Properties

Label 873.1.br.a.451.1
Level $873$
Weight $1$
Character 873.451
Analytic conductor $0.436$
Analytic rank $0$
Dimension $16$
Projective image $D_{32}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [873,1,Mod(19,873)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(873, base_ring=CyclotomicField(32))
 
chi = DirichletCharacter(H, H._module([0, 27]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("873.19");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 873 = 3^{2} \cdot 97 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 873.br (of order \(32\), degree \(16\), minimal)

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: no
Analytic conductor: \(0.435683756029\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{32})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{32}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{32} - \cdots)\)

Embedding invariants

Embedding label 451.1
Root \(0.555570 - 0.831470i\) of defining polynomial
Character \(\chi\) \(=\) 873.451
Dual form 873.1.br.a.271.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.923880 + 0.382683i) q^{4} +(-0.151537 - 0.124363i) q^{7} +O(q^{10})\) \(q+(0.923880 + 0.382683i) q^{4} +(-0.151537 - 0.124363i) q^{7} +(0.187593 - 1.90466i) q^{13} +(0.707107 + 0.707107i) q^{16} +(0.980785 + 1.19509i) q^{19} +(-0.195090 + 0.980785i) q^{25} +(-0.0924099 - 0.172887i) q^{28} +(-1.81225 + 0.360480i) q^{31} +(-1.68789 - 0.512016i) q^{37} +(1.53636 + 0.636379i) q^{43} +(-0.187593 - 0.943094i) q^{49} +(0.902197 - 1.68789i) q^{52} -1.11114 q^{61} +(0.382683 + 0.923880i) q^{64} +(0.0924099 - 0.938254i) q^{67} +(-1.30656 + 0.541196i) q^{73} +(0.448786 + 1.47945i) q^{76} +(-0.0761205 - 0.382683i) q^{79} +(-0.265297 + 0.265297i) q^{91} +(0.195090 - 0.980785i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/873\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(389\)
\(\chi(n)\) \(e\left(\frac{25}{32}\right)\) \(1\)

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 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(3\) 0 0
\(4\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(5\) 0 0 −0.634393 0.773010i \(-0.718750\pi\)
0.634393 + 0.773010i \(0.281250\pi\)
\(6\) 0 0
\(7\) −0.151537 0.124363i −0.151537 0.124363i 0.555570 0.831470i \(-0.312500\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(12\) 0 0
\(13\) 0.187593 1.90466i 0.187593 1.90466i −0.195090 0.980785i \(-0.562500\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(17\) 0 0 −0.995185 0.0980171i \(-0.968750\pi\)
0.995185 + 0.0980171i \(0.0312500\pi\)
\(18\) 0 0
\(19\) 0.980785 + 1.19509i 0.980785 + 1.19509i 0.980785 + 0.195090i \(0.0625000\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.471397 0.881921i \(-0.343750\pi\)
−0.471397 + 0.881921i \(0.656250\pi\)
\(24\) 0 0
\(25\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(26\) 0 0
\(27\) 0 0
\(28\) −0.0924099 0.172887i −0.0924099 0.172887i
\(29\) 0 0 0.471397 0.881921i \(-0.343750\pi\)
−0.471397 + 0.881921i \(0.656250\pi\)
\(30\) 0 0
\(31\) −1.81225 + 0.360480i −1.81225 + 0.360480i −0.980785 0.195090i \(-0.937500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.68789 0.512016i −1.68789 0.512016i −0.707107 0.707107i \(-0.750000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.956940 0.290285i \(-0.0937500\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(42\) 0 0
\(43\) 1.53636 + 0.636379i 1.53636 + 0.636379i 0.980785 0.195090i \(-0.0625000\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(48\) 0 0
\(49\) −0.187593 0.943094i −0.187593 0.943094i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.902197 1.68789i 0.902197 1.68789i
\(53\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.881921 0.471397i \(-0.156250\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(60\) 0 0
\(61\) −1.11114 −1.11114 −0.555570 0.831470i \(-0.687500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.0924099 0.938254i 0.0924099 0.938254i −0.831470 0.555570i \(-0.812500\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.956940 0.290285i \(-0.906250\pi\)
