Properties

Label 1323.2.a.c.1.1
Level $1323$
Weight $2$
Character 1323.1
Self dual yes
Analytic conductor $10.564$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,2,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1323.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.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: \(10.5642081874\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 1323.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{5} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{5} -6.00000 q^{10} -2.00000 q^{11} -6.00000 q^{13} -4.00000 q^{16} +3.00000 q^{17} +6.00000 q^{19} +6.00000 q^{20} +4.00000 q^{22} +8.00000 q^{23} +4.00000 q^{25} +12.0000 q^{26} +2.00000 q^{29} -6.00000 q^{31} +8.00000 q^{32} -6.00000 q^{34} +9.00000 q^{37} -12.0000 q^{38} +9.00000 q^{41} -9.00000 q^{43} -4.00000 q^{44} -16.0000 q^{46} -3.00000 q^{47} -8.00000 q^{50} -12.0000 q^{52} -4.00000 q^{53} -6.00000 q^{55} -4.00000 q^{58} -3.00000 q^{59} +6.00000 q^{61} +12.0000 q^{62} -8.00000 q^{64} -18.0000 q^{65} +4.00000 q^{67} +6.00000 q^{68} +4.00000 q^{71} +12.0000 q^{73} -18.0000 q^{74} +12.0000 q^{76} -1.00000 q^{79} -12.0000 q^{80} -18.0000 q^{82} +15.0000 q^{83} +9.00000 q^{85} +18.0000 q^{86} -6.00000 q^{89} +16.0000 q^{92} +6.00000 q^{94} +18.0000 q^{95} +6.00000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) −6.00000 −1.89737
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 6.00000 1.34164
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 12.0000 2.35339
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 9.00000 1.47959 0.739795 0.672832i \(-0.234922\pi\)
0.739795 + 0.672832i \(0.234922\pi\)
\(38\) −12.0000 −1.94666
\(39\) 0 0
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −16.0000 −2.35907
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −8.00000 −1.13137
\(51\) 0 0
\(52\) −12.0000 −1.66410
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 12.0000 1.52400
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −18.0000 −2.23263
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) −18.0000 −2.09246
\(75\) 0 0
\(76\) 12.0000 1.37649
\(77\) 0 0
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −12.0000 −1.34164
\(81\) 0 0
\(82\) −18.0000 −1.98777
\(83\) 15.0000 1.64646 0.823232 0.567705i \(-0.192169\pi\)
0.823232 + 0.567705i \(0.192169\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 18.0000 1.94099
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 16.0000 1.66812
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 18.0000 1.84676
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 8.00000 0.800000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −9.00000 −0.862044 −0.431022 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) 12.0000 1.14416
\(111\) 0 0
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 24.0000 2.23801
\(116\) 4.00000 0.371391
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −12.0000 −1.08643
\(123\) 0 0
\(124\) −12.0000 −1.07763
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −11.0000 −0.976092 −0.488046 0.872818i \(-0.662290\pi\)
−0.488046 + 0.872818i \(0.662290\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 36.0000 3.15741
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) −24.0000 −1.98625
\(147\) 0 0
\(148\) 18.0000 1.47959
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 9.00000 0.732410 0.366205 0.930534i \(-0.380657\pi\)
0.366205 + 0.930534i \(0.380657\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −18.0000 −1.44579
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 2.00000 0.159111
