Properties

Label 1755.1.o.a.298.8
Level $1755$
Weight $1$
Character 1755.298
Analytic conductor $0.876$
Analytic rank $0$
Dimension $16$
Projective image $D_{24}$
CM discriminant -39
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1755,1,Mod(298,1755)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1755, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1755.298");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1755.o (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.875859097171\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{24}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

Embedding invariants

Embedding label 298.8
Root \(0.991445 - 0.130526i\) of defining polynomial
Character \(\chi\) \(=\) 1755.298
Dual form 1755.1.o.a.1702.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.40211 + 1.40211i) q^{2} +2.93185i q^{4} +(0.991445 + 0.130526i) q^{5} +(-2.70868 + 2.70868i) q^{8} +O(q^{10})\) \(q+(1.40211 + 1.40211i) q^{2} +2.93185i q^{4} +(0.991445 + 0.130526i) q^{5} +(-2.70868 + 2.70868i) q^{8} +(1.20711 + 1.57313i) q^{10} -0.765367i q^{11} +(-0.707107 - 0.707107i) q^{13} -4.66390 q^{16} +(-0.382683 + 2.90677i) q^{20} +(1.07313 - 1.07313i) q^{22} +(0.965926 + 0.258819i) q^{25} -1.98289i q^{26} +(-3.83065 - 3.83065i) q^{32} +(-3.03906 + 2.33195i) q^{40} -1.84776i q^{41} +(1.36603 + 1.36603i) q^{43} +2.24394 q^{44} +(-0.184592 - 0.184592i) q^{47} -1.00000i q^{49} +(0.991445 + 1.71723i) q^{50} +(2.07313 - 2.07313i) q^{52} +(0.0999004 - 0.758819i) q^{55} -1.58671 q^{59} +0.517638 q^{61} -6.07812i q^{64} +(-0.608761 - 0.793353i) q^{65} -0.261052i q^{71} +1.41421i q^{79} +(-4.62400 - 0.608761i) q^{80} +(2.59077 - 2.59077i) q^{82} +(-1.12197 + 1.12197i) q^{83} +3.83065i q^{86} +(2.07313 + 2.07313i) q^{88} -1.21752 q^{89} -0.517638i q^{94} +(1.40211 - 1.40211i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{10} - 16 q^{16} - 8 q^{22} - 8 q^{40} + 8 q^{43} + 8 q^{52} + 8 q^{82} + 8 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1755\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(1081\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.40211 + 1.40211i 1.40211 + 1.40211i 0.793353 + 0.608761i \(0.208333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(3\) 0 0
\(4\) 2.93185i 2.93185i
\(5\) 0.991445 + 0.130526i 0.991445 + 0.130526i
\(6\) 0 0
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) −2.70868 + 2.70868i −2.70868 + 2.70868i
\(9\) 0 0
\(10\) 1.20711 + 1.57313i 1.20711 + 1.57313i
\(11\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(12\) 0 0
\(13\) −0.707107 0.707107i −0.707107 0.707107i
\(14\) 0 0
\(15\) 0 0
\(16\) −4.66390 −4.66390
\(17\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −0.382683 + 2.90677i −0.382683 + 2.90677i
\(21\) 0 0
\(22\) 1.07313 1.07313i 1.07313 1.07313i
\(23\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(26\) 1.98289i 1.98289i
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −3.83065 3.83065i −3.83065 3.83065i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −3.03906 + 2.33195i −3.03906 + 2.33195i
\(41\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(42\) 0 0
\(43\) 1.36603 + 1.36603i 1.36603 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 2.24394 2.24394
\(45\) 0 0
\(46\) 0 0
\(47\) −0.184592 0.184592i −0.184592 0.184592i 0.608761 0.793353i \(-0.291667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(48\) 0 0
\(49\) 1.00000i 1.00000i
\(50\) 0.991445 + 1.71723i 0.991445 + 1.71723i
\(51\) 0 0
\(52\) 2.07313 2.07313i 2.07313 2.07313i
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0.0999004 0.758819i 0.0999004 0.758819i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.58671 −1.58671 −0.793353 0.608761i \(-0.791667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(60\) 0 0
\(61\) 0.517638 0.517638 0.258819 0.965926i \(-0.416667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 6.07812i 6.07812i
\(65\) −0.608761 0.793353i −0.608761 0.793353i
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.261052i 0.261052i −0.991445 0.130526i \(-0.958333\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(72\) 0 0
\(73\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(80\) −4.62400 0.608761i −4.62400 0.608761i
\(81\) 0 0
\(82\) 2.59077 2.59077i 2.59077 2.59077i
\(83\) −1.12197 + 1.12197i −1.12197 + 1.12197i −0.130526 + 0.991445i \(0.541667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.83065i 3.83065i
\(87\) 0 0
\(88\) 2.07313 + 2.07313i 2.07313 + 2.07313i
