Properties

Label 2205.1.bi.b.2089.3
Level $2205$
Weight $1$
Character 2205.2089
Analytic conductor $1.100$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,1,Mod(19,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.19");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2205.bi (of order \(6\), 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: \(1.10043835286\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.8338372875.1

Embedding invariants

Embedding label 2089.3
Root \(0.662827 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 2205.2089
Dual form 2205.1.bi.b.19.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.662827 - 0.382683i) q^{2} +(-0.207107 + 0.358719i) q^{4} +(0.500000 + 0.866025i) q^{5} +1.08239i q^{8} +O(q^{10})\) \(q+(0.662827 - 0.382683i) q^{2} +(-0.207107 + 0.358719i) q^{4} +(0.500000 + 0.866025i) q^{5} +1.08239i q^{8} +(0.662827 + 0.382683i) q^{10} +(0.207107 + 0.358719i) q^{16} +(-0.707107 + 1.22474i) q^{17} +(-0.662827 + 0.382683i) q^{19} -0.414214 q^{20} +(1.60021 - 0.923880i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-1.60021 - 0.923880i) q^{31} +(-0.662827 - 0.382683i) q^{32} +1.08239i q^{34} +(-0.292893 + 0.507306i) q^{38} +(-0.937379 + 0.541196i) q^{40} +(0.707107 - 1.22474i) q^{46} +(0.707107 + 1.22474i) q^{47} +0.765367i q^{50} +(1.60021 + 0.923880i) q^{53} +(1.60021 - 0.923880i) q^{61} -1.41421 q^{62} -1.00000 q^{64} +(-0.292893 - 0.507306i) q^{68} -0.317025i q^{76} +(-0.707107 - 1.22474i) q^{79} +(-0.207107 + 0.358719i) q^{80} +1.41421 q^{83} -1.41421 q^{85} +0.765367i q^{92} +(0.937379 + 0.541196i) q^{94} +(-0.662827 - 0.382683i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 4 q^{5} - 4 q^{16} + 8 q^{20} - 4 q^{25} - 8 q^{38} - 8 q^{64} - 8 q^{68} + 4 q^{80}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(1081\) \(1226\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.662827 0.382683i 0.662827 0.382683i −0.130526 0.991445i \(-0.541667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(3\) 0 0
\(4\) −0.207107 + 0.358719i −0.207107 + 0.358719i
\(5\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(6\) 0 0
\(7\) 0 0
\(8\) 1.08239i 1.08239i
\(9\) 0 0
\(10\) 0.662827 + 0.382683i 0.662827 + 0.382683i
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.207107 + 0.358719i 0.207107 + 0.358719i
\(17\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(18\) 0 0
\(19\) −0.662827 + 0.382683i −0.662827 + 0.382683i −0.793353 0.608761i \(-0.791667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(20\) −0.414214 −0.414214
\(21\) 0 0
\(22\) 0 0
\(23\) 1.60021 0.923880i 1.60021 0.923880i 0.608761 0.793353i \(-0.291667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −1.60021 0.923880i −1.60021 0.923880i −0.991445 0.130526i \(-0.958333\pi\)
−0.608761 0.793353i \(-0.708333\pi\)
\(32\) −0.662827 0.382683i −0.662827 0.382683i
\(33\) 0 0
\(34\) 1.08239i 1.08239i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) −0.292893 + 0.507306i −0.292893 + 0.507306i
\(39\) 0 0
\(40\) −0.937379 + 0.541196i −0.937379 + 0.541196i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.707107 1.22474i 0.707107 1.22474i
\(47\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0.765367i 0.765367i
\(51\) 0 0
\(52\) 0 0
\(53\) 1.60021 + 0.923880i 1.60021 + 0.923880i 0.991445 + 0.130526i \(0.0416667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 1.60021 0.923880i 1.60021 0.923880i 0.608761 0.793353i \(-0.291667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(62\) −1.41421 −1.41421
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) −0.292893 0.507306i −0.292893 0.507306i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0.317025i 0.317025i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(80\) −0.207107 + 0.358719i −0.207107 + 0.358719i
\(81\) 0 0
\(82\) 0 0
\(83\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) −1.41421 −1.41421
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.765367i 0.765367i
\(93\) 0 0
\(94\) 0.937379 + 0.541196i 0.937379 + 0.541196i
\(95\) −0.662827 0.382683i −0.662827 0.382683i
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −0.207107 0.358719i −0.207107 0.358719i
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 0 0
