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Properties

Label 2704.2.a.c.1.1
Level $2704$
Weight $2$
Character 2704.1
Self dual yes
Analytic conductor $21.592$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2704,2,Mod(1,2704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2704, 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("2704.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.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: \(21.5915487066\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2704.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +1.00000 q^{11} +2.00000 q^{15} +3.00000 q^{17} +7.00000 q^{19} +1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{25} +5.00000 q^{27} +3.00000 q^{29} +8.00000 q^{31} -1.00000 q^{33} +2.00000 q^{35} +1.00000 q^{37} -11.0000 q^{41} -11.0000 q^{43} +4.00000 q^{45} +12.0000 q^{47} -6.00000 q^{49} -3.00000 q^{51} -6.00000 q^{53} -2.00000 q^{55} -7.00000 q^{57} -9.00000 q^{59} -9.00000 q^{61} +2.00000 q^{63} -3.00000 q^{67} +1.00000 q^{69} -5.00000 q^{71} +2.00000 q^{73} +1.00000 q^{75} -1.00000 q^{77} +12.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} -6.00000 q^{85} -3.00000 q^{87} +1.00000 q^{89} -8.00000 q^{93} -14.0000 q^{95} +1.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 − 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 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
\(19\) 7.00000 1.60591 0.802955 − 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 − 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 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 \(0.244865\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 − 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.0000 −1.71791 −0.858956 − 0.512050i \(-0.828886\pi\)
−0.858956 + 0.512050i \(0.828886\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 − 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 0 0
\(45\) 4.00000 0.596285
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 − 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) −9.00000 −1.15233 −0.576166 − 0.817333i \(-0.695452\pi\)
−0.576166 + 0.817333i \(0.695452\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 − 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 − 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 1.00000 0.106000 0.0529999 − 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) −14.0000 −1.43637
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −17.0000 −1.69156 −0.845782 − 0.533529i \(-0.820865\pi\)
−0.845782 + 0.533529i \(0.820865\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) −17.0000 −1.64345 −0.821726 − 0.569883i \(-0.806989\pi\)
−0.821726 + 0.569883i \(0.806989\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 0 0
\(113\) 11.0000 1.03479 0.517396 − 0.855746i \(-0.326901\pi\)
0.517396 + 0.855746i \(0.326901\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 11.0000 0.991837
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −13.0000 −1.15356 −0.576782 − 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) 0 0
\(129\) 11.0000 0.968496
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 0 0
\(135\) −10.0000 −0.860663
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 − 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) 9.00000 0.763370 0.381685 − 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) 6.00000 0.494872
\(148\) 0 0
\(149\) 1.00000 0.0819232 0.0409616 − 0.999161i \(-0.486958\pi\)
0.0409616 + 0.999161i \(0.486958\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −16.0000 −1.28515
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −5.00000 −0.391630 −0.195815 − 0.980641i \(-0.562735\pi\)
−0.195815 + 0.980641i \(0.562735\pi\)
\(164\) 0 0
\(165\) 2.00000 0.155700
\(166\) 0 0
\(167\) −7.00000 −0.541676 −0.270838 − 0.962625i \(-0.587301\pi\)
