Properties

Label 5776.2.a.o.1.1
Level $5776$
Weight $2$
Character 5776.1
Self dual yes
Analytic conductor $46.122$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5776,2,Mod(1,5776)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5776, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5776.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5776 = 2^{4} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5776.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: \(46.1215922075\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5776.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.00000 q^{5} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.00000 q^{5} -2.00000 q^{9} +4.00000 q^{11} +5.00000 q^{13} +3.00000 q^{15} -5.00000 q^{17} +1.00000 q^{23} +4.00000 q^{25} -5.00000 q^{27} -3.00000 q^{29} +4.00000 q^{31} +4.00000 q^{33} -2.00000 q^{37} +5.00000 q^{39} +5.00000 q^{41} +11.0000 q^{43} -6.00000 q^{45} +5.00000 q^{47} -7.00000 q^{49} -5.00000 q^{51} +9.00000 q^{53} +12.0000 q^{55} +13.0000 q^{59} -1.00000 q^{61} +15.0000 q^{65} -5.00000 q^{67} +1.00000 q^{69} +1.00000 q^{71} -9.00000 q^{73} +4.00000 q^{75} +17.0000 q^{79} +1.00000 q^{81} -16.0000 q^{83} -15.0000 q^{85} -3.00000 q^{87} -3.00000 q^{89} +4.00000 q^{93} +13.0000 q^{97} -8.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 5.00000 0.800641
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 11.0000 1.67748 0.838742 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) 0 0
\(45\) −6.00000 −0.894427
\(46\) 0 0
\(47\) 5.00000 0.729325 0.364662 0.931140i \(-0.381184\pi\)
0.364662 + 0.931140i \(0.381184\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 13.0000 1.69246 0.846228 0.532821i \(-0.178868\pi\)
0.846228 + 0.532821i \(0.178868\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 15.0000 1.86052
\(66\) 0 0
\(67\) −5.00000 −0.610847 −0.305424 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 1.00000 0.118678 0.0593391 0.998238i \(-0.481101\pi\)
0.0593391 + 0.998238i \(0.481101\pi\)
\(72\) 0 0
\(73\) −9.00000 −1.05337 −0.526685 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 17.0000 1.91265 0.956325 0.292306i \(-0.0944227\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) −15.0000 −1.62698
\(86\) 0 0
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.0000 1.31995 0.659975 0.751288i \(-0.270567\pi\)
0.659975 + 0.751288i \(0.270567\pi\)
\(98\) 0 0
\(99\) −8.00000 −0.804030
\(100\) 0 0
\(101\) 19.0000 1.89057 0.945285 0.326245i \(-0.105783\pi\)
0.945285 + 0.326245i \(0.105783\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) −10.0000 −0.924500
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 5.00000 0.450835
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 15.0000 1.33103 0.665517 0.746382i \(-0.268211\pi\)
0.665517 + 0.746382i \(0.268211\pi\)
\(128\) 0 0
\(129\) 11.0000 0.968496
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −15.0000 −1.29099
\(136\) 0 0
\(137\) −5.00000 −0.427179 −0.213589 0.976924i \(-0.568515\pi\)
−0.213589 + 0.976924i \(0.568515\pi\)
\(138\) 0 0
\(139\) −15.0000 −1.27228 −0.636142 0.771572i \(-0.719471\pi\)
−0.636142 + 0.771572i \(0.719471\pi\)
\(140\) 0 0
\(141\) 5.00000 0.421076
\(142\) 0 0
\(143\) 20.0000 1.67248
\(144\) 0 0
\(145\) −9.00000 −0.747409
\(146\) 0 0
\(147\) −7.00000 −0.577350
\(148\) 0 0
\(149\) −17.0000 −1.39269 −0.696347 0.717705i \(-0.745193\pi\)