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1.41421 1.41421
\(107\) −0.662827 + 0.382683i −0.662827 + 0.382683i −0.793353 0.608761i \(-0.791667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(108\) 0 0
\(109\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(114\) 0 0
\(115\) 1.60021 + 0.923880i 1.60021 + 0.923880i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(122\) 0.707107 1.22474i 0.707107 1.22474i
\(123\) 0 0
\(124\) 0.662827 0.382683i 0.662827 0.382683i
\(125\) −1.00000 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −1.32565 0.765367i −1.32565 0.765367i
\(137\) 0.662827 + 0.382683i 0.662827 + 0.382683i 0.793353 0.608761i \(-0.208333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(138\) 0 0
\(139\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(152\) −0.414214 0.717439i −0.414214 0.717439i
\(153\) 0 0
\(154\) 0 0
\(155\) 1.84776i 1.84776i
\(156\) 0 0
\(157\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(158\) −0.937379 0.541196i −0.937379 0.541196i
\(159\) 0 0
\(160\) 0.765367i 0.765367i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0.937379 0.541196i 0.937379 0.541196i
\(167\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(168\) 0 0
\(169\) −1.00000 −1.00000
\(170\) −0.937379 + 0.541196i −0.937379 + 0.541196i
\(171\) 0 0
\(172\) 0 0
\(173\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(180\) 0 0
\(181\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −0.585786 −0.585786
\(189\) 0 0
\(190\) −0.585786 −0.585786
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) 0 0
\(193\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(198\) 0 0
\(199\) 0.662827 + 0.382683i 0.662827 + 0.382683i 0.793353 0.608761i \(-0.208333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(200\) −0.937379 0.541196i −0.937379 0.541196i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −0.662827 + 0.382683i −0.662827 + 0.382683i
\(213\) 0 0
\(214\) −0.292893 + 0.507306i −0.292893 + 0.507306i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 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
\(225\) 0 0
\(226\) −0.292893 0.507306i −0.292893 0.507306i
\(227\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(228\) 0 0
\(229\) −1.60021 + 0.923880i −1.60021 + 0.923880i −0.608761 + 0.793353i \(0.708333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(230\) 1.41421 1.41421
\(231\) 0 0
\(232\) 0 0
\(233\) −1.60021 + 0.923880i −1.60021 + 0.923880i −0.608761 + 0.793353i \(0.708333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(234\) 0 0
\(235\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −0.662827 0.382683i −0.662827 0.382683i 0.130526 0.991445i \(-0.458333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(242\) 0.662827 + 0.382683i 0.662827 + 0.382683i
\(243\) 0 0
\(244\) 0.765367i 0.765367i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 1.00000 1.73205i 1.00000 1.73205i
\(249\) 0 0
\(250\) −0.662827 + 0.382683i −0.662827 + 0.382683i
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.500000 0.866025i 0.500000 0.866025i
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −0.662827 0.382683i −0.662827 0.382683i 0.130526 0.991445i \(-0.458333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(264\) 0 0
\(265\) 1.84776i 1.84776i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(270\) 0 0
\(271\) 0.662827 0.382683i 0.662827 0.382683i −0.130526 0.991445i \(-0.541667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(272\) −0.585786 −0.585786
\(273\) 0 0
\(274\) 0.585786 0.585786
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) −0.707107 1.22474i −0.707107 1.22474i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.500000 0.866025i −0.500000 0.866025i
\(290\) 0 0
\(291\) 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
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 1.08239i 1.08239i
\(303\) 0 0
\(304\) −0.274552 0.158513i −0.274552 0.158513i
\(305\) 1.60021 + 0.923880i 1.60021 + 0.923880i
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −0.707107 1.22474i −0.707107 1.22474i
\(311\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) 0 0
\(313\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0.585786 0.585786
