Properties

Label 2979.1.c.d.2647.1
Level $2979$
Weight $1$
Character 2979.2647
Analytic conductor $1.487$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2979,1,Mod(2647,2979)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2979, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2979.2647");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2979 = 3^{2} \cdot 331 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2979.c (of order \(2\), degree \(1\), 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: \(1.48671467263\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 331)
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.331.1
Artin image: $\GL(2,3)$
Artin field: Galois closure of 8.2.2937439971.1

Embedding invariants

Embedding label 2647.1
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 2979.2647
Dual form 2979.1.c.d.2647.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.41421i q^{7} +O(q^{10})\) \(q-1.41421i q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.41421i q^{7} -1.41421i q^{10} -1.41421i q^{11} -2.00000 q^{14} -1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} +1.41421i q^{28} +1.41421i q^{29} +1.00000 q^{31} +1.41421i q^{32} +1.41421i q^{34} -1.41421i q^{35} +1.41421i q^{37} -1.41421i q^{38} +1.41421i q^{41} -1.00000 q^{43} +1.41421i q^{44} -1.00000 q^{49} +1.00000 q^{53} -1.41421i q^{55} +2.00000 q^{58} -1.41421i q^{61} -1.41421i q^{62} +1.00000 q^{64} +1.00000 q^{67} +1.00000 q^{68} -2.00000 q^{70} -1.00000 q^{71} +2.00000 q^{74} -1.00000 q^{76} -2.00000 q^{77} +1.00000 q^{79} -1.00000 q^{80} +2.00000 q^{82} -1.00000 q^{85} +1.41421i q^{86} +1.00000 q^{95} -1.41421i q^{97} +1.41421i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{5} - 4 q^{14} - 2 q^{16} - 2 q^{17} + 2 q^{19} - 2 q^{20} - 4 q^{22} + 2 q^{31} - 2 q^{43} - 2 q^{49} + 2 q^{53} + 4 q^{58} + 2 q^{64} + 2 q^{67} + 2 q^{68} - 4 q^{70} - 2 q^{71} + 4 q^{74} - 2 q^{76} - 4 q^{77} + 2 q^{79} - 2 q^{80} + 4 q^{82} - 2 q^{85} + 2 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2979\mathbb{Z}\right)^\times\).

\(n\) \(334\) \(1325\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(3\) 0 0
\(4\) −1.00000 −1.00000
\(5\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) − 1.41421i − 1.41421i
\(11\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −2.00000 −2.00000
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) −1.00000 −1.00000
\(21\) 0 0
\(22\) −2.00000 −2.00000
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 1.41421i 1.41421i
\(29\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) 0 0
\(31\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 1.41421i 1.41421i
\(33\) 0 0
\(34\) 1.41421i 1.41421i
\(35\) − 1.41421i − 1.41421i
\(36\) 0 0
\(37\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) − 1.41421i − 1.41421i
\(39\) 0 0
\(40\) 0 0
\(41\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(42\) 0 0
\(43\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 1.41421i 1.41421i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −1.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) − 1.41421i − 1.41421i
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000 2.00000
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(62\) − 1.41421i − 1.41421i
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 1.00000 1.00000
\(69\) 0 0
\(70\) −2.00000 −2.00000
\(71\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 2.00000 2.00000
\(75\) 0 0
\(76\) −1.00000 −1.00000
\(77\) −2.00000 −2.00000
\(78\) 0 0
\(79\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) −1.00000 −1.00000
\(81\) 0 0
\(82\) 2.00000 2.00000
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −1.00000 −1.00000
\(86\) 1.41421i 1.41421i
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 1.00000
\(96\) 0 0
\(97\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(98\) 1.41421i 1.41421i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) − 1.41421i − 1.41421i
