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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} -3.00000 q^{5} +O(q^{10})\) \(q-2.00000 q^{4} -3.00000 q^{5} +6.00000 q^{11} +4.00000 q^{13} +4.00000 q^{16} -3.00000 q^{17} -2.00000 q^{19} +6.00000 q^{20} -6.00000 q^{23} +4.00000 q^{25} -6.00000 q^{29} +4.00000 q^{31} -7.00000 q^{37} +3.00000 q^{41} -1.00000 q^{43} -12.0000 q^{44} -9.00000 q^{47} -8.00000 q^{52} -6.00000 q^{53} -18.0000 q^{55} -9.00000 q^{59} +10.0000 q^{61} -8.00000 q^{64} -12.0000 q^{65} -4.00000 q^{67} +6.00000 q^{68} -2.00000 q^{73} +4.00000 q^{76} -1.00000 q^{79} -12.0000 q^{80} -3.00000 q^{83} +9.00000 q^{85} -6.00000 q^{89} +12.0000 q^{92} +6.00000 q^{95} +10.0000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 6.00000 1.34164
\(21\) 0 0
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −12.0000 −1.80907
\(45\) 0 0
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) −8.00000 −1.10940
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −18.0000 −2.42712
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 − 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 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
\(82\) 0 0
\(83\) −3.00000 −0.329293 −0.164646 − 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 0 0
\(87\) 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
\(92\) 12.0000 1.25109
\(93\) 0 0
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −8.00000 −0.800000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 − 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 − 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 − 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 18.0000 1.67851
\(116\) 12.0000 1.11417
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 − 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 22.0000 1.86602 0.933008 − 0.359856i \(-0.117174\pi\)
0.933008 + 0.359856i \(0.117174\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 24.0000 2.00698
\(144\) 0 0
\(145\) 18.0000 1.49482
\(146\) 0 0
\(147\) 0 0
\(148\) 14.0000 1.15079
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 − 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −12.0000 −0.963863
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −7.00000 −0.548282 −0.274141 − 0.961689i \(-0.588394\pi\)
−0.274141 + 0.961689i \(0.588394\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 3.00000 0.232147 0.116073 − 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 24.0000 1.80907
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 − 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 21.0000 1.54395
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) 18.0000 1.31278
\(189\) 0 0
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 − 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) −13.0000 −0.935760 −0.467880 − 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 − 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 0 0
\(207\) 0 0
\(208\) 16.0000 1.10940
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) 0 0
\(215\) 3.00000 0.204598
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 36.0000 2.42712
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 − 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 − 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) 27.0000 1.76129
\(236\) 18.0000 1.17170
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 − 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −20.0000 −1.28037
\(245\) 0 0
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −27.0000 −1.70422 −0.852112 − 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) −36.0000 −2.26330
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 24.0000 1.48842
\(261\) 0 0
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 18.0000 1.10573
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) −3.00000 −0.182913 −0.0914566 − 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 − 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) −12.0000 −0.727607
\(273\) 0 0
\(274\) 0 0
\(275\) 24.0000 1.44725
\(276\) 0 0
\(277\) 17.0000 1.02143 0.510716 − 0.859750i \(-0.329381\pi\)
0.510716 + 0.859750i \(0.329381\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 − 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) −33.0000 −1.92788 −0.963940 − 0.266119i \(-0.914259\pi\)
−0.963940 + 0.266119i \(0.914259\pi\)
\(294\) 0 0
\(295\) 27.0000 1.57200
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −8.00000 −0.458831
\(305\) −30.0000 −1.71780
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.00000 −0.510343 −0.255172 − 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 − 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −36.0000 −2.01561
\(320\) 24.0000 1.34164
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) 16.0000 0.887520