−0.696347 + 0.717705i \(0.745193\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 10.0000 0.808452
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 12.0000 0.934199
\(166\) 0 0
\(167\) −5.00000 −0.386912 −0.193456 0.981109i \(-0.561970\pi\)
−0.193456 + 0.981109i \(0.561970\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 5.00000 0.380143 0.190071 0.981770i \(-0.439128\pi\)
0.190071 + 0.981770i \(0.439128\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 13.0000 0.977140
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) −20.0000 −1.46254
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −15.0000 −1.07972 −0.539862 0.841754i \(-0.681524\pi\)
−0.539862 + 0.841754i \(0.681524\pi\)
\(194\) 0 0
\(195\) 15.0000 1.07417
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) −3.00000 −0.212664 −0.106332 0.994331i \(-0.533911\pi\)
−0.106332 + 0.994331i \(0.533911\pi\)
\(200\) 0 0
\(201\) −5.00000 −0.352673
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 15.0000 1.04765
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 9.00000 0.619586 0.309793 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(212\) 0 0
\(213\) 1.00000 0.0685189
\(214\) 0 0
\(215\) 33.0000 2.25058
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −9.00000 −0.608164
\(220\) 0 0
\(221\) −25.0000 −1.68168
\(222\) 0 0
\(223\) −11.0000 −0.736614 −0.368307 0.929704i \(-0.620063\pi\)
−0.368307 + 0.929704i \(0.620063\pi\)
\(224\) 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
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 11.0000 0.720634 0.360317 0.932830i \(-0.382669\pi\)
0.360317 + 0.932830i \(0.382669\pi\)
\(234\) 0 0
\(235\) 15.0000 0.978492
\(236\) 0 0
\(237\) 17.0000 1.10427
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 21.0000 1.35273 0.676364 0.736567i \(-0.263554\pi\)
0.676364 + 0.736567i \(0.263554\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) −21.0000 −1.34164
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) 9.00000 0.568075 0.284037 0.958813i \(-0.408326\pi\)
0.284037 + 0.958813i \(0.408326\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) −15.0000 −0.939336
\(256\) 0 0
\(257\) 25.0000 1.55946 0.779729 0.626118i \(-0.215357\pi\)
0.779729 + 0.626118i \(0.215357\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 19.0000 1.17159 0.585795 0.810459i \(-0.300782\pi\)
0.585795 + 0.810459i \(0.300782\pi\)
\(264\) 0 0
\(265\) 27.0000 1.65860
\(266\) 0 0
\(267\) −3.00000 −0.183597
\(268\) 0 0
\(269\) 17.0000 1.03651 0.518254 0.855227i \(-0.326582\pi\)
0.518254 + 0.855227i \(0.326582\pi\)
\(270\) 0 0
\(271\) −29.0000 −1.76162 −0.880812 0.473466i \(-0.843003\pi\)
−0.880812 + 0.473466i \(0.843003\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.0000 0.964836
\(276\) 0 0
\(277\) −30.0000 −1.80253 −0.901263 0.433273i \(-0.857359\pi\)
−0.901263 + 0.433273i \(0.857359\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 0 0
\(283\) 15.0000 0.891657 0.445829 0.895118i \(-0.352909\pi\)
0.445829 + 0.895118i \(0.352909\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 13.0000 0.762073
\(292\) 0 0
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 0 0
\(295\) 39.0000 2.27067
\(296\) 0 0
\(297\) −20.0000 −1.16052
\(298\) 0 0
\(299\) 5.00000 0.289157
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 19.0000 1.09152
\(304\) 0 0
\(305\) −3.00000 −0.171780
\(306\) 0 0
