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Properties

Label 5408.2.a.k.1.1
Level $5408$
Weight $2$
Character 5408.1
Self dual yes
Analytic conductor $43.183$
Analytic rank $0$
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] = [5408,2,Mod(1,5408)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5408, 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("5408.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5408 = 2^{5} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5408.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: \(43.1830974131\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 416)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5408.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +4.00000 q^{11} +2.00000 q^{15} +3.00000 q^{17} -2.00000 q^{19} +2.00000 q^{23} -4.00000 q^{25} -4.00000 q^{27} +5.00000 q^{29} +2.00000 q^{31} +8.00000 q^{33} +5.00000 q^{37} +3.00000 q^{41} -4.00000 q^{43} +1.00000 q^{45} +6.00000 q^{47} -7.00000 q^{49} +6.00000 q^{51} +13.0000 q^{53} +4.00000 q^{55} -4.00000 q^{57} +12.0000 q^{59} -7.00000 q^{61} -14.0000 q^{67} +4.00000 q^{69} +6.00000 q^{71} +7.00000 q^{73} -8.00000 q^{75} +8.00000 q^{79} -11.0000 q^{81} +4.00000 q^{83} +3.00000 q^{85} +10.0000 q^{87} +14.0000 q^{89} +4.00000 q^{93} -2.00000 q^{95} -2.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 8.00000 1.39262
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 − 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 13.0000 1.78569 0.892844 − 0.450367i \(-0.148707\pi\)
0.892844 + 0.450367i \(0.148707\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −14.0000 −1.71037 −0.855186 − 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 − 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) −8.00000 −0.923760
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) 0 0
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −11.0000 −1.09454 −0.547270 − 0.836956i \(-0.684333\pi\)
−0.547270 + 0.836956i \(0.684333\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 − 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) −9.00000 −0.846649 −0.423324 − 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −22.0000 −1.95218 −0.976092 − 0.217357i \(-0.930256\pi\)
−0.976092 + 0.217357i \(0.930256\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 − 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) −1.00000 −0.0854358 −0.0427179 − 0.999087i \(-0.513602\pi\)
−0.0427179 + 0.999087i \(0.513602\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 − 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 5.00000 0.415227
\(146\) 0 0
\(147\) −14.0000 −1.15470
\(148\) 0 0
\(149\) 21.0000 1.72039 0.860194 − 0.509968i \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) 0 0
\(151\) 22.0000 1.79033 0.895167 − 0.445730i \(-0.147056\pi\)
0.895167 + 0.445730i \(0.147056\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 5.00000 0.399043 0.199522 − 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) 0 0
\(159\) 26.0000 2.06193
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 0 0
\(165\) 8.00000 0.622799
\(166\) 0 0
\(167\) −4.00000 −0.309529 −0.154765 − 0.987951i \(-0.549462\pi\)
−0.154765 + 0.987951i \(0.549462\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) −2.00000 −0.152057 −0.0760286 − 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 24.0000 1.80395
\(178\) 0 0
\(179\) 16.0000 1.19590 0.597948 − 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 − 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) 0 0
\(185\) 5.00000 0.367607
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 − 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) −21.0000 −1.51161 −0.755807 − 0.654795i \(-0.772755\pi\)
−0.755807 + 0.654795i \(0.772755\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 − 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) −28.0000 −1.97497
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3.00000 0.209529
\(206\) 0 0
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) 18.0000 1.23917 0.619586 − 0.784929i \(-0.287301\pi\)
0.619586 + 0.784929i \(0.287301\pi\)
\(212\) 0 0
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 18.0000 1.20537 0.602685 − 0.797980i \(-0.294098\pi\)
0.602685 + 0.797980i \(0.294098\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 − 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) −18.0000 −1.18947 −0.594737 − 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 0 0