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(110\) −2.00000 −2.00000
\(111\) 0 0
\(112\) 1.41421i 1.41421i
\(113\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(114\) 0 0
\(115\) 0 0
\(116\) − 1.41421i − 1.41421i
\(117\) 0 0
\(118\) 0 0
\(119\) 1.41421i 1.41421i
\(120\) 0 0
\(121\) −1.00000 −1.00000
\(122\) −2.00000 −2.00000
\(123\) 0 0
\(124\) −1.00000 −1.00000
\(125\) −1.00000 −1.00000
\(126\) 0 0
\(127\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) − 1.41421i − 1.41421i
\(134\) − 1.41421i − 1.41421i
\(135\) 0 0
\(136\) 0 0
\(137\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 1.41421i 1.41421i
\(141\) 0 0
\(142\) 1.41421i 1.41421i
\(143\) 0 0
\(144\) 0 0
\(145\) 1.41421i 1.41421i
\(146\) 0 0
\(147\) 0 0
\(148\) − 1.41421i − 1.41421i
\(149\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 2.82843i 2.82843i
\(155\) 1.00000 1.00000
\(156\) 0 0
\(157\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(158\) − 1.41421i − 1.41421i
\(159\) 0 0
\(160\) 1.41421i 1.41421i
\(161\) 0 0
\(162\) 0 0
\(163\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(164\) − 1.41421i − 1.41421i
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 1.41421i 1.41421i
\(171\) 0 0
\(172\) 1.00000 1.00000
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.41421i 1.41421i
\(177\) 0 0
\(178\) 0 0
\(179\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.41421i 1.41421i
\(186\) 0 0
\(187\) 1.41421i 1.41421i
\(188\) 0 0
\(189\) 0 0
\(190\) − 1.41421i − 1.41421i
\(191\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) −2.00000 −2.00000
\(195\) 0 0
\(196\) 1.00000 1.00000
\(197\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(198\) 0 0
\(199\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.00000 2.00000
\(204\) 0 0
\(205\) 1.41421i 1.41421i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 1.41421i − 1.41421i
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −1.00000 −1.00000
\(213\) 0 0
\(214\) 0 0
\(215\) −1.00000 −1.00000
\(216\) 0 0
\(217\) − 1.41421i − 1.41421i
\(218\) 1.41421i 1.41421i
\(219\) 0 0
\(220\) 1.41421i 1.41421i
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(224\) 2.00000 2.00000
\(225\) 0 0
\(226\) − 2.82843i − 2.82843i
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 2.00000 2.00000
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) 1.41421i 1.41421i
\(243\) 0 0
\(244\) 1.41421i 1.41421i
\(245\) −1.00000 −1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 1.41421i 1.41421i
\(251\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 1.41421i 1.41421i
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 2.00000 2.00000
\(260\) 0 0
\(261\) 0 0
\(262\) 1.41421i 1.41421i
\(263\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(264\) 0 0
\(265\) 1.00000 1.00000
\(266\) −2.00000 −2.00000
\(267\) 0 0
\(268\) −1.00000 −1.00000
\(269\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(272\) 1.00000 1.00000
\(273\) 0 0
\(274\) −2.00000 −2.00000
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(282\) 0 0
\(283\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(284\) 1.00000 1.00000
\(285\) 0 0
\(286\) 0 0
\(287\) 2.00000 2.00000
\(288\) 0 0
\(289\) 0 0
\(290\) 2.00000 2.00000
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) − 1.41421i − 1.41421i
\(299\) 0 0
\(300\) 0 0
\(301\) 1.41421i 1.41421i
\(302\) 0 0
\(303\) 0 0
\(304\) −1.00000 −1.00000
\(305\) − 1.41421i − 1.41421i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 2.00000 2.00000
\(309\) 0 0
\(310\) − 1.41421i − 1.41421i
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(314\) 1.41421i 1.41421i
\(315\) 0 0
\(316\) −1.00000 −1.00000
\(317\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 2.00000 2.00000
\(320\) 1.00000 1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) −1.00000 −1.00000
\(324\) 0 0
\(325\) 0 0