\(307\) −3.00000 −0.171219 −0.0856095 0.996329i \(-0.527284\pi\)
−0.0856095 + 0.996329i \(0.527284\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −5.00000 −0.282617 −0.141308 0.989966i \(-0.545131\pi\)
−0.141308 + 0.989966i \(0.545131\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −27.0000 −1.51647 −0.758236 0.651981i \(-0.773938\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 20.0000 1.10940
\(326\) 0 0
\(327\) −7.00000 −0.387101
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) −15.0000 −0.819538
\(336\) 0 0
\(337\) −3.00000 −0.163420 −0.0817102 0.996656i \(-0.526038\pi\)
−0.0817102 + 0.996656i \(0.526038\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 3.00000 0.161515
\(346\) 0 0
\(347\) −5.00000 −0.268414 −0.134207 0.990953i \(-0.542849\pi\)
−0.134207 + 0.990953i \(0.542849\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −25.0000 −1.33440
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 3.00000 0.159223
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5.00000 −0.263890 −0.131945 0.991257i \(-0.542122\pi\)
−0.131945 + 0.991257i \(0.542122\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −27.0000 −1.41324
\(366\) 0 0
\(367\) 5.00000 0.260998 0.130499 0.991448i \(-0.458342\pi\)
0.130499 + 0.991448i \(0.458342\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) −15.0000 −0.772539
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) 15.0000 0.768473
\(382\) 0 0
\(383\) −15.0000 −0.766464 −0.383232 0.923652i \(-0.625189\pi\)
−0.383232 + 0.923652i \(0.625189\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −22.0000 −1.11832
\(388\) 0 0
\(389\) −17.0000 −0.861934 −0.430967 0.902368i \(-0.641828\pi\)
−0.430967 + 0.902368i \(0.641828\pi\)
\(390\) 0 0
\(391\) −5.00000 −0.252861
\(392\) 0 0
\(393\) 15.0000 0.756650
\(394\) 0 0
\(395\) 51.0000 2.56609
\(396\) 0 0
\(397\) 35.0000 1.75660 0.878300 0.478110i \(-0.158678\pi\)
0.878300 + 0.478110i \(0.158678\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.0000 −0.549314 −0.274657 0.961542i \(-0.588564\pi\)
−0.274657 + 0.961542i \(0.588564\pi\)
\(402\) 0 0
\(403\) 20.0000 0.996271
\(404\) 0 0
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −7.00000 −0.346128 −0.173064 0.984911i \(-0.555367\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 0 0
\(411\) −5.00000 −0.246632
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −48.0000 −2.35623
\(416\) 0 0
\(417\) −15.0000 −0.734553
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 29.0000 1.41337 0.706687 0.707527i \(-0.250189\pi\)
0.706687 + 0.707527i \(0.250189\pi\)
\(422\) 0 0
\(423\) −10.0000 −0.486217
\(424\) 0 0
\(425\) −20.0000 −0.970143
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 20.0000 0.965609
\(430\) 0 0
\(431\) −9.00000 −0.433515 −0.216757 0.976226i \(-0.569548\pi\)
−0.216757 + 0.976226i \(0.569548\pi\)
\(432\) 0 0
\(433\) 25.0000 1.20142 0.600712 0.799466i \(-0.294884\pi\)
0.600712 + 0.799466i \(0.294884\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −7.00000 −0.334092 −0.167046 0.985949i \(-0.553423\pi\)
−0.167046 + 0.985949i \(0.553423\pi\)
\(440\) 0 0
\(441\) 14.0000 0.666667
\(442\) 0 0
\(443\) −15.0000 −0.712672 −0.356336 0.934358i \(-0.615974\pi\)
−0.356336 + 0.934358i \(0.615974\pi\)
\(444\) 0 0
\(445\) −9.00000 −0.426641
\(446\) 0 0
\(447\) −17.0000 −0.804072
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 20.0000 0.941763