−0.270838 + 0.962625i \(0.587301\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −14.0000 −1.07061
\(172\) 0 0
\(173\) −9.00000 −0.684257 −0.342129 − 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 9.00000 0.676481
\(178\) 0 0
\(179\) −9.00000 −0.672692 −0.336346 − 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 − 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 9.00000 0.665299
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 3.00000 0.219382
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) −7.00000 −0.506502 −0.253251 − 0.967401i \(-0.581500\pi\)
−0.253251 + 0.967401i \(0.581500\pi\)
\(192\) 0 0
\(193\) 5.00000 0.359908 0.179954 − 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.0000 0.926212 0.463106 − 0.886303i \(-0.346735\pi\)
0.463106 + 0.886303i \(0.346735\pi\)
\(198\) 0 0
\(199\) −3.00000 −0.212664 −0.106332 − 0.994331i \(-0.533911\pi\)
−0.106332 + 0.994331i \(0.533911\pi\)
\(200\) 0 0
\(201\) 3.00000 0.211604
\(202\) 0 0
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) 22.0000 1.53655
\(206\) 0 0
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) 7.00000 0.484200
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 − 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 0 0
\(213\) 5.00000 0.342594
\(214\) 0 0
\(215\) 22.0000 1.50039
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.00000 0.0669650 0.0334825 − 0.999439i \(-0.489340\pi\)
0.0334825 + 0.999439i \(0.489340\pi\)
\(224\) 0 0
\(225\) 2.00000 0.133333
\(226\) 0 0
\(227\) −1.00000 −0.0663723 −0.0331862 − 0.999449i \(-0.510565\pi\)
−0.0331862 + 0.999449i \(0.510565\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) −26.0000 −1.70332 −0.851658 − 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) 0 0
\(237\) −12.0000 −0.779484
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 − 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −11.0000 −0.708572 −0.354286 − 0.935137i \(-0.615276\pi\)
−0.354286 + 0.935137i \(0.615276\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 12.0000 0.766652
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 13.0000 0.820553 0.410276 − 0.911961i \(-0.365432\pi\)
0.410276 + 0.911961i \(0.365432\pi\)
\(252\) 0 0
\(253\) −1.00000 −0.0628695
\(254\) 0 0
\(255\) 6.00000 0.375735
\(256\) 0 0
\(257\) −13.0000 −0.810918 −0.405459 − 0.914113i \(-0.632888\pi\)
−0.405459 + 0.914113i \(0.632888\pi\)
\(258\) 0 0
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 31.0000 1.91154 0.955771 − 0.294112i \(-0.0950239\pi\)
0.955771 + 0.294112i \(0.0950239\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) −1.00000 −0.0611990
\(268\) 0 0
\(269\) −5.00000 −0.304855 −0.152428 − 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(270\) 0 0
\(271\) −7.00000 −0.425220 −0.212610 − 0.977137i \(-0.568196\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −21.0000 −1.26177 −0.630884 − 0.775877i \(-0.717308\pi\)
−0.630884 + 0.775877i \(0.717308\pi\)
\(278\) 0 0
\(279\) −16.0000 −0.957895
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 − 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) −25.0000 −1.48610 −0.743048 − 0.669238i \(-0.766621\pi\)
−0.743048 + 0.669238i \(0.766621\pi\)
\(284\) 0 0
\(285\) 14.0000 0.829288
\(286\) 0 0
\(287\) 11.0000 0.649309
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) 0 0
\(293\) 9.00000 0.525786 0.262893 − 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 18.0000 1.04800
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 11.0000 0.634029
\(302\) 0 0
\(303\) 17.0000 0.976624
\(304\) 0 0
\(305\) 18.0000 1.03068
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 − 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 − 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 30.0000 1.69570 0.847850 − 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) 0 0