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.00000 −0.0549650 −0.0274825 − 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 − 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −18.0000 −0.976187
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 − 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 21.0000 1.11772 0.558859 − 0.829263i \(-0.311239\pi\)
0.558859 + 0.829263i \(0.311239\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 12.0000 0.635999
\(357\) 0 0
\(358\) 0 0
\(359\) 6.00000 0.316668 0.158334 − 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 10.0000 0.521996 0.260998 − 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) −24.0000 −1.25109
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 23.0000 1.19089 0.595447 − 0.803394i \(-0.296975\pi\)
0.595447 + 0.803394i \(0.296975\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) 29.0000 1.48963 0.744815 − 0.667271i \(-0.232538\pi\)
0.744815 + 0.667271i \(0.232538\pi\)
\(380\) −12.0000 −0.615587
\(381\) 0 0
\(382\) 0 0
\(383\) 15.0000 0.766464 0.383232 − 0.923652i \(-0.374811\pi\)
0.383232 + 0.923652i \(0.374811\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −20.0000 −1.01535
\(389\) 6.00000 0.304212 0.152106 − 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.00000 0.150946
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 − 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 16.0000 0.800000
\(401\) 12.0000 0.599251 0.299626 − 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) −42.0000 −2.08186
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 − 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 28.0000 1.37946
\(413\) 0 0
\(414\) 0 0
\(415\) 9.00000 0.441793
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 21.0000 1.02592 0.512959 − 0.858413i \(-0.328549\pi\)
0.512959 + 0.858413i \(0.328549\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 − 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 − 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −22.0000 −1.05361
\(437\) 12.0000 0.574038
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 − 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.0000 0.855206 0.427603 − 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) 0 0
\(445\) 18.0000 0.853282
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) 24.0000 1.12887
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000 1.21623 0.608114 − 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −36.0000 −1.67851
\(461\) 15.0000 0.698620 0.349310 − 0.937007i \(-0.386416\pi\)
0.349310 + 0.937007i \(0.386416\pi\)
\(462\) 0 0
\(463\) 5.00000 0.232370 0.116185 − 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) −24.0000 −1.11417
\(465\) 0 0
\(466\) 0 0
\(467\) −24.0000 −1.11059 −0.555294 − 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 21.0000 0.959514 0.479757 − 0.877401i \(-0.340725\pi\)
0.479757 + 0.877401i \(0.340725\pi\)
\(480\) 0 0
\(481\) −28.0000 −1.27669
\(482\) 0 0
\(483\) 0 0
\(484\) −50.0000 −2.27273
\(485\) −30.0000 −1.36223
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 − 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 − 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) 0 0
\(495\) 0 0
\(496\) 16.0000 0.718421
\(497\) 0 0
\(498\) 0 0
\(499\) 5.00000 0.223831 0.111915 − 0.993718i \(-0.464301\pi\)
0.111915 + 0.993718i \(0.464301\pi\)
\(500\) −6.00000 −0.268328
\(501\) 0 0
\(502\) 0 0
\(503\) 27.0000 1.20387 0.601935 − 0.798545i \(-0.294397\pi\)
0.601935 + 0.798545i \(0.294397\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 14.0000 0.621150
\(509\) −39.0000 −1.72864 −0.864322 − 0.502938i \(-0.832252\pi\)
−0.864322 + 0.502938i \(0.832252\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 42.0000 1.85074
\(516\) 0 0
\(517\) −54.0000 −2.37492
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 − 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 24.0000 1.04844
\(525\) 0 0
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) −18.0000 −0.778208
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −43.0000 −1.84871 −0.924357 − 0.381528i \(-0.875398\pi\)
−0.924357 + 0.381528i \(0.875398\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −33.0000 −1.41356
\(546\) 0 0
\(547\) −1.00000 −0.0427569 −0.0213785 − 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) 24.0000 1.02523
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −44.0000 −1.86602
\(557\) 12.0000 0.508456 0.254228 − 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 36.0000 1.51453
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 − 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 23.0000 0.962520 0.481260 − 0.876578i \(-0.340179\pi\)
0.481260 + 0.876578i \(0.340179\pi\)
\(572\) −48.0000 −2.00698