\(159\) 0 0
\(160\) 24.0000 1.89737
\(161\) 0 0
\(162\) 0 0
\(163\) −11.0000 −0.861586 −0.430793 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 18.0000 1.40556
\(165\) 0 0
\(166\) −30.0000 −2.32845
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −18.0000 −1.38054
\(171\) 0 0
\(172\) −18.0000 −1.37249
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 8.00000 0.603023
\(177\) 0 0
\(178\) 12.0000 0.899438
\(179\) −14.0000 −1.04641 −0.523205 0.852207i \(-0.675264\pi\)
−0.523205 + 0.852207i \(0.675264\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 27.0000 1.98508
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) −36.0000 −2.61171
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −9.00000 −0.647834 −0.323917 0.946085i \(-0.605000\pi\)
−0.323917 + 0.946085i \(0.605000\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −12.0000 −0.844317
\(203\) 0 0
\(204\) 0 0
\(205\) 27.0000 1.88576
\(206\) −24.0000 −1.67216
\(207\) 0 0
\(208\) 24.0000 1.66410
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −8.00000 −0.549442
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) −27.0000 −1.84138
\(216\) 0 0
\(217\) 0 0
\(218\) 18.0000 1.21911
\(219\) 0 0
\(220\) −12.0000 −0.809040
\(221\) −18.0000 −1.21081
\(222\) 0 0
\(223\) −18.0000 −1.20537 −0.602685 0.797980i \(-0.705902\pi\)
−0.602685 + 0.797980i \(0.705902\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 20.0000 1.33038
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) −48.0000 −3.16503
\(231\) 0 0
\(232\) 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
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) 0 0
\(239\) 10.0000 0.646846 0.323423 0.946254i \(-0.395166\pi\)
0.323423 + 0.946254i \(0.395166\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 14.0000 0.899954
\(243\) 0 0
\(244\) 12.0000 0.768221
\(245\) 0 0
\(246\) 0 0
\(247\) −36.0000 −2.29063
\(248\) 0 0
\(249\) 0 0
\(250\) 6.00000 0.379473
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) 22.0000 1.38040
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −36.0000 −2.23263
\(261\) 0 0
\(262\) 0 0
\(263\) 2.00000 0.123325 0.0616626 0.998097i \(-0.480360\pi\)
0.0616626 + 0.998097i \(0.480360\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 15.0000 0.914566 0.457283 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) −12.0000 −0.727607
\(273\) 0 0
\(274\) −4.00000 −0.241649
\(275\) −8.00000 −0.482418
\(276\) 0 0
\(277\) −19.0000 −1.14160 −0.570800 0.821089i \(-0.693367\pi\)
−0.570800 + 0.821089i \(0.693367\pi\)
\(278\) −12.0000 −0.719712
\(279\) 0 0
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) 0 0
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) −12.0000 −0.704664
\(291\) 0 0
\(292\) 24.0000 1.40449
\(293\) 21.0000 1.22683 0.613417 0.789760i \(-0.289795\pi\)
0.613417 + 0.789760i \(0.289795\pi\)
\(294\) 0 0
\(295\) −9.00000 −0.524000
\(296\) 0 0
\(297\) 0 0
\(298\) −20.0000 −1.15857
\(299\) −48.0000 −2.77591
\(300\) 0 0
\(301\) 0 0
\(302\) −18.0000 −1.03578
\(303\) 0 0
\(304\) −24.0000 −1.37649
\(305\) 18.0000 1.03068
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 36.0000 2.04466
\(311\) −27.0000 −1.53103 −0.765515 0.643418i \(-0.777516\pi\)
−0.765515 + 0.643418i \(0.777516\pi\)
\(312\) 0 0
\(313\) 24.0000 1.35656 0.678280 0.734803i \(-0.262726\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(314\) −36.0000 −2.03160
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) 20.0000 1.12331 0.561656 0.827371i \(-0.310164\pi\)
0.561656 + 0.827371i \(0.310164\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) −24.0000 −1.34164