\(315\) −4.00000 −0.225374
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 − 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) 3.00000 0.167968
\(320\) 0 0
\(321\) 17.0000 0.948847
\(322\) 0 0
\(323\) 21.0000 1.16847
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 10.0000 0.553001
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 27.0000 1.48405 0.742027 − 0.670370i \(-0.233865\pi\)
0.742027 + 0.670370i \(0.233865\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 − 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 0 0
\(339\) −11.0000 −0.597438
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) −2.00000 −0.107676
\(346\) 0 0
\(347\) −27.0000 −1.44944 −0.724718 − 0.689046i \(-0.758030\pi\)
−0.724718 + 0.689046i \(0.758030\pi\)
\(348\) 0 0
\(349\) 9.00000 0.481759 0.240879 − 0.970555i \(-0.422564\pi\)
0.240879 + 0.970555i \(0.422564\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 25.0000 1.33062 0.665308 − 0.746569i \(-0.268300\pi\)
0.665308 + 0.746569i \(0.268300\pi\)
\(354\) 0 0
\(355\) 10.0000 0.530745
\(356\) 0 0
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) 15.0000 0.782994 0.391497 − 0.920179i \(-0.371957\pi\)
0.391497 + 0.920179i \(0.371957\pi\)
\(368\) 0 0
\(369\) 22.0000 1.14527
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 11.0000 0.569558 0.284779 − 0.958593i \(-0.408080\pi\)
0.284779 + 0.958593i \(0.408080\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −15.0000 −0.770498 −0.385249 − 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) 13.0000 0.666010
\(382\) 0 0
\(383\) −33.0000 −1.68622 −0.843111 − 0.537740i \(-0.819278\pi\)
−0.843111 + 0.537740i \(0.819278\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 0 0
\(387\) 22.0000 1.11832
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 − 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) −24.0000 −1.20757
\(396\) 0 0
\(397\) −19.0000 −0.953583 −0.476791 − 0.879017i \(-0.658200\pi\)
−0.476791 + 0.879017i \(0.658200\pi\)
\(398\) 0 0
\(399\) 7.00000 0.350438
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 − 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 1.00000 0.0495682
\(408\) 0 0
\(409\) −35.0000 −1.73064 −0.865319 − 0.501221i \(-0.832884\pi\)
−0.865319 + 0.501221i \(0.832884\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 0 0
\(413\) 9.00000 0.442861
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) −9.00000 −0.440732
\(418\) 0 0
\(419\) −21.0000 −1.02592 −0.512959 − 0.858413i \(-0.671451\pi\)
−0.512959 + 0.858413i \(0.671451\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 − 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) −24.0000 −1.16692
\(424\) 0 0
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) 9.00000 0.435541
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −11.0000 −0.529851 −0.264926 − 0.964269i \(-0.585347\pi\)
−0.264926 + 0.964269i \(0.585347\pi\)
\(432\) 0 0
\(433\) 19.0000 0.913082 0.456541 − 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 0 0
\(437\) −7.00000 −0.334855
\(438\) 0 0
\(439\) 35.0000 1.67046 0.835229 − 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −2.00000 −0.0948091
\(446\) 0 0
\(447\) −1.00000 −0.0472984
\(448\) 0 0
\(449\) 21.0000 0.991051 0.495526 − 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 0 0
\(451\) −11.0000 −0.517970
\(452\) 0 0
\(453\) −16.0000 −0.751746
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7.00000 −0.327446 −0.163723 − 0.986506i \(-0.552350\pi\)
−0.163723 + 0.986506i \(0.552350\pi\)
\(458\) 0 0
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) −3.00000 −0.139724 −0.0698620 − 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 − 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 0 0
\(465\) 16.0000 0.741982
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 − 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 3.00000 0.138527
