148
\(389\) 15.0000 0.760530 0.380265 − 0.924878i \(-0.375833\pi\)
0.380265 + 0.924878i \(0.375833\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 0 0
\(394\) −4.00000 −0.201517
\(395\) −30.0000 −1.50946
\(396\) −24.0000 −1.20605
\(397\) 13.0000 0.652451 0.326226 − 0.945292i \(-0.394223\pi\)
0.326226 + 0.945292i \(0.394223\pi\)
\(398\) −34.0000 −1.70427
\(399\) 0 0
\(400\) −16.0000 −0.800000
\(401\) 22.0000 1.09863 0.549314 − 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) −18.0000 −0.895533
\(405\) −27.0000 −1.34164
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 17.0000 0.840596 0.420298 − 0.907386i \(-0.361926\pi\)
0.420298 + 0.907386i \(0.361926\pi\)
\(410\) 72.0000 3.55583
\(411\) 0 0
\(412\) 12.0000 0.591198
\(413\) 0 0
\(414\) −36.0000 −1.76930
\(415\) 51.0000 2.50349
\(416\) −8.00000 −0.392232
\(417\) 0 0
\(418\) 24.0000 1.17388
\(419\) −11.0000 −0.537385 −0.268693 − 0.963226i \(-0.586592\pi\)
−0.268693 + 0.963226i \(0.586592\pi\)
\(420\) 0 0
\(421\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) −16.0000 −0.778868
\(423\) 3.00000 0.145865
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) 66.0000 3.18280
\(431\) −22.0000 −1.05970 −0.529851 − 0.848091i \(-0.677752\pi\)
−0.529851 + 0.848091i \(0.677752\pi\)
\(432\) 0 0
\(433\) 21.0000 1.00920 0.504598 − 0.863355i \(-0.331641\pi\)
0.504598 + 0.863355i \(0.331641\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −28.0000 −1.34096
\(437\) 18.0000 0.861057
\(438\) 0 0
\(439\) −27.0000 −1.28864 −0.644320 − 0.764756i \(-0.722859\pi\)
−0.644320 + 0.764756i \(0.722859\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.00000 −0.0951303
\(443\) 3.00000 0.142534 0.0712672 − 0.997457i \(-0.477296\pi\)
0.0712672 + 0.997457i \(0.477296\pi\)
\(444\) 0 0
\(445\) −3.00000 −0.142214
\(446\) 52.0000 2.46227
\(447\) 0 0
\(448\) 0 0
\(449\) 8.00000 0.377543 0.188772 − 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) −24.0000 −1.13137
\(451\) −48.0000 −2.26023
\(452\) −8.00000 −0.376288
\(453\) 0 0
\(454\) −42.0000 −1.97116
\(455\) 0 0
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 − 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −16.0000 −0.747631
\(459\) 0 0
\(460\) −36.0000 −1.67851
\(461\) −42.0000 −1.95614 −0.978068 − 0.208288i \(-0.933211\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −12.0000 −0.557086
\(465\) 0 0
\(466\) 48.0000 2.22356
\(467\) −16.0000 −0.740392 −0.370196 − 0.928954i \(-0.620709\pi\)
−0.370196 + 0.928954i \(0.620709\pi\)
\(468\) −6.00000 −0.277350
\(469\) 0 0
\(470\) 6.00000 0.276759
\(471\) 0 0
\(472\) 0 0
\(473\) −44.0000 −2.02312
\(474\) 0 0
\(475\) 12.0000 0.550598
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −33.0000 −1.50781 −0.753904 − 0.656984i \(-0.771832\pi\)
−0.753904 + 0.656984i \(0.771832\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) −8.00000 −0.364390
\(483\) 0 0
\(484\) 10.0000 0.454545
\(485\) −42.0000 −1.90712
\(486\) 0 0
\(487\) 27.0000 1.22349 0.611743 − 0.791056i \(-0.290469\pi\)
0.611743 + 0.791056i \(0.290469\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 − 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 0 0
\(493\) −3.00000 −0.135113
\(494\) 6.00000 0.269953
\(495\) 36.0000 1.61808
\(496\) −32.0000 −1.43684
\(497\) 0 0
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 − 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 6.00000 0.268328
\(501\) 0 0
\(502\) 0 0
\(503\) −40.0000 −1.78351 −0.891756 − 0.452517i \(-0.850526\pi\)
−0.891756 + 0.452517i \(0.850526\pi\)
\(504\) 0 0
\(505\) 27.0000 1.20148