\(321\) 0 0
\(322\) 0 0
\(323\) 18.0000 1.00155
\(324\) 0 0
\(325\) −24.0000 −1.33128
\(326\) 22.0000 1.21847
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −9.00000 −0.494685 −0.247342 0.968928i \(-0.579557\pi\)
−0.247342 + 0.968928i \(0.579557\pi\)
\(332\) 30.0000 1.64646
\(333\) 0 0
\(334\) −18.0000 −0.984916
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 27.0000 1.47078 0.735392 0.677642i \(-0.236998\pi\)
0.735392 + 0.677642i \(0.236998\pi\)
\(338\) −46.0000 −2.50207
\(339\) 0 0
\(340\) 18.0000 0.976187
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −36.0000 −1.93537
\(347\) −26.0000 −1.39575 −0.697877 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) 0 0
\(349\) −12.0000 −0.642345 −0.321173 0.947021i \(-0.604077\pi\)
−0.321173 + 0.947021i \(0.604077\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −16.0000 −0.852803
\(353\) 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
\(356\) −12.0000 −0.635999
\(357\) 0 0
\(358\) 28.0000 1.47985
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −24.0000 −1.26141
\(363\) 0 0
\(364\) 0 0
\(365\) 36.0000 1.88433
\(366\) 0 0
\(367\) −36.0000 −1.87918 −0.939592 0.342296i \(-0.888796\pi\)
−0.939592 + 0.342296i \(0.888796\pi\)
\(368\) −32.0000 −1.66812
\(369\) 0 0
\(370\) −54.0000 −2.80733
\(371\) 0 0
\(372\) 0 0
\(373\) −5.00000 −0.258890 −0.129445 0.991587i \(-0.541320\pi\)
−0.129445 + 0.991587i \(0.541320\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 9.00000 0.462299 0.231149 0.972918i \(-0.425751\pi\)
0.231149 + 0.972918i \(0.425751\pi\)
\(380\) 36.0000 1.84676
\(381\) 0 0
\(382\) 32.0000 1.63726
\(383\) −3.00000 −0.153293 −0.0766464 0.997058i \(-0.524421\pi\)
−0.0766464 + 0.997058i \(0.524421\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 0 0
\(388\) 12.0000 0.609208
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 0 0
\(394\) −44.0000 −2.21669
\(395\) −3.00000 −0.150946
\(396\) 0 0
\(397\) −12.0000 −0.602263 −0.301131 0.953583i \(-0.597364\pi\)
−0.301131 + 0.953583i \(0.597364\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −16.0000 −0.800000
\(401\) −28.0000 −1.39825 −0.699127 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) 0 0
\(403\) 36.0000 1.79329
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) −18.0000 −0.892227
\(408\) 0 0
\(409\) −12.0000 −0.593362 −0.296681 0.954977i \(-0.595880\pi\)
−0.296681 + 0.954977i \(0.595880\pi\)
\(410\) −54.0000 −2.66687
\(411\) 0 0
\(412\) 24.0000 1.18240
\(413\) 0 0
\(414\) 0 0
\(415\) 45.0000 2.20896
\(416\) −48.0000 −2.35339
\(417\) 0 0
\(418\) 24.0000 1.17388
\(419\) −21.0000 −1.02592 −0.512959 0.858413i \(-0.671451\pi\)
−0.512959 + 0.858413i \(0.671451\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 54.0000 2.60411
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) 0 0
\(433\) −18.0000 −0.865025 −0.432512 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) 48.0000 2.29615
\(438\) 0 0
\(439\) −12.0000 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 36.0000 1.71235
\(443\) −2.00000 −0.0950229 −0.0475114 0.998871i \(-0.515129\pi\)
−0.0475114 + 0.998871i \(0.515129\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 36.0000 1.70465
\(447\) 0 0
\(448\) 0 0
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) −20.0000 −0.940721
\(453\) 0 0
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 48.0000 2.23801
\(461\) −3.00000 −0.139724 −0.0698620 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 0 0
\(463\) 5.00000 0.232370 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) −44.0000 −2.03826