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) −11.0000 −0.505781
\(474\) 0 0
\(475\) −7.00000 −0.321182
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) −27.0000 −1.23366 −0.616831 − 0.787096i \(-0.711584\pi\)
−0.616831 + 0.787096i \(0.711584\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −13.0000 −0.589086 −0.294543 − 0.955638i \(-0.595167\pi\)
−0.294543 + 0.955638i \(0.595167\pi\)
\(488\) 0 0
\(489\) 5.00000 0.226108
\(490\) 0 0
\(491\) 3.00000 0.135388 0.0676941 − 0.997706i \(-0.478436\pi\)
0.0676941 + 0.997706i \(0.478436\pi\)
\(492\) 0 0
\(493\) 9.00000 0.405340
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 5.00000 0.224281
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 − 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) 7.00000 0.312737
\(502\) 0 0
\(503\) 25.0000 1.11469 0.557347 − 0.830279i \(-0.311819\pi\)
0.557347 + 0.830279i \(0.311819\pi\)
\(504\) 0 0
\(505\) 34.0000 1.51298
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.00000 0.0443242 0.0221621 − 0.999754i \(-0.492945\pi\)
0.0221621 + 0.999754i \(0.492945\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 0 0
\(513\) 35.0000 1.54529
\(514\) 0 0
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 9.00000 0.395056
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 − 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) 19.0000 0.830812 0.415406 − 0.909636i \(-0.363640\pi\)
0.415406 + 0.909636i \(0.363640\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 18.0000 0.781133
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 34.0000 1.46995
\(536\) 0 0
\(537\) 9.00000 0.388379
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 − 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 0 0
\(543\) 14.0000 0.600798
\(544\) 0 0
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 − 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) 18.0000 0.768221
\(550\) 0 0
\(551\) 21.0000 0.894630
\(552\) 0 0
\(553\) −12.0000 −0.510292
\(554\) 0 0
\(555\) 2.00000 0.0848953
\(556\) 0 0
\(557\) −19.0000 −0.805056 −0.402528 − 0.915408i \(-0.631868\pi\)
−0.402528 + 0.915408i \(0.631868\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) 0 0
\(563\) 9.00000 0.379305 0.189652 − 0.981851i \(-0.439264\pi\)
0.189652 + 0.981851i \(0.439264\pi\)
\(564\) 0 0
\(565\) −22.0000 −0.925547
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −17.0000 −0.712677 −0.356339 − 0.934357i \(-0.615975\pi\)
−0.356339 + 0.934357i \(0.615975\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 7.00000 0.292429
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −22.0000 −0.915872 −0.457936 − 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 0 0
\(579\) −5.00000 −0.207793
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −6.00000 −0.248495
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21.0000 0.866763 0.433381 − 0.901211i \(-0.357320\pi\)
0.433381 + 0.901211i \(0.357320\pi\)
\(588\) 0 0
\(589\) 56.0000 2.30744
\(590\) 0 0
\(591\) −13.0000 −0.534749
\(592\) 0 0
\(593\) −14.0000 −0.574911 −0.287456 − 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) 0 0
\(597\) 3.00000 0.122782
\(598\) 0 0
\(599\) 44.0000 1.79779 0.898896 − 0.438163i \(-0.144371\pi\)
0.898896 + 0.438163i \(0.144371\pi\)
\(600\) 0 0
\(601\) −21.0000 −0.856608 −0.428304 − 0.903635i \(-0.640889\pi\)
−0.428304 + 0.903635i \(0.640889\pi\)
\(602\) 0 0
\(603\) 6.00000 0.244339
\(604\) 0 0
\(605\) 20.0000 0.813116
\(606\) 0 0
\(607\) 13.0000 0.527654 0.263827 − 0.964570i \(-0.415015\pi\)
0.263827 + 0.964570i \(0.415015\pi\)
\(608\) 0 0
\(609\) 3.00000 0.121566
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −39.0000 −1.57520 −0.787598 − 0.616190i \(-0.788675\pi\)
−0.787598 + 0.616190i \(0.788675\pi\)
\(614\) 0 0
\(615\) −22.0000 −0.887126
