Properties

Label 912.2.a.a.1.1
Level $912$
Weight $2$
Character 912.1
Self dual yes
Analytic conductor $7.282$
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] = [912,2,Mod(1,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.28235666434\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 456)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 912.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -2.00000 q^{13} +3.00000 q^{15} -5.00000 q^{17} -1.00000 q^{19} -3.00000 q^{21} +4.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} +2.00000 q^{31} -1.00000 q^{33} -9.00000 q^{35} +8.00000 q^{37} +2.00000 q^{39} -8.00000 q^{41} -13.0000 q^{43} -3.00000 q^{45} -13.0000 q^{47} +2.00000 q^{49} +5.00000 q^{51} -6.00000 q^{53} -3.00000 q^{55} +1.00000 q^{57} -4.00000 q^{59} -13.0000 q^{61} +3.00000 q^{63} +6.00000 q^{65} -4.00000 q^{67} -4.00000 q^{69} +8.00000 q^{71} -3.00000 q^{73} -4.00000 q^{75} +3.00000 q^{77} +4.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +15.0000 q^{85} +6.00000 q^{87} -6.00000 q^{89} -6.00000 q^{91} -2.00000 q^{93} +3.00000 q^{95} +2.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −9.00000 −1.52128
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) −13.0000 −1.98248 −0.991241 0.132068i \(-0.957838\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) −13.0000 −1.89624 −0.948122 0.317905i \(-0.897021\pi\)
−0.948122 + 0.317905i \(0.897021\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 5.00000 0.700140
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 3.00000 0.377964
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −3.00000 −0.351123 −0.175562 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 15.0000 1.62698
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 3.00000 0.307794
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 9.00000 0.878310
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −12.0000 −1.11901
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −15.0000 −1.37505
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 0 0
\(129\) 13.0000 1.14459
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) 0 0
\(135\) 3.00000 0.258199
\(136\) 0 0
\(137\) −5.00000 −0.427179 −0.213589 0.976924i \(-0.568515\pi\)
−0.213589 + 0.976924i \(0.568515\pi\)
\(138\) 0 0
\(139\) 15.0000 1.27228 0.636142 0.771572i \(-0.280529\pi\)
0.636142 + 0.771572i \(0.280529\pi\)
\(140\) 0 0
\(141\) 13.0000 1.09480
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 18.0000 1.49482
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) −5.00000 −0.409616 −0.204808 0.978802i \(-0.565657\pi\)
−0.204808 + 0.978802i \(0.565657\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 0 0
\(165\) 3.00000 0.233550
\(166\) 0 0
\(167\) 14.0000 1.08335 0.541676 0.840587i \(-0.317790\pi\)
0.541676 + 0.840587i \(0.317790\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −26.0000 −1.97674 −0.988372 0.152057i \(-0.951410\pi\)
−0.988372 + 0.152057i \(0.951410\pi\)
\(174\) 0 0
\(175\) 12.0000 0.907115
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 13.0000 0.960988
\(184\) 0 0
\(185\) −24.0000 −1.76452
\(186\) 0 0
\(187\) −5.00000 −0.365636
\(188\) 0 0
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) −11.0000 −0.795932 −0.397966 0.917400i \(-0.630284\pi\)
−0.397966 + 0.917400i \(0.630284\pi\)
\(192\) 0 0
\(193\) −24.0000 −1.72756 −0.863779 0.503871i \(-0.831909\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 0 0
\(195\) −6.00000 −0.429669
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) 3.00000 0.212664 0.106332 0.994331i \(-0.466089\pi\)