\(326\) − 1.41421i − 1.41421i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.00000 −1.00000
\(332\) 0 0
\(333\) 0 0
\(334\) − 1.41421i − 1.41421i
\(335\) 1.00000 1.00000
\(336\) 0 0
\(337\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) − 1.41421i − 1.41421i
\(339\) 0 0
\(340\) 1.00000 1.00000
\(341\) − 1.41421i − 1.41421i
\(342\) 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
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000 2.00000
\(353\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) −1.00000 −1.00000
\(356\) 0 0
\(357\) 0 0
\(358\) − 1.41421i − 1.41421i
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) −2.00000 −2.00000
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 2.00000 2.00000
\(371\) − 1.41421i − 1.41421i
\(372\) 0 0
\(373\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(374\) 2.00000 2.00000
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) −1.00000 −1.00000
\(381\) 0 0
\(382\) − 1.41421i − 1.41421i
\(383\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(384\) 0 0
\(385\) −2.00000 −2.00000
\(386\) 0 0
\(387\) 0 0
\(388\) 1.41421i 1.41421i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 1.41421i 1.41421i
\(395\) 1.00000 1.00000
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 2.00000 2.00000
\(399\) 0 0
\(400\) 0 0
\(401\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) − 2.82843i − 2.82843i
\(407\) 2.00000 2.00000
\(408\) 0 0
\(409\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(410\) 2.00000 2.00000
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) −2.00000 −2.00000
\(419\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(420\) 0 0
\(421\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.00000 −2.00000
\(428\) 0 0
\(429\) 0 0
\(430\) 1.41421i 1.41421i
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(434\) −2.00000 −2.00000
\(435\) 0 0
\(436\) 1.00000 1.00000
\(437\) 0 0
\(438\) 0 0
\(439\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.41421i 1.41421i
\(447\) 0 0
\(448\) − 1.41421i − 1.41421i
\(449\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(450\) 0 0
\(451\) 2.00000 2.00000
\(452\) −2.00000 −2.00000
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 2.00000 2.00000
\(459\) 0 0
\(460\) 0 0
\(461\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) − 1.41421i − 1.41421i
\(465\) 0 0
\(466\) 1.41421i 1.41421i
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) − 1.41421i − 1.41421i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.41421i 1.41421i
\(474\) 0 0
\(475\) 0 0
\(476\) − 1.41421i − 1.41421i
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 1.41421i − 1.41421i
\(483\) 0 0
\(484\) 1.00000 1.00000
\(485\) − 1.41421i − 1.41421i
\(486\) 0 0
\(487\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 1.41421i 1.41421i
\(491\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(492\) 0 0
\(493\) − 1.41421i − 1.41421i
\(494\) 0 0
\(495\) 0 0
\(496\) −1.00000 −1.00000
\(497\) 1.41421i 1.41421i
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 1.00000 1.00000
\(501\) 0 0
\(502\) 2.00000 2.00000
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 1.00000 1.00000
\(509\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.41421i − 1.41421i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) − 2.82843i − 2.82843i
\(519\) 0 0
\(520\) 0 0
\(521\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(522\) 0 0
\(523\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(524\) 1.00000 1.00000
\(525\) 0 0
\(526\) −2.00000 −2.00000
\(527\) −1.00000 −1.00000
\(528\) 0 0
\(529\) 1.00000 1.00000
\(530\) − 1.41421i − 1.41421i
\(531\) 0 0
\(532\) 1.41421i 1.41421i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) − 1.41421i − 1.41421i
\(539\) 1.41421i 1.41421i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) − 1.41421i − 1.41421i
\(543\) 0 0
\(544\) − 1.41421i − 1.41421i
\(545\) −1.00000 −1.00000
\(546\) 0 0