\(89\) −1.21752 −1.21752 −0.608761 0.793353i \(-0.708333\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0.517638i 0.517638i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 1.40211 1.40211i 1.40211 1.40211i
\(99\) 0 0
\(100\) −0.758819 + 2.83195i −0.758819 + 2.83195i
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0.366025 + 0.366025i 0.366025 + 0.366025i 0.866025 0.500000i \(-0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(104\) 3.83065 3.83065
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 1.20402 0.923880i 1.20402 0.923880i
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −2.22474 2.22474i −2.22474 2.22474i
\(119\) 0 0
\(120\) 0 0
\(121\) 0.414214 0.414214
\(122\) 0.725788 + 0.725788i 0.725788 + 0.725788i
\(123\) 0 0
\(124\) 0 0
\(125\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(126\) 0 0
\(127\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(128\) 4.69157 4.69157i 4.69157 4.69157i
\(129\) 0 0
\(130\) 0.258819 1.96593i 0.258819 1.96593i
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(138\) 0 0
\(139\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.366025 0.366025i 0.366025 0.366025i
\(143\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.58671 1.58671 0.793353 0.608761i \(-0.208333\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.366025 + 0.366025i −0.366025 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(158\) −1.98289 + 1.98289i −1.98289 + 1.98289i
\(159\) 0 0
\(160\) −3.29788 4.29788i −3.29788 4.29788i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 5.41736 5.41736
\(165\) 0 0
\(166\) −3.14626 −3.14626
\(167\) −0.860919 0.860919i −0.860919 0.860919i 0.130526 0.991445i \(-0.458333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(168\) 0 0
\(169\) 1.00000i 1.00000i
\(170\) 0 0
\(171\) 0 0
\(172\) −4.00498 + 4.00498i −4.00498 + 4.00498i
\(173\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.56960i 3.56960i
\(177\) 0 0
\(178\) −1.70711 1.70711i −1.70711 1.70711i
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0.541196 0.541196i 0.541196 0.541196i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2.93185 2.93185
\(197\) 0.184592 + 0.184592i 0.184592 + 0.184592i 0.793353 0.608761i \(-0.208333\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(198\) 0 0
\(199\) 0.517638i 0.517638i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(200\) −3.31744 + 1.91532i −3.31744 + 1.91532i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0.241181 1.83195i 0.241181 1.83195i
\(206\) 1.02642i 1.02642i
\(207\) 0 0
\(208\) 3.29788 + 3.29788i 3.29788 + 3.29788i
\(209\) 0 0
\(210\) 0 0
\(211\) −0.517638 −0.517638 −0.258819 0.965926i \(-0.583333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.17604 + 1.53264i 1.17604 + 1.53264i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 2.22474 + 0.292893i 2.22474 + 0.292893i
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.860919 + 0.860919i 0.860919 + 0.860919i 0.991445 0.130526i \(-0.0416667\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(234\) 0 0
\(235\) −0.158919 0.207107i −0.158919 0.207107i
\(236\) 4.65199i 4.65199i
\(237\) 0 0
\(238\) 0 0
\(239\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0.580775 + 0.580775i 0.580775 + 0.580775i
\(243\) 0 0
\(244\) 1.51764i 1.51764i
\(245\) 0.130526 0.991445i 0.130526 0.991445i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0.758819 + 1.83195i 0.758819 + 1.83195i
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.98289 −1.98289
\(255\) 0 0
\(256\) 7.07812 7.07812
\(257\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2.32599 1.78480i 2.32599 1.78480i
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 1.51764i 1.51764i
\(275\) 0.198092 0.739288i 0.198092 0.739288i
\(276\) 0 0
\(277\) −1.36603 + 1.36603i −1.36603 + 1.36603i −0.500000 + 0.866025i \(0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(278\) 2.42853 2.42853i 2.42853 2.42853i
\(279\) 0 0
\(280\) 0 0
\(281\) 1.98289i 1.98289i −0.130526 0.991445i \(-0.541667\pi\)
0.130526 0.991445i \(-0.458333\pi\)
\(282\) 0 0
\(283\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(284\) 0.765367 0.765367
\(285\) 0 0
\(286\) −1.51764 −1.51764
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000i 1.00000i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(294\) 0 0
\(295\) −1.57313 0.207107i −1.57313 0.207107i
\(296\) 0 0
\(297\) 0 0