\(573\) 0 0
\(574\) 0 0
\(575\) −24.0000 −1.00087
\(576\) 0 0
\(577\) 16.0000 0.666089 0.333044 − 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −36.0000 −1.49482
\(581\) 0 0
\(582\) 0 0
\(583\) −36.0000 −1.49097
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) −28.0000 −1.15079
\(593\) −15.0000 −0.615976 −0.307988 − 0.951390i \(-0.599656\pi\)
−0.307988 + 0.951390i \(0.599656\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) 0 0
\(598\) 0 0
\(599\) 18.0000 0.735460 0.367730 − 0.929933i \(-0.380135\pi\)
0.367730 + 0.929933i \(0.380135\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 − 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 38.0000 1.54620
\(605\) −75.0000 −3.04918
\(606\) 0 0
\(607\) 40.0000 1.62355 0.811775 − 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −36.0000 −1.45640
\(612\) 0 0
\(613\) −34.0000 −1.37325 −0.686624 − 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 − 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 − 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 24.0000 0.963863
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 28.0000 1.11732
\(629\) 21.0000 0.837325
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 − 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 21.0000 0.833360
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 − 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) −54.0000 −2.11969
\(650\) 0 0
\(651\) 0 0
\(652\) 14.0000 0.548282
\(653\) −24.0000 −0.939193 −0.469596 − 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) 36.0000 1.40664
\(656\) 12.0000 0.468521
\(657\) 0 0
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 − 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 − 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) −6.00000 −0.232147
\(669\) 0 0
\(670\) 0 0
\(671\) 60.0000 2.31627
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −6.00000 −0.230769
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 36.0000 1.37549
\(686\) 0 0
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 − 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) −36.0000 −1.36851
\(693\) 0 0
\(694\) 0 0
\(695\) −66.0000 −2.50352
\(696\) 0 0
\(697\) −9.00000 −0.340899
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 − 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) 14.0000 0.528020
\(704\) −48.0000 −1.80907
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.00000 −0.0375558 −0.0187779 − 0.999824i \(-0.505978\pi\)
−0.0187779 + 0.999824i \(0.505978\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) −72.0000 −2.69265
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) 0 0
\(719\) −33.0000 −1.23069 −0.615346 − 0.788257i \(-0.710984\pi\)
−0.615346 + 0.788257i \(0.710984\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 40.0000 1.48659
\(725\) −24.0000 −0.891338
\(726\) 0 0
\(727\) 10.0000 0.370879 0.185440 − 0.982656i \(-0.440629\pi\)
0.185440 + 0.982656i \(0.440629\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) −32.0000 −1.18195 −0.590973 − 0.806691i \(-0.701256\pi\)
−0.590973 + 0.806691i \(0.701256\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −42.0000 −1.54395
\(741\) 0 0
\(742\) 0 0
\(743\) 6.00000 0.220119 0.110059 − 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 0 0
\(747\) 0 0
\(748\) 36.0000 1.31629
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 − 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −36.0000 −1.31278
\(753\) 0 0
\(754\) 0 0
\(755\) 57.0000 2.07444
\(756\) 0 0
\(757\) 47.0000 1.70824 0.854122 − 0.520073i \(-0.174095\pi\)
0.854122 + 0.520073i \(0.174095\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 51.0000 1.84875 0.924374 − 0.381487i \(-0.124588\pi\)
0.924374 + 0.381487i \(0.124588\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −36.0000 −1.30243
\(765\) 0 0
\(766\) 0 0
\(767\) −36.0000 −1.29988
\(768\) 0 0
\(769\) 40.0000 1.44244 0.721218 − 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 26.0000 0.935760
\(773\) −21.0000 −0.755318 −0.377659 − 0.925945i \(-0.623271\pi\)
−0.377659 + 0.925945i \(0.623271\pi\)
\(774\) 0 0
\(775\) 16.0000 0.574737
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 42.0000 1.49904
\(786\) 0 0
\(787\) −14.0000 −0.499046 −0.249523 − 0.968369i \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) 36.0000 1.28245
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) 0 0
\(795\) 0 0
\(796\) 40.0000 1.41776
\(797\) −6.00000 −0.212531 −0.106265 − 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 27.0000 0.955191
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 52.0000 1.82597 0.912983 − 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 21.0000 0.735598
\(816\) 0 0
\(817\) 2.00000 0.0699711
\(818\) 0 0
\(819\) 0 0
\(820\) 18.0000 0.628587