0.956940 + 0.290285i \(0.0937500\pi\)
\(72\) 0 0
\(73\) −1.30656 + 0.541196i −1.30656 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0.448786 + 1.47945i 0.448786 + 1.47945i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.0761205 0.382683i −0.0761205 0.382683i 0.923880 0.382683i \(-0.125000\pi\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.773010 0.634393i \(-0.218750\pi\)
−0.773010 + 0.634393i \(0.781250\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(90\) 0 0
\(91\) −0.265297 + 0.265297i −0.265297 + 0.265297i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.195090 0.980785i 0.195090 0.980785i
\(98\) 0 0
\(99\) 0 0
\(100\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(101\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(102\) 0 0
\(103\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.881921 0.471397i \(-0.156250\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(108\) 0 0
\(109\) −0.636379 + 0.425215i −0.636379 + 0.425215i −0.831470 0.555570i \(-0.812500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.0192147 0.195090i −0.0192147 0.195090i
\(113\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.382683 0.923880i −0.382683 0.923880i
\(122\) 0 0
\(123\) 0 0
\(124\) −1.81225 0.360480i −1.81225 0.360480i
\(125\) 0 0
\(126\) 0 0
\(127\) 1.26268 + 0.124363i 1.26268 + 0.124363i 0.707107 0.707107i \(-0.250000\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.290285 0.956940i \(-0.406250\pi\)
−0.290285 + 0.956940i \(0.593750\pi\)
\(132\) 0 0
\(133\) 0.303073i 0.303073i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.995185 0.0980171i \(-0.0312500\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(138\) 0 0
\(139\) −0.0924099 0.938254i −0.0924099 0.938254i −0.923880 0.382683i \(-0.875000\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −1.36347 1.11897i −1.36347 1.11897i
\(149\) 0 0 −0.881921 0.471397i \(-0.843750\pi\)
0.881921 + 0.471397i \(0.156250\pi\)
\(150\) 0 0
\(151\) −0.750661 + 1.81225i −0.750661 + 1.81225i −0.195090 + 0.980785i \(0.562500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.368309 + 1.21415i −0.368309 + 1.21415i 0.555570 + 0.831470i \(0.312500\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.382683 + 1.92388i 0.382683 + 1.92388i 0.382683 + 0.923880i \(0.375000\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(168\) 0 0
\(169\) −2.61177 0.519514i −2.61177 0.519514i
\(170\) 0 0
\(171\) 0 0
\(172\) 1.17588 + 1.17588i 1.17588 + 1.17588i
\(173\) 0 0 0.956940 0.290285i \(-0.0937500\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(174\) 0 0
\(175\) 0.151537 0.124363i 0.151537 0.124363i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.0980171 0.995185i \(-0.531250\pi\)
0.0980171 + 0.995185i \(0.468750\pi\)
\(180\) 0 0
\(181\) −0.577774 0.0569057i −0.577774 0.0569057i −0.195090 0.980785i \(-0.562500\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(192\) 0 0
\(193\) 0.390181i 0.390181i 0.980785 + 0.195090i \(0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.187593 0.943094i 0.187593 0.943094i
\(197\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(198\) 0 0
\(199\) 1.36347 1.11897i 1.36347 1.11897i 0.382683 0.923880i \(-0.375000\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1.47945 1.21415i 1.47945 1.21415i
\(209\) 0 0
\(210\) 0 0
\(211\) −0.124363 + 1.26268i −0.124363 + 1.26268i 0.707107 + 0.707107i \(0.250000\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.319453 + 0.170751i 0.319453 + 0.170751i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −0.512016 0.273678i −0.512016 0.273678i 0.195090 0.980785i \(-0.437500\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(228\) 0 0
\(229\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.634393 0.773010i \(-0.281250\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.471397 0.881921i \(-0.343750\pi\)
−0.471397 + 0.881921i \(0.656250\pi\)
\(240\) 0 0