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 18.0000 0.830278
\(471\) 0 0
\(472\) 0 0
\(473\) 18.0000 0.827641
\(474\) 0 0
\(475\) 24.0000 1.10120
\(476\) 0 0
\(477\) 0 0
\(478\) −20.0000 −0.914779
\(479\) 15.0000 0.685367 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) −54.0000 −2.46219
\(482\) 36.0000 1.63976
\(483\) 0 0
\(484\) −14.0000 −0.636364
\(485\) 18.0000 0.817338
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −10.0000 −0.451294 −0.225647 0.974209i \(-0.572450\pi\)
−0.225647 + 0.974209i \(0.572450\pi\)
\(492\) 0 0
\(493\) 6.00000 0.270226
\(494\) 72.0000 3.23943
\(495\) 0 0
\(496\) 24.0000 1.07763
\(497\) 0 0
\(498\) 0 0
\(499\) 9.00000 0.402895 0.201448 0.979499i \(-0.435435\pi\)
0.201448 + 0.979499i \(0.435435\pi\)
\(500\) −6.00000 −0.268328
\(501\) 0 0
\(502\) −6.00000 −0.267793
\(503\) 33.0000 1.47140 0.735699 0.677309i \(-0.236854\pi\)
0.735699 + 0.677309i \(0.236854\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 32.0000 1.42257
\(507\) 0 0
\(508\) −22.0000 −0.976092
\(509\) 3.00000 0.132973 0.0664863 0.997787i \(-0.478821\pi\)
0.0664863 + 0.997787i \(0.478821\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) 36.0000 1.58635
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15.0000 0.657162 0.328581 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) 0 0
\(523\) 6.00000 0.262362 0.131181 0.991358i \(-0.458123\pi\)
0.131181 + 0.991358i \(0.458123\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) −18.0000 −0.784092
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 24.0000 1.04249
\(531\) 0 0
\(532\) 0 0
\(533\) −54.0000 −2.33900
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 0 0
\(538\) −30.0000 −1.29339
\(539\) 0 0
\(540\) 0 0
\(541\) −7.00000 −0.300954 −0.150477 0.988614i \(-0.548081\pi\)
−0.150477 + 0.988614i \(0.548081\pi\)
\(542\) 24.0000 1.03089
\(543\) 0 0
\(544\) 24.0000 1.02899
\(545\) −27.0000 −1.15655
\(546\) 0 0
\(547\) −37.0000 −1.58201 −0.791003 0.611812i \(-0.790441\pi\)
−0.791003 + 0.611812i \(0.790441\pi\)
\(548\) 4.00000 0.170872
\(549\) 0 0
\(550\) 16.0000 0.682242
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) 38.0000 1.61447
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −32.0000 −1.35588 −0.677942 0.735116i \(-0.737128\pi\)
−0.677942 + 0.735116i \(0.737128\pi\)
\(558\) 0 0
\(559\) 54.0000 2.28396
\(560\) 0 0
\(561\) 0 0
\(562\) 40.0000 1.68730
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) −30.0000 −1.26211
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) 0 0
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 0 0
\(571\) −41.0000 −1.71580 −0.857898 0.513820i \(-0.828230\pi\)
−0.857898 + 0.513820i \(0.828230\pi\)
\(572\) 24.0000 1.00349
\(573\) 0 0
\(574\) 0 0
\(575\) 32.0000 1.33449
\(576\) 0 0
\(577\) 12.0000 0.499567 0.249783 0.968302i \(-0.419641\pi\)
0.249783 + 0.968302i \(0.419641\pi\)
\(578\) 16.0000 0.665512
\(579\) 0 0
\(580\) 12.0000 0.498273
\(581\) 0 0
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) 0 0
\(585\) 0 0
\(586\) −42.0000 −1.73500
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) −36.0000 −1.48335
\(590\) 18.0000 0.741048
\(591\) 0 0
\(592\) −36.0000 −1.47959
\(593\) 39.0000 1.60154 0.800769 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 20.0000 0.819232
\(597\) 0 0
\(598\) 96.0000 3.92573
\(599\) −46.0000 −1.87951 −0.939755 0.341850i \(-0.888947\pi\)
−0.939755 + 0.341850i \(0.888947\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 18.0000 0.732410
\(605\) −21.0000 −0.853771
\(606\) 0 0
\(607\) −42.0000 −1.70473 −0.852364 0.522949i \(-0.824832\pi\)