\(506\) 48.0000 2.13386
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) 15.0000 0.664863 0.332432 − 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 32.0000 1.41421
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) −18.0000 −0.793175
\(516\) 0 0
\(517\) −4.00000 −0.175920
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −18.0000 −0.787839
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −15.0000 −0.650945
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 0 0
\(538\) −36.0000 −1.55207
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 − 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) 8.00000 0.342997
\(545\) 42.0000 1.79908
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −20.0000 −0.854358
\(549\) 42.0000 1.79252
\(550\) 32.0000 1.36448
\(551\) 9.00000 0.383413
\(552\) 0 0
\(553\) 0 0
\(554\) −28.0000 −1.18961
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) −29.0000 −1.22877 −0.614385 − 0.789007i \(-0.710596\pi\)
−0.614385 + 0.789007i \(0.710596\pi\)
\(558\) −48.0000 −2.03200
\(559\) −11.0000 −0.465250
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 38.0000 1.60151 0.800755 − 0.598993i \(-0.204432\pi\)
0.800755 + 0.598993i \(0.204432\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 0 0
\(569\) 8.00000 0.335377 0.167689 − 0.985840i \(-0.446370\pi\)
0.167689 + 0.985840i \(0.446370\pi\)
\(570\) 0 0
\(571\) 18.0000 0.753277 0.376638 − 0.926360i \(-0.377080\pi\)
0.376638 + 0.926360i \(0.377080\pi\)
\(572\) 8.00000 0.334497
\(573\) 0 0
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 24.0000 1.00000
\(577\) −32.0000 −1.33218 −0.666089 − 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) −32.0000 −1.33102
\(579\) 0 0
\(580\) −18.0000 −0.747409
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 9.00000 0.372104
\(586\) 16.0000 0.660954
\(587\) 34.0000 1.40333 0.701665 − 0.712507i \(-0.252440\pi\)
0.701665 + 0.712507i \(0.252440\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) −30.0000 −1.23508
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) −26.0000 −1.06769 −0.533846 − 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −36.0000 −1.47462
\(597\) 0 0
\(598\) 12.0000 0.490716
\(599\) 20.0000 0.817178 0.408589 − 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 − 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 0 0
\(603\) 27.0000 1.09952
\(604\) 42.0000 1.70896
\(605\) −15.0000 −0.609837
\(606\) 0 0
\(607\) −1.00000 −0.0405887 −0.0202944 − 0.999794i \(-0.506460\pi\)
−0.0202944 + 0.999794i \(0.506460\pi\)
\(608\) −24.0000 −0.973329
\(609\) 0 0
\(610\) 84.0000 3.40106
\(611\) −1.00000 −0.0404557
\(612\) 6.00000 0.242536
\(613\) 31.0000 1.25208 0.626039 − 0.779792i \(-0.284675\pi\)
0.626039 + 0.779792i \(0.284675\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) −13.0000 −0.523360 −0.261680 − 0.965155i \(-0.584277\pi\)
−0.261680 + 0.965155i \(0.584277\pi\)
\(618\) 0 0
\(619\) −35.0000 −1.40677 −0.703384 − 0.710810i \(-0.748329\pi\)
−0.703384 + 0.710810i \(0.748329\pi\)
\(620\) −48.0000 −1.92773
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 16.0000 0.639489
\(627\) 0 0
\(628\) −8.00000 −0.319235
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) 18.0000 0.716569 0.358284 − 0.933613i \(-0.383362\pi\)
0.358284 + 0.933613i \(0.383362\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 58.0000 2.30347
\(635\) 24.0000 0.952411
\(636\) 0 0
\(637\) 0 0
\(638\) 24.0000 0.950169
\(639\) 0 0
\(640\) 0 0
\(641\) 28.0000 1.10593 0.552967 − 0.833203i \(-0.313496\pi\)
0.552967 + 0.833203i \(0.313496\pi\)