\(317\) −1.60021 + 0.923880i −1.60021 + 0.923880i −0.608761 + 0.793353i \(0.708333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.500000 0.866025i −0.500000 0.866025i
\(321\) 0 0
\(322\) 0 0
\(323\) 1.08239i 1.08239i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(332\) −0.292893 + 0.507306i −0.292893 + 0.507306i
\(333\) 0 0
\(334\) 1.32565 0.765367i 1.32565 0.765367i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −0.662827 + 0.382683i −0.662827 + 0.382683i
\(339\) 0 0
\(340\) 0.292893 0.507306i 0.292893 0.507306i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −0.937379 0.541196i −0.937379 0.541196i
\(347\) −1.60021 0.923880i −1.60021 0.923880i −0.991445 0.130526i \(-0.958333\pi\)
−0.608761 0.793353i \(-0.708333\pi\)
\(348\) 0 0
\(349\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(360\) 0 0
\(361\) −0.207107 + 0.358719i −0.207107 + 0.358719i
\(362\) −0.292893 0.507306i −0.292893 0.507306i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(368\) 0.662827 + 0.382683i 0.662827 + 0.382683i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −1.32565 + 0.765367i −1.32565 + 0.765367i
\(377\) 0 0
\(378\) 0 0
\(379\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(380\) 0.274552 0.158513i 0.274552 0.158513i
\(381\) 0 0
\(382\) 0 0
\(383\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0 0
\(391\) 2.61313i 2.61313i
\(392\) 0 0
\(393\) 0 0
\(394\) −0.292893 0.507306i −0.292893 0.507306i
\(395\) 0.707107 1.22474i 0.707107 1.22474i
\(396\) 0 0
\(397\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(398\) 0.585786 0.585786
\(399\) 0 0
\(400\) −0.414214 −0.414214
\(401\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.662827 + 0.382683i 0.662827 + 0.382683i 0.793353 0.608761i \(-0.208333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(425\) −0.707107 1.22474i −0.707107 1.22474i
\(426\) 0 0
\(427\) 0 0
\(428\) 0.317025i 0.317025i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(438\) 0 0
\(439\) −1.60021 + 0.923880i −1.60021 + 0.923880i −0.608761 + 0.793353i \(0.708333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −0.662827 + 0.382683i −0.662827 + 0.382683i −0.793353 0.608761i \(-0.791667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0.274552 + 0.158513i 0.274552 + 0.158513i
\(453\) 0 0
\(454\) 1.53073i 1.53073i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(458\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(459\) 0 0
\(460\) −0.662827 + 0.382683i −0.662827 + 0.382683i
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(467\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1.08239i 1.08239i
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0.765367i 0.765367i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −0.585786 −0.585786
\(483\) 0 0
\(484\) −0.414214 −0.414214
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(488\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0.765367i 0.765367i
\(497\) 0 0
\(498\) 0 0
\(499\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(500\) 0.207107 0.358719i 0.207107 0.358719i
\(501\) 0 0
\(502\) 0 0
\(503\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.765367i 0.765367i
\(513\) 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 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −0.585786 −0.585786
\(527\) 2.26303 1.30656i 2.26303 1.30656i
\(528\) 0 0
\(529\) 1.20711 2.09077i 1.20711 2.09077i
\(530\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.662827 0.382683i −0.662827 0.382683i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(542\) 0.292893 0.507306i 0.292893 0.507306i
\(543\) 0 0
\(544\) 0.937379 0.541196i 0.937379 0.541196i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −0.274552 + 0.158513i −0.274552 + 0.158513i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0.662827 + 0.382683i 0.662827 + 0.382683i
\(557\) 0.662827 + 0.382683i 0.662827 + 0.382683i 0.793353 0.608761i \(-0.208333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(564\) 0 0
\(565\) 0.662827 0.382683i 0.662827 0.382683i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(570\) 0 0
\(571\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.84776i 1.84776i
\(576\) 0 0