−0.852364 + 0.522949i \(0.824832\pi\)
\(608\) 48.0000 1.94666
\(609\) 0 0
\(610\) −36.0000 −1.45760
\(611\) 18.0000 0.728202
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 24.0000 0.968561
\(615\) 0 0
\(616\) 0 0
\(617\) −8.00000 −0.322068 −0.161034 0.986949i \(-0.551483\pi\)
−0.161034 + 0.986949i \(0.551483\pi\)
\(618\) 0 0
\(619\) 6.00000 0.241160 0.120580 0.992704i \(-0.461525\pi\)
0.120580 + 0.992704i \(0.461525\pi\)
\(620\) −36.0000 −1.44579
\(621\) 0 0
\(622\) 54.0000 2.16520
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −48.0000 −1.91847
\(627\) 0 0
\(628\) 36.0000 1.43656
\(629\) 27.0000 1.07656
\(630\) 0 0
\(631\) 25.0000 0.995234 0.497617 0.867397i \(-0.334208\pi\)
0.497617 + 0.867397i \(0.334208\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −40.0000 −1.58860
\(635\) −33.0000 −1.30957
\(636\) 0 0
\(637\) 0 0
\(638\) 8.00000 0.316723
\(639\) 0 0
\(640\) 0 0
\(641\) −38.0000 −1.50091 −0.750455 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) 0 0
\(643\) −12.0000 −0.473234 −0.236617 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −36.0000 −1.41640
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) 0 0
\(649\) 6.00000 0.235521
\(650\) 48.0000 1.88271
\(651\) 0 0
\(652\) −22.0000 −0.861586
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −36.0000 −1.40556
\(657\) 0 0
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 18.0000 0.699590
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.0000 0.619522
\(668\) 18.0000 0.696441
\(669\) 0 0
\(670\) −24.0000 −0.927201
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) −54.0000 −2.08000
\(675\) 0 0
\(676\) 46.0000 1.76923
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −24.0000 −0.919007
\(683\) 14.0000 0.535695 0.267848 0.963461i \(-0.413688\pi\)
0.267848 + 0.963461i \(0.413688\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 0 0
\(687\) 0 0
\(688\) 36.0000 1.37249
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) −18.0000 −0.684752 −0.342376 0.939563i \(-0.611232\pi\)
−0.342376 + 0.939563i \(0.611232\pi\)
\(692\) 36.0000 1.36851
\(693\) 0 0
\(694\) 52.0000 1.97389
\(695\) 18.0000 0.682779
\(696\) 0 0
\(697\) 27.0000 1.02270
\(698\) 24.0000 0.908413
\(699\) 0 0
\(700\) 0 0
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 0 0
\(703\) 54.0000 2.03665
\(704\) 16.0000 0.603023
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) 27.0000 1.01401 0.507003 0.861944i \(-0.330753\pi\)
0.507003 + 0.861944i \(0.330753\pi\)
\(710\) −24.0000 −0.900704
\(711\) 0 0
\(712\) 0 0
\(713\) −48.0000 −1.79761
\(714\) 0 0
\(715\) 36.0000 1.34632
\(716\) −28.0000 −1.04641
\(717\) 0 0
\(718\) 8.00000 0.298557
\(719\) −51.0000 −1.90198 −0.950990 0.309223i \(-0.899931\pi\)
−0.950990 + 0.309223i \(0.899931\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −34.0000 −1.26535
\(723\) 0 0
\(724\) 24.0000 0.891953
\(725\) 8.00000 0.297113
\(726\) 0 0
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −72.0000 −2.66484
\(731\) −27.0000 −0.998631
\(732\) 0 0
\(733\) 48.0000 1.77292 0.886460 0.462805i \(-0.153157\pi\)
0.886460 + 0.462805i \(0.153157\pi\)
\(734\) 72.0000 2.65757
\(735\) 0 0
\(736\) 64.0000 2.35907
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 54.0000 1.98508
\(741\) 0 0
\(742\) 0 0
\(743\) −10.0000 −0.366864 −0.183432 0.983032i \(-0.558721\pi\)
−0.183432 + 0.983032i \(0.558721\pi\)
\(744\) 0 0
\(745\) 30.0000 1.09911
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) −12.0000 −0.438763
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 24.0000 0.874028