\(642\) 0 0
\(643\) 19.0000 0.749287 0.374643 − 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) −32.0000 −1.25805 −0.629025 − 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 8.00000 0.313786
\(651\) 0 0
\(652\) −40.0000 −1.56652
\(653\) 1.00000 0.0391330 0.0195665 − 0.999809i \(-0.493771\pi\)
0.0195665 + 0.999809i \(0.493771\pi\)
\(654\) 0 0
\(655\) −18.0000 −0.703318
\(656\) 48.0000 1.87409
\(657\) 30.0000 1.17041
\(658\) 0 0
\(659\) −42.0000 −1.63609 −0.818044 − 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 0 0
\(661\) −29.0000 −1.12797 −0.563985 − 0.825785i \(-0.690732\pi\)
−0.563985 + 0.825785i \(0.690732\pi\)
\(662\) 40.0000 1.55464
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −12.0000 −0.464991
\(667\) 18.0000 0.696963
\(668\) −20.0000 −0.773823
\(669\) 0 0
\(670\) 54.0000 2.08620
\(671\) −56.0000 −2.16186
\(672\) 0 0
\(673\) −18.0000 −0.693849 −0.346925 − 0.937893i \(-0.612774\pi\)
−0.346925 + 0.937893i \(0.612774\pi\)
\(674\) −68.0000 −2.61926
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 64.0000 2.45069
\(683\) 9.00000 0.344375 0.172188 − 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) −18.0000 −0.688247
\(685\) 30.0000 1.14624
\(686\) 0 0
\(687\) 0 0
\(688\) 44.0000 1.67748
\(689\) 0 0
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 − 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 28.0000 1.06440
\(693\) 0 0
\(694\) 30.0000 1.13878
\(695\) 21.0000 0.796575
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 36.0000 1.36262
\(699\) 0 0
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 − 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 0 0
\(703\) 6.00000 0.226294
\(704\) −32.0000 −1.20605
\(705\) 0 0
\(706\) 48.0000 1.80650
\(707\) 0 0
\(708\) 0 0
\(709\) 31.0000 1.16423 0.582115 − 0.813107i \(-0.302225\pi\)
0.582115 + 0.813107i \(0.302225\pi\)
\(710\) 0 0
\(711\) −30.0000 −1.12509
\(712\) 0 0
\(713\) 48.0000 1.79761
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 2.00000 0.0747435
\(717\) 0 0
\(718\) 64.0000 2.38846
\(719\) −19.0000 −0.708580 −0.354290 − 0.935136i \(-0.615277\pi\)
−0.354290 + 0.935136i \(0.615277\pi\)
\(720\) −36.0000 −1.34164
\(721\) 0 0
\(722\) −20.0000 −0.744323
\(723\) 0 0
\(724\) −4.00000 −0.148659
\(725\) 12.0000 0.445669
\(726\) 0 0
\(727\) 24.0000 0.890111 0.445055 − 0.895503i \(-0.353184\pi\)
0.445055 + 0.895503i \(0.353184\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 60.0000 2.22070
\(731\) 11.0000 0.406850
\(732\) 0 0
\(733\) 13.0000 0.480166 0.240083 − 0.970752i \(-0.422825\pi\)
0.240083 + 0.970752i \(0.422825\pi\)
\(734\) 22.0000 0.812035
\(735\) 0 0
\(736\) −48.0000 −1.76930
\(737\) −36.0000 −1.32608
\(738\) 72.0000 2.65036
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) 0 0
\(743\) −33.0000 −1.21065 −0.605326 − 0.795977i \(-0.706957\pi\)
−0.605326 + 0.795977i \(0.706957\pi\)
\(744\) 0 0
\(745\) 54.0000 1.97841
\(746\) 74.0000 2.70933
\(747\) 51.0000 1.86599
\(748\) −8.00000 −0.292509
\(749\) 0 0
\(750\) 0 0
\(751\) −30.0000 −1.09472 −0.547358 − 0.836899i \(-0.684366\pi\)
−0.547358 + 0.836899i \(0.684366\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) 6.00000 0.218507
\(755\) −63.0000 −2.29280
\(756\) 0 0
\(757\) 44.0000 1.59921 0.799604 − 0.600528i \(-0.205043\pi\)
0.799604 + 0.600528i \(0.205043\pi\)
\(758\) −36.0000 −1.30758
\(759\) 0 0
\(760\) 0 0
\(761\) 35.0000 1.26875 0.634375 − 0.773026i \(-0.281258\pi\)
0.634375 + 0.773026i \(0.281258\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −26.0000 −0.940647
\(765\) −9.00000 −0.325396
