Properties

Label 924.2.a.e.1.1
Level $924$
Weight $2$
Character 924.1
Self dual yes
Analytic conductor $7.378$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [924,2,Mod(1,924)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(924, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("924.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 924 = 2^{2} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 924.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: \(7.37817714677\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 924.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +3.00000 q^{13} -3.00000 q^{15} -2.00000 q^{17} -3.00000 q^{19} -1.00000 q^{21} -4.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} -9.00000 q^{29} -2.00000 q^{31} -1.00000 q^{33} +3.00000 q^{35} -11.0000 q^{37} +3.00000 q^{39} -4.00000 q^{41} -4.00000 q^{43} -3.00000 q^{45} -3.00000 q^{47} +1.00000 q^{49} -2.00000 q^{51} -4.00000 q^{53} +3.00000 q^{55} -3.00000 q^{57} -3.00000 q^{59} +10.0000 q^{61} -1.00000 q^{63} -9.00000 q^{65} +11.0000 q^{67} -4.00000 q^{69} +4.00000 q^{71} +9.00000 q^{73} +4.00000 q^{75} +1.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +10.0000 q^{83} +6.00000 q^{85} -9.00000 q^{87} +6.00000 q^{89} -3.00000 q^{91} -2.00000 q^{93} +9.00000 q^{95} +12.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(4\) 0 0
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) −11.0000 −1.80839 −0.904194 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −9.00000 −1.11631
\(66\) 0 0
\(67\) 11.0000 1.34386 0.671932 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) −9.00000 −0.964901
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 9.00000 0.923381
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) −19.0000 −1.83680 −0.918400 0.395654i \(-0.870518\pi\)
−0.918400 + 0.395654i \(0.870518\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −11.0000 −1.04407
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 12.0000 1.11901
\(116\) 0 0
\(117\) 3.00000 0.277350
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −4.00000 −0.360668
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 3.00000 0.260133
\(134\) 0 0
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) 27.0000 2.24223
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −13.0000 −1.01824 −0.509119 0.860696i \(-0.670029\pi\)
−0.509119 + 0.860696i \(0.670029\pi\)
\(164\) 0 0
\(165\) 3.00000 0.233550
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) 0 0
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) −3.00000 −0.225494
\(178\) 0 0
\(179\) −8.00000 −0.597948 −0.298974 0.954261i \(-0.596644\pi\)
−0.298974 + 0.954261i \(0.596644\pi\)
\(180\) 0 0
\(181\) −4.00000 −0.297318 −0.148659 0.988889i \(-0.547496\pi\)
−0.148659 + 0.988889i \(0.547496\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 33.0000 2.42621
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) 24.0000 1.72756 0.863779 0.503871i \(-0.168091\pi\)
0.863779 + 0.503871i \(0.168091\pi\)
\(194\) 0 0
\(195\) −9.00000 −0.644503
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 11.0000 0.775880
\(202\) 0 0
\(203\) 9.00000 0.631676
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 0 0
\(213\) 4.00000 0.274075
\(214\) 0 0
\(215\) 12.0000 0.818393
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 0 0
\(219\) 9.00000 0.608164
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 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
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 11.0000 0.711531 0.355765 0.934575i \(-0.384220\pi\)
0.355765 + 0.934575i \(0.384220\pi\)
\(240\) 0 0
\(241\) 19.0000 1.22390 0.611949 0.790897i \(-0.290386\pi\)
0.611949 + 0.790897i \(0.290386\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) −9.00000 −0.572656
\(248\) 0 0
\(249\) 10.0000 0.633724
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 6.00000 0.375735
\(256\) 0 0
\(257\) 25.0000 1.55946 0.779729 0.626118i \(-0.215357\pi\)
0.779729 + 0.626118i \(0.215357\pi\)
\(258\) 0 0
\(259\) 11.0000 0.683507