\(452\) 0 0
\(453\) 16.0000 0.751746
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) 25.0000 1.16690
\(460\) 0 0
\(461\) −1.00000 −0.0465746 −0.0232873 0.999729i \(-0.507413\pi\)
−0.0232873 + 0.999729i \(0.507413\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 0 0
\(465\) 12.0000 0.556487
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −13.0000 −0.599008
\(472\) 0 0
\(473\) 44.0000 2.02312
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −18.0000 −0.824163
\(478\) 0 0
\(479\) −27.0000 −1.23366 −0.616831 0.787096i \(-0.711584\pi\)
−0.616831 + 0.787096i \(0.711584\pi\)
\(480\) 0 0
\(481\) −10.0000 −0.455961
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 39.0000 1.77090
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) 15.0000 0.675566
\(494\) 0 0
\(495\) −24.0000 −1.07872
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 35.0000 1.56682 0.783408 0.621508i \(-0.213480\pi\)
0.783408 + 0.621508i \(0.213480\pi\)
\(500\) 0 0
\(501\) −5.00000 −0.223384
\(502\) 0 0
\(503\) −15.0000 −0.668817 −0.334408 0.942428i \(-0.608537\pi\)
−0.334408 + 0.942428i \(0.608537\pi\)
\(504\) 0 0
\(505\) 57.0000 2.53647
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) 5.00000 0.221621 0.110811 0.993842i \(-0.464655\pi\)
0.110811 + 0.993842i \(0.464655\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 20.0000 0.879599
\(518\) 0 0
\(519\) 5.00000 0.219476
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −29.0000 −1.26808 −0.634041 0.773300i \(-0.718605\pi\)
−0.634041 + 0.773300i \(0.718605\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −20.0000 −0.871214
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) −26.0000 −1.12830
\(532\) 0 0
\(533\) 25.0000 1.08287
\(534\) 0 0
\(535\) −36.0000 −1.55642
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) −28.0000 −1.20605
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 0 0
\(543\) −7.00000 −0.300399
\(544\) 0 0
\(545\) −21.0000 −0.899541
\(546\) 0 0
\(547\) 7.00000 0.299298 0.149649 0.988739i \(-0.452186\pi\)
0.149649 + 0.988739i \(0.452186\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −6.00000 −0.254686
\(556\) 0 0
\(557\) −33.0000 −1.39825 −0.699127 0.714997i \(-0.746428\pi\)
−0.699127 + 0.714997i \(0.746428\pi\)
\(558\) 0 0
\(559\) 55.0000 2.32625
\(560\) 0 0
\(561\) −20.0000 −0.844401
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) −15.0000 −0.623379
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) 0 0
\(585\) −30.0000 −1.24035
\(586\) 0 0
\(587\) −25.0000 −1.03186 −0.515930 0.856631i \(-0.672554\pi\)
−0.515930 + 0.856631i \(0.672554\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 0 0
\(593\) 35.0000 1.43728 0.718639 0.695383i \(-0.244765\pi\)
0.718639 + 0.695383i \(0.244765\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.00000 −0.122782
\(598\) 0 0
\(599\) −25.0000 −1.02147 −0.510736 0.859738i \(-0.670627\pi\)
−0.510736 + 0.859738i \(0.670627\pi\)
\(600\) 0 0
\(601\) 34.0000 1.38689 0.693444 0.720510i \(-0.256092\pi\)
0.693444 + 0.720510i \(0.256092\pi\)
\(602\) 0 0
\(603\) 10.0000 0.407231
\(604\) 0 0
\(605\) 15.0000 0.609837
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 25.0000 1.01139
\(612\) 0 0
\(613\) −1.00000 −0.0403896 −0.0201948 0.999796i \(-0.506429\pi\)
−0.0201948 + 0.999796i \(0.506429\pi\)
\(614\) 0 0
\(615\) 15.0000 0.604858
\(616\) 0 0
\(617\) 3.00000 0.120775 0.0603877 0.998175i \(-0.480766\pi\)