\(766\) −42.0000 −1.51752
\(767\) 5.00000 0.180540
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −12.0000 −0.431889
\(773\) −26.0000 −0.935155 −0.467578 − 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 66.0000 2.37232
\(775\) 32.0000 1.14947
\(776\) 0 0
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) 0 0
\(782\) −12.0000 −0.429119
\(783\) 0 0
\(784\) 0 0
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) −26.0000 −0.926800 −0.463400 − 0.886149i \(-0.653371\pi\)
−0.463400 + 0.886149i \(0.653371\pi\)
\(788\) −4.00000 −0.142494
\(789\) 0 0
\(790\) −60.0000 −2.13470
\(791\) 0 0
\(792\) 0 0
\(793\) −14.0000 −0.497155
\(794\) 26.0000 0.922705
\(795\) 0 0
\(796\) −34.0000 −1.20510
\(797\) −6.00000 −0.212531 −0.106265 − 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 1.00000 0.0353775
\(800\) −32.0000 −1.13137
\(801\) −3.00000 −0.106000
\(802\) 44.0000 1.55369
\(803\) −40.0000 −1.41157
\(804\) 0 0
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) 0 0
\(809\) 41.0000 1.44148 0.720742 − 0.693204i \(-0.243801\pi\)
0.720742 + 0.693204i \(0.243801\pi\)
\(810\) −54.0000 −1.89737
\(811\) 19.0000 0.667180 0.333590 − 0.942718i \(-0.391740\pi\)
0.333590 + 0.942718i \(0.391740\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 16.0000 0.560800
\(815\) 60.0000 2.10171
\(816\) 0 0
\(817\) −33.0000 −1.15452
\(818\) 34.0000 1.18878
\(819\) 0 0
\(820\) 72.0000 2.51435
\(821\) −42.0000 −1.46581 −0.732905 − 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) −7.00000 −0.244005 −0.122002 − 0.992530i \(-0.538932\pi\)
−0.122002 + 0.992530i \(0.538932\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8.00000 0.278187 0.139094 − 0.990279i \(-0.455581\pi\)
0.139094 + 0.990279i \(0.455581\pi\)
\(828\) −36.0000 −1.25109
\(829\) −30.0000 −1.04194 −0.520972 − 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 102.000 3.54047
\(831\) 0 0
\(832\) −8.00000 −0.277350
\(833\) 0 0
\(834\) 0 0
\(835\) 30.0000 1.03819
\(836\) 24.0000 0.830057
\(837\) 0 0
\(838\) −22.0000 −0.759977
\(839\) −30.0000 −1.03572 −0.517858 − 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 0 0
\(844\) −16.0000 −0.550743
\(845\) 36.0000 1.23844
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) −8.00000 −0.274398
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) 8.00000 0.273915 0.136957 − 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) 0 0
\(855\) 27.0000 0.923381
\(856\) 0 0
\(857\) 5.00000 0.170797 0.0853984 − 0.996347i \(-0.472784\pi\)
0.0853984 + 0.996347i \(0.472784\pi\)
\(858\) 0 0
\(859\) −16.0000 −0.545913 −0.272956 − 0.962026i \(-0.588002\pi\)
−0.272956 + 0.962026i \(0.588002\pi\)
\(860\) 66.0000 2.25058
\(861\) 0 0
\(862\) −44.0000 −1.49865
\(863\) 33.0000 1.12333 0.561667 − 0.827364i \(-0.310160\pi\)
0.561667 + 0.827364i \(0.310160\pi\)
\(864\) 0 0
\(865\) −42.0000 −1.42804
\(866\) 42.0000 1.42722
\(867\) 0 0
\(868\) 0 0
\(869\) 40.0000 1.35691
\(870\) 0 0
\(871\) −9.00000 −0.304953
\(872\) 0 0
\(873\) −42.0000 −1.42148
\(874\) 36.0000 1.21772
\(875\) 0 0
\(876\) 0 0
\(877\) 5.00000 0.168838 0.0844190 − 0.996430i \(-0.473097\pi\)
0.0844190 + 0.996430i \(0.473097\pi\)
\(878\) −54.0000 −1.82241
\(879\) 0 0
\(880\) 48.0000 1.61808
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −10.0000 −0.336527 −0.168263 − 0.985742i \(-0.553816\pi\)
−0.168263 + 0.985742i \(0.553816\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) 15.0000 0.503651 0.251825 − 0.967773i \(-0.418969\pi\)
0.251825 + 0.967773i \(0.418969\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −6.00000 −0.201120