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) 0 0
\(263\) −3.00000 −0.184988 −0.0924940 0.995713i \(-0.529484\pi\)
−0.0924940 + 0.995713i \(0.529484\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) −3.00000 −0.182237 −0.0911185 0.995840i \(-0.529044\pi\)
−0.0911185 + 0.995840i \(0.529044\pi\)
\(272\) 0 0
\(273\) −3.00000 −0.181568
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 23.0000 1.37206 0.686032 0.727571i \(-0.259351\pi\)
0.686032 + 0.727571i \(0.259351\pi\)
\(282\) 0 0
\(283\) 21.0000 1.24832 0.624160 0.781296i \(-0.285441\pi\)
0.624160 + 0.781296i \(0.285441\pi\)
\(284\) 0 0
\(285\) 9.00000 0.533114
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) 0 0
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) 0 0
\(295\) 9.00000 0.524000
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) 10.0000 0.574485
\(304\) 0 0
\(305\) −30.0000 −1.71780
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 18.0000 1.01742 0.508710 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(314\) 0 0
\(315\) 3.00000 0.169031
\(316\) 0 0
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 0 0
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) −19.0000 −1.06048
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) 12.0000 0.665640
\(326\) 0 0
\(327\) −2.00000 −0.110600
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) −11.0000 −0.602796
\(334\) 0 0
\(335\) −33.0000 −1.80298
\(336\) 0 0
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) 0 0
\(339\) 10.0000 0.543125
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 0 0
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) 3.00000 0.160128
\(352\) 0 0
\(353\) 3.00000 0.159674 0.0798369 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 2.00000 0.105851
\(358\) 0 0
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −27.0000 −1.41324
\(366\) 0 0
\(367\) −38.0000 −1.98358 −0.991792 0.127862i \(-0.959188\pi\)
−0.991792 + 0.127862i \(0.959188\pi\)
\(368\) 0 0
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) −27.0000 −1.39057
\(378\) 0 0
\(379\) −35.0000 −1.79783 −0.898915 0.438124i \(-0.855643\pi\)
−0.898915 + 0.438124i \(0.855643\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 0 0
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 24.0000 1.21685 0.608424 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 3.00000 0.150188
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) −6.00000 −0.298881
\(404\) 0 0
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) 11.0000 0.545250
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −14.0000 −0.690569
\(412\) 0 0
\(413\) 3.00000 0.147620
\(414\) 0 0
\(415\) −30.0000 −1.47264
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −5.00000 −0.244266 −0.122133 0.992514i \(-0.538973\pi\)
−0.122133 + 0.992514i \(0.538973\pi\)
\(420\) 0 0
\(421\) −37.0000 −1.80327 −0.901635 0.432498i \(-0.857632\pi\)
−0.901635 + 0.432498i \(0.857632\pi\)
\(422\) 0 0
\(423\) −3.00000 −0.145865
\(424\) 0 0
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 0 0
\(429\) −3.00000 −0.144841
\(430\) 0 0
\(431\) −11.0000 −0.529851 −0.264926 0.964269i \(-0.585347\pi\)
−0.264926 + 0.964269i \(0.585347\pi\)
\(432\) 0 0
\(433\) 8.00000 0.384455 0.192228 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) 0 0
\(435\) 27.0000 1.29455
\(436\) 0 0
\(437\) 12.0000 0.574038
\(438\) 0 0
\(439\) 37.0000 1.76591 0.882957 0.469454i \(-0.155549\pi\)
0.882957 + 0.469454i \(0.155549\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) 3.00000 0.141895
\(448\) 0 0
\(449\) 32.0000 1.51017 0.755087 0.655625i \(-0.227595\pi\)
0.755087 + 0.655625i \(0.227595\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 0 0
\(453\) 18.0000 0.845714
\(454\) 0 0
\(455\) 9.00000 0.421927
\(456\) 0 0
\(457\) −32.0000 −1.49690 −0.748448 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) 0 0
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 8.00000 0.372597 0.186299 0.982493i \(-0.440351\pi\)
0.186299 + 0.982493i \(0.440351\pi\)
\(462\) 0 0
