Properties

Label 2898.2.a.m.1.1
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
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] = [2898,2,Mod(1,2898)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2898, 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("2898.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.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: \(23.1406465058\)
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 \(\chi\) \(=\) 2898.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -2.00000 q^{11} -5.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -2.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} -1.00000 q^{23} -4.00000 q^{25} -5.00000 q^{26} +1.00000 q^{28} -5.00000 q^{29} +1.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} -7.00000 q^{37} -2.00000 q^{38} -1.00000 q^{40} +3.00000 q^{41} -11.0000 q^{43} -2.00000 q^{44} -1.00000 q^{46} +9.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} -5.00000 q^{52} +10.0000 q^{53} +2.00000 q^{55} +1.00000 q^{56} -5.00000 q^{58} -12.0000 q^{59} -2.00000 q^{61} +1.00000 q^{64} +5.00000 q^{65} -4.00000 q^{67} +2.00000 q^{68} -1.00000 q^{70} -6.00000 q^{71} +6.00000 q^{73} -7.00000 q^{74} -2.00000 q^{76} -2.00000 q^{77} -12.0000 q^{79} -1.00000 q^{80} +3.00000 q^{82} +4.00000 q^{83} -2.00000 q^{85} -11.0000 q^{86} -2.00000 q^{88} -5.00000 q^{91} -1.00000 q^{92} +9.00000 q^{94} +2.00000 q^{95} -3.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) −5.00000 −0.980581
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) −5.00000 −0.693375
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −5.00000 −0.656532
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 5.00000 0.620174
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −7.00000 −0.813733
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 3.00000 0.331295
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) −11.0000 −1.18616
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −5.00000 −0.524142
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 9.00000 0.928279
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −3.00000 −0.304604 −0.152302 0.988334i \(-0.548669\pi\)
−0.152302 + 0.988334i \(0.548669\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) 19.0000 1.87213 0.936063 0.351833i \(-0.114441\pi\)
0.936063 + 0.351833i \(0.114441\pi\)
\(104\) −5.00000 −0.490290
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −14.0000 −1.35343 −0.676716 0.736245i \(-0.736597\pi\)
−0.676716 + 0.736245i \(0.736597\pi\)
\(108\) 0 0
\(109\) −3.00000 −0.287348 −0.143674 0.989625i \(-0.545892\pi\)
−0.143674 + 0.989625i \(0.545892\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 11.0000 1.03479 0.517396 0.855746i \(-0.326901\pi\)
0.517396 + 0.855746i \(0.326901\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) −5.00000 −0.464238
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −11.0000 −0.976092 −0.488046 0.872818i \(-0.662290\pi\)
−0.488046 + 0.872818i \(0.662290\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 5.00000 0.438529
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 10.0000 0.836242
\(144\) 0 0
\(145\) 5.00000 0.415227
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) −11.0000 −0.895167 −0.447584 0.894242i \(-0.647715\pi\)
−0.447584 + 0.894242i \(0.647715\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −12.0000 −0.954669
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −11.0000 −0.838742
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 0 0
\(179\) 25.0000 1.86859 0.934294 0.356504i \(-0.116031\pi\)
0.934294 + 0.356504i \(0.116031\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) −5.00000 −0.370625
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) 7.00000 0.514650
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 9.00000 0.656392
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) −13.0000 −0.935760 −0.467880 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) −3.00000 −0.215387
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −15.0000 −1.06871 −0.534353 0.845262i \(-0.679445\pi\)
−0.534353 + 0.845262i \(0.679445\pi\)
\(198\) 0 0
\(199\) 17.0000 1.20510 0.602549 0.798082i \(-0.294152\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) −14.0000 −0.985037
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) −3.00000 −0.209529
\(206\) 19.0000 1.32379
\(207\) 0 0
\(208\) −5.00000 −0.346688
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) −14.0000 −0.957020
\(215\) 11.0000 0.750194
\(216\) 0 0
\(217\) 0 0
\(218\) −3.00000 −0.203186