0.106332 + 0.994331i \(0.466089\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) −18.0000 −1.26335
\(204\) 0 0
\(205\) 24.0000 1.67623
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 24.0000 1.65223 0.826114 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) 0 0
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) 39.0000 2.65978
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) 3.00000 0.202721
\(220\) 0 0
\(221\) 10.0000 0.672673
\(222\) 0 0
\(223\) 20.0000 1.33930 0.669650 0.742677i \(-0.266444\pi\)
0.669650 + 0.742677i \(0.266444\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −19.0000 −1.25556 −0.627778 0.778393i \(-0.716035\pi\)
−0.627778 + 0.778393i \(0.716035\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) −19.0000 −1.24473 −0.622366 0.782727i \(-0.713828\pi\)
−0.622366 + 0.782727i \(0.713828\pi\)
\(234\) 0 0
\(235\) 39.0000 2.54408
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) 12.0000 0.772988 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −9.00000 −0.568075 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) −15.0000 −0.939336
\(256\) 0 0
\(257\) −16.0000 −0.998053 −0.499026 0.866587i \(-0.666309\pi\)
−0.499026 + 0.866587i \(0.666309\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −5.00000 −0.308313 −0.154157 0.988046i \(-0.549266\pi\)
−0.154157 + 0.988046i \(0.549266\pi\)
\(264\) 0 0
\(265\) 18.0000 1.10573
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) 6.00000 0.363137
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −15.0000 −0.901263 −0.450631 0.892710i \(-0.648801\pi\)
−0.450631 + 0.892710i \(0.648801\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −33.0000 −1.96165 −0.980823 0.194900i \(-0.937562\pi\)
−0.980823 + 0.194900i \(0.937562\pi\)
\(284\) 0 0
\(285\) −3.00000 −0.177705
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 0 0
\(293\) 28.0000 1.63578 0.817889 0.575376i \(-0.195144\pi\)
0.817889 + 0.575376i \(0.195144\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) −39.0000 −2.24792
\(302\) 0 0
\(303\) −10.0000 −0.574485
\(304\) 0 0
\(305\) 39.0000 2.23313
\(306\) 0 0
\(307\) 24.0000 1.36975 0.684876 0.728659i \(-0.259856\pi\)
0.684876 + 0.728659i \(0.259856\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) −9.00000 −0.510343 −0.255172 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(312\) 0 0
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 0 0
\(315\) −9.00000 −0.507093
\(316\) 0 0
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) 5.00000 0.278207
\(324\) 0 0
\(325\) −8.00000 −0.443760
\(326\) 0 0
\(327\) −16.0000 −0.884802
\(328\) 0 0
\(329\) −39.0000 −2.15014
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 8.00000 0.438397
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 30.0000 1.63420 0.817102 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) 7.00000 0.375780 0.187890 0.982190i \(-0.439835\pi\)
0.187890 + 0.982190i \(0.439835\pi\)
\(348\) 0 0
\(349\) −11.0000 −0.588817 −0.294408 0.955680i \(-0.595123\pi\)
−0.294408 + 0.955680i \(0.595123\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −34.0000 −1.80964 −0.904819 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 0 0
\(355\) −24.0000 −1.27379
\(356\) 0 0
\(357\) 15.0000 0.793884
\(358\) 0 0
\(359\) 25.0000 1.31945 0.659725 0.751507i \(-0.270673\pi\)
0.659725 + 0.751507i \(0.270673\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 9.00000 0.471082
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) −18.0000 −0.934513