\(463\) 7.00000 0.325318 0.162659 0.986682i \(-0.447993\pi\)
0.162659 + 0.986682i \(0.447993\pi\)
\(464\) 0 0
\(465\) 6.00000 0.278243
\(466\) 0 0
\(467\) −23.0000 −1.06431 −0.532157 0.846646i \(-0.678618\pi\)
−0.532157 + 0.846646i \(0.678618\pi\)
\(468\) 0 0
\(469\) −11.0000 −0.507933
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) −12.0000 −0.550598
\(476\) 0 0
\(477\) −4.00000 −0.183147
\(478\) 0 0
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) −33.0000 −1.50467
\(482\) 0 0
\(483\) 4.00000 0.182006
\(484\) 0 0
\(485\) −36.0000 −1.63468
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 0 0
\(489\) −13.0000 −0.587880
\(490\) 0 0
\(491\) −3.00000 −0.135388 −0.0676941 0.997706i \(-0.521564\pi\)
−0.0676941 + 0.997706i \(0.521564\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) 0 0
\(495\) 3.00000 0.134840
\(496\) 0 0
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) −27.0000 −1.20869 −0.604343 0.796724i \(-0.706564\pi\)
−0.604343 + 0.796724i \(0.706564\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −30.0000 −1.33498
\(506\) 0 0
\(507\) −4.00000 −0.177646
\(508\) 0 0
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) −9.00000 −0.398137
\(512\) 0 0
\(513\) −3.00000 −0.132453
\(514\) 0 0
\(515\) 42.0000 1.85074
\(516\) 0 0
\(517\) 3.00000 0.131940
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) −31.0000 −1.35813 −0.679067 0.734076i \(-0.737616\pi\)
−0.679067 + 0.734076i \(0.737616\pi\)
\(522\) 0 0
\(523\) −21.0000 −0.918266 −0.459133 0.888368i \(-0.651840\pi\)
−0.459133 + 0.888368i \(0.651840\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −3.00000 −0.130189
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 57.0000 2.46432
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) −4.00000 −0.171656
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 27.0000 1.15024
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) 33.0000 1.40077
\(556\) 0 0
\(557\) −5.00000 −0.211857 −0.105928 0.994374i \(-0.533781\pi\)
−0.105928 + 0.994374i \(0.533781\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) 0 0
\(563\) 32.0000 1.34864 0.674320 0.738440i \(-0.264437\pi\)
0.674320 + 0.738440i \(0.264437\pi\)
\(564\) 0 0
\(565\) −30.0000 −1.26211
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) −18.0000 −0.751961
\(574\) 0 0
\(575\) −16.0000 −0.667246
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) −10.0000 −0.414870
\(582\) 0 0
\(583\) 4.00000 0.165663
\(584\) 0 0
\(585\) −9.00000 −0.372104
\(586\) 0 0
\(587\) 15.0000 0.619116 0.309558 0.950881i \(-0.399819\pi\)
0.309558 + 0.950881i \(0.399819\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) 20.0000 0.821302 0.410651 0.911793i \(-0.365302\pi\)
0.410651 + 0.911793i \(0.365302\pi\)
\(594\) 0 0
\(595\) −6.00000 −0.245976
\(596\) 0 0
\(597\) −20.0000 −0.818546
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 37.0000 1.50926 0.754631 0.656150i \(-0.227816\pi\)
0.754631 + 0.656150i \(0.227816\pi\)
\(602\) 0 0
\(603\) 11.0000 0.447955
\(604\) 0 0
\(605\) −3.00000 −0.121967
\(606\) 0 0
\(607\) −27.0000 −1.09590 −0.547948 0.836512i \(-0.684591\pi\)
−0.547948 + 0.836512i \(0.684591\pi\)
\(608\) 0 0
\(609\) 9.00000 0.364698
\(610\) 0 0
\(611\) −9.00000 −0.364101
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 0 0
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 3.00000 0.119808
\(628\) 0 0
\(629\) 22.0000 0.877197
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) −10.0000 −0.397464
\(634\) 0 0
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 0 0
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 0 0
\(645\) 12.0000 0.472500
\(646\) 0 0
\(647\) 23.0000 0.904223 0.452112 0.891961i \(-0.350671\pi\)
0.452112 + 0.891961i \(0.350671\pi\)
\(648\) 0 0
\(649\) 3.00000 0.117760
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 0 0
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) 9.00000 0.351123
\(658\) 0 0
\(659\) −23.0000 −0.895953 −0.447976 0.894045i \(-0.647855\pi\)
−0.447976 + 0.894045i \(0.647855\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 0 0