\(298\) 2.22474 + 2.22474i 2.22474 + 2.22474i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.513210 + 0.0675653i 0.513210 + 0.0675653i
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(314\) −1.02642 −1.02642
\(315\) 0 0
\(316\) −4.14626 −4.14626
\(317\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.793353 6.02612i 0.793353 6.02612i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −0.500000 0.866025i −0.500000 0.866025i
\(326\) 0 0
\(327\) 0 0
\(328\) 5.00498 + 5.00498i 5.00498 + 5.00498i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −3.28945 3.28945i −3.28945 3.28945i
\(333\) 0 0
\(334\) 2.41421i 2.41421i
\(335\) 0 0
\(336\) 0 0
\(337\) −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i \(0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(338\) −1.40211 + 1.40211i −1.40211 + 1.40211i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −7.40025 −7.40025
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.93185 + 2.93185i −2.93185 + 2.93185i
\(353\) −0.860919 + 0.860919i −0.860919 + 0.860919i −0.991445 0.130526i \(-0.958333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(354\) 0 0
\(355\) 0.0340742 0.258819i 0.0340742 0.258819i
\(356\) 3.56960i 3.56960i
\(357\) 0 0
\(358\) 0 0
\(359\) −1.98289 −1.98289 −0.991445 0.130526i \(-0.958333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(360\) 0 0
\(361\) −1.00000 −1.00000
\(362\) 2.42853 + 2.42853i 2.42853 + 2.42853i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.22474 1.22474i −1.22474 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 1.00000 1.00000
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 2.70868 + 2.70868i 2.70868 + 2.70868i
\(393\) 0 0
\(394\) 0.517638i 0.517638i
\(395\) −0.184592 + 1.40211i −0.184592 + 1.40211i
\(396\) 0 0
\(397\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(398\) −0.725788 + 0.725788i −0.725788 + 0.725788i
\(399\) 0 0
\(400\) −4.50498 1.20711i −4.50498 1.20711i
\(401\) 1.58671i 1.58671i −0.608761 0.793353i \(-0.708333\pi\)
0.608761 0.793353i \(-0.291667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 2.90677 2.23044i 2.90677 2.23044i
\(411\) 0 0
\(412\) −1.07313 + 1.07313i −1.07313 + 1.07313i
\(413\) 0 0
\(414\) 0 0
\(415\) −1.25882 + 0.965926i −1.25882 + 0.965926i
\(416\) 5.41736i 5.41736i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −0.725788 0.725788i −0.725788 0.725788i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) −0.500000 + 3.79788i −0.500000 + 3.79788i
\(431\) 1.21752i 1.21752i 0.793353 + 0.608761i \(0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(432\) 0 0
\(433\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(440\) 1.78480 + 2.32599i 1.78480 + 2.32599i
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 0 0
\(445\) −1.20711 0.158919i −1.20711 0.158919i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.261052 −0.261052 −0.130526 0.991445i \(-0.541667\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(450\) 0 0
\(451\) −1.41421 −1.41421
\(452\) 0 0
\(453\) 0 0
\(454\) 2.41421i 2.41421i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.21752i 1.21752i −0.793353 0.608761i \(-0.791667\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0.0675653 0.513210i 0.0675653 0.513210i
\(471\) 0 0
\(472\) 4.29788 4.29788i 4.29788 4.29788i
\(473\) 1.04551 1.04551i 1.04551 1.04551i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −2.59077 2.59077i −2.59077 2.59077i
\(479\) 1.21752 1.21752 0.608761 0.793353i \(-0.291667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.21441i 1.21441i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) −1.40211 + 1.40211i −1.40211 + 1.40211i
\(489\) 0 0
\(490\) 1.57313 1.20711i 1.57313 1.20711i
\(491\) 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 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) −1.12197 + 2.70868i −1.12197 + 2.70868i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −2.07313 2.07313i −2.07313 2.07313i
\(509\) 1.21752 1.21752 0.608761 0.793353i \(-0.291667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 5.23276 + 5.23276i 5.23276 + 5.23276i
\(513\) 0 0
\(514\) 0 0
\(515\) 0.315118 + 0.410670i 0.315118 + 0.410670i
\(516\) 0 0
\(517\) −0.141281 + 0.141281i −0.141281 + 0.141281i
\(518\) 0 0
\(519\) 0 0
\(520\) 3.79788 + 0.500000i 3.79788 + 0.500000i
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0.366025 + 0.366025i 0.366025 + 0.366025i 0.866025 0.500000i \(-0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.765367 −0.765367
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(548\) 1.58671 1.58671i 1.58671 1.58671i