\(577\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(578\) −0.662827 0.382683i −0.662827 0.382683i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 0 0
\(589\) 1.41421 1.41421
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 0 0
\(601\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0.292893 + 0.507306i 0.292893 + 0.507306i
\(605\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(606\) 0 0
\(607\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(608\) 0.585786 0.585786
\(609\) 0 0
\(610\) 1.41421 1.41421
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(618\) 0 0
\(619\) −0.662827 0.382683i −0.662827 0.382683i 0.130526 0.991445i \(-0.458333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(620\) 0.662827 + 0.382683i 0.662827 + 0.382683i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(632\) 1.32565 0.765367i 1.32565 0.765367i
\(633\) 0 0
\(634\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.414214 0.717439i −0.414214 0.717439i
\(647\) −1.00000 + 1.73205i −1.00000 + 1.73205i −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
\(652\) 0 0
\(653\) 0.662827 0.382683i 0.662827 0.382683i −0.130526 0.991445i \(-0.541667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 1.60021 + 0.923880i 1.60021 + 0.923880i 0.991445 + 0.130526i \(0.0416667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 1.53073i 1.53073i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −0.414214 + 0.717439i −0.414214 + 0.717439i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0.207107 0.358719i 0.207107 0.358719i
\(677\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1.53073i 1.53073i
\(681\) 0 0
\(682\) 0 0
\(683\) −1.60021 0.923880i −1.60021 0.923880i −0.991445 0.130526i \(-0.958333\pi\)
−0.608761 0.793353i \(-0.708333\pi\)
\(684\) 0 0
\(685\) 0.765367i 0.765367i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0.662827 0.382683i 0.662827 0.382683i −0.130526 0.991445i \(-0.541667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(692\) 0.585786 0.585786
\(693\) 0 0
\(694\) −1.41421 −1.41421
\(695\) 1.60021 0.923880i 1.60021 0.923880i
\(696\) 0 0
\(697\) 0 0
\(698\) 0.292893 + 0.507306i 0.292893 + 0.507306i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3.41421 −3.41421
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.317025i 0.317025i
\(723\) 0 0
\(724\) 0.274552 + 0.158513i 0.274552 + 0.158513i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −1.41421 −1.41421
\(737\) 0 0
\(738\) 0 0
\(739\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(752\) −0.292893 + 0.507306i −0.292893 + 0.507306i
\(753\) 0 0
\(754\) 0 0
\(755\) 1.41421 1.41421
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0.937379 0.541196i 0.937379 0.541196i
\(759\) 0 0
\(760\) 0.414214 0.717439i 0.414214 0.717439i
\(761\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0.937379 + 0.541196i 0.937379 + 0.541196i
\(767\) 0 0
\(768\) 0 0
\(769\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(774\) 0 0
\(775\) 1.60021 0.923880i 1.60021 0.923880i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(788\) 0.274552 + 0.158513i 0.274552 + 0.158513i
\(789\) 0 0
\(790\) 1.08239i 1.08239i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −0.274552 + 0.158513i −0.274552 + 0.158513i
\(797\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) −2.00000 −2.00000
\(800\) 0.662827 0.382683i 0.662827 0.382683i
\(801\) 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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(810\) 0 0
\(811\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0.585786 0.585786
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(822\) 0 0
\(823\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(828\) 0 0
\(829\) −1.60021 0.923880i −1.60021 0.923880i −0.991445 0.130526i \(-0.958333\pi\)
−0.608761 0.793353i \(-0.708333\pi\)
\(830\) 0.937379 + 0.541196i 0.937379 + 0.541196i
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −1.00000 −1.00000
\(842\) 0.937379 0.541196i 0.937379 0.541196i
\(843\) 0 0
\(844\) 0 0
\(845\) −0.500000 0.866025i −0.500000 0.866025i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.765367i 0.765367i
\(849\) 0 0