0.0603877 + 0.998175i \(0.480766\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10.0000 0.398726
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 0 0
\(633\) 9.00000 0.357718
\(634\) 0 0
\(635\) 45.0000 1.78577
\(636\) 0 0
\(637\) −35.0000 −1.38675
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 9.00000 0.355479 0.177739 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) 0 0
\(643\) −5.00000 −0.197181 −0.0985904 0.995128i \(-0.531433\pi\)
−0.0985904 + 0.995128i \(0.531433\pi\)
\(644\) 0 0
\(645\) 33.0000 1.29937
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 52.0000 2.04118
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 45.0000 1.75830
\(656\) 0 0
\(657\) 18.0000 0.702247
\(658\) 0 0
\(659\) 19.0000 0.740135 0.370067 0.929005i \(-0.379335\pi\)
0.370067 + 0.929005i \(0.379335\pi\)
\(660\) 0 0
\(661\) 29.0000 1.12797 0.563985 0.825785i \(-0.309268\pi\)
0.563985 + 0.825785i \(0.309268\pi\)
\(662\) 0 0
\(663\) −25.0000 −0.970920
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.00000 −0.116160
\(668\) 0 0
\(669\) −11.0000 −0.425285
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 0 0
\(675\) −20.0000 −0.769800
\(676\) 0 0
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −15.0000 −0.573121
\(686\) 0 0
\(687\) −22.0000 −0.839352
\(688\) 0 0
\(689\) 45.0000 1.71436
\(690\) 0 0
\(691\) −36.0000 −1.36950 −0.684752 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −45.0000 −1.70695
\(696\) 0 0
\(697\) −25.0000 −0.946943
\(698\) 0 0
\(699\) 11.0000 0.416058
\(700\) 0 0
\(701\) 23.0000 0.868698 0.434349 0.900745i \(-0.356978\pi\)
0.434349 + 0.900745i \(0.356978\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 15.0000 0.564933
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −17.0000 −0.638448 −0.319224 0.947679i \(-0.603422\pi\)
−0.319224 + 0.947679i \(0.603422\pi\)
\(710\) 0 0
\(711\) −34.0000 −1.27510
\(712\) 0 0
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 60.0000 2.24387
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) 27.0000 1.00693 0.503465 0.864016i \(-0.332058\pi\)
0.503465 + 0.864016i \(0.332058\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 21.0000 0.780998
\(724\) 0 0
\(725\) −12.0000 −0.445669
\(726\) 0 0
\(727\) −13.0000 −0.482143 −0.241072 0.970507i \(-0.577499\pi\)
−0.241072 + 0.970507i \(0.577499\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −55.0000 −2.03425
\(732\) 0 0
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 0 0
\(735\) −21.0000 −0.774597
\(736\) 0 0
\(737\) −20.0000 −0.736709
\(738\) 0 0
\(739\) −37.0000 −1.36107 −0.680534 0.732717i \(-0.738252\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 29.0000 1.06391 0.531953 0.846774i \(-0.321458\pi\)
0.531953 + 0.846774i \(0.321458\pi\)
\(744\) 0 0
\(745\) −51.0000 −1.86850
\(746\) 0 0
\(747\) 32.0000 1.17082
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 31.0000 1.13121 0.565603 0.824678i \(-0.308643\pi\)
0.565603 + 0.824678i \(0.308643\pi\)
\(752\) 0 0
\(753\) 9.00000 0.327978
\(754\) 0 0
\(755\) 48.0000 1.74690
\(756\) 0 0
\(757\) 35.0000 1.27210 0.636048 0.771649i \(-0.280568\pi\)
0.636048 + 0.771649i \(0.280568\pi\)
\(758\) 0 0
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 38.0000 1.37750 0.688749 0.724999i \(-0.258160\pi\)
0.688749 + 0.724999i \(0.258160\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 30.0000 1.08465