\(616\) 0 0
\(617\) −47.0000 −1.89215 −0.946074 − 0.323949i \(-0.894989\pi\)
−0.946074 + 0.323949i \(0.894989\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) −1.00000 −0.0400642
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −7.00000 −0.279553
\(628\) 0 0
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) 27.0000 1.07485 0.537427 − 0.843311i \(-0.319397\pi\)
0.537427 + 0.843311i \(0.319397\pi\)
\(632\) 0 0
\(633\) 1.00000 0.0397464
\(634\) 0 0
\(635\) 26.0000 1.03178
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 10.0000 0.395594
\(640\) 0 0
\(641\) 11.0000 0.434474 0.217237 − 0.976119i \(-0.430296\pi\)
0.217237 + 0.976119i \(0.430296\pi\)
\(642\) 0 0
\(643\) 19.0000 0.749287 0.374643 − 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) 0 0
\(645\) −22.0000 −0.866249
\(646\) 0 0
\(647\) 3.00000 0.117942 0.0589711 − 0.998260i \(-0.481218\pi\)
0.0589711 + 0.998260i \(0.481218\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 0 0
\(653\) −9.00000 −0.352197 −0.176099 − 0.984373i \(-0.556348\pi\)
−0.176099 + 0.984373i \(0.556348\pi\)
\(654\) 0 0
\(655\) 24.0000 0.937758
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) 29.0000 1.12968 0.564840 − 0.825201i \(-0.308938\pi\)
0.564840 + 0.825201i \(0.308938\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 − 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 14.0000 0.542897
\(666\) 0 0
\(667\) −3.00000 −0.116160
\(668\) 0 0
\(669\) −1.00000 −0.0386622
\(670\) 0 0
\(671\) −9.00000 −0.347441
\(672\) 0 0
\(673\) 39.0000 1.50334 0.751670 − 0.659540i \(-0.229249\pi\)
0.751670 + 0.659540i \(0.229249\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) 1.00000 0.0383201
\(682\) 0 0
\(683\) −9.00000 −0.344375 −0.172188 − 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) −6.00000 −0.228914
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 25.0000 0.951045 0.475522 − 0.879704i \(-0.342259\pi\)
0.475522 + 0.879704i \(0.342259\pi\)
\(692\) 0 0
\(693\) 2.00000 0.0759737
\(694\) 0 0
\(695\) −18.0000 −0.682779
\(696\) 0 0
\(697\) −33.0000 −1.24996
\(698\) 0 0
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 − 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) 7.00000 0.264010
\(704\) 0 0
\(705\) 24.0000 0.903892
\(706\) 0 0
\(707\) 17.0000 0.639351
\(708\) 0 0
\(709\) 29.0000 1.08912 0.544559 − 0.838723i \(-0.316697\pi\)
0.544559 + 0.838723i \(0.316697\pi\)
\(710\) 0 0
\(711\) −24.0000 −0.900070
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 24.0000 0.896296
\(718\) 0 0
\(719\) −39.0000 −1.45445 −0.727227 − 0.686397i \(-0.759191\pi\)
−0.727227 + 0.686397i \(0.759191\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 11.0000 0.409094
\(724\) 0 0
\(725\) −3.00000 −0.111417
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 − 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −33.0000 −1.22055
\(732\) 0 0
\(733\) −46.0000 −1.69905 −0.849524 − 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) 0 0
\(735\) −12.0000 −0.442627
\(736\) 0 0
\(737\) −3.00000 −0.110506
\(738\) 0 0
\(739\) −11.0000 −0.404642 −0.202321 − 0.979319i \(-0.564848\pi\)
−0.202321 + 0.979319i \(0.564848\pi\)
\(740\) 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
\(745\) −2.00000 −0.0732743
\(746\) 0 0
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) 17.0000 0.621166
\(750\) 0 0
\(751\) −17.0000 −0.620339 −0.310169 − 0.950681i \(-0.600386\pi\)
−0.310169 + 0.950681i \(0.600386\pi\)
\(752\) 0 0
\(753\) −13.0000 −0.473746
\(754\) 0 0
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) 47.0000 1.70824 0.854122 − 0.520073i \(-0.174095\pi\)
0.854122 + 0.520073i \(0.174095\pi\)
\(758\) 0 0
\(759\) 1.00000 0.0362977
\(760\) 0 0
\(761\) −3.00000 −0.108750 −0.0543750 − 0.998521i \(-0.517317\pi\)
−0.0543750 + 0.998521i \(0.517317\pi\)
\(762\) 0 0
\(763\) 10.0000 0.362024