\(663\) −6.00000 −0.233021
\(664\) 0 0
\(665\) −9.00000 −0.349005
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) 0 0
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) −32.0000 −1.23351 −0.616755 0.787155i \(-0.711553\pi\)
−0.616755 + 0.787155i \(0.711553\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) −12.0000 −0.460518
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) 30.0000 1.14792 0.573959 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(684\) 0 0
\(685\) 42.0000 1.60474
\(686\) 0 0
\(687\) 8.00000 0.305219
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −42.0000 −1.59776 −0.798878 0.601494i \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) 0 0
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) 24.0000 0.910372
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) 0 0
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) 33.0000 1.24462
\(704\) 0 0
\(705\) 9.00000 0.338960
\(706\) 0 0
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 9.00000 0.336581
\(716\) 0 0
\(717\) 11.0000 0.410803
\(718\) 0 0
\(719\) −29.0000 −1.08152 −0.540759 0.841178i \(-0.681863\pi\)
−0.540759 + 0.841178i \(0.681863\pi\)
\(720\) 0 0
\(721\) 14.0000 0.521387
\(722\) 0 0
\(723\) 19.0000 0.706618
\(724\) 0 0
\(725\) −36.0000 −1.33701
\(726\) 0 0
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 0 0
\(735\) −3.00000 −0.110657
\(736\) 0 0
\(737\) −11.0000 −0.405190
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 0 0
\(741\) −9.00000 −0.330623
\(742\) 0 0
\(743\) 13.0000 0.476924 0.238462 0.971152i \(-0.423357\pi\)
0.238462 + 0.971152i \(0.423357\pi\)
\(744\) 0 0
\(745\) −9.00000 −0.329734
\(746\) 0 0
\(747\) 10.0000 0.365881
\(748\) 0 0
\(749\) 19.0000 0.694245
\(750\) 0 0
\(751\) −41.0000 −1.49611 −0.748056 0.663636i \(-0.769012\pi\)
−0.748056 + 0.663636i \(0.769012\pi\)
\(752\) 0 0
\(753\) −21.0000 −0.765283
\(754\) 0 0
\(755\) −54.0000 −1.96526
\(756\) 0 0
\(757\) 13.0000 0.472493 0.236247 0.971693i \(-0.424083\pi\)
0.236247 + 0.971693i \(0.424083\pi\)
\(758\) 0 0
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 20.0000 0.724999 0.362500 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) −9.00000 −0.324971
\(768\) 0 0
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 25.0000 0.900353
\(772\) 0 0
\(773\) −31.0000 −1.11499 −0.557496 0.830179i \(-0.688238\pi\)
−0.557496 + 0.830179i \(0.688238\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) 11.0000 0.394623
\(778\) 0 0
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) −9.00000 −0.321634
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −5.00000 −0.178231 −0.0891154 0.996021i \(-0.528404\pi\)
−0.0891154 + 0.996021i \(0.528404\pi\)
\(788\) 0 0
\(789\) −3.00000 −0.106803
\(790\) 0 0
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) 30.0000 1.06533
\(794\) 0 0
\(795\) 12.0000 0.425596
\(796\) 0 0
\(797\) 7.00000 0.247953 0.123976 0.992285i \(-0.460435\pi\)
0.123976 + 0.992285i \(0.460435\pi\)
\(798\) 0 0
\(799\) 6.00000 0.212265
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) −9.00000 −0.317603
\(804\) 0 0
\(805\) −12.0000 −0.422944
\(806\) 0 0
\(807\) −30.0000 −1.05605
\(808\) 0 0
\(809\) 27.0000 0.949269 0.474635 0.880183i \(-0.342580\pi\)
0.474635 + 0.880183i \(0.342580\pi\)
\(810\) 0 0
\(811\) −21.0000 −0.737410 −0.368705 0.929547i \(-0.620199\pi\)
−0.368705 + 0.929547i \(0.620199\pi\)
\(812\) 0 0
\(813\) −3.00000 −0.105215
\(814\) 0 0
\(815\) 39.0000 1.36611
\(816\) 0 0
\(817\) 12.0000 0.419827
\(818\) 0 0
\(819\) −3.00000 −0.104828
\(820\) 0 0
\(821\) 13.0000 0.453703 0.226852 0.973929i \(-0.427157\pi\)
0.226852 + 0.973929i \(0.427157\pi\)
\(822\) 0 0
\(823\) 55.0000 1.91718 0.958590 0.284791i \(-0.0919241\pi\)
0.958590 + 0.284791i \(0.0919241\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) −3.00000 −0.104320 −0.0521601 0.998639i \(-0.516611\pi\)
−0.0521601 + 0.998639i \(0.516611\pi\)
\(828\) 0 0
\(829\) −24.0000 −0.833554 −0.416777 0.909009i \(-0.636840\pi\)
−0.416777 + 0.909009i \(0.636840\pi\)
\(830\) 0 0