\(766\) 0 0
\(767\) 65.0000 2.34701
\(768\) 0 0
\(769\) 23.0000 0.829401 0.414701 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) 25.0000 0.900353
\(772\) 0 0
\(773\) 25.0000 0.899188 0.449594 0.893233i \(-0.351569\pi\)
0.449594 + 0.893233i \(0.351569\pi\)
\(774\) 0 0
\(775\) 16.0000 0.574737
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 15.0000 0.536056
\(784\) 0 0
\(785\) −39.0000 −1.39197
\(786\) 0 0
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) 0 0
\(789\) 19.0000 0.676418
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −5.00000 −0.177555
\(794\) 0 0
\(795\) 27.0000 0.957591
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) −25.0000 −0.884436
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) −36.0000 −1.27041
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 17.0000 0.598428
\(808\) 0 0
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) −21.0000 −0.737410 −0.368705 0.929547i \(-0.620199\pi\)
−0.368705 + 0.929547i \(0.620199\pi\)
\(812\) 0 0
\(813\) −29.0000 −1.01707
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 11.0000 0.383903 0.191951 0.981404i \(-0.438518\pi\)
0.191951 + 0.981404i \(0.438518\pi\)
\(822\) 0 0
\(823\) 5.00000 0.174289 0.0871445 0.996196i \(-0.472226\pi\)
0.0871445 + 0.996196i \(0.472226\pi\)
\(824\) 0 0
\(825\) 16.0000 0.557048
\(826\) 0 0
\(827\) 15.0000 0.521601 0.260801 0.965393i \(-0.416014\pi\)
0.260801 + 0.965393i \(0.416014\pi\)
\(828\) 0 0
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 0 0
\(831\) −30.0000 −1.04069
\(832\) 0 0
\(833\) 35.0000 1.21268
\(834\) 0 0
\(835\) −15.0000 −0.519096
\(836\) 0 0
\(837\) −20.0000 −0.691301
\(838\) 0 0
\(839\) −35.0000 −1.20833 −0.604167 0.796858i \(-0.706494\pi\)
−0.604167 + 0.796858i \(0.706494\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −15.0000 −0.516627
\(844\) 0 0
\(845\) 36.0000 1.23844
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 15.0000 0.514799
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) −21.0000 −0.719026 −0.359513 0.933140i \(-0.617057\pi\)
−0.359513 + 0.933140i \(0.617057\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −7.00000 −0.239115 −0.119558 0.992827i \(-0.538148\pi\)
−0.119558 + 0.992827i \(0.538148\pi\)
\(858\) 0 0
\(859\) 13.0000 0.443554 0.221777 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) 15.0000 0.510015
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 68.0000 2.30674
\(870\) 0 0
\(871\) −25.0000 −0.847093
\(872\) 0 0
\(873\) −26.0000 −0.879967
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −47.0000 −1.58708 −0.793539 0.608520i \(-0.791764\pi\)
−0.793539 + 0.608520i \(0.791764\pi\)
\(878\) 0 0
\(879\) −10.0000 −0.337292
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) −55.0000 −1.85090 −0.925449 0.378873i \(-0.876312\pi\)
−0.925449 + 0.378873i \(0.876312\pi\)
\(884\) 0 0
\(885\) 39.0000 1.31097
\(886\) 0 0
\(887\) −45.0000 −1.51095 −0.755476 0.655176i \(-0.772594\pi\)
−0.755476 + 0.655176i \(0.772594\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −36.0000 −1.20335
\(896\) 0 0
\(897\) 5.00000 0.166945
\(898\) 0 0
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −45.0000 −1.49917
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −21.0000 −0.698064
\(906\) 0 0
\(907\) −35.0000 −1.16216 −0.581078 0.813848i \(-0.697369\pi\)
−0.581078 + 0.813848i \(0.697369\pi\)
\(908\) 0 0