\(764\) 0 0
\(765\) 12.0000 0.433861
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 − 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 13.0000 0.468184
\(772\) 0 0
\(773\) −3.00000 −0.107903 −0.0539513 − 0.998544i \(-0.517182\pi\)
−0.0539513 + 0.998544i \(0.517182\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) 1.00000 0.0358748
\(778\) 0 0
\(779\) −77.0000 −2.75881
\(780\) 0 0
\(781\) −5.00000 −0.178914
\(782\) 0 0
\(783\) 15.0000 0.536056
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) −21.0000 −0.748569 −0.374285 − 0.927314i \(-0.622112\pi\)
−0.374285 + 0.927314i \(0.622112\pi\)
\(788\) 0 0
\(789\) −31.0000 −1.10363
\(790\) 0 0
\(791\) −11.0000 −0.391115
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −12.0000 −0.425596
\(796\) 0 0
\(797\) 23.0000 0.814702 0.407351 − 0.913272i \(-0.366453\pi\)
0.407351 + 0.913272i \(0.366453\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 0 0
\(803\) 2.00000 0.0705785
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) 0 0
\(807\) 5.00000 0.176008
\(808\) 0 0
\(809\) 27.0000 0.949269 0.474635 − 0.880183i \(-0.342580\pi\)
0.474635 + 0.880183i \(0.342580\pi\)
\(810\) 0 0
\(811\) 32.0000 1.12367 0.561836 − 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 0 0
\(813\) 7.00000 0.245501
\(814\) 0 0
\(815\) 10.0000 0.350285
\(816\) 0 0
\(817\) −77.0000 −2.69389
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 5.00000 0.174501 0.0872506 − 0.996186i \(-0.472192\pi\)
0.0872506 + 0.996186i \(0.472192\pi\)
\(822\) 0 0
\(823\) 29.0000 1.01088 0.505438 − 0.862863i \(-0.331331\pi\)
0.505438 + 0.862863i \(0.331331\pi\)
\(824\) 0 0
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 − 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 0 0
\(829\) 3.00000 0.104194 0.0520972 − 0.998642i \(-0.483409\pi\)
0.0520972 + 0.998642i \(0.483409\pi\)
\(830\) 0 0
\(831\) 21.0000 0.728482
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 14.0000 0.484490
\(836\) 0 0
\(837\) 40.0000 1.38260
\(838\) 0 0
\(839\) 27.0000 0.932144 0.466072 − 0.884747i \(-0.345669\pi\)
0.466072 + 0.884747i \(0.345669\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 14.0000 0.482186
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) 0 0
\(849\) 25.0000 0.857998
\(850\) 0 0
\(851\) −1.00000 −0.0342796
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 − 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 28.0000 0.957580
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 − 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) −11.0000 −0.374879
\(862\) 0 0
\(863\) −16.0000 −0.544646 −0.272323 − 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) 1.00000 0.0337676 0.0168838 − 0.999857i \(-0.494625\pi\)
0.0168838 + 0.999857i \(0.494625\pi\)
\(878\) 0 0
\(879\) −9.00000 −0.303562
\(880\) 0 0
\(881\) −45.0000 −1.51609 −0.758044 − 0.652203i \(-0.773845\pi\)
−0.758044 + 0.652203i \(0.773845\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 − 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) −18.0000 −0.605063
\(886\) 0 0
\(887\) 27.0000 0.906571 0.453286 − 0.891365i \(-0.350252\pi\)
0.453286 + 0.891365i \(0.350252\pi\)
\(888\) 0 0
\(889\) 13.0000 0.436006
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 84.0000 2.81095
\(894\) 0 0
\(895\) 18.0000 0.601674
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 0 0
\(903\) −11.0000 −0.366057
\(904\) 0 0
\(905\) 28.0000 0.930751
\(906\) 0 0
\(907\) −25.0000 −0.830111 −0.415056 − 0.909796i \(-0.636238\pi\)
−0.415056 + 0.909796i \(0.636238\pi\)
\(908\) 0 0
\(909\) 34.0000 1.12771
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 − 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) 0 0
\(915\) −18.0000 −0.595062
\(916\) 0 0
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −15.0000 −0.494804 −0.247402 − 0.968913i \(-0.579577\pi\)