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −30.0000 −1.54100 −0.770498 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) 0 0
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) −9.00000 −0.458682
\(386\) 0 0
\(387\) −13.0000 −0.660827
\(388\) 0 0
\(389\) 13.0000 0.659126 0.329563 0.944134i \(-0.393099\pi\)
0.329563 + 0.944134i \(0.393099\pi\)
\(390\) 0 0
\(391\) −20.0000 −1.01144
\(392\) 0 0
\(393\) −15.0000 −0.756650
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −19.0000 −0.953583 −0.476791 0.879017i \(-0.658200\pi\)
−0.476791 + 0.879017i \(0.658200\pi\)
\(398\) 0 0
\(399\) 3.00000 0.150188
\(400\) 0 0
\(401\) −16.0000 −0.799002 −0.399501 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 0 0
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 5.00000 0.246632
\(412\) 0 0
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) −15.0000 −0.734553
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) 0 0
\(423\) −13.0000 −0.632082
\(424\) 0 0
\(425\) −20.0000 −0.970143
\(426\) 0 0
\(427\) −39.0000 −1.88734
\(428\) 0 0
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) −18.0000 −0.863034
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 3.00000 0.142534 0.0712672 0.997457i \(-0.477296\pi\)
0.0712672 + 0.997457i \(0.477296\pi\)
\(444\) 0 0
\(445\) 18.0000 0.853282
\(446\) 0 0
\(447\) 5.00000 0.236492
\(448\) 0 0
\(449\) 28.0000 1.32140 0.660701 0.750649i \(-0.270259\pi\)
0.660701 + 0.750649i \(0.270259\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) 4.00000 0.187936
\(454\) 0 0
\(455\) 18.0000 0.843853
\(456\) 0 0
\(457\) 19.0000 0.888783 0.444391 0.895833i \(-0.353420\pi\)
0.444391 + 0.895833i \(0.353420\pi\)
\(458\) 0 0
\(459\) 5.00000 0.233380
\(460\) 0 0
\(461\) 35.0000 1.63011 0.815056 0.579382i \(-0.196706\pi\)
0.815056 + 0.579382i \(0.196706\pi\)
\(462\) 0 0
\(463\) −7.00000 −0.325318 −0.162659 0.986682i \(-0.552007\pi\)
−0.162659 + 0.986682i \(0.552007\pi\)
\(464\) 0 0
\(465\) 6.00000 0.278243
\(466\) 0 0
\(467\) 3.00000 0.138823 0.0694117 0.997588i \(-0.477888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 0 0
\(473\) −13.0000 −0.597741
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) −12.0000 −0.546019
\(484\) 0 0
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 30.0000 1.35113
\(494\) 0 0
\(495\) −3.00000 −0.134840
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) 41.0000 1.83541 0.917706 0.397260i \(-0.130039\pi\)
0.917706 + 0.397260i \(0.130039\pi\)
\(500\) 0 0
\(501\) −14.0000 −0.625474
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −30.0000 −1.33498
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) −9.00000 −0.398137
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) −18.0000 −0.793175
\(516\) 0 0
\(517\) −13.0000 −0.571739
\(518\) 0 0
\(519\) 26.0000 1.14127
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 38.0000 1.66162 0.830812 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) 0 0
\(525\) −12.0000 −0.523723
\(526\) 0 0
\(527\) −10.0000 −0.435607
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 16.0000 0.693037
\(534\) 0 0
\(535\) −18.0000 −0.778208
\(536\) 0 0
\(537\) 6.00000 0.258919
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −1.00000 −0.0429934 −0.0214967 0.999769i \(-0.506843\pi\)
−0.0214967 + 0.999769i \(0.506843\pi\)
\(542\) 0 0
\(543\) 14.0000 0.600798
\(544\) 0 0
\(545\) −48.0000 −2.05609
\(546\) 0 0
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) 0 0
\(549\) −13.0000 −0.554826
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 0 0
