Properties

Label 2023.1.f.a.1483.2
Level $2023$
Weight $1$
Character 2023.1483
Analytic conductor $1.010$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2023,1,Mod(251,2023)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2023, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2023.251");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2023 = 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2023.f (of order \(4\), degree \(2\), not 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: \(1.00960852056\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2023.1
Artin image: $S_3\times C_8$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

Embedding invariants

Embedding label 1483.2
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2023.1483
Dual form 2023.1.f.a.251.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +(0.707107 + 0.707107i) q^{7} -1.00000i q^{8} -1.00000i q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +(0.707107 + 0.707107i) q^{7} -1.00000i q^{8} -1.00000i q^{9} +(-1.41421 - 1.41421i) q^{11} +(0.707107 - 0.707107i) q^{14} -1.00000 q^{16} -1.00000 q^{18} +(-1.41421 + 1.41421i) q^{22} +(0.707107 + 0.707107i) q^{23} -1.00000i q^{25} +(-0.707107 + 0.707107i) q^{29} +(0.707107 - 0.707107i) q^{37} +1.00000i q^{43} +(0.707107 - 0.707107i) q^{46} +1.00000i q^{49} -1.00000 q^{50} -1.00000i q^{53} +(0.707107 - 0.707107i) q^{56} +(0.707107 + 0.707107i) q^{58} +(0.707107 - 0.707107i) q^{63} -1.00000 q^{64} +2.00000 q^{67} +(0.707107 - 0.707107i) q^{71} -1.00000 q^{72} +(-0.707107 - 0.707107i) q^{74} -2.00000i q^{77} +(0.707107 + 0.707107i) q^{79} -1.00000 q^{81} +1.00000 q^{86} +(-1.41421 + 1.41421i) q^{88} +1.00000 q^{98} +(-1.41421 + 1.41421i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{16} - 4 q^{18} - 4 q^{50} - 4 q^{64} + 8 q^{67} - 4 q^{72} - 4 q^{81} + 4 q^{86} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2023\mathbb{Z}\right)^\times\).

\(n\) \(290\) \(1737\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(3\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0 0
\(7\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(8\) 1.00000i 1.00000i
\(9\) 1.00000i 1.00000i
\(10\) 0 0
\(11\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0.707107 0.707107i 0.707107 0.707107i
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) 0 0
\(18\) −1.00000 −1.00000
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(23\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(24\) 0 0
\(25\) 1.00000i 1.00000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(30\) 0 0
\(31\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(42\) 0 0
\(43\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.707107 0.707107i 0.707107 0.707107i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.00000i 1.00000i
\(50\) −1.00000 −1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.707107 0.707107i 0.707107 0.707107i
\(57\) 0 0
\(58\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(62\) 0 0
\(63\) 0.707107 0.707107i 0.707107 0.707107i
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(72\) −1.00000 −1.00000
\(73\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) −0.707107 0.707107i −0.707107 0.707107i
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000i 2.00000i
\(78\) 0 0
\(79\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(80\) 0 0
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.00000 1.00000
\(87\) 0 0
\(88\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 1.00000 1.00000
\(99\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1.00000 −1.00000
\(107\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(108\) 0 0
\(109\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.707107 0.707107i −0.707107 0.707107i
\(113\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.00000i 3.00000i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −0.707107 0.707107i −0.707107 0.707107i
\(127\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(128\) 1.00000i 1.00000i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2.00000i 2.00000i
\(135\) 0 0
\(136\) 0 0
\(137\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.707107 0.707107i −0.707107 0.707107i
\(143\) 0 0
\(144\) 1.00000i 1.00000i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −2.00000 −2.00000
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0.707107 0.707107i 0.707107 0.707107i
\(159\) 0 0
\(160\) 0 0
\(161\) 1.00000i 1.00000i
\(162\) 1.00000i 1.00000i
\(163\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0.707107 0.707107i 0.707107 0.707107i
\(176\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(177\) 0 0
\(178\) 0 0
\(179\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 0 0
\(181\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0.707107 0.707107i 0.707107 0.707107i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 0 0
\(193\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(199\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(200\) −1.00000 −1.00000
\(201\) 0 0
\(202\) 0 0
\(203\) −1.00000 −1.00000
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.707107 0.707107i 0.707107 0.707107i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −1.00000 −1.00000
\(226\) 0.707107 0.707107i 0.707107 0.707107i
\(227\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(233\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(240\) 0 0
\(241\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(242\) 3.00000 3.00000
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 2.00000i 2.00000i
\(254\) 1.00000 1.00000
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 1.00000 1.00000
\(260\) 0 0