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) −10.0000 −0.672673
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 11.0000 0.731709
\(227\) −15.0000 −0.995585 −0.497792 0.867296i \(-0.665856\pi\)
−0.497792 + 0.867296i \(0.665856\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 1.00000 0.0659380
\(231\) 0 0
\(232\) −5.00000 −0.328266
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) 30.0000 1.94054 0.970269 0.242028i \(-0.0778125\pi\)
0.970269 + 0.242028i \(0.0778125\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 10.0000 0.636285
\(248\) 0 0
\(249\) 0 0
\(250\) 9.00000 0.569210
\(251\) 1.00000 0.0631194 0.0315597 0.999502i \(-0.489953\pi\)
0.0315597 + 0.999502i \(0.489953\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) −11.0000 −0.690201
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −7.00000 −0.434959
\(260\) 5.00000 0.310087
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) 1.00000 0.0616626 0.0308313 0.999525i \(-0.490185\pi\)
0.0308313 + 0.999525i \(0.490185\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 3.00000 0.181237
\(275\) 8.00000 0.482418
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 5.00000 0.299880
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 0 0
\(283\) −32.0000 −1.90220 −0.951101 0.308879i \(-0.900046\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 10.0000 0.591312
\(287\) 3.00000 0.177084
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 5.00000 0.293610
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) −7.00000 −0.406867
\(297\) 0 0
\(298\) 4.00000 0.231714
\(299\) 5.00000 0.289157
\(300\) 0 0
\(301\) −11.0000 −0.634029
\(302\) −11.0000 −0.632979
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −25.0000 −1.42683 −0.713413 0.700744i \(-0.752851\pi\)
−0.713413 + 0.700744i \(0.752851\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 25.0000 1.40414 0.702070 0.712108i \(-0.252259\pi\)
0.702070 + 0.712108i \(0.252259\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) 20.0000 1.10940
\(326\) 10.0000 0.553849
\(327\) 0 0
\(328\) 3.00000 0.165647
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) 26.0000 1.42909 0.714545 0.699590i \(-0.246634\pi\)
0.714545 + 0.699590i \(0.246634\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −11.0000 −0.593080
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 25.0000 1.34207 0.671035 0.741426i \(-0.265850\pi\)
0.671035 + 0.741426i \(0.265850\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) −23.0000 −1.22417 −0.612083 0.790793i \(-0.709668\pi\)
−0.612083 + 0.790793i \(0.709668\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 0 0
\(357\) 0 0
\(358\) 25.0000 1.32129
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 18.0000 0.946059
\(363\) 0 0
\(364\) −5.00000 −0.262071
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −15.0000 −0.782994 −0.391497 0.920179i \(-0.628043\pi\)
−0.391497 + 0.920179i \(0.628043\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 7.00000 0.363913
\(371\) 10.0000 0.519174
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 9.00000 0.464140
\(377\) 25.0000 1.28757
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) −13.0000 −0.661683
\(387\) 0 0
\(388\) −3.00000 −0.152302
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −15.0000 −0.755689
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 17.0000 0.852133
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −5.00000 −0.248146
\(407\) 14.0000 0.693954
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) −3.00000 −0.148159
\(411\) 0 0
\(412\) 19.0000 0.936063
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 4.00000 0.195646
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) 13.0000 0.633581 0.316791 0.948495i \(-0.397395\pi\)
0.316791 + 0.948495i \(0.397395\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) −14.0000 −0.676716
\(429\) 0 0
\(430\) 11.0000 0.530467
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 0 0
\(433\) 19.0000 0.913082 0.456541 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.00000 −0.143674
\(437\) 2.00000 0.0956730
\(438\) 0 0
\(439\) 2.00000 0.0954548 0.0477274 0.998860i \(-0.484802\pi\)
0.0477274 + 0.998860i \(0.484802\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) −10.0000 −0.475651
\(443\) −29.0000 −1.37783 −0.688916 0.724841i \(-0.741913\pi\)
−0.688916 + 0.724841i \(0.741913\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −32.0000 −1.51017 −0.755087 0.655625i \(-0.772405\pi\)
−0.755087 + 0.655625i \(0.772405\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 11.0000 0.517396