\(555\) 24.0000 1.01874
\(556\) 0 0
\(557\) 15.0000 0.635570 0.317785 0.948163i \(-0.397061\pi\)
0.317785 + 0.948163i \(0.397061\pi\)
\(558\) 0 0
\(559\) 26.0000 1.09968
\(560\) 0 0
\(561\) 5.00000 0.211100
\(562\) 0 0
\(563\) 44.0000 1.85438 0.927189 0.374593i \(-0.122217\pi\)
0.927189 + 0.374593i \(0.122217\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) 0 0
\(567\) 3.00000 0.125988
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) 11.0000 0.459532
\(574\) 0 0
\(575\) 16.0000 0.667246
\(576\) 0 0
\(577\) −5.00000 −0.208153 −0.104076 0.994569i \(-0.533189\pi\)
−0.104076 + 0.994569i \(0.533189\pi\)
\(578\) 0 0
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) −6.00000 −0.248495
\(584\) 0 0
\(585\) 6.00000 0.248069
\(586\) 0 0
\(587\) −25.0000 −1.03186 −0.515930 0.856631i \(-0.672554\pi\)
−0.515930 + 0.856631i \(0.672554\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) −22.0000 −0.904959
\(592\) 0 0
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) 45.0000 1.84482
\(596\) 0 0
\(597\) −3.00000 −0.122782
\(598\) 0 0
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) 20.0000 0.815817 0.407909 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 30.0000 1.21967
\(606\) 0 0
\(607\) −42.0000 −1.70473 −0.852364 0.522949i \(-0.824832\pi\)
−0.852364 + 0.522949i \(0.824832\pi\)
\(608\) 0 0
\(609\) 18.0000 0.729397
\(610\) 0 0
\(611\) 26.0000 1.05185
\(612\) 0 0
\(613\) −19.0000 −0.767403 −0.383701 0.923457i \(-0.625351\pi\)
−0.383701 + 0.923457i \(0.625351\pi\)
\(614\) 0 0
\(615\) −24.0000 −0.967773
\(616\) 0 0
\(617\) −33.0000 −1.32853 −0.664265 0.747497i \(-0.731255\pi\)
−0.664265 + 0.747497i \(0.731255\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) −18.0000 −0.721155
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 1.00000 0.0399362
\(628\) 0 0
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) −1.00000 −0.0398094 −0.0199047 0.999802i \(-0.506336\pi\)
−0.0199047 + 0.999802i \(0.506336\pi\)
\(632\) 0 0
\(633\) −24.0000 −0.953914
\(634\) 0 0
\(635\) 18.0000 0.714308
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) −29.0000 −1.14365 −0.571824 0.820376i \(-0.693764\pi\)
−0.571824 + 0.820376i \(0.693764\pi\)
\(644\) 0 0
\(645\) −39.0000 −1.53562
\(646\) 0 0
\(647\) −27.0000 −1.06148 −0.530740 0.847535i \(-0.678086\pi\)
−0.530740 + 0.847535i \(0.678086\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 0 0
\(653\) 27.0000 1.05659 0.528296 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) 0 0
\(655\) −45.0000 −1.75830
\(656\) 0 0
\(657\) −3.00000 −0.117041
\(658\) 0 0
\(659\) 50.0000 1.94772 0.973862 0.227142i \(-0.0729380\pi\)
0.973862 + 0.227142i \(0.0729380\pi\)
\(660\) 0 0
\(661\) −44.0000 −1.71140 −0.855701 0.517471i \(-0.826874\pi\)
−0.855701 + 0.517471i \(0.826874\pi\)
\(662\) 0 0
\(663\) −10.0000 −0.388368
\(664\) 0 0
\(665\) 9.00000 0.349005
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) −20.0000 −0.773245
\(670\) 0 0
\(671\) −13.0000 −0.501859
\(672\) 0 0
\(673\) −12.0000 −0.462566 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) 6.00000 0.229584 0.114792 0.993390i \(-0.463380\pi\)
0.114792 + 0.993390i \(0.463380\pi\)
\(684\) 0 0
\(685\) 15.0000 0.573121
\(686\) 0 0
\(687\) 19.0000 0.724895
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 21.0000 0.798878 0.399439 0.916760i \(-0.369205\pi\)
0.399439 + 0.916760i \(0.369205\pi\)
\(692\) 0 0
\(693\) 3.00000 0.113961
\(694\) 0 0
\(695\) −45.0000 −1.70695
\(696\) 0 0
\(697\) 40.0000 1.51511
\(698\) 0 0