\(261\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(262\) 0 0
\(263\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 1.00000i 1.00000i
\(275\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(276\) 0 0
\(277\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(282\) 0 0
\(283\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.707107 0.707107i −0.707107 0.707107i
\(297\) 0 0
\(298\) 1.00000i 1.00000i
\(299\) 0 0
\(300\) 0 0
\(301\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(302\) 2.00000 2.00000
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(312\) 0 0
\(313\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(318\) 0 0
\(319\) 2.00000 2.00000
\(320\) 0 0
\(321\) 0 0
\(322\) 1.00000 1.00000
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(332\) 0 0
\(333\) −0.707107 0.707107i −0.707107 0.707107i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(338\) 1.00000i 1.00000i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(344\) 1.00000 1.00000
\(345\) 0 0
\(346\) 0 0
\(347\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) −0.707107 0.707107i −0.707107 0.707107i
\(351\) 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
\(358\) 1.00000 1.00000
\(359\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(360\) 0 0
\(361\) −1.00000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(368\) −0.707107 0.707107i −0.707107 0.707107i
\(369\) 0 0
\(370\) 0 0
\(371\) 0.707107 0.707107i 0.707107 0.707107i
\(372\) 0 0
\(373\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.00000i 1.00000i
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(387\) 1.00000 1.00000
\(388\) 0 0
\(389\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 1.00000
\(393\) 0 0
\(394\) 1.41421 1.41421i 1.41421 1.41421i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000i 1.00000i
\(401\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 1.00000i 1.00000i
\(407\) −2.00000 −2.00000
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −0.707107 0.707107i −0.707107 0.707107i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(420\) 0 0
\(421\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(422\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(423\) 0 0
\(424\) −1.00000 −1.00000
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(440\) 0 0
\(441\) 1.00000 1.00000
\(442\) 0 0
\(443\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.707107 0.707107i −0.707107 0.707107i
\(449\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(450\) 1.00000i 1.00000i
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(464\) 0.707107 0.707107i 0.707107 0.707107i
\(465\) 0 0
\(466\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.41421 1.41421i 1.41421 1.41421i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.00000 −1.00000
\(478\) 2.00000i 2.00000i
\(479\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.00000 1.00000
\(498\) 0 0
\(499\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) −0.707107 0.707107i −0.707107 0.707107i
\(505\) 0 0
\(506\) −2.00000 −2.00000
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 1.00000i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 1.00000i 1.00000i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(522\) 0.707107 0.707107i 0.707107 0.707107i
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 1.00000 1.00000
\(527\) 0 0
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 2.00000i 2.00000i
\(537\) 0 0
\(538\) 0 0
\(539\) 1.41421 1.41421i 1.41421 1.41421i
\(540\) 0 0
\(541\) −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i \(0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(551\) 0 0
\(552\) 0 0
\(553\) 1.00000i 1.00000i
\(554\) −1.41421 1.41421i −1.41421 1.41421i
\(555\) 0 0
\(556\) 0 0
\(557\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 1.00000 1.00000
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.707107 0.707107i −0.707107 0.707107i
\(568\) −0.707107 0.707107i −0.707107 0.707107i
\(569\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(570\) 0 0
\(571\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.707107 0.707107i 0.707107 0.707107i
\(576\) 1.00000i 1.00000i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.41421 + 1.41421i −1.41421 + 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
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(602\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(603\) 2.00000i 2.00000i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −2.00000 −2.00000
\(617\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(618\) 0 0
\(619\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(632\) 0.707107 0.707107i 0.707107 0.707107i
\(633\) 0 0
\(634\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 2.00000i 2.00000i
\(639\) −0.707107 0.707107i −0.707107 0.707107i
\(640\) 0 0
\(641\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(642\) 0 0
\(643\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 1.00000i 1.00000i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 1.00000 1.00000
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(667\) −1.00000 −1.00000
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(674\) −0.707107 0.707107i −0.707107 0.707107i
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(687\) 0 0
\(688\) 1.00000i 1.00000i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(692\) 0 0
\(693\) −2.00000 −2.00000
\(694\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(710\) 0 0
\(711\) 0.707107 0.707107i 0.707107 0.707107i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) −1.00000 −1.00000
\(719\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.00000i 1.00000i
\(723\) 0 0
\(724\) 0 0