\(909\) −38.0000 −1.26038
\(910\) 0 0
\(911\) −56.0000 −1.85536 −0.927681 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(912\) 0 0
\(913\) −64.0000 −2.11809
\(914\) 0 0
\(915\) −3.00000 −0.0991769
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) −3.00000 −0.0988534
\(922\) 0 0
\(923\) 5.00000 0.164577
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.00000 0.229663 0.114831 0.993385i \(-0.463367\pi\)
0.114831 + 0.993385i \(0.463367\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −12.0000 −0.392862
\(934\) 0 0
\(935\) −60.0000 −1.96221
\(936\) 0 0
\(937\) −17.0000 −0.555366 −0.277683 0.960673i \(-0.589566\pi\)
−0.277683 + 0.960673i \(0.589566\pi\)
\(938\) 0 0
\(939\) −5.00000 −0.163169
\(940\) 0 0
\(941\) −39.0000 −1.27136 −0.635682 0.771951i \(-0.719281\pi\)
−0.635682 + 0.771951i \(0.719281\pi\)
\(942\) 0 0
\(943\) 5.00000 0.162822
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 35.0000 1.13735 0.568674 0.822563i \(-0.307457\pi\)
0.568674 + 0.822563i \(0.307457\pi\)
\(948\) 0 0
\(949\) −45.0000 −1.46076
\(950\) 0 0
\(951\) −27.0000 −0.875535
\(952\) 0 0
\(953\) −15.0000 −0.485898 −0.242949 0.970039i \(-0.578115\pi\)
−0.242949 + 0.970039i \(0.578115\pi\)
\(954\) 0 0
\(955\) 48.0000 1.55324
\(956\) 0 0
\(957\) −12.0000 −0.387905
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 24.0000 0.773389
\(964\) 0 0
\(965\) −45.0000 −1.44860
\(966\) 0 0
\(967\) 23.0000 0.739630 0.369815 0.929105i \(-0.379421\pi\)
0.369815 + 0.929105i \(0.379421\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.00000 −0.0962746 −0.0481373 0.998841i \(-0.515328\pi\)
−0.0481373 + 0.998841i \(0.515328\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 20.0000 0.640513
\(976\) 0 0
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) 25.0000 0.797376 0.398688 0.917087i \(-0.369466\pi\)
0.398688 + 0.917087i \(0.369466\pi\)
\(984\) 0 0
\(985\) 30.0000 0.955879
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 11.0000 0.349780
\(990\) 0 0
\(991\) −55.0000 −1.74713 −0.873566 0.486705i \(-0.838199\pi\)
−0.873566 + 0.486705i \(0.838199\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) −9.00000 −0.285319
\(996\) 0 0
\(997\) −25.0000 −0.791758 −0.395879 0.918303i \(-0.629560\pi\)
−0.395879 + 0.918303i \(0.629560\pi\)
\(998\) 0 0
\(999\) 10.0000 0.316386
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 5776.2.a.o.1.1 1
4.3 odd 2 2888.2.a.c.1.1 1
19.8 odd 6 304.2.i.b.273.1 2
19.12 odd 6 304.2.i.b.49.1 2
19.18 odd 2 5776.2.a.h.1.1 1
57.8 even 6 2736.2.s.q.577.1 2
57.50 even 6 2736.2.s.q.1873.1 2
76.27 even 6 152.2.i.a.121.1 yes 2
76.31 even 6 152.2.i.a.49.1 2
76.75 even 2 2888.2.a.e.1.1 1
152.27 even 6 1216.2.i.i.577.1 2
152.69 odd 6 1216.2.i.e.961.1 2
152.107 even 6 1216.2.i.i.961.1 2
152.141 odd 6 1216.2.i.e.577.1 2
228.107 odd 6 1368.2.s.g.505.1 2
228.179 odd 6 1368.2.s.g.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.i.a.49.1 2 76.31 even 6
152.2.i.a.121.1 yes 2 76.27 even 6
304.2.i.b.49.1 2 19.12 odd 6
304.2.i.b.273.1 2 19.8 odd 6
1216.2.i.e.577.1 2 152.141 odd 6
1216.2.i.e.961.1 2 152.69 odd 6
1216.2.i.i.577.1 2 152.27 even 6
1216.2.i.i.961.1 2 152.107 even 6
1368.2.s.g.505.1 2 228.107 odd 6
1368.2.s.g.577.1 2 228.179 odd 6
2736.2.s.q.577.1 2 57.8 even 6
2736.2.s.q.1873.1 2 57.50 even 6
2888.2.a.c.1.1 1 4.3 odd 2
2888.2.a.e.1.1 1 76.75 even 2
5776.2.a.h.1.1 1 19.18 odd 2
5776.2.a.o.1.1 1 1.1 even 1 trivial