\(549\) 0 0
\(550\) 1.31431 0.758819i 1.31431 0.758819i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −3.83065 −3.83065
\(555\) 0 0
\(556\) 5.07812 5.07812
\(557\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(558\) 0 0
\(559\) 1.93185i 1.93185i
\(560\) 0 0
\(561\) 0 0
\(562\) 2.78024 2.78024i 2.78024 2.78024i
\(563\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.98289i 1.98289i
\(567\) 0 0
\(568\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 1.93185 1.93185 0.965926 0.258819i \(-0.0833333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(572\) −1.58671 1.58671i −1.58671 1.58671i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(578\) −1.40211 + 1.40211i −1.40211 + 1.40211i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 1.51764 1.51764
\(587\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −1.91532 2.49610i −1.91532 2.49610i
\(591\) 0 0
\(592\) 0 0
\(593\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.65199i 4.65199i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.93185 −1.93185 −0.965926 0.258819i \(-0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.410670 + 0.0540657i 0.410670 + 0.0540657i
\(606\) 0 0
\(607\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0.624844 + 0.814313i 0.624844 + 0.814313i
\(611\) 0.261052i 0.261052i
\(612\) 0 0
\(613\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.12197 + 1.12197i 1.12197 + 1.12197i 0.991445 + 0.130526i \(0.0416667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(626\) 1.98289i 1.98289i
\(627\) 0 0
\(628\) −1.07313 1.07313i −1.07313 1.07313i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) −3.83065 3.83065i −3.83065 3.83065i
\(633\) 0 0
\(634\) 3.66390i 3.66390i
\(635\) −0.793353 + 0.608761i −0.793353 + 0.608761i
\(636\) 0 0
\(637\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(638\) 0 0
\(639\) 0 0
\(640\) 5.26380 4.03906i 5.26380 4.03906i
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(648\) 0 0
\(649\) 1.21441i 1.21441i
\(650\) 0.513210 1.91532i 0.513210 1.91532i
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 8.61777i 8.61777i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 6.07812i 6.07812i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 2.52409 2.52409i 2.52409 2.52409i
\(669\) 0 0
\(670\) 0 0
\(671\) 0.396183i 0.396183i
\(672\) 0 0
\(673\) −1.36603 1.36603i −1.36603 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(674\) −3.43447 −3.43447
\(675\) 0 0
\(676\) −2.93185 −2.93185
\(677\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.860919 0.860919i 0.860919 0.860919i −0.130526 0.991445i \(-0.541667\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(684\) 0 0
\(685\) −0.465926 0.607206i −0.465926 0.607206i
\(686\) 0 0
\(687\) 0 0
\(688\) −6.37101 6.37101i −6.37101 6.37101i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.226078 1.71723i 0.226078 1.71723i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −4.65199 −4.65199
\(705\) 0 0
\(706\) −2.41421 −2.41421
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 0.410670 0.315118i 0.410670 0.315118i
\(711\) 0 0
\(712\) 3.29788 3.29788i 3.29788 3.29788i
\(713\) 0 0
\(714\) 0 0
\(715\) −0.607206 + 0.465926i −0.607206 + 0.465926i
\(716\) 0 0
\(717\) 0 0
\(718\) −2.78024 2.78024i −2.78024 2.78024i
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.40211 1.40211i −1.40211 1.40211i
\(723\) 0 0
\(724\) 5.07812i 5.07812i
\(725\) 0 0
\(726\) 0 0
\(727\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(734\) 2.80423 2.80423
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.12197 1.12197i 1.12197 1.12197i 0.130526 0.991445i \(-0.458333\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(744\) 0 0
\(745\) 1.57313 + 0.207107i 1.57313 + 0.207107i
\(746\) 3.43447i 3.43447i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0.860919 + 0.860919i 0.860919 + 0.860919i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.98289i 1.98289i 0.130526 + 0.991445i \(0.458333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −3.66390 −3.66390
\(767\) 1.12197 + 1.12197i 1.12197 + 1.12197i
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −0.184592 + 0.184592i −0.184592 + 0.184592i −0.793353 0.608761i \(-0.791667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −0.199801 −0.199801
\(782\) 0 0
\(783\) 0 0
\(784\) 4.66390i 4.66390i
\(785\) −0.410670 + 0.315118i −0.410670 + 0.315118i
\(786\) 0 0
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(789\) 0 0