\(547\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(548\) 1.41421i 1.41421i
\(549\) 0 0
\(550\) 0 0
\(551\) 1.41421i 1.41421i
\(552\) 0 0
\(553\) − 1.41421i − 1.41421i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1.41421i 1.41421i
\(561\) 0 0
\(562\) − 1.41421i − 1.41421i
\(563\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 0 0
\(565\) 2.00000 2.00000
\(566\) 1.41421i 1.41421i
\(567\) 0 0
\(568\) 0 0
\(569\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 2.82843i − 2.82843i
\(575\) 0 0
\(576\) 0 0
\(577\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) − 1.41421i − 1.41421i
\(581\) 0 0
\(582\) 0 0
\(583\) − 1.41421i − 1.41421i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 1.00000 1.00000
\(590\) 0 0
\(591\) 0 0
\(592\) − 1.41421i − 1.41421i
\(593\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 1.41421i 1.41421i
\(596\) −1.00000 −1.00000
\(597\) 0 0
\(598\) 0 0
\(599\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(602\) 2.00000 2.00000
\(603\) 0 0
\(604\) 0 0
\(605\) −1.00000 −1.00000
\(606\) 0 0
\(607\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 1.41421i 1.41421i
\(609\) 0 0
\(610\) −2.00000 −2.00000
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) −1.00000 −1.00000
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −1.00000
\(626\) 1.41421i 1.41421i
\(627\) 0 0
\(628\) 1.00000 1.00000
\(629\) − 1.41421i − 1.41421i
\(630\) 0 0
\(631\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 2.00000 2.00000
\(635\) −1.00000 −1.00000
\(636\) 0 0
\(637\) 0 0
\(638\) − 2.82843i − 2.82843i
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1.41421i 1.41421i
\(647\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −1.00000 −1.00000
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −1.00000 −1.00000
\(656\) − 1.41421i − 1.41421i
\(657\) 0 0
\(658\) 0 0
\(659\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(660\) 0 0
\(661\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(662\) 1.41421i 1.41421i
\(663\) 0 0
\(664\) 0 0
\(665\) − 1.41421i − 1.41421i
\(666\) 0 0
\(667\) 0 0
\(668\) −1.00000 −1.00000
\(669\) 0 0
\(670\) − 1.41421i − 1.41421i
\(671\) −2.00000 −2.00000
\(672\) 0 0
\(673\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(674\) 1.41421i 1.41421i
\(675\) 0 0
\(676\) −1.00000 −1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −2.00000 −2.00000
\(680\) 0 0
\(681\) 0 0
\(682\) −2.00000 −2.00000
\(683\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(684\) 0 0
\(685\) − 1.41421i − 1.41421i
\(686\) 0 0
\(687\) 0 0
\(688\) 1.00000 1.00000
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 1.41421i − 1.41421i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(702\) 0 0
\(703\) 1.41421i 1.41421i
\(704\) − 1.41421i − 1.41421i
\(705\) 0 0
\(706\) 1.41421i 1.41421i
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 1.41421i 1.41421i
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −1.00000 −1.00000
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 1.41421i 1.41421i
\(725\) 0 0
\(726\) 0 0
\(727\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.00000 1.00000
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 1.41421i − 1.41421i
\(738\) 0 0
\(739\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(740\) − 1.41421i − 1.41421i
\(741\) 0 0
\(742\) −2.00000 −2.00000
\(743\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(744\) 0 0
\(745\) 1.00000 1.00000
\(746\) −2.00000 −2.00000
\(747\) 0 0
\(748\) − 1.41421i − 1.41421i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.41421i 1.41421i
\(764\) −1.00000 −1.00000
\(765\) 0 0
\(766\) −2.00000 −2.00000
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 2.82843i 2.82843i
\(771\) 0 0
\(772\) 0 0
\(773\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.41421i 1.41421i
\(780\) 0 0
\(781\) 1.41421i 1.41421i
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 1.00000
\(785\) −1.00000 −1.00000