−0.247402 + 0.968913i \(0.579577\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) 5.00000 0.164045 0.0820223 − 0.996630i \(-0.473862\pi\)
0.0820223 + 0.996630i \(0.473862\pi\)
\(930\) 0 0
\(931\) −42.0000 −1.37649
\(932\) 0 0
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) −6.00000 −0.196221
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 − 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) −30.0000 −0.979013
\(940\) 0 0
\(941\) −14.0000 −0.456387 −0.228193 − 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 0 0
\(943\) 11.0000 0.358209
\(944\) 0 0
\(945\) 10.0000 0.325300
\(946\) 0 0
\(947\) −39.0000 −1.26733 −0.633665 − 0.773608i \(-0.718450\pi\)
−0.633665 + 0.773608i \(0.718450\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) −9.00000 −0.291539 −0.145769 − 0.989319i \(-0.546566\pi\)
−0.145769 + 0.989319i \(0.546566\pi\)
\(954\) 0 0
\(955\) 14.0000 0.453029
\(956\) 0 0
\(957\) −3.00000 −0.0969762
\(958\) 0 0
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 34.0000 1.09563
\(964\) 0 0
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 − 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 0 0
\(969\) −21.0000 −0.674617
\(970\) 0 0
\(971\) −27.0000 −0.866471 −0.433236 − 0.901281i \(-0.642628\pi\)
−0.433236 + 0.901281i \(0.642628\pi\)
\(972\) 0 0
\(973\) −9.00000 −0.288527
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.00000 0.287936 0.143968 − 0.989582i \(-0.454014\pi\)
0.143968 + 0.989582i \(0.454014\pi\)
\(978\) 0 0
\(979\) 1.00000 0.0319601
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) 0 0
\(983\) 32.0000 1.02064 0.510321 − 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) 0 0
\(985\) −26.0000 −0.828429
\(986\) 0 0
\(987\) 12.0000 0.381964
\(988\) 0 0
\(989\) 11.0000 0.349780
\(990\) 0 0
\(991\) −13.0000 −0.412959 −0.206479 − 0.978451i \(-0.566201\pi\)
−0.206479 + 0.978451i \(0.566201\pi\)
\(992\) 0 0
\(993\) −27.0000 −0.856819
\(994\) 0 0
\(995\) 6.00000 0.190213
\(996\) 0 0
\(997\) −25.0000 −0.791758 −0.395879 − 0.918303i \(-0.629560\pi\)
−0.395879 + 0.918303i \(0.629560\pi\)
\(998\) 0 0
\(999\) 5.00000 0.158193
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 2704.2.a.c.1.1 1
4.3 odd 2 1352.2.a.a.1.1 1
13.4 even 6 208.2.i.c.81.1 2
13.5 odd 4 2704.2.f.c.337.2 2
13.8 odd 4 2704.2.f.c.337.1 2
13.10 even 6 208.2.i.c.113.1 2
13.12 even 2 2704.2.a.e.1.1 1
39.17 odd 6 1872.2.t.d.289.1 2
39.23 odd 6 1872.2.t.d.1153.1 2
52.3 odd 6 1352.2.i.a.529.1 2
52.7 even 12 1352.2.o.b.361.1 4
52.11 even 12 1352.2.o.b.1161.1 4
52.15 even 12 1352.2.o.b.1161.2 4
52.19 even 12 1352.2.o.b.361.2 4
52.23 odd 6 104.2.i.a.9.1 ✓ 2
52.31 even 4 1352.2.f.a.337.2 2
52.35 odd 6 1352.2.i.a.1329.1 2
52.43 odd 6 104.2.i.a.81.1 yes 2
52.47 even 4 1352.2.f.a.337.1 2
52.51 odd 2 1352.2.a.c.1.1 1
104.43 odd 6 832.2.i.g.705.1 2
104.69 even 6 832.2.i.d.705.1 2
104.75 odd 6 832.2.i.g.321.1 2
104.101 even 6 832.2.i.d.321.1 2
156.23 even 6 936.2.t.c.217.1 2
156.95 even 6 936.2.t.c.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
104.2.i.a.9.1 ✓ 2 52.23 odd 6
104.2.i.a.81.1 yes 2 52.43 odd 6
208.2.i.c.81.1 2 13.4 even 6
208.2.i.c.113.1 2 13.10 even 6
832.2.i.d.321.1 2 104.101 even 6
832.2.i.d.705.1 2 104.69 even 6
832.2.i.g.321.1 2 104.75 odd 6
832.2.i.g.705.1 2 104.43 odd 6
936.2.t.c.217.1 2 156.23 even 6
936.2.t.c.289.1 2 156.95 even 6
1352.2.a.a.1.1 1 4.3 odd 2
1352.2.a.c.1.1 1 52.51 odd 2
1352.2.f.a.337.1 2 52.47 even 4
1352.2.f.a.337.2 2 52.31 even 4
1352.2.i.a.529.1 2 52.3 odd 6
1352.2.i.a.1329.1 2 52.35 odd 6
1352.2.o.b.361.1 4 52.7 even 12
1352.2.o.b.361.2 4 52.19 even 12
1352.2.o.b.1161.1 4 52.11 even 12
1352.2.o.b.1161.2 4 52.15 even 12
1872.2.t.d.289.1 2 39.17 odd 6
1872.2.t.d.1153.1 2 39.23 odd 6
2704.2.a.c.1.1 1 1.1 even 1 trivial
2704.2.a.e.1.1 1 13.12 even 2
2704.2.f.c.337.1 2 13.8 odd 4
2704.2.f.c.337.2 2 13.5 odd 4