\(755\) 27.0000 0.982631
\(756\) 0 0
\(757\) −45.0000 −1.63555 −0.817776 0.575536i \(-0.804793\pi\)
−0.817776 + 0.575536i \(0.804793\pi\)
\(758\) −18.0000 −0.653789
\(759\) 0 0
\(760\) 0 0
\(761\) 21.0000 0.761249 0.380625 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −32.0000 −1.15772
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) 18.0000 0.649942
\(768\) 0 0
\(769\) 36.0000 1.29819 0.649097 0.760706i \(-0.275147\pi\)
0.649097 + 0.760706i \(0.275147\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −18.0000 −0.647834
\(773\) −27.0000 −0.971123 −0.485561 0.874203i \(-0.661385\pi\)
−0.485561 + 0.874203i \(0.661385\pi\)
\(774\) 0 0
\(775\) −24.0000 −0.862105
\(776\) 0 0
\(777\) 0 0
\(778\) −68.0000 −2.43792
\(779\) 54.0000 1.93475
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) −48.0000 −1.71648
\(783\) 0 0
\(784\) 0 0
\(785\) 54.0000 1.92734
\(786\) 0 0
\(787\) −48.0000 −1.71102 −0.855508 0.517790i \(-0.826755\pi\)
−0.855508 + 0.517790i \(0.826755\pi\)
\(788\) 44.0000 1.56744
\(789\) 0 0
\(790\) 6.00000 0.213470
\(791\) 0 0
\(792\) 0 0
\(793\) −36.0000 −1.27840
\(794\) 24.0000 0.851728
\(795\) 0 0
\(796\) 0 0
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) −9.00000 −0.318397
\(800\) 32.0000 1.13137
\(801\) 0 0
\(802\) 56.0000 1.97743
\(803\) −24.0000 −0.846942
\(804\) 0 0
\(805\) 0 0
\(806\) −72.0000 −2.53609
\(807\) 0 0
\(808\) 0 0
\(809\) 4.00000 0.140633 0.0703163 0.997525i \(-0.477599\pi\)
0.0703163 + 0.997525i \(0.477599\pi\)
\(810\) 0 0
\(811\) −42.0000 −1.47482 −0.737410 0.675446i \(-0.763951\pi\)
−0.737410 + 0.675446i \(0.763951\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 36.0000 1.26180
\(815\) −33.0000 −1.15594
\(816\) 0 0
\(817\) −54.0000 −1.88922
\(818\) 24.0000 0.839140
\(819\) 0 0
\(820\) 54.0000 1.88576
\(821\) −20.0000 −0.698005 −0.349002 0.937122i \(-0.613479\pi\)
−0.349002 + 0.937122i \(0.613479\pi\)
\(822\) 0 0
\(823\) −31.0000 −1.08059 −0.540296 0.841475i \(-0.681688\pi\)
−0.540296 + 0.841475i \(0.681688\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −22.0000 −0.765015 −0.382507 0.923952i \(-0.624939\pi\)
−0.382507 + 0.923952i \(0.624939\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) −90.0000 −3.12395
\(831\) 0 0
\(832\) 48.0000 1.66410
\(833\) 0 0
\(834\) 0 0
\(835\) 27.0000 0.934374
\(836\) −24.0000 −0.830057
\(837\) 0 0
\(838\) 42.0000 1.45087
\(839\) −3.00000 −0.103572 −0.0517858 0.998658i \(-0.516491\pi\)
−0.0517858 + 0.998658i \(0.516491\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 52.0000 1.79204
\(843\) 0 0
\(844\) 0 0
\(845\) 69.0000 2.37367
\(846\) 0 0
\(847\) 0 0
\(848\) 16.0000 0.549442
\(849\) 0 0
\(850\) −24.0000 −0.823193
\(851\) 72.0000 2.46813
\(852\) 0 0
\(853\) 12.0000 0.410872 0.205436 0.978671i \(-0.434139\pi\)
0.205436 + 0.978671i \(0.434139\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.00000 0.307434 0.153717 0.988115i \(-0.450876\pi\)
0.153717 + 0.988115i \(0.450876\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) −54.0000 −1.84138
\(861\) 0 0
\(862\) 8.00000 0.272481
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 0 0
\(865\) 54.0000 1.83606
\(866\) 36.0000 1.22333
\(867\) 0 0
\(868\) 0 0
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) 0 0
\(874\) −96.0000 −3.24725
\(875\) 0 0
\(876\) 0 0
\(877\) −31.0000 −1.04680 −0.523398 0.852088i \(-0.675336\pi\)
−0.523398 + 0.852088i \(0.675336\pi\)
\(878\) 24.0000 0.809961
\(879\) 0 0
\(880\) 24.0000 0.809040
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) 45.0000 1.51437 0.757185 0.653200i \(-0.226574\pi\)