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 1.00000 1.00000
\(789\) 0 0
\(790\) − 1.41421i − 1.41421i
\(791\) − 2.82843i − 2.82843i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) − 1.41421i − 1.41421i
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 1.41421i 1.41421i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(812\) −2.00000 −2.00000
\(813\) 0 0
\(814\) − 2.82843i − 2.82843i
\(815\) 1.00000 1.00000
\(816\) 0 0
\(817\) −1.00000 −1.00000
\(818\) 2.00000 2.00000
\(819\) 0 0
\(820\) − 1.41421i − 1.41421i
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(828\) 0 0
\(829\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.00000 1.00000
\(834\) 0 0
\(835\) 1.00000 1.00000
\(836\) 1.41421i 1.41421i
\(837\) 0 0
\(838\) − 1.41421i − 1.41421i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −1.00000 −1.00000
\(842\) 2.00000 2.00000
\(843\) 0 0
\(844\) 0 0
\(845\) 1.00000 1.00000
\(846\) 0 0
\(847\) 1.41421i 1.41421i
\(848\) −1.00000 −1.00000
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(854\) 2.82843i 2.82843i
\(855\) 0 0
\(856\) 0 0
\(857\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(858\) 0 0
\(859\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(860\) 1.00000 1.00000
\(861\) 0 0
\(862\) 0 0
\(863\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) − 1.41421i − 1.41421i
\(867\) 0 0
\(868\) 1.41421i 1.41421i
\(869\) − 1.41421i − 1.41421i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.41421i 1.41421i
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 2.00000 2.00000
\(879\) 0 0
\(880\) 1.41421i 1.41421i
\(881\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(882\) 0 0
\(883\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(888\) 0 0
\(889\) 1.41421i 1.41421i
\(890\) 0 0
\(891\) 0 0
\(892\) 1.00000 1.00000
\(893\) 0 0
\(894\) 0 0
\(895\) 1.00000 1.00000
\(896\) 0 0
\(897\) 0 0
\(898\) 1.41421i 1.41421i
\(899\) 1.41421i 1.41421i
\(900\) 0 0
\(901\) −1.00000 −1.00000
\(902\) − 2.82843i − 2.82843i
\(903\) 0 0
\(904\) 0 0
\(905\) − 1.41421i − 1.41421i
\(906\) 0 0
\(907\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1.41421i 1.41421i
\(915\) 0 0
\(916\) − 1.41421i − 1.41421i
\(917\) 1.41421i 1.41421i
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.41421i 1.41421i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) −2.00000 −2.00000
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −1.00000 −1.00000
\(932\) 1.00000 1.00000
\(933\) 0 0
\(934\) 0 0
\(935\) 1.41421i 1.41421i
\(936\) 0 0
\(937\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(938\) −2.00000 −2.00000
\(939\) 0 0
\(940\) 0 0
\(941\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 2.00000 2.00000
\(947\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 1.00000 1.00000
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.00000 −2.00000
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) −1.00000 −1.00000
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −2.00000 −2.00000
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 1.41421i − 1.41421i
\(975\) 0 0
\(976\) 1.41421i 1.41421i
\(977\) − 1.41421i − 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.00000 1.00000
\(981\) 0 0
\(982\) −2.00000 −2.00000
\(983\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(984\) 0 0
\(985\) −1.00000 −1.00000
\(986\) −2.00000 −2.00000
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 1.41421i 1.41421i
\(993\) 0 0
\(994\) 2.00000 2.00000
\(995\) 1.41421i 1.41421i
\(996\) 0 0
\(997\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(998\) 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 2979.1.c.d.2647.1 2
3.2 odd 2 331.1.b.b.330.2 yes 2
331.330 odd 2 inner 2979.1.c.d.2647.2 2
993.992 even 2 331.1.b.b.330.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
331.1.b.b.330.1 2 993.992 even 2
331.1.b.b.330.2 yes 2 3.2 odd 2
2979.1.c.d.2647.1 2 1.1 even 1 trivial
2979.1.c.d.2647.2 2 331.330 odd 2 inner