\(831\) 12.0000 0.416275
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) −27.0000 −0.932144 −0.466072 0.884747i \(-0.654331\pi\)
−0.466072 + 0.884747i \(0.654331\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 23.0000 0.792162
\(844\) 0 0
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) 21.0000 0.720718
\(850\) 0 0
\(851\) 44.0000 1.50830
\(852\) 0 0
\(853\) 54.0000 1.84892 0.924462 0.381273i \(-0.124514\pi\)
0.924462 + 0.381273i \(0.124514\pi\)
\(854\) 0 0
\(855\) 9.00000 0.307794
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 4.00000 0.136320
\(862\) 0 0
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) 36.0000 1.22404
\(866\) 0 0
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 33.0000 1.11816
\(872\) 0 0
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −24.0000 −0.810422 −0.405211 0.914223i \(-0.632802\pi\)
−0.405211 + 0.914223i \(0.632802\pi\)
\(878\) 0 0
\(879\) −16.0000 −0.539667
\(880\) 0 0
\(881\) 25.0000 0.842271 0.421136 0.906998i \(-0.361632\pi\)
0.421136 + 0.906998i \(0.361632\pi\)
\(882\) 0 0
\(883\) 29.0000 0.975928 0.487964 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) 0 0
\(885\) 9.00000 0.302532
\(886\) 0 0
\(887\) 18.0000 0.604381 0.302190 0.953248i \(-0.402282\pi\)
0.302190 + 0.953248i \(0.402282\pi\)
\(888\) 0 0
\(889\) −2.00000 −0.0670778
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 9.00000 0.301174
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) −12.0000 −0.400668
\(898\) 0 0
\(899\) 18.0000 0.600334
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) 12.0000 0.398893
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 42.0000 1.39152 0.695761 0.718273i \(-0.255067\pi\)
0.695761 + 0.718273i \(0.255067\pi\)
\(912\) 0 0
\(913\) −10.0000 −0.330952
\(914\) 0 0
\(915\) −30.0000 −0.991769
\(916\) 0 0
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) −44.0000 −1.44671
\(926\) 0 0
\(927\) −14.0000 −0.459820
\(928\) 0 0
\(929\) −29.0000 −0.951459 −0.475730 0.879592i \(-0.657816\pi\)
−0.475730 + 0.879592i \(0.657816\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.00000 −0.196221
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 0 0
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) 0 0
\(945\) 3.00000 0.0975900
\(946\) 0 0
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 0 0
\(949\) 27.0000 0.876457
\(950\) 0 0
\(951\) 24.0000 0.778253
\(952\) 0 0
\(953\) −45.0000 −1.45769 −0.728846 0.684677i \(-0.759943\pi\)
−0.728846 + 0.684677i \(0.759943\pi\)
\(954\) 0 0
\(955\) 54.0000 1.74740
\(956\) 0 0
\(957\) 9.00000 0.290929
\(958\) 0 0
\(959\) 14.0000 0.452084
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −19.0000 −0.612266
\(964\) 0 0
\(965\) −72.0000 −2.31776
\(966\) 0 0
\(967\) 18.0000 0.578841 0.289420 0.957202i \(-0.406537\pi\)
0.289420 + 0.957202i \(0.406537\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) 23.0000 0.738105 0.369053 0.929409i \(-0.379682\pi\)
0.369053 + 0.929409i \(0.379682\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) 0 0
\(975\) 12.0000 0.384308
\(976\) 0 0
\(977\) 8.00000 0.255943 0.127971 0.991778i \(-0.459153\pi\)
0.127971 + 0.991778i \(0.459153\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) −56.0000 −1.78612 −0.893061 0.449935i \(-0.851447\pi\)
−0.893061 + 0.449935i \(0.851447\pi\)
\(984\) 0 0
\(985\) 54.0000 1.72058
\(986\) 0 0
\(987\) 3.00000 0.0954911
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −17.0000 −0.540023 −0.270011 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(992\) 0 0
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) 60.0000 1.90213
\(996\) 0 0
\(997\) −58.0000 −1.83688 −0.918439 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) 0 0
\(999\) −11.0000 −0.348025
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 924.2.a.e.1.1 1
3.2 odd 2 2772.2.a.j.1.1 1
4.3 odd 2 3696.2.a.d.1.1 1
7.6 odd 2 6468.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
924.2.a.e.1.1 1 1.1 even 1 trivial
2772.2.a.j.1.1 1 3.2 odd 2
3696.2.a.d.1.1 1 4.3 odd 2
6468.2.a.h.1.1 1 7.6 odd 2