\(453\) 0 0
\(454\) −15.0000 −0.703985
\(455\) 5.00000 0.234404
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 1.00000 0.0466252
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) −17.0000 −0.790057 −0.395029 0.918669i \(-0.629265\pi\)
−0.395029 + 0.918669i \(0.629265\pi\)
\(464\) −5.00000 −0.232119
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −1.00000 −0.0462745 −0.0231372 0.999732i \(-0.507365\pi\)
−0.0231372 + 0.999732i \(0.507365\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) −9.00000 −0.415139
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) 22.0000 1.01156
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) 30.0000 1.37217
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 0 0
\(481\) 35.0000 1.59586
\(482\) −5.00000 −0.227744
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 3.00000 0.136223
\(486\) 0 0
\(487\) 7.00000 0.317200 0.158600 0.987343i \(-0.449302\pi\)
0.158600 + 0.987343i \(0.449302\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −10.0000 −0.450377
\(494\) 10.0000 0.449921
\(495\) 0 0
\(496\) 0 0
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 9.00000 0.402492
\(501\) 0 0
\(502\) 1.00000 0.0446322
\(503\) 18.0000 0.802580 0.401290 0.915951i \(-0.368562\pi\)
0.401290 + 0.915951i \(0.368562\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 2.00000 0.0889108
\(507\) 0 0
\(508\) −11.0000 −0.488046
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) −19.0000 −0.837240
\(516\) 0 0
\(517\) −18.0000 −0.791639
\(518\) −7.00000 −0.307562
\(519\) 0 0
\(520\) 5.00000 0.219265
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) 0 0
\(523\) 38.0000 1.66162 0.830812 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) 1.00000 0.0436021
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −10.0000 −0.434372
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) −15.0000 −0.649722
\(534\) 0 0
\(535\) 14.0000 0.605273
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 14.0000 0.603583
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −10.0000 −0.429537
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) 42.0000 1.79579 0.897895 0.440209i \(-0.145096\pi\)
0.897895 + 0.440209i \(0.145096\pi\)
\(548\) 3.00000 0.128154
\(549\) 0 0
\(550\) 8.00000 0.341121
\(551\) 10.0000 0.426014
\(552\) 0 0
\(553\) −12.0000 −0.510292
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) 5.00000 0.212047
\(557\) −22.0000 −0.932170 −0.466085 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) 0 0
\(559\) 55.0000 2.32625
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −15.0000 −0.632737
\(563\) 17.0000 0.716465 0.358232 0.933632i \(-0.383380\pi\)
0.358232 + 0.933632i \(0.383380\pi\)
\(564\) 0 0
\(565\) −11.0000 −0.462773
\(566\) −32.0000 −1.34506
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −11.0000 −0.461144 −0.230572 0.973055i \(-0.574060\pi\)
−0.230572 + 0.973055i \(0.574060\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 10.0000 0.418121
\(573\) 0 0
\(574\) 3.00000 0.125218
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −12.0000 −0.499567 −0.249783 0.968302i \(-0.580359\pi\)
−0.249783 + 0.968302i \(0.580359\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 5.00000 0.207614
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −20.0000 −0.828315
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 26.0000 1.07405
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 12.0000 0.494032
\(591\) 0 0
\(592\) −7.00000 −0.287698
\(593\) 15.0000 0.615976 0.307988 0.951390i \(-0.400344\pi\)
0.307988 + 0.951390i \(0.400344\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) 5.00000 0.204465
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −11.0000 −0.448327
\(603\) 0 0
\(604\) −11.0000 −0.447584
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) −45.0000 −1.82051
\(612\) 0 0
\(613\) 11.0000 0.444286 0.222143 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) −25.0000 −1.00892
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 22.0000 0.884255 0.442127 0.896952i \(-0.354224\pi\)
0.442127 + 0.896952i \(0.354224\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16.0000 0.641542
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) −14.0000 −0.558217
\(630\) 0 0
\(631\) 26.0000 1.03504 0.517522 0.855670i \(-0.326855\pi\)
0.517522 + 0.855670i \(0.326855\pi\)
\(632\) −12.0000 −0.477334
\(633\) 0 0
\(634\) 25.0000 0.992877
\(635\) 11.0000 0.436522
\(636\) 0 0
\(637\) −5.00000 −0.198107
\(638\) 10.0000 0.395904
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −23.0000 −0.908445 −0.454223 0.890888i \(-0.650083\pi\)