\(241\) 0.636379 1.53636i 0.636379 1.53636i −0.195090 0.980785i \(-0.562500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −1.02656 0.425215i −1.02656 0.425215i
\(245\) 0 0
\(246\) 0 0
\(247\) 2.46024 1.64388i 2.46024 1.64388i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.0980171 0.995185i \(-0.531250\pi\)
0.0980171 + 0.995185i \(0.468750\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000i 1.00000i
\(257\) 0 0 −0.956940 0.290285i \(-0.906250\pi\)
0.956940 + 0.290285i \(0.0937500\pi\)
\(258\) 0 0
\(259\) 0.192102 + 0.287500i 0.192102 + 0.287500i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.471397 0.881921i \(-0.656250\pi\)
0.471397 + 0.881921i \(0.343750\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0.444430 0.831470i 0.444430 0.831470i
\(269\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(270\) 0 0
\(271\) 0.512016 + 1.68789i 0.512016 + 1.68789i 0.707107 + 0.707107i \(0.250000\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.11897 + 1.36347i 1.11897 + 1.36347i 0.923880 + 0.382683i \(0.125000\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 −0.471397 0.881921i \(-0.656250\pi\)
0.471397 + 0.881921i \(0.343750\pi\)
\(282\) 0 0
\(283\) 1.38268 + 0.923880i 1.38268 + 0.923880i 1.00000 \(0\)
0.382683 + 0.923880i \(0.375000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(290\) 0 0
\(291\) 0 0
\(292\) −1.41421 −1.41421
\(293\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −0.153672 0.287500i −0.153672 0.287500i
\(302\) 0 0
\(303\) 0 0
\(304\) −0.151537 + 1.53858i −0.151537 + 1.53858i
\(305\) 0 0
\(306\) 0 0
\(307\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.290285 0.956940i \(-0.593750\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(312\) 0 0
\(313\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0.0761205 0.382683i 0.0761205 0.382683i
\(317\) 0 0 0.290285 0.956940i \(-0.406250\pi\)
−0.290285 + 0.956940i \(0.593750\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.83147 + 0.555570i 1.83147 + 0.555570i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.0569057 + 0.577774i 0.0569057 + 0.577774i 0.980785 + 0.195090i \(0.0625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.26268 1.53858i 1.26268 1.53858i 0.555570 0.831470i \(-0.312500\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.181269 + 0.339130i −0.181269 + 0.339130i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.290285 0.956940i \(-0.593750\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(348\) 0 0
\(349\) 1.26268 1.53858i 1.26268 1.53858i 0.555570 0.831470i \(-0.312500\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.881921 0.471397i \(-0.843750\pi\)
0.881921 + 0.471397i \(0.156250\pi\)
\(360\) 0 0
\(361\) −0.271211 + 1.36347i −0.271211 + 1.36347i
\(362\) 0 0
\(363\) 0 0
\(364\) −0.346627 + 0.143578i −0.346627 + 0.143578i
\(365\) 0 0
\(366\) 0 0
\(367\) −0.577774 1.90466i −0.577774 1.90466i −0.382683 0.923880i \(-0.625000\pi\)
−0.195090 0.980785i \(-0.562500\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.53858 + 0.151537i −1.53858 + 0.151537i −0.831470 0.555570i \(-0.812500\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.70711 + 0.707107i −1.70711 + 0.707107i −0.707107 + 0.707107i \(0.750000\pi\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.773010 0.634393i \(-0.218750\pi\)
−0.773010 + 0.634393i \(0.781250\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0.555570 0.831470i 0.555570 0.831470i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −0.149316 0.360480i −0.149316 0.360480i 0.831470 0.555570i \(-0.187500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(401\) 0 0 −0.995185 0.0980171i \(-0.968750\pi\)
0.995185 + 0.0980171i \(0.0312500\pi\)
\(402\) 0 0
\(403\) 0.346627 + 3.51936i 0.346627 + 3.51936i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.21415 0.368309i 1.21415 0.368309i 0.382683 0.923880i \(-0.375000\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.541196 1.30656i −0.541196 1.30656i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(420\) 0 0