\(821\) −36.0000 −1.25641 −0.628204 − 0.778048i \(-0.716210\pi\)
−0.628204 + 0.778048i \(0.716210\pi\)
\(822\) 0 0
\(823\) 5.00000 0.174289 0.0871445 − 0.996196i \(-0.472226\pi\)
0.0871445 + 0.996196i \(0.472226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 − 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −32.0000 −1.10940
\(833\) 0 0
\(834\) 0 0
\(835\) −9.00000 −0.311458
\(836\) 24.0000 0.830057
\(837\) 0 0
\(838\) 0 0
\(839\) −21.0000 −0.725001 −0.362500 − 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) −24.0000 −0.824163
\(849\) 0 0
\(850\) 0 0
\(851\) 42.0000 1.43974
\(852\) 0 0
\(853\) −44.0000 −1.50653 −0.753266 − 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.00000 0.102478 0.0512390 − 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) 22.0000 0.750630 0.375315 − 0.926897i \(-0.377534\pi\)
0.375315 + 0.926897i \(0.377534\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 − 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) −54.0000 −1.83606
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6.00000 −0.203536
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −31.0000 −1.04680 −0.523398 − 0.852088i \(-0.675336\pi\)
−0.523398 + 0.852088i \(0.675336\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −72.0000 −2.42712
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −7.00000 −0.235569 −0.117784 − 0.993039i \(-0.537579\pi\)
−0.117784 + 0.993039i \(0.537579\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 0 0
\(887\) −21.0000 −0.705111 −0.352555 − 0.935791i \(-0.614687\pi\)
−0.352555 + 0.935791i \(0.614687\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) 18.0000 0.602347
\(894\) 0 0
\(895\) −36.0000 −1.20335
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 18.0000 0.599667
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 60.0000 1.99447
\(906\) 0 0
\(907\) −55.0000 −1.82625 −0.913123 − 0.407685i \(-0.866336\pi\)
−0.913123 + 0.407685i \(0.866336\pi\)
\(908\) −24.0000 −0.796468
\(909\) 0 0
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 − 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) −18.0000 −0.595713
\(914\) 0 0
\(915\) 0 0
\(916\) −8.00000 −0.264327
\(917\) 0 0
\(918\) 0 0
\(919\) 29.0000 0.956622 0.478311 − 0.878191i \(-0.341249\pi\)
0.478311 + 0.878191i \(0.341249\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 27.0000 0.885841 0.442921 − 0.896561i \(-0.353942\pi\)
0.442921 + 0.896561i \(0.353942\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −48.0000 −1.57229
\(933\) 0 0
\(934\) 0 0
\(935\) 54.0000 1.76599
\(936\) 0 0
\(937\) 34.0000 1.11073 0.555366 − 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −54.0000 −1.76129
\(941\) −27.0000 −0.880175 −0.440087 − 0.897955i \(-0.645053\pi\)
−0.440087 + 0.897955i \(0.645053\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) −36.0000 −1.17170
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 − 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 54.0000 1.74923 0.874616 − 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 0 0
\(955\) −54.0000 −1.74740
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 16.0000 0.515325
\(965\) 39.0000 1.25545
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 − 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 21.0000 0.673922 0.336961 − 0.941519i \(-0.390601\pi\)
0.336961 + 0.941519i \(0.390601\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 40.0000 1.28037
\(977\) −36.0000 −1.15174 −0.575871 − 0.817541i \(-0.695337\pi\)
−0.575871 + 0.817541i \(0.695337\pi\)
\(978\) 0 0
\(979\) −36.0000 −1.15056
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.00000 0.0956851 0.0478426 − 0.998855i \(-0.484765\pi\)
0.0478426 + 0.998855i \(0.484765\pi\)
\(984\) 0 0
\(985\) 54.0000 1.72058
\(986\) 0 0
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 − 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 60.0000 1.90213
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 − 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 0 0
\(999\) 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.h.1.1 1
3.2 odd 2 1323.2.a.l.1.1 1
7.6 odd 2 189.2.a.c.1.1 yes 1
21.20 even 2 189.2.a.b.1.1 ✓ 1
28.27 even 2 3024.2.a.y.1.1 1
35.34 odd 2 4725.2.a.k.1.1 1
63.13 odd 6 567.2.f.d.379.1 2
63.20 even 6 567.2.f.e.190.1 2
63.34 odd 6 567.2.f.d.190.1 2
63.41 even 6 567.2.f.e.379.1 2
84.83 odd 2 3024.2.a.f.1.1 1
105.104 even 2 4725.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.a.b.1.1 ✓ 1 21.20 even 2
189.2.a.c.1.1 yes 1 7.6 odd 2
567.2.f.d.190.1 2 63.34 odd 6
567.2.f.d.379.1 2 63.13 odd 6
567.2.f.e.190.1 2 63.20 even 6
567.2.f.e.379.1 2 63.41 even 6
1323.2.a.h.1.1 1 1.1 even 1 trivial
1323.2.a.l.1.1 1 3.2 odd 2
3024.2.a.f.1.1 1 84.83 odd 2
3024.2.a.y.1.1 1 28.27 even 2
4725.2.a.i.1.1 1 105.104 even 2
4725.2.a.k.1.1 1 35.34 odd 2