−0.454223 + 0.890888i \(0.650083\pi\)
\(642\) 0 0
\(643\) −2.00000 −0.0788723 −0.0394362 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) −20.0000 −0.786281 −0.393141 0.919478i \(-0.628611\pi\)
−0.393141 + 0.919478i \(0.628611\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 20.0000 0.784465
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) −33.0000 −1.29139 −0.645695 0.763596i \(-0.723432\pi\)
−0.645695 + 0.763596i \(0.723432\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) 3.00000 0.117130
\(657\) 0 0
\(658\) 9.00000 0.350857
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 24.0000 0.933492 0.466746 0.884391i \(-0.345426\pi\)
0.466746 + 0.884391i \(0.345426\pi\)
\(662\) 26.0000 1.01052
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 2.00000 0.0775567
\(666\) 0 0
\(667\) 5.00000 0.193601
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) −29.0000 −1.11787 −0.558934 0.829212i \(-0.688789\pi\)
−0.558934 + 0.829212i \(0.688789\pi\)
\(674\) 34.0000 1.30963
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) −3.00000 −0.115129
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) −3.00000 −0.114624
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −11.0000 −0.419371
\(689\) −50.0000 −1.90485
\(690\) 0 0
\(691\) 23.0000 0.874961 0.437481 0.899228i \(-0.355871\pi\)
0.437481 + 0.899228i \(0.355871\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 25.0000 0.948987
\(695\) −5.00000 −0.189661
\(696\) 0 0
\(697\) 6.00000 0.227266
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) 14.0000 0.528020
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −23.0000 −0.865616
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 6.00000 0.225176
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −10.0000 −0.373979
\(716\) 25.0000 0.934294
\(717\) 0 0
\(718\) 15.0000 0.559795
\(719\) −39.0000 −1.45445 −0.727227 0.686397i \(-0.759191\pi\)
−0.727227 + 0.686397i \(0.759191\pi\)
\(720\) 0 0
\(721\) 19.0000 0.707597
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) 18.0000 0.668965
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) −5.00000 −0.185312
\(729\) 0 0
\(730\) −6.00000 −0.222070
\(731\) −22.0000 −0.813699
\(732\) 0 0
\(733\) 16.0000 0.590973 0.295487 0.955347i \(-0.404518\pi\)
0.295487 + 0.955347i \(0.404518\pi\)
\(734\) −15.0000 −0.553660
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) −14.0000 −0.514998 −0.257499 0.966279i \(-0.582898\pi\)
−0.257499 + 0.966279i \(0.582898\pi\)
\(740\) 7.00000 0.257325
\(741\) 0 0
\(742\) 10.0000 0.367112
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) −4.00000 −0.146549
\(746\) 26.0000 0.951928
\(747\) 0 0
\(748\) −4.00000 −0.146254
\(749\) −14.0000 −0.511549
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 9.00000 0.328196
\(753\) 0 0
\(754\) 25.0000 0.910446
\(755\) 11.0000 0.400331
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) −5.00000 −0.181608
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) −3.00000 −0.108607
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) 60.0000 2.16647
\(768\) 0 0
\(769\) −35.0000 −1.26213 −0.631066 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) 2.00000 0.0720750
\(771\) 0 0
\(772\) −13.0000 −0.467880
\(773\) 7.00000 0.251773 0.125886 0.992045i \(-0.459823\pi\)
0.125886 + 0.992045i \(0.459823\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3.00000 −0.107694
\(777\) 0 0
\(778\) 12.0000 0.430221
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) −2.00000 −0.0715199
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) −24.0000 −0.855508 −0.427754 0.903895i \(-0.640695\pi\)
−0.427754 + 0.903895i \(0.640695\pi\)
\(788\) −15.0000 −0.534353
\(789\) 0 0
\(790\) 12.0000 0.426941
\(791\) 11.0000 0.391115
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 17.0000 0.602549
\(797\) −5.00000 −0.177109 −0.0885545 0.996071i \(-0.528225\pi\)
−0.0885545 + 0.996071i \(0.528225\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) −6.00000 −0.211867
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) 1.00000 0.0352454
\(806\) 0 0
\(807\) 0 0
\(808\) −14.0000 −0.492518
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 29.0000 1.01833 0.509164 0.860670i \(-0.329955\pi\)
0.509164 + 0.860670i \(0.329955\pi\)
\(812\) −5.00000 −0.175466
\(813\) 0 0
\(814\) 14.0000 0.490700
\(815\) −10.0000 −0.350285
\(816\) 0 0
\(817\) 22.0000 0.769683
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) −3.00000 −0.104765
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) −27.0000 −0.941161 −0.470580 0.882357i \(-0.655955\pi\)