\(850\) −0.937379 0.541196i −0.937379 0.541196i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −0.414214 0.717439i −0.414214 0.717439i
\(857\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(858\) 0 0
\(859\) 1.60021 0.923880i 1.60021 0.923880i 0.608761 0.793353i \(-0.291667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0.662827 0.382683i 0.662827 0.382683i −0.130526 0.991445i \(-0.541667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(864\) 0 0
\(865\) 0.707107 1.22474i 0.707107 1.22474i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 1.08239i 1.08239i
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(878\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.292893 + 0.507306i −0.292893 + 0.507306i
\(887\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −0.937379 0.541196i −0.937379 0.541196i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −2.26303 + 1.30656i −2.26303 + 1.30656i
\(902\) 0 0
\(903\) 0 0
\(904\) 0.828427 0.828427
\(905\) 0.662827 0.382683i 0.662827 0.382683i
\(906\) 0 0
\(907\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(908\) 0.414214 + 0.717439i 0.414214 + 0.717439i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0.765367i 0.765367i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(920\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0.765367i 0.765367i
\(933\) 0 0
\(934\) −1.32565 0.765367i −1.32565 0.765367i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −0.292893 0.507306i −0.292893 0.507306i
\(941\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −0.662827 + 0.382683i −0.662827 + 0.382683i −0.793353 0.608761i \(-0.791667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −0.292893 0.507306i −0.292893 0.507306i
\(951\) 0 0
\(952\) 0 0
\(953\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.20711 + 2.09077i 1.20711 + 2.09077i
\(962\) 0 0
\(963\) 0 0
\(964\) 0.274552 0.158513i 0.274552 0.158513i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −0.937379 + 0.541196i −0.937379 + 0.541196i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0.662827 + 0.382683i 0.662827 + 0.382683i
\(977\) 1.60021 + 0.923880i 1.60021 + 0.923880i 0.991445 + 0.130526i \(0.0416667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(984\) 0 0
\(985\) 0.662827 0.382683i 0.662827 0.382683i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(993\) 0 0
\(994\) 0 0
\(995\) 0.765367i 0.765367i
\(996\) 0 0
\(997\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(998\) 0.937379 + 0.541196i 0.937379 + 0.541196i
\(999\) 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 2205.1.bi.b.2089.3 8
3.2 odd 2 2205.1.bi.a.2089.2 8
5.4 even 2 2205.1.bi.a.2089.2 8
7.2 even 3 inner 2205.1.bi.b.19.2 8
7.3 odd 6 2205.1.e.b.244.3 yes 4
7.4 even 3 2205.1.e.a.244.3 yes 4
7.5 odd 6 2205.1.bi.a.19.2 8
7.6 odd 2 2205.1.bi.a.2089.3 8
15.14 odd 2 CM 2205.1.bi.b.2089.3 8
21.2 odd 6 2205.1.bi.a.19.3 8
21.5 even 6 inner 2205.1.bi.b.19.3 8
21.11 odd 6 2205.1.e.b.244.2 yes 4
21.17 even 6 2205.1.e.a.244.2 4
21.20 even 2 inner 2205.1.bi.b.2089.2 8
35.4 even 6 2205.1.e.b.244.2 yes 4
35.9 even 6 2205.1.bi.a.19.3 8
35.19 odd 6 inner 2205.1.bi.b.19.3 8
35.24 odd 6 2205.1.e.a.244.2 4
35.34 odd 2 inner 2205.1.bi.b.2089.2 8
105.44 odd 6 inner 2205.1.bi.b.19.2 8
105.59 even 6 2205.1.e.b.244.3 yes 4
105.74 odd 6 2205.1.e.a.244.3 yes 4
105.89 even 6 2205.1.bi.a.19.2 8
105.104 even 2 2205.1.bi.a.2089.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2205.1.e.a.244.2 4 21.17 even 6
2205.1.e.a.244.2 4 35.24 odd 6
2205.1.e.a.244.3 yes 4 7.4 even 3
2205.1.e.a.244.3 yes 4 105.74 odd 6
2205.1.e.b.244.2 yes 4 21.11 odd 6
2205.1.e.b.244.2 yes 4 35.4 even 6
2205.1.e.b.244.3 yes 4 7.3 odd 6
2205.1.e.b.244.3 yes 4 105.59 even 6
2205.1.bi.a.19.2 8 7.5 odd 6
2205.1.bi.a.19.2 8 105.89 even 6
2205.1.bi.a.19.3 8 21.2 odd 6
2205.1.bi.a.19.3 8 35.9 even 6
2205.1.bi.a.2089.2 8 3.2 odd 2
2205.1.bi.a.2089.2 8 5.4 even 2
2205.1.bi.a.2089.3 8 7.6 odd 2
2205.1.bi.a.2089.3 8 105.104 even 2
2205.1.bi.b.19.2 8 7.2 even 3 inner
2205.1.bi.b.19.2 8 105.44 odd 6 inner
2205.1.bi.b.19.3 8 21.5 even 6 inner
2205.1.bi.b.19.3 8 35.19 odd 6 inner
2205.1.bi.b.2089.2 8 21.20 even 2 inner
2205.1.bi.b.2089.2 8 35.34 odd 2 inner
2205.1.bi.b.2089.3 8 1.1 even 1 trivial
2205.1.bi.b.2089.3 8 15.14 odd 2 CM