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 7.00000 0.450910 0.225455 − 0.974254i \(-0.427613\pi\)
0.225455 + 0.974254i \(0.427613\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) −7.00000 −0.447214
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) −22.0000 −1.38863 −0.694314 − 0.719672i \(-0.744292\pi\)
−0.694314 + 0.719672i \(0.744292\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) 6.00000 0.375735
\(256\) 0 0
\(257\) 11.0000 0.686161 0.343081 − 0.939306i \(-0.388530\pi\)
0.343081 + 0.939306i \(0.388530\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 13.0000 0.798584
\(266\) 0 0
\(267\) 28.0000 1.71357
\(268\) 0 0
\(269\) 14.0000 0.853595 0.426798 − 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 − 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −16.0000 −0.964836
\(276\) 0 0
\(277\) 1.00000 0.0600842 0.0300421 − 0.999549i \(-0.490436\pi\)
0.0300421 + 0.999549i \(0.490436\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −21.0000 −1.25275 −0.626377 − 0.779520i \(-0.715463\pi\)
−0.626377 + 0.779520i \(0.715463\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) 0 0
\(293\) −19.0000 −1.10999 −0.554996 − 0.831853i \(-0.687280\pi\)
−0.554996 + 0.831853i \(0.687280\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) −16.0000 −0.928414
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −22.0000 −1.26387
\(304\) 0 0
\(305\) −7.00000 −0.400819
\(306\) 0 0
\(307\) 26.0000 1.48390 0.741949 − 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) 0 0
\(309\) 28.0000 1.59286
\(310\) 0 0
\(311\) 10.0000 0.567048 0.283524 − 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 0 0
\(313\) −34.0000 −1.92179 −0.960897 − 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.00000 −0.168497 −0.0842484 − 0.996445i \(-0.526849\pi\)
−0.0842484 + 0.996445i \(0.526849\pi\)
\(318\) 0 0
\(319\) 20.0000 1.11979
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −20.0000 −1.10600
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 − 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 5.00000 0.273998
\(334\) 0 0
\(335\) −14.0000 −0.764902
\(336\) 0 0
\(337\) −1.00000 −0.0544735 −0.0272367 − 0.999629i \(-0.508671\pi\)
−0.0272367 + 0.999629i \(0.508671\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) 10.0000 0.536828 0.268414 − 0.963304i \(-0.413500\pi\)
0.268414 + 0.963304i \(0.413500\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.00000 0.159674 0.0798369 − 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.0000 1.05556 0.527780 − 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 7.00000 0.366397
\(366\) 0 0
\(367\) −22.0000 −1.14839 −0.574195 − 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 0 0
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3.00000 −0.155334 −0.0776671 − 0.996979i \(-0.524747\pi\)
−0.0776671 + 0.996979i \(0.524747\pi\)
\(374\) 0 0
\(375\) −18.0000 −0.929516
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 − 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) −44.0000 −2.25419
\(382\) 0 0
\(383\) 4.00000 0.204390 0.102195 − 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 33.0000 1.67317 0.836583 − 0.547840i \(-0.184550\pi\)
0.836583 + 0.547840i \(0.184550\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 0 0
\(393\) −36.0000 −1.81596
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 − 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −33.0000 −1.64794 −0.823971 − 0.566632i \(-0.808246\pi\)
−0.823971 + 0.566632i \(0.808246\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) −9.00000 −0.445021 −0.222511 − 0.974930i \(-0.571425\pi\)
−0.222511 + 0.974930i \(0.571425\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) −32.0000 −1.56705
\(418\) 0 0
\(419\) 14.0000 0.683945 0.341972 − 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) 9.00000 0.438633 0.219317 − 0.975654i \(-0.429617\pi\)
0.219317 + 0.975654i \(0.429617\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 − 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −1.00000 −0.0480569 −0.0240285 − 0.999711i \(-0.507649\pi\)
−0.0240285 + 0.999711i \(0.507649\pi\)
\(434\) 0 0
\(435\) 10.0000 0.479463
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 − 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 14.0000 0.663664