−0.470580 + 0.882357i \(0.655955\pi\)
\(824\) 19.0000 0.661896
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 0 0
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) −5.00000 −0.173344
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) 24.0000 0.829066
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 13.0000 0.448010
\(843\) 0 0
\(844\) −8.00000 −0.275371
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) −7.00000 −0.240523
\(848\) 10.0000 0.343401
\(849\) 0 0
\(850\) −8.00000 −0.274398
\(851\) 7.00000 0.239957
\(852\) 0 0
\(853\) −11.0000 −0.376633 −0.188316 0.982108i \(-0.560303\pi\)
−0.188316 + 0.982108i \(0.560303\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) −14.0000 −0.478510
\(857\) −45.0000 −1.53717 −0.768585 0.639747i \(-0.779039\pi\)
−0.768585 + 0.639747i \(0.779039\pi\)
\(858\) 0 0
\(859\) 49.0000 1.67186 0.835929 0.548837i \(-0.184929\pi\)
0.835929 + 0.548837i \(0.184929\pi\)
\(860\) 11.0000 0.375097
\(861\) 0 0
\(862\) −3.00000 −0.102180
\(863\) 20.0000 0.680808 0.340404 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) 19.0000 0.645646
\(867\) 0 0
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) −3.00000 −0.101593
\(873\) 0 0
\(874\) 2.00000 0.0676510
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) −48.0000 −1.62084 −0.810422 0.585846i \(-0.800762\pi\)
−0.810422 + 0.585846i \(0.800762\pi\)
\(878\) 2.00000 0.0674967
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) 24.0000 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(884\) −10.0000 −0.336336
\(885\) 0 0
\(886\) −29.0000 −0.974274
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) −11.0000 −0.368928
\(890\) 0 0
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) −18.0000 −0.602347
\(894\) 0 0
\(895\) −25.0000 −0.835658
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −32.0000 −1.06785
\(899\) 0 0
\(900\) 0 0
\(901\) 20.0000 0.666297
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) 11.0000 0.365855
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) 7.00000 0.232431 0.116216 0.993224i \(-0.462924\pi\)
0.116216 + 0.993224i \(0.462924\pi\)
\(908\) −15.0000 −0.497792
\(909\) 0 0
\(910\) 5.00000 0.165748
\(911\) −17.0000 −0.563235 −0.281618 0.959527i \(-0.590871\pi\)
−0.281618 + 0.959527i \(0.590871\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 6.00000 0.198137
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 1.00000 0.0329690
\(921\) 0 0
\(922\) −24.0000 −0.790398
\(923\) 30.0000 0.987462
\(924\) 0 0
\(925\) 28.0000 0.920634
\(926\) −17.0000 −0.558655
\(927\) 0 0
\(928\) −5.00000 −0.164133
\(929\) −21.0000 −0.688988 −0.344494 0.938789i \(-0.611949\pi\)
−0.344494 + 0.938789i \(0.611949\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) −1.00000 −0.0327210
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) 37.0000 1.20874 0.604369 0.796705i \(-0.293425\pi\)
0.604369 + 0.796705i \(0.293425\pi\)
\(938\) −4.00000 −0.130605
\(939\) 0 0
\(940\) −9.00000 −0.293548
\(941\) −45.0000 −1.46696 −0.733479 0.679712i \(-0.762105\pi\)
−0.733479 + 0.679712i \(0.762105\pi\)
\(942\) 0 0
\(943\) −3.00000 −0.0976934
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 22.0000 0.715282
\(947\) −57.0000 −1.85225 −0.926126 0.377215i \(-0.876882\pi\)
−0.926126 + 0.377215i \(0.876882\pi\)
\(948\) 0 0
\(949\) −30.0000 −0.973841
\(950\) 8.00000 0.259554
\(951\) 0 0
\(952\) 2.00000 0.0648204
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 0 0
\(955\) 8.00000 0.258874
\(956\) 30.0000 0.970269
\(957\) 0 0
\(958\) 32.0000 1.03387
\(959\) 3.00000 0.0968751
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 35.0000 1.12845
\(963\) 0 0
\(964\) −5.00000 −0.161039
\(965\) 13.0000 0.418485
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 3.00000 0.0963242
\(971\) −52.0000 −1.66876 −0.834380 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) 0 0
\(973\) 5.00000 0.160293
\(974\) 7.00000 0.224294
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −21.0000 −0.671850 −0.335925 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 15.0000 0.477940
\(986\) −10.0000 −0.318465
\(987\) 0 0
\(988\) 10.0000 0.318142
\(989\) 11.0000 0.349780
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −6.00000 −0.190308
\(995\) −17.0000 −0.538936
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) −20.0000 −0.633089
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2898.2.a.m.1.1 yes 1
3.2 odd 2 2898.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2898.2.a.e.1.1 1 3.2 odd 2
2898.2.a.m.1.1 yes 1 1.1 even 1 trivial