\(699\) 19.0000 0.718646
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) −39.0000 −1.46882
\(706\) 0 0
\(707\) 30.0000 1.12827
\(708\) 0 0
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) 0 0
\(717\) −9.00000 −0.336111
\(718\) 0 0
\(719\) −3.00000 −0.111881 −0.0559406 0.998434i \(-0.517816\pi\)
−0.0559406 + 0.998434i \(0.517816\pi\)
\(720\) 0 0
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) −12.0000 −0.446285
\(724\) 0 0
\(725\) −24.0000 −0.891338
\(726\) 0 0
\(727\) 17.0000 0.630495 0.315248 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 65.0000 2.40411
\(732\) 0 0
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) 23.0000 0.846069 0.423034 0.906114i \(-0.360965\pi\)
0.423034 + 0.906114i \(0.360965\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) 15.0000 0.549557
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 9.00000 0.327978
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 11.0000 0.399802 0.199901 0.979816i \(-0.435938\pi\)
0.199901 + 0.979816i \(0.435938\pi\)
\(758\) 0 0
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) 11.0000 0.398750 0.199375 0.979923i \(-0.436109\pi\)
0.199375 + 0.979923i \(0.436109\pi\)
\(762\) 0 0
\(763\) 48.0000 1.73772
\(764\) 0 0
\(765\) 15.0000 0.542326
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 35.0000 1.26213 0.631066 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(770\) 0 0
\(771\) 16.0000 0.576226
\(772\) 0 0
\(773\) −40.0000 −1.43870 −0.719350 0.694648i \(-0.755560\pi\)
−0.719350 + 0.694648i \(0.755560\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) −24.0000 −0.860995
\(778\) 0 0
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) −42.0000 −1.49904
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 0 0
\(789\) 5.00000 0.178005
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 26.0000 0.923287
\(794\) 0 0
\(795\) −18.0000 −0.638394
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 65.0000 2.29953
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) −3.00000 −0.105868
\(804\) 0 0
\(805\) −36.0000 −1.26883
\(806\) 0 0
\(807\) −10.0000 −0.352017
\(808\) 0 0
\(809\) −27.0000 −0.949269 −0.474635 0.880183i \(-0.657420\pi\)
−0.474635 + 0.880183i \(0.657420\pi\)
\(810\) 0 0
\(811\) 30.0000 1.05344 0.526721 0.850038i \(-0.323421\pi\)
0.526721 + 0.850038i \(0.323421\pi\)
\(812\) 0 0
\(813\) −28.0000 −0.982003
\(814\) 0 0
\(815\) 24.0000 0.840683
\(816\) 0 0
\(817\) 13.0000 0.454812
\(818\) 0 0
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 11.0000 0.383903 0.191951 0.981404i \(-0.438518\pi\)
0.191951 + 0.981404i \(0.438518\pi\)
\(822\) 0 0
\(823\) −37.0000 −1.28974 −0.644869 0.764293i \(-0.723088\pi\)
−0.644869 + 0.764293i \(0.723088\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 24.0000 0.834562 0.417281 0.908778i \(-0.362983\pi\)
0.417281 + 0.908778i \(0.362983\pi\)
\(828\) 0 0
\(829\) 36.0000 1.25033 0.625166 0.780492i \(-0.285031\pi\)
0.625166 + 0.780492i \(0.285031\pi\)
\(830\) 0 0
\(831\) 15.0000 0.520344
\(832\) 0 0
\(833\) −10.0000 −0.346479
\(834\) 0 0
\(835\) −42.0000 −1.45347
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) 38.0000 1.31191 0.655953 0.754802i \(-0.272267\pi\)
0.655953 + 0.754802i \(0.272267\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −18.0000 −0.619953
\(844\) 0 0
\(845\) 27.0000 0.928828
\(846\) 0 0
\(847\) −30.0000 −1.03081
\(848\) 0 0
\(849\) 33.0000 1.13256
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 0 0
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 0 0
\(855\) 3.00000 0.102598