\(421\) 1.81225 0.750661i 1.81225 0.750661i 0.831470 0.555570i \(-0.187500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.168378 + 0.138185i 0.168378 + 0.138185i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(432\) 0 0
\(433\) −0.831470 0.444430i −0.831470 0.444430i 1.00000i \(-0.5\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.750661 + 0.149316i −0.750661 + 0.149316i
\(437\) 0 0
\(438\) 0 0
\(439\) 1.53858 + 1.26268i 1.53858 + 1.26268i 0.831470 + 0.555570i \(0.187500\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 −0.0980171 0.995185i \(-0.531250\pi\)
0.0980171 + 0.995185i \(0.468750\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0.0569057 0.187593i 0.0569057 0.187593i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.831470 + 0.444430i −0.831470 + 0.444430i −0.831470 0.555570i \(-0.812500\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(462\) 0 0
\(463\) −0.785695 0.785695i −0.785695 0.785695i 0.195090 0.980785i \(-0.437500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(468\) 0 0
\(469\) −0.130687 + 0.130687i −0.130687 + 0.130687i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1.36347 + 0.728789i −1.36347 + 0.728789i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(480\) 0 0
\(481\) −1.29186 + 3.11882i −1.29186 + 3.11882i
\(482\) 0 0
\(483\) 0 0
\(484\) 1.00000i 1.00000i
\(485\) 0 0
\(486\) 0 0
\(487\) 0.216773 1.08979i 0.216773 1.08979i −0.707107 0.707107i \(-0.750000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.53636 1.02656i −1.53636 1.02656i
\(497\) 0 0
\(498\) 0 0
\(499\) 1.53858 1.26268i 1.53858 1.26268i 0.707107 0.707107i \(-0.250000\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 1.11897 + 0.598102i 1.11897 + 0.598102i
\(509\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(510\) 0 0
\(511\) 0.265297 + 0.0804769i 0.265297 + 0.0804769i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −0.512016 + 0.273678i −0.512016 + 0.273678i −0.707107 0.707107i \(-0.750000\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.555570 0.831470i −0.555570 0.831470i
\(530\) 0 0
\(531\) 0 0
\(532\) 0.115981 0.280003i 0.115981 0.280003i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.555570 0.168530i 0.555570 0.168530i 1.00000i \(-0.5\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −0.0360566 + 0.0674571i −0.0360566 + 0.0674571i
\(554\) 0 0
\(555\) 0 0
\(556\) 0.273678 0.902197i 0.273678 0.902197i
\(557\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(558\) 0 0
\(559\) 1.50030 2.80686i 1.50030 2.80686i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.634393 0.773010i \(-0.718750\pi\)
0.634393 + 0.773010i \(0.281250\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.0980171 0.995185i \(-0.468750\pi\)
−0.0980171 + 0.995185i \(0.531250\pi\)
\(570\) 0 0
\(571\) −0.785695 + 1.17588i −0.785695 + 1.17588i 0.195090 + 0.980785i \(0.437500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −0.980785 1.19509i −0.980785 1.19509i −0.980785 0.195090i \(-0.937500\pi\)
1.00000i \(-0.5\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.634393 0.773010i \(-0.718750\pi\)
0.634393 + 0.773010i \(0.281250\pi\)
\(588\) 0 0
\(589\) −2.20824 1.81225i −2.20824 1.81225i
\(590\) 0 0
\(591\) 0 0
\(592\) −0.831470 1.55557i −0.831470 1.55557i
\(593\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.995185 0.0980171i \(-0.968750\pi\)
0.995185 + 0.0980171i \(0.0312500\pi\)
\(600\) 0 0
\(601\) 1.21415 + 1.47945i 1.21415 + 1.47945i 0.831470 + 0.555570i \(0.187500\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.38704 + 1.38704i −1.38704 + 1.38704i
\(605\) 0 0
\(606\) 0 0
\(607\) 0.149316 0.750661i 0.149316 0.750661i −0.831470 0.555570i \(-0.812500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −0.750661 + 0.149316i −0.750661 + 0.149316i −0.555570 0.831470i \(-0.687500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 1.21415 + 0.368309i 1.21415 + 0.368309i 0.831470 0.555570i \(-0.187500\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.923880 0.382683i −0.923880 0.382683i
\(626\) 0 0
\(627\) 0 0