\(790\) −2.22474 + 1.70711i −2.22474 + 1.70711i
\(791\) 0 0
\(792\) 0 0
\(793\) −0.366025 0.366025i −0.366025 0.366025i
\(794\) 0 0
\(795\) 0 0
\(796\) −1.51764 −1.51764
\(797\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −2.70868 4.69157i −2.70868 4.69157i
\(801\) 0 0
\(802\) 2.22474 2.22474i 2.22474 2.22474i
\(803\) 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
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 5.37101 + 0.707107i 5.37101 + 0.707107i
\(821\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(822\) 0 0
\(823\) 1.36603 + 1.36603i 1.36603 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(824\) −1.98289 −1.98289
\(825\) 0 0
\(826\) 0 0
\(827\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(828\) 0 0
\(829\) 1.93185i 1.93185i 0.258819 + 0.965926i \(0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(830\) −3.11935 0.410670i −3.11935 0.410670i
\(831\) 0 0
\(832\) −4.29788 + 4.29788i −4.29788 + 4.29788i
\(833\) 0 0
\(834\) 0 0
\(835\) −0.741181 0.965926i −0.741181 0.965926i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.98289 1.98289 0.991445 0.130526i \(-0.0416667\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 1.51764i 1.51764i
\(845\) −0.130526 + 0.991445i −0.130526 + 0.991445i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(860\) −4.49348 + 3.44797i −4.49348 + 3.44797i
\(861\) 0 0
\(862\) −1.70711 + 1.70711i −1.70711 + 1.70711i
\(863\) −0.860919 + 0.860919i −0.860919 + 0.860919i −0.991445 0.130526i \(-0.958333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.08239 1.08239
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(878\) 2.42853 2.42853i 2.42853 2.42853i
\(879\) 0 0
\(880\) −0.465926 + 3.53906i −0.465926 + 3.53906i
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −1.46968 1.91532i −1.46968 1.91532i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −0.366025 0.366025i −0.366025 0.366025i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −1.98289 1.98289i −1.98289 1.98289i
\(903\) 0 0
\(904\) 0 0
\(905\) 1.71723 + 0.226078i 1.71723 + 0.226078i
\(906\) 0 0
\(907\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(908\) −2.52409 + 2.52409i −2.52409 + 2.52409i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0.858719 + 0.858719i 0.858719 + 0.858719i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.93185i 1.93185i −0.258819 0.965926i \(-0.583333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.70711 1.70711i 1.70711 1.70711i
\(923\) −0.184592 + 0.184592i −0.184592 + 0.184592i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0.607206 0.465926i 0.607206 0.465926i
\(941\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 7.40025 7.40025
\(945\) 0 0
\(946\) 2.93185 2.93185
\(947\) 0.184592 + 0.184592i 0.184592 + 0.184592i 0.793353 0.608761i \(-0.208333\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 5.41736i 5.41736i
\(957\) 0 0
\(958\) 1.70711 + 1.70711i 1.70711 + 1.70711i
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(968\) −1.12197 + 1.12197i −1.12197 + 1.12197i
\(969\) 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 0
\(975\) 0 0
\(976\) −2.41421 −2.41421
\(977\) −1.12197 1.12197i −1.12197 1.12197i −0.991445 0.130526i \(-0.958333\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(978\) 0 0
\(979\) 0.931852i 0.931852i
\(980\) 2.90677 + 0.382683i 2.90677 + 0.382683i
\(981\) 0 0
\(982\) 0 0
\(983\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(984\) 0 0
\(985\) 0.158919 + 0.207107i 0.158919 + 0.207107i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −0.0675653 + 0.513210i −0.0675653 + 0.513210i
\(996\) 0 0
\(997\) −0.366025 + 0.366025i −0.366025 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
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 1755.1.o.a.298.8 yes 16
3.2 odd 2 inner 1755.1.o.a.298.1 16
5.2 odd 4 inner 1755.1.o.a.1702.1 yes 16
13.12 even 2 inner 1755.1.o.a.298.1 16
15.2 even 4 inner 1755.1.o.a.1702.8 yes 16
39.38 odd 2 CM 1755.1.o.a.298.8 yes 16
65.12 odd 4 inner 1755.1.o.a.1702.8 yes 16
195.77 even 4 inner 1755.1.o.a.1702.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1755.1.o.a.298.1 16 3.2 odd 2 inner
1755.1.o.a.298.1 16 13.12 even 2 inner
1755.1.o.a.298.8 yes 16 1.1 even 1 trivial
1755.1.o.a.298.8 yes 16 39.38 odd 2 CM
1755.1.o.a.1702.1 yes 16 5.2 odd 4 inner
1755.1.o.a.1702.1 yes 16 195.77 even 4 inner
1755.1.o.a.1702.8 yes 16 15.2 even 4 inner
1755.1.o.a.1702.8 yes 16 65.12 odd 4 inner