\(446\) 0 0
\(447\) 42.0000 1.98653
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 − 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 44.0000 2.06730
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −25.0000 −1.16945 −0.584725 − 0.811231i \(-0.698798\pi\)
−0.584725 + 0.811231i \(0.698798\pi\)
\(458\) 0 0
\(459\) −12.0000 −0.560112
\(460\) 0 0
\(461\) 21.0000 0.978068 0.489034 − 0.872265i \(-0.337349\pi\)
0.489034 + 0.872265i \(0.337349\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 − 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) 24.0000 1.11059 0.555294 − 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) 13.0000 0.595229
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 − 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 − 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 0 0
\(489\) 32.0000 1.44709
\(490\) 0 0
\(491\) −16.0000 −0.722070 −0.361035 − 0.932552i \(-0.617576\pi\)
−0.361035 + 0.932552i \(0.617576\pi\)
\(492\) 0 0
\(493\) 15.0000 0.675566
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −14.0000 −0.626726 −0.313363 − 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) 0 0
\(503\) −28.0000 −1.24846 −0.624229 − 0.781241i \(-0.714587\pi\)
−0.624229 + 0.781241i \(0.714587\pi\)
\(504\) 0 0
\(505\) −11.0000 −0.489494
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.00000 −0.132973 −0.0664863 − 0.997787i \(-0.521179\pi\)
−0.0664863 + 0.997787i \(0.521179\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 8.00000 0.353209
\(514\) 0 0
\(515\) 14.0000 0.616914
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) −25.0000 −1.09527 −0.547635 − 0.836717i \(-0.684472\pi\)
−0.547635 + 0.836717i \(0.684472\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000 0.261364
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) 32.0000 1.38090
\(538\) 0 0
\(539\) −28.0000 −1.20605
\(540\) 0 0
\(541\) −23.0000 −0.988847 −0.494424 − 0.869221i \(-0.664621\pi\)
−0.494424 + 0.869221i \(0.664621\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 − 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) 0 0
\(549\) −7.00000 −0.298753
\(550\) 0 0
\(551\) −10.0000 −0.426014
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 10.0000 0.424476
\(556\) 0 0
\(557\) −23.0000 −0.974541 −0.487271 − 0.873251i \(-0.662007\pi\)
−0.487271 + 0.873251i \(0.662007\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) −9.00000 −0.378633
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 − 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 − 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) −29.0000 −1.20729 −0.603643 − 0.797255i \(-0.706285\pi\)
−0.603643 + 0.797255i \(0.706285\pi\)
\(578\) 0 0
\(579\) −42.0000 −1.74546
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 52.0000 2.15362
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.0000 0.660391 0.330195 − 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) 0 0
\(593\) −1.00000 −0.0410651 −0.0205325 − 0.999789i \(-0.506536\pi\)
−0.0205325 + 0.999789i \(0.506536\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −28.0000 −1.14596
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 − 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −9.00000 −0.367118 −0.183559 − 0.983009i \(-0.558762\pi\)
−0.183559 + 0.983009i \(0.558762\pi\)
\(602\) 0 0
\(603\) −14.0000 −0.570124
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) −34.0000 −1.38002 −0.690009 − 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.00000 0.0403896 0.0201948 − 0.999796i \(-0.493571\pi\)
0.0201948 + 0.999796i \(0.493571\pi\)
\(614\) 0 0
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) −37.0000 −1.48956 −0.744782 − 0.667308i \(-0.767447\pi\)
−0.744782 + 0.667308i \(0.767447\pi\)
\(618\) 0 0
\(619\) −36.0000 −1.44696 −0.723481 − 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −16.0000 −0.638978
\(628\) 0 0
\(629\) 15.0000 0.598089
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 − 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 0 0
\(633\) 36.0000 1.43087
\(634\) 0 0
\(635\) −22.0000 −0.873043
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) −17.0000 −0.671460 −0.335730 − 0.941958i \(-0.608983\pi\)
−0.335730 + 0.941958i \(0.608983\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 − 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) 38.0000 1.49393 0.746967 − 0.664861i \(-0.231509\pi\)