0.757185 + 0.653200i \(0.226574\pi\)
\(884\) −36.0000 −1.21081
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) −15.0000 −0.503651 −0.251825 0.967773i \(-0.581031\pi\)
−0.251825 + 0.967773i \(0.581031\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 36.0000 1.20672
\(891\) 0 0
\(892\) −36.0000 −1.20537
\(893\) −18.0000 −0.602347
\(894\) 0 0
\(895\) −42.0000 −1.40391
\(896\) 0 0
\(897\) 0 0
\(898\) 44.0000 1.46830
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 36.0000 1.19867
\(903\) 0 0
\(904\) 0 0
\(905\) 36.0000 1.19668
\(906\) 0 0
\(907\) −27.0000 −0.896520 −0.448260 0.893903i \(-0.647956\pi\)
−0.448260 + 0.893903i \(0.647956\pi\)
\(908\) 24.0000 0.796468
\(909\) 0 0
\(910\) 0 0
\(911\) −2.00000 −0.0662630 −0.0331315 0.999451i \(-0.510548\pi\)
−0.0331315 + 0.999451i \(0.510548\pi\)
\(912\) 0 0
\(913\) −30.0000 −0.992855
\(914\) −36.0000 −1.19077
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −27.0000 −0.890648 −0.445324 0.895370i \(-0.646911\pi\)
−0.445324 + 0.895370i \(0.646911\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 6.00000 0.197599
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) 36.0000 1.18367
\(926\) −10.0000 −0.328620
\(927\) 0 0
\(928\) 16.0000 0.525226
\(929\) 9.00000 0.295280 0.147640 0.989041i \(-0.452832\pi\)
0.147640 + 0.989041i \(0.452832\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 44.0000 1.44127
\(933\) 0 0
\(934\) −24.0000 −0.785304
\(935\) −18.0000 −0.588663
\(936\) 0 0
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −18.0000 −0.587095
\(941\) −9.00000 −0.293392 −0.146696 0.989182i \(-0.546864\pi\)
−0.146696 + 0.989182i \(0.546864\pi\)
\(942\) 0 0
\(943\) 72.0000 2.34464
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −36.0000 −1.17046
\(947\) −56.0000 −1.81976 −0.909878 0.414876i \(-0.863825\pi\)
−0.909878 + 0.414876i \(0.863825\pi\)
\(948\) 0 0
\(949\) −72.0000 −2.33722
\(950\) −48.0000 −1.55733
\(951\) 0 0
\(952\) 0 0
\(953\) 40.0000 1.29573 0.647864 0.761756i \(-0.275663\pi\)
0.647864 + 0.761756i \(0.275663\pi\)
\(954\) 0 0
\(955\) −48.0000 −1.55324
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) −30.0000 −0.969256
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 108.000 3.48206
\(963\) 0 0
\(964\) −36.0000 −1.15948
\(965\) −27.0000 −0.869161
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −36.0000 −1.15589
\(971\) 27.0000 0.866471 0.433236 0.901281i \(-0.357372\pi\)
0.433236 + 0.901281i \(0.357372\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −24.0000 −0.768221
\(977\) −8.00000 −0.255943 −0.127971 0.991778i \(-0.540847\pi\)
−0.127971 + 0.991778i \(0.540847\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) 0 0
\(982\) 20.0000 0.638226
\(983\) 33.0000 1.05254 0.526268 0.850319i \(-0.323591\pi\)
0.526268 + 0.850319i \(0.323591\pi\)
\(984\) 0 0
\(985\) 66.0000 2.10293
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) −72.0000 −2.29063
\(989\) −72.0000 −2.28947
\(990\) 0 0
\(991\) −9.00000 −0.285894 −0.142947 0.989730i \(-0.545658\pi\)
−0.142947 + 0.989730i \(0.545658\pi\)
\(992\) −48.0000 −1.52400
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 18.0000 0.570066 0.285033 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) −18.0000 −0.569780
\(999\) 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 1323.2.a.c.1.1 yes 1
3.2 odd 2 1323.2.a.q.1.1 yes 1
7.6 odd 2 1323.2.a.a.1.1 1
21.20 even 2 1323.2.a.s.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.2.a.a.1.1 1 7.6 odd 2
1323.2.a.c.1.1 yes 1 1.1 even 1 trivial
1323.2.a.q.1.1 yes 1 3.2 odd 2
1323.2.a.s.1.1 yes 1 21.20 even 2