\(891\) 36.0000 1.20605
\(892\) 52.0000 1.74109
\(893\) −3.00000 −0.100391
\(894\) 0 0
\(895\) −3.00000 −0.100279
\(896\) 0 0
\(897\) 0 0
\(898\) 16.0000 0.533927
\(899\) 24.0000 0.800445
\(900\) −24.0000 −0.800000
\(901\) 0 0
\(902\) −96.0000 −3.19645
\(903\) 0 0
\(904\) 0 0
\(905\) 6.00000 0.199447
\(906\) 0 0
\(907\) 43.0000 1.42779 0.713896 − 0.700252i \(-0.246929\pi\)
0.713896 + 0.700252i \(0.246929\pi\)
\(908\) −42.0000 −1.39382
\(909\) 27.0000 0.895533
\(910\) 0 0
\(911\) −27.0000 −0.894550 −0.447275 − 0.894397i \(-0.647605\pi\)
−0.447275 + 0.894397i \(0.647605\pi\)
\(912\) 0 0
\(913\) −68.0000 −2.25047
\(914\) −16.0000 −0.529233
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) 0 0
\(918\) 0 0
\(919\) −14.0000 −0.461817 −0.230909 − 0.972975i \(-0.574170\pi\)
−0.230909 + 0.972975i \(0.574170\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −84.0000 −2.76639
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) −32.0000 −1.05159
\(927\) −18.0000 −0.591198
\(928\) −24.0000 −0.787839
\(929\) −4.00000 −0.131236 −0.0656179 − 0.997845i \(-0.520902\pi\)
−0.0656179 + 0.997845i \(0.520902\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 48.0000 1.57229
\(933\) 0 0
\(934\) −32.0000 −1.04707
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) −53.0000 −1.73143 −0.865717 − 0.500533i \(-0.833137\pi\)
−0.865717 + 0.500533i \(0.833137\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 6.00000 0.195698
\(941\) 46.0000 1.49956 0.749779 − 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 0 0
\(943\) −72.0000 −2.34464
\(944\) −20.0000 −0.650945
\(945\) 0 0
\(946\) −88.0000 −2.86113
\(947\) 7.00000 0.227469 0.113735 − 0.993511i \(-0.463719\pi\)
0.113735 + 0.993511i \(0.463719\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) 24.0000 0.778663
\(951\) 0 0
\(952\) 0 0
\(953\) −52.0000 −1.68445 −0.842223 − 0.539130i \(-0.818753\pi\)
−0.842223 + 0.539130i \(0.818753\pi\)
\(954\) 0 0
\(955\) 39.0000 1.26201
\(956\) 0 0
\(957\) 0 0
\(958\) −66.0000 −2.13236
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 4.00000 0.128965
\(963\) 12.0000 0.386695
\(964\) −8.00000 −0.257663
\(965\) 18.0000 0.579441
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 − 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −84.0000 −2.69708
\(971\) −20.0000 −0.641831 −0.320915 − 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 54.0000 1.73027
\(975\) 0 0
\(976\) 56.0000 1.79252
\(977\) 54.0000 1.72761 0.863807 − 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) 4.00000 0.127841
\(980\) 0 0
\(981\) 42.0000 1.34096
\(982\) −12.0000 −0.382935
\(983\) 5.00000 0.159475 0.0797376 − 0.996816i \(-0.474592\pi\)
0.0797376 + 0.996816i \(0.474592\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) −6.00000 −0.191079
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) −66.0000 −2.09868
\(990\) 72.0000 2.28831
\(991\) −20.0000 −0.635321 −0.317660 − 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) −64.0000 −2.03200
\(993\) 0 0
\(994\) 0 0
\(995\) 51.0000 1.61681
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 − 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −56.0000 −1.77265
\(999\) 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 8771.2.a.b.1.1 1
7.6 odd 2 179.2.a.a.1.1 ✓ 1
21.20 even 2 1611.2.a.a.1.1 1
28.27 even 2 2864.2.a.b.1.1 1
35.34 odd 2 4475.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
179.2.a.a.1.1 ✓ 1 7.6 odd 2
1611.2.a.a.1.1 1 21.20 even 2
2864.2.a.b.1.1 1 28.27 even 2
4475.2.a.a.1.1 1 35.34 odd 2
8771.2.a.b.1.1 1 1.1 even 1 trivial