Properties

Label 3381.1.o.b.2138.7
Level $3381$
Weight $1$
Character 3381.2138
Analytic conductor $1.687$
Analytic rank $0$
Dimension $16$
Projective image $D_{24}$
CM discriminant -23
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3381,1,Mod(68,3381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3381.68");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3381.o (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: \(1.68733880771\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2, a_3]\)
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 2138.7
Root \(-0.991445 - 0.130526i\) of defining polynomial
Character \(\chi\) \(=\) 3381.2138
Dual form 3381.1.o.b.68.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.67303 - 0.965926i) q^{2} +(-0.793353 - 0.608761i) q^{3} +(1.36603 - 2.36603i) q^{4} +(-1.91532 - 0.252157i) q^{6} -3.34607i q^{8} +(0.258819 + 0.965926i) q^{9} +O(q^{10})\) \(q+(1.67303 - 0.965926i) q^{2} +(-0.793353 - 0.608761i) q^{3} +(1.36603 - 2.36603i) q^{4} +(-1.91532 - 0.252157i) q^{6} -3.34607i q^{8} +(0.258819 + 0.965926i) q^{9} +(-2.52409 + 1.04551i) q^{12} -1.21752i q^{13} +(-1.86603 - 3.23205i) q^{16} +(1.36603 + 1.36603i) q^{18} +(-0.866025 + 0.500000i) q^{23} +(-2.03696 + 2.65461i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(-1.17604 - 2.03696i) q^{26} +(0.382683 - 0.923880i) q^{27} -1.73205i q^{29} +(1.37413 + 0.793353i) q^{31} +(-3.34607 - 1.93185i) q^{32} +(2.63896 + 0.707107i) q^{36} +(-0.741181 + 0.965926i) q^{39} -1.21752 q^{41} +(-0.965926 + 1.67303i) q^{46} +(0.991445 + 1.71723i) q^{47} +(-0.487130 + 3.70012i) q^{48} +1.93185i q^{50} +(-2.88069 - 1.66317i) q^{52} +(-0.252157 - 1.91532i) q^{54} +(-1.67303 - 2.89778i) q^{58} +(0.923880 - 1.60021i) q^{59} +3.06528 q^{62} -3.73205 q^{64} +(0.991445 + 0.130526i) q^{69} +1.00000i q^{71} +(3.23205 - 0.866025i) q^{72} +(0.226078 + 0.130526i) q^{73} +(0.923880 - 0.382683i) q^{75} +(-0.307007 + 2.33195i) q^{78} +(-0.866025 + 0.500000i) q^{81} +(-2.03696 + 1.17604i) q^{82} +(-1.05441 + 1.37413i) q^{87} +2.73205i q^{92} +(-0.607206 - 1.46593i) q^{93} +(3.31744 + 1.91532i) q^{94} +(1.47858 + 3.56960i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 16 q^{16} + 8 q^{18} - 8 q^{25} - 16 q^{39} - 32 q^{64} + 24 q^{72} + 8 q^{78}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3381\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(2255\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.67303 0.965926i 1.67303 0.965926i 0.707107 0.707107i \(-0.250000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(3\) −0.793353 0.608761i −0.793353 0.608761i
\(4\) 1.36603 2.36603i 1.36603 2.36603i
\(5\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) −1.91532 0.252157i −1.91532 0.252157i
\(7\) 0 0
\(8\) 3.34607i 3.34607i
\(9\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(10\) 0 0
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) −2.52409 + 1.04551i −2.52409 + 1.04551i
\(13\) 1.21752i 1.21752i −0.793353 0.608761i \(-0.791667\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.86603 3.23205i −1.86603 3.23205i
\(17\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 1.36603 + 1.36603i 1.36603 + 1.36603i
\(19\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(24\) −2.03696 + 2.65461i −2.03696 + 2.65461i
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) −1.17604 2.03696i −1.17604 2.03696i
\(27\) 0.382683 0.923880i 0.382683 0.923880i
\(28\) 0 0
\(29\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(30\) 0 0
\(31\) 1.37413 + 0.793353i 1.37413 + 0.793353i 0.991445 0.130526i \(-0.0416667\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(32\) −3.34607 1.93185i −3.34607 1.93185i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 2.63896 + 0.707107i 2.63896 + 0.707107i
\(37\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) −0.741181 + 0.965926i −0.741181 + 0.965926i
\(40\) 0 0
\(41\) −1.21752 −1.21752 −0.608761 0.793353i \(-0.708333\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.965926 + 1.67303i −0.965926 + 1.67303i
\(47\) 0.991445 + 1.71723i 0.991445 + 1.71723i 0.608761 + 0.793353i \(0.291667\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(48\) −0.487130 + 3.70012i −0.487130 + 3.70012i
\(49\) 0 0
\(50\) 1.93185i 1.93185i
\(51\) 0 0
\(52\) −2.88069 1.66317i −2.88069 1.66317i
\(53\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(54\) −0.252157 1.91532i −0.252157 1.91532i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −1.67303 2.89778i −1.67303 2.89778i
\(59\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 3.06528 3.06528
\(63\) 0 0
\(64\) −3.73205 −3.73205
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 0.991445 + 0.130526i 0.991445 + 0.130526i
\(70\) 0 0
\(71\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 3.23205 0.866025i 3.23205 0.866025i