\(628\) −0.804910 + 0.980785i −0.804910 + 0.980785i
\(629\) 0 0
\(630\) 0 0
\(631\) 0.275899 + 1.38704i 0.275899 + 1.38704i 0.831470 + 0.555570i \(0.187500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.83147 + 0.180384i −1.83147 + 0.180384i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.881921 0.471397i \(-0.156250\pi\)
−0.881921 + 0.471397i \(0.843750\pi\)
\(642\) 0 0
\(643\) 1.66294 1.66294 0.831470 0.555570i \(-0.187500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −0.382683 + 1.92388i −0.382683 + 1.92388i
\(653\) 0 0 −0.956940 0.290285i \(-0.906250\pi\)
0.956940 + 0.290285i \(0.0937500\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.956940 0.290285i \(-0.0937500\pi\)
−0.956940 + 0.290285i \(0.906250\pi\)
\(660\) 0 0
\(661\) 0.324423 + 1.63099i 0.324423 + 1.63099i 0.707107 + 0.707107i \(0.250000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −2.21415 1.47945i −2.21415 1.47945i
\(677\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(678\) 0 0
\(679\) −0.151537 + 0.124363i −0.151537 + 0.124363i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0.636379 + 1.53636i 0.636379 + 1.53636i
\(689\) 0 0
\(690\) 0 0
\(691\) −0.923880 + 0.617317i −0.923880 + 0.617317i −0.923880 0.382683i \(-0.875000\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.187593 0.0569057i 0.187593 0.0569057i
\(701\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(702\) 0 0
\(703\) −1.04355 2.51936i −1.04355 2.51936i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.98079 0.195090i −1.98079 0.195090i −0.980785 0.195090i \(-0.937500\pi\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.995185 0.0980171i \(-0.0312500\pi\)
−0.995185 + 0.0980171i \(0.968750\pi\)
\(720\) 0 0
\(721\) 0.0271737 + 0.275899i 0.0271737 + 0.275899i
\(722\) 0 0
\(723\) 0 0
\(724\) −0.512016 0.273678i −0.512016 0.273678i
\(725\) 0 0
\(726\) 0 0
\(727\) −1.92388 + 0.382683i −1.92388 + 0.382683i −0.923880 + 0.382683i \(0.875000\pi\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0.149316 0.360480i 0.149316 0.360480i −0.831470 0.555570i \(-0.812500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0.555570 1.83147i 0.555570 1.83147i 1.00000i \(-0.5\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0.750661 + 0.149316i 0.750661 + 0.149316i 0.555570 0.831470i \(-0.312500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.448786 0.368309i 0.448786 0.368309i −0.382683 0.923880i \(-0.625000\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.0980171 0.995185i \(-0.531250\pi\)
0.0980171 + 0.995185i \(0.468750\pi\)
\(762\) 0 0
\(763\) 0.149316 + 0.0147063i 0.149316 + 0.0147063i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −0.368309 + 0.448786i −0.368309 + 0.448786i −0.923880 0.382683i \(-0.875000\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.149316 + 0.360480i −0.149316 + 0.360480i
\(773\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(774\) 0 0
\(775\) 1.84776i 1.84776i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.534220 0.799517i 0.534220 0.799517i
\(785\) 0 0
\(786\) 0 0
\(787\) −1.63099 1.08979i −1.63099 1.08979i −0.923880 0.382683i \(-0.875000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.208442 + 2.11635i −0.208442 + 2.11635i
\(794\) 0 0
\(795\) 0 0
\(796\) 1.68789 0.512016i 1.68789 0.512016i
\(797\) 0 0 −0.290285 0.956940i \(-0.593750\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(810\) 0 0
\(811\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.746304 + 2.46024i 0.746304 + 2.46024i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.471397 0.881921i \(-0.343750\pi\)
−0.471397 + 0.881921i \(0.656250\pi\)
\(822\) 0 0
\(823\) −0.750661 + 1.81225i −0.750661 + 1.81225i −0.195090 + 0.980785i \(0.562500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.634393 0.773010i \(-0.281250\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(828\) 0 0
\(829\) 0.324423 0.216773i 0.324423 0.216773i −0.382683 0.923880i \(-0.625000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.83147 0.555570i 1.83147 0.555570i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.956940 0.290285i \(-0.906250\pi\)