\(856\) 0 0
\(857\) 28.0000 0.956462 0.478231 0.878234i \(-0.341278\pi\)
0.478231 + 0.878234i \(0.341278\pi\)
\(858\) 0 0
\(859\) 31.0000 1.05771 0.528853 0.848713i \(-0.322622\pi\)
0.528853 + 0.848713i \(0.322622\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) 0 0
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) 78.0000 2.65208
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 0 0
\(879\) −28.0000 −0.944417
\(880\) 0 0
\(881\) −1.00000 −0.0336909 −0.0168454 0.999858i \(-0.505362\pi\)
−0.0168454 + 0.999858i \(0.505362\pi\)
\(882\) 0 0
\(883\) −1.00000 −0.0336527 −0.0168263 0.999858i \(-0.505356\pi\)
−0.0168263 + 0.999858i \(0.505356\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) 0 0
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 13.0000 0.435028
\(894\) 0 0
\(895\) 18.0000 0.601674
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 0 0
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 30.0000 0.999445
\(902\) 0 0
\(903\) 39.0000 1.29784
\(904\) 0 0
\(905\) 42.0000 1.39613
\(906\) 0 0
\(907\) 14.0000 0.464862 0.232431 0.972613i \(-0.425332\pi\)
0.232431 + 0.972613i \(0.425332\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −22.0000 −0.728893 −0.364446 0.931224i \(-0.618742\pi\)
−0.364446 + 0.931224i \(0.618742\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) 0 0
\(915\) −39.0000 −1.28930
\(916\) 0 0
\(917\) 45.0000 1.48603
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) −24.0000 −0.790827
\(922\) 0 0
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 32.0000 1.05215
\(926\) 0 0
\(927\) 6.00000 0.197066
\(928\) 0 0
\(929\) 46.0000 1.50921 0.754606 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 0 0
\(933\) 9.00000 0.294647
\(934\) 0 0
\(935\) 15.0000 0.490552
\(936\) 0 0
\(937\) −35.0000 −1.14340 −0.571700 0.820463i \(-0.693716\pi\)
−0.571700 + 0.820463i \(0.693716\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) −32.0000 −1.04206
\(944\) 0 0
\(945\) 9.00000 0.292770
\(946\) 0 0
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) 6.00000 0.194768
\(950\) 0 0
\(951\) −24.0000 −0.778253
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 33.0000 1.06785
\(956\) 0 0
\(957\) 6.00000 0.193952
\(958\) 0 0
\(959\) −15.0000 −0.484375
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 0 0
\(965\) 72.0000 2.31776
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) −5.00000 −0.160623
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 45.0000 1.44263
\(974\) 0 0
\(975\) 8.00000 0.256205
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 16.0000 0.510841
\(982\) 0 0
\(983\) −28.0000 −0.893061 −0.446531 0.894768i \(-0.647341\pi\)
−0.446531 + 0.894768i \(0.647341\pi\)
\(984\) 0 0
\(985\) −66.0000 −2.10293
\(986\) 0 0
\(987\) 39.0000 1.24138
\(988\) 0 0
\(989\) −52.0000 −1.65350
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) −9.00000 −0.285319
\(996\) 0 0
\(997\) 53.0000 1.67853 0.839263 0.543725i \(-0.182987\pi\)
0.839263 + 0.543725i \(0.182987\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
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 912.2.a.a.1.1 1
3.2 odd 2 2736.2.a.u.1.1 1
4.3 odd 2 456.2.a.c.1.1 1
8.3 odd 2 3648.2.a.p.1.1 1
8.5 even 2 3648.2.a.bj.1.1 1
12.11 even 2 1368.2.a.i.1.1 1
76.75 even 2 8664.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.2.a.c.1.1 1 4.3 odd 2
912.2.a.a.1.1 1 1.1 even 1 trivial
1368.2.a.i.1.1 1 12.11 even 2
2736.2.a.u.1.1 1 3.2 odd 2
3648.2.a.p.1.1 1 8.3 odd 2
3648.2.a.bj.1.1 1 8.5 even 2
8664.2.a.a.1.1 1 76.75 even 2