0.746967 + 0.664861i \(0.231509\pi\)
\(648\) 0 0
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 14.0000 0.547862 0.273931 − 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) −18.0000 −0.703318
\(656\) 0 0
\(657\) 7.00000 0.273096
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 − 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) 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
\(667\) 10.0000 0.387202
\(668\) 0 0
\(669\) 36.0000 1.39184
\(670\) 0 0
\(671\) −28.0000 −1.08093
\(672\) 0 0
\(673\) −5.00000 −0.192736 −0.0963679 − 0.995346i \(-0.530723\pi\)
−0.0963679 + 0.995346i \(0.530723\pi\)
\(674\) 0 0
\(675\) 16.0000 0.615840
\(676\) 0 0
\(677\) 26.0000 0.999261 0.499631 − 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 36.0000 1.37952
\(682\) 0 0
\(683\) −38.0000 −1.45403 −0.727015 − 0.686622i \(-0.759093\pi\)
−0.727015 + 0.686622i \(0.759093\pi\)
\(684\) 0 0
\(685\) −1.00000 −0.0382080
\(686\) 0 0
\(687\) −36.0000 −1.37349
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.0000 −0.606915
\(696\) 0 0
\(697\) 9.00000 0.340899
\(698\) 0 0
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 − 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) −10.0000 −0.377157
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 21.0000 0.788672 0.394336 − 0.918966i \(-0.370975\pi\)
0.394336 + 0.918966i \(0.370975\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) 0 0
\(719\) −48.0000 −1.79010 −0.895049 − 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 14.0000 0.520666
\(724\) 0 0
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 − 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 1.00000 0.0369358 0.0184679 − 0.999829i \(-0.494121\pi\)
0.0184679 + 0.999829i \(0.494121\pi\)
\(734\) 0 0
\(735\) −14.0000 −0.516398
\(736\) 0 0
\(737\) −56.0000 −2.06279
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −48.0000 −1.76095 −0.880475 − 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) 21.0000 0.769380
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 − 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 0 0
\(753\) −44.0000 −1.60345
\(754\) 0 0
\(755\) 22.0000 0.800662
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 − 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) 16.0000 0.580763
\(760\) 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
\(765\) 3.00000 0.108465
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 38.0000 1.37032 0.685158 − 0.728395i \(-0.259733\pi\)
0.685158 + 0.728395i \(0.259733\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) 0 0
\(773\) −6.00000 −0.215805 −0.107903 − 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 0 0
\(783\) −20.0000 −0.714742
\(784\) 0 0
\(785\) 5.00000 0.178458
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 − 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 0 0
\(789\) 48.0000 1.70885
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 26.0000 0.922125
\(796\) 0 0
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) 0 0
\(803\) 28.0000 0.988099
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 28.0000 0.985647
\(808\) 0 0
\(809\) −25.0000 −0.878953 −0.439477 − 0.898254i \(-0.644836\pi\)
−0.439477 + 0.898254i \(0.644836\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 − 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 24.0000 0.841717
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 8.00000 0.279885
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 − 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 52.0000 1.81261 0.906303 − 0.422628i \(-0.138892\pi\)
0.906303 + 0.422628i \(0.138892\pi\)
\(824\) 0 0
\(825\) −32.0000 −1.11410
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −31.0000 −1.07667 −0.538337 − 0.842729i \(-0.680947\pi\)
−0.538337 + 0.842729i \(0.680947\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) −21.0000 −0.727607
\(834\) 0 0
\(835\) −4.00000 −0.138426
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 0 0
\(839\) −26.0000 −0.897620 −0.448810 − 0.893627i \(-0.648152\pi\)
−0.448810 + 0.893627i \(0.648152\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) −42.0000 −1.44656
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −40.0000 −1.37280
\(850\) 0 0
\(851\) 10.0000 0.342796
\(852\) 0 0
\(853\) −39.0000 −1.33533 −0.667667 − 0.744460i \(-0.732707\pi\)