0.956940 + 0.290285i \(0.0937500\pi\)
\(840\) 0 0
\(841\) −0.555570 0.831470i −0.555570 0.831470i
\(842\) 0 0
\(843\) 0 0
\(844\) −0.598102 + 1.11897i −0.598102 + 1.11897i
\(845\) 0 0
\(846\) 0 0
\(847\) −0.0569057 + 0.187593i −0.0569057 + 0.187593i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0.555570 + 1.83147i 0.555570 + 1.83147i 0.555570 + 0.831470i \(0.312500\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) 0.804910 + 0.980785i 0.804910 + 0.980785i 1.00000 \(0\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.471397 0.881921i \(-0.656250\pi\)
0.471397 + 0.881921i \(0.343750\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0.229793 + 0.280003i 0.229793 + 0.280003i
\(869\) 0 0
\(870\) 0 0
\(871\) −1.76972 0.352020i −1.76972 0.352020i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.02656 + 0.425215i 1.02656 + 0.425215i 0.831470 0.555570i \(-0.187500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(882\) 0 0
\(883\) −0.902197 1.68789i −0.902197 1.68789i −0.707107 0.707107i \(-0.750000\pi\)
−0.195090 0.980785i \(-0.562500\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.634393 0.773010i \(-0.718750\pi\)
0.634393 + 0.773010i \(0.281250\pi\)
\(888\) 0 0
\(889\) −0.175876 0.175876i −0.175876 0.175876i
\(890\) 0 0
\(891\) 0 0
\(892\) −0.368309 0.448786i −0.368309 0.448786i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.90466 + 0.577774i 1.90466 + 0.577774i 0.980785 + 0.195090i \(0.0625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.471397 0.881921i \(-0.656250\pi\)
0.471397 + 0.881921i \(0.343750\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(917\) 0 0
\(918\) 0 0
\(919\) 1.11897 1.36347i 1.11897 1.36347i 0.195090 0.980785i \(-0.437500\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0.831470 1.55557i 0.831470 1.55557i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 −0.290285 0.956940i \(-0.593750\pi\)
0.290285 + 0.956940i \(0.406250\pi\)
\(930\) 0 0
\(931\) 0.943094 1.14916i 0.943094 1.14916i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.149316 + 0.360480i 0.149316 + 0.360480i 0.980785 0.195090i \(-0.0625000\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.881921 0.471397i \(-0.843750\pi\)
0.881921 + 0.471397i \(0.156250\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.881921 0.471397i \(-0.843750\pi\)
0.881921 + 0.471397i \(0.156250\pi\)
\(948\) 0 0
\(949\) 0.785695 + 2.59009i 0.785695 + 2.59009i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.0980171 0.995185i \(-0.468750\pi\)
−0.0980171 + 0.995185i \(0.531250\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.23044 0.923880i 2.23044 0.923880i
\(962\) 0 0
\(963\) 0 0
\(964\) 1.17588 1.17588i 1.17588 1.17588i
\(965\) 0 0
\(966\) 0 0
\(967\) −0.324423 0.216773i −0.324423 0.216773i 0.382683 0.923880i \(-0.375000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −0.102680 + 0.153672i −0.102680 + 0.153672i
\(974\) 0 0
\(975\) 0 0
\(976\) −0.785695 0.785695i −0.785695 0.785695i
\(977\) 0 0 0.634393 0.773010i \(-0.281250\pi\)
−0.634393 + 0.773010i \(0.718750\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 −0.995185 0.0980171i \(-0.968750\pi\)
0.995185 + 0.0980171i \(0.0312500\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 2.90205 0.577253i 2.90205 0.577253i
\(989\) 0 0
\(990\) 0 0
\(991\) 1.47945 0.448786i 1.47945 0.448786i 0.555570 0.831470i \(-0.312500\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.382683 + 0.0761205i 0.382683 + 0.0761205i 0.382683 0.923880i \(-0.375000\pi\)
1.00000i \(0.5\pi\)
\(998\) 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 873.1.br.a.451.1 yes 16
3.2 odd 2 CM 873.1.br.a.451.1 yes 16
97.77 odd 32 inner 873.1.br.a.271.1 16
291.77 even 32 inner 873.1.br.a.271.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
873.1.br.a.271.1 16 97.77 odd 32 inner
873.1.br.a.271.1 16 291.77 even 32 inner
873.1.br.a.451.1 yes 16 1.1 even 1 trivial
873.1.br.a.451.1 yes 16 3.2 odd 2 CM