\(725\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.00000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.82843 2.82843i −2.82843 2.82843i
\(738\) 0 0
\(739\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.707107 0.707107i −0.707107 0.707107i
\(743\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.00000i 1.00000i
\(747\) 0 0
\(748\) 0 0
\(749\) −1.00000 −1.00000
\(750\) 0 0
\(751\) −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i \(0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(758\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.00000i 1.00000i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 1.00000i 1.00000i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −1.00000 −1.00000
\(779\) 0 0
\(780\) 0 0
\(781\) −2.00000 −2.00000
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000i 1.00000i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.00000i 1.00000i
\(792\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 1.41421 1.41421i 1.41421 1.41421i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(810\) 0 0
\(811\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 2.00000i 2.00000i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(822\) 0 0
\(823\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 2.00000i 2.00000i
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −2.12132 + 2.12132i −2.12132 + 2.12132i
\(848\) 1.00000i 1.00000i
\(849\) 0 0
\(850\) 0 0
\(851\) 1.00000 1.00000
\(852\) 0 0
\(853\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(857\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0.707107 0.707107i 0.707107 0.707107i
\(863\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.00000i 2.00000i
\(870\) 0 0
\(871\) 0 0
\(872\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(882\) 1.00000i 1.00000i
\(883\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.00000i 1.00000i
\(887\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(888\) 0 0
\(889\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(890\) 0 0
\(891\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(897\) 0 0
\(898\) 1.41421 1.41421i 1.41421 1.41421i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0.707107 0.707107i 0.707107 0.707107i
\(905\) 0 0
\(906\) 0 0
\(907\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1.00000 −1.00000
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −0.707107 0.707107i −0.707107 0.707107i
\(926\) 1.00000i 1.00000i
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 1.41421 1.41421i 1.41421 1.41421i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −1.41421 1.41421i −1.41421 1.41421i
\(947\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(954\) 1.00000i 1.00000i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −0.707107 0.707107i −0.707107 0.707107i
\(960\) 0 0
\(961\) 1.00000i 1.00000i
\(962\) 0 0
\(963\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(968\) 3.00000 3.00000
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0.707107 0.707107i 0.707107 0.707107i
\(975\) 0 0
\(976\) 0 0
\(977\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(982\) 2.00000 2.00000
\(983\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(990\) 0 0
\(991\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 1.00000i 1.00000i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(998\) 0.707107 0.707107i 0.707107 0.707107i
\(999\) 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 2023.1.f.a.1483.2 4
7.6 odd 2 CM 2023.1.f.a.1483.2 4
17.2 even 8 2023.1.d.a.2022.1 2
17.3 odd 16 2023.1.l.a.1889.2 8
17.4 even 4 inner 2023.1.f.a.251.1 4
17.5 odd 16 2023.1.l.a.1868.1 8
17.6 odd 16 2023.1.l.a.1266.1 8
17.7 odd 16 2023.1.l.a.468.1 8
17.8 even 8 2023.1.c.b.1735.1 yes 1
17.9 even 8 2023.1.c.a.1735.1 1
17.10 odd 16 2023.1.l.a.468.2 8
17.11 odd 16 2023.1.l.a.1266.2 8
17.12 odd 16 2023.1.l.a.1868.2 8
17.13 even 4 inner 2023.1.f.a.251.2 4
17.14 odd 16 2023.1.l.a.1889.1 8
17.15 even 8 2023.1.d.a.2022.2 2
17.16 even 2 inner 2023.1.f.a.1483.1 4
119.6 even 16 2023.1.l.a.1266.1 8
119.13 odd 4 inner 2023.1.f.a.251.2 4
119.20 even 16 2023.1.l.a.1889.2 8
119.27 even 16 2023.1.l.a.468.2 8
119.41 even 16 2023.1.l.a.468.1 8
119.48 even 16 2023.1.l.a.1889.1 8
119.55 odd 4 inner 2023.1.f.a.251.1 4
119.62 even 16 2023.1.l.a.1266.2 8
119.76 odd 8 2023.1.c.b.1735.1 yes 1
119.83 odd 8 2023.1.d.a.2022.2 2
119.90 even 16 2023.1.l.a.1868.1 8
119.97 even 16 2023.1.l.a.1868.2 8
119.104 odd 8 2023.1.d.a.2022.1 2
119.111 odd 8 2023.1.c.a.1735.1 1
119.118 odd 2 inner 2023.1.f.a.1483.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2023.1.c.a.1735.1 1 17.9 even 8
2023.1.c.a.1735.1 1 119.111 odd 8
2023.1.c.b.1735.1 yes 1 17.8 even 8
2023.1.c.b.1735.1 yes 1 119.76 odd 8
2023.1.d.a.2022.1 2 17.2 even 8
2023.1.d.a.2022.1 2 119.104 odd 8
2023.1.d.a.2022.2 2 17.15 even 8
2023.1.d.a.2022.2 2 119.83 odd 8
2023.1.f.a.251.1 4 17.4 even 4 inner
2023.1.f.a.251.1 4 119.55 odd 4 inner
2023.1.f.a.251.2 4 17.13 even 4 inner
2023.1.f.a.251.2 4 119.13 odd 4 inner
2023.1.f.a.1483.1 4 17.16 even 2 inner
2023.1.f.a.1483.1 4 119.118 odd 2 inner
2023.1.f.a.1483.2 4 1.1 even 1 trivial
2023.1.f.a.1483.2 4 7.6 odd 2 CM
2023.1.l.a.468.1 8 17.7 odd 16
2023.1.l.a.468.1 8 119.41 even 16
2023.1.l.a.468.2 8 17.10 odd 16
2023.1.l.a.468.2 8 119.27 even 16
2023.1.l.a.1266.1 8 17.6 odd 16
2023.1.l.a.1266.1 8 119.6 even 16
2023.1.l.a.1266.2 8 17.11 odd 16
2023.1.l.a.1266.2 8 119.62 even 16
2023.1.l.a.1868.1 8 17.5 odd 16
2023.1.l.a.1868.1 8 119.90 even 16
2023.1.l.a.1868.2 8 17.12 odd 16
2023.1.l.a.1868.2 8 119.97 even 16
2023.1.l.a.1889.1 8 17.14 odd 16
2023.1.l.a.1889.1 8 119.48 even 16
2023.1.l.a.1889.2 8 17.3 odd 16
2023.1.l.a.1889.2 8 119.20 even 16