−0.667667 + 0.744460i \(0.732707\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) 0 0
\(857\) −21.0000 −0.717346 −0.358673 − 0.933463i \(-0.616771\pi\)
−0.358673 + 0.933463i \(0.616771\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 − 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 0 0
\(861\) 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
\(865\) −2.00000 −0.0680020
\(866\) 0 0
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 41.0000 1.38447 0.692236 − 0.721671i \(-0.256626\pi\)
0.692236 + 0.721671i \(0.256626\pi\)
\(878\) 0 0
\(879\) −38.0000 −1.28171
\(880\) 0 0
\(881\) −29.0000 −0.977035 −0.488517 − 0.872554i \(-0.662462\pi\)
−0.488517 + 0.872554i \(0.662462\pi\)
\(882\) 0 0
\(883\) −58.0000 −1.95186 −0.975928 − 0.218094i \(-0.930016\pi\)
−0.975928 + 0.218094i \(0.930016\pi\)
\(884\) 0 0
\(885\) 24.0000 0.806751
\(886\) 0 0
\(887\) −56.0000 −1.88030 −0.940148 − 0.340766i \(-0.889313\pi\)
−0.940148 + 0.340766i \(0.889313\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −44.0000 −1.47406
\(892\) 0 0
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) 16.0000 0.534821
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 10.0000 0.333519
\(900\) 0 0
\(901\) 39.0000 1.29928
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.00000 0.166206
\(906\) 0 0
\(907\) −16.0000 −0.531271 −0.265636 − 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 0 0
\(909\) −11.0000 −0.364847
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 − 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 0 0
\(915\) −14.0000 −0.462826
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −30.0000 −0.989609 −0.494804 − 0.869004i \(-0.664760\pi\)
−0.494804 + 0.869004i \(0.664760\pi\)
\(920\) 0 0
\(921\) 52.0000 1.71346
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −20.0000 −0.657596
\(926\) 0 0
\(927\) 14.0000 0.459820
\(928\) 0 0
\(929\) 3.00000 0.0984268 0.0492134 − 0.998788i \(-0.484329\pi\)
0.0492134 + 0.998788i \(0.484329\pi\)
\(930\) 0 0
\(931\) 14.0000 0.458831
\(932\) 0 0
\(933\) 20.0000 0.654771
\(934\) 0 0
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) 55.0000 1.79677 0.898386 − 0.439207i \(-0.144741\pi\)
0.898386 + 0.439207i \(0.144741\pi\)
\(938\) 0 0
\(939\) −68.0000 −2.21910
\(940\) 0 0
\(941\) −46.0000 −1.49956 −0.749779 − 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −38.0000 −1.23483 −0.617417 − 0.786636i \(-0.711821\pi\)
−0.617417 + 0.786636i \(0.711821\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 − 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 0 0
\(955\) 6.00000 0.194155
\(956\) 0 0
\(957\) 40.0000 1.29302
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 0 0
\(965\) −21.0000 −0.676014
\(966\) 0 0
\(967\) −38.0000 −1.22200 −0.610999 − 0.791632i \(-0.709232\pi\)
−0.610999 + 0.791632i \(0.709232\pi\)
\(968\) 0 0
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 − 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −17.0000 −0.543878 −0.271939 − 0.962314i \(-0.587665\pi\)
−0.271939 + 0.962314i \(0.587665\pi\)
\(978\) 0 0
\(979\) 56.0000 1.78977
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 0 0
\(983\) 18.0000 0.574111 0.287055 − 0.957914i \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −2.00000 −0.0635321 −0.0317660 − 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 0 0
\(993\) −16.0000 −0.507745
\(994\) 0 0
\(995\) −14.0000 −0.443830
\(996\) 0 0
\(997\) −55.0000 −1.74187 −0.870934 − 0.491400i \(-0.836485\pi\)
−0.870934 + 0.491400i \(0.836485\pi\)
\(998\) 0 0
\(999\) −20.0000 −0.632772
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 5408.2.a.k.1.1 1
4.3 odd 2 5408.2.a.d.1.1 1
13.3 even 3 416.2.i.a.321.1 yes 2
13.9 even 3 416.2.i.a.289.1 ✓ 2
13.12 even 2 5408.2.a.j.1.1 1
52.3 odd 6 416.2.i.b.321.1 yes 2
52.35 odd 6 416.2.i.b.289.1 yes 2
52.51 odd 2 5408.2.a.c.1.1 1
104.3 odd 6 832.2.i.b.321.1 2
104.29 even 6 832.2.i.h.321.1 2
104.35 odd 6 832.2.i.b.705.1 2
104.61 even 6 832.2.i.h.705.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
416.2.i.a.289.1 ✓ 2 13.9 even 3
416.2.i.a.321.1 yes 2 13.3 even 3
416.2.i.b.289.1 yes 2 52.35 odd 6
416.2.i.b.321.1 yes 2 52.3 odd 6
832.2.i.b.321.1 2 104.3 odd 6
832.2.i.b.705.1 2 104.35 odd 6
832.2.i.h.321.1 2 104.29 even 6
832.2.i.h.705.1 2 104.61 even 6
5408.2.a.c.1.1 1 52.51 odd 2
5408.2.a.d.1.1 1 4.3 odd 2
5408.2.a.j.1.1 1 13.12 even 2
5408.2.a.k.1.1 1 1.1 even 1 trivial