\(73\) 0.226078 + 0.130526i 0.226078 + 0.130526i 0.608761 0.793353i \(-0.291667\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(74\) 0 0
\(75\) 0.923880 0.382683i 0.923880 0.382683i
\(76\) 0 0
\(77\) 0 0
\(78\) −0.307007 + 2.33195i −0.307007 + 2.33195i
\(79\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(82\) −2.03696 + 1.17604i −2.03696 + 1.17604i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.05441 + 1.37413i −1.05441 + 1.37413i
\(88\) 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
\(92\) 2.73205i 2.73205i
\(93\) −0.607206 1.46593i −0.607206 1.46593i
\(94\) 3.31744 + 1.91532i 3.31744 + 1.91532i
\(95\) 0 0
\(96\) 1.47858 + 3.56960i 1.47858 + 3.56960i
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.36603 + 2.36603i 1.36603 + 2.36603i
\(101\) −0.382683 + 0.662827i −0.382683 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(102\) 0 0
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) −4.07391 −4.07391
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(108\) −1.66317 2.16748i −1.66317 2.16748i
\(109\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.09808 2.36603i −4.09808 2.36603i
\(117\) 1.17604 0.315118i 1.17604 0.315118i
\(118\) 3.56960i 3.56960i
\(119\) 0 0
\(120\) 0 0
\(121\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0.965926 + 0.741181i 0.965926 + 0.741181i
\(124\) 3.75419 2.16748i 3.75419 2.16748i
\(125\) 0 0
\(126\) 0 0
\(127\) 1.93185 1.93185 0.965926 0.258819i \(-0.0833333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(128\) −2.89778 + 1.67303i −2.89778 + 1.67303i
\(129\) 0 0
\(130\) 0 0
\(131\) 0.793353 + 1.37413i 0.793353 + 1.37413i 0.923880 + 0.382683i \(0.125000\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(138\) 1.78480 0.739288i 1.78480 0.739288i
\(139\) 0.261052i 0.261052i −0.991445 0.130526i \(-0.958333\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(140\) 0 0
\(141\) 0.258819 1.96593i 0.258819 1.96593i
\(142\) 0.965926 + 1.67303i 0.965926 + 1.67303i
\(143\) 0 0
\(144\) 2.63896 2.63896i 2.63896 2.63896i
\(145\) 0 0
\(146\) 0.504314 0.504314
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 1.17604 1.53264i 1.17604 1.53264i
\(151\) 0.258819 0.448288i 0.258819 0.448288i −0.707107 0.707107i \(-0.750000\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.27293 + 3.07313i 1.27293 + 3.07313i
\(157\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −0.965926 + 1.67303i −0.965926 + 1.67303i
\(163\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(164\) −1.66317 + 2.88069i −1.66317 + 2.88069i
\(165\) 0 0
\(166\) 0 0
\(167\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(168\) 0 0
\(169\) −0.482362 −0.482362
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.923880 1.60021i −0.923880 1.60021i −0.793353 0.608761i \(-0.791667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(174\) −0.436749 + 3.31744i −0.436749 + 3.31744i
\(175\) 0 0
\(176\) 0 0
\(177\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(178\) 0 0
\(179\) 0.448288 + 0.258819i 0.448288 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.67303 + 2.89778i 1.67303 + 2.89778i
\(185\) 0 0
\(186\) −2.43185 1.86603i −2.43185 1.86603i
\(187\) 0 0
\(188\) 5.41736 5.41736
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) 2.96083 + 2.27193i 2.96083 + 2.27193i
\(193\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(198\) 0 0
\(199\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(200\) 2.89778 + 1.67303i 2.89778 + 1.67303i
\(201\) 0 0
\(202\) 1.47858i 1.47858i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.707107 0.707107i −0.707107 0.707107i
\(208\) −3.93510 + 2.27193i −3.93510 + 2.27193i
\(209\) 0 0
\(210\) 0 0
\(211\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(212\) 0 0
\(213\) 0.608761 0.793353i 0.608761 0.793353i
\(214\) 0 0
\(215\) 0 0
\(216\) −3.09136 1.28048i −3.09136 1.28048i
\(217\) 0 0
\(218\) 0 0
\(219\) −0.0999004 0.241181i −0.0999004 0.241181i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(224\) 0 0
\(225\) −0.965926 0.258819i −0.965926 0.258819i
\(226\) 0 0
\(227\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(228\) 0 0
\(229\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −5.79555 −5.79555
\(233\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(234\) 1.66317 1.66317i 1.66317 1.66317i
\(235\) 0 0
\(236\) −2.52409 4.37184i −2.52409 4.37184i
\(237\) 0 0
\(238\) 0 0
\(239\) 0.517638i 0.517638i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(240\) 0 0
\(241\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(242\) 1.67303 + 0.965926i 1.67303 + 0.965926i
\(243\) 0.991445 + 0.130526i 0.991445 + 0.130526i
\(244\) 0 0
\(245\) 0 0
\(246\) 2.33195 + 0.307007i 2.33195 + 0.307007i
\(247\) 0 0
\(248\) 2.65461 4.59792i 2.65461 4.59792i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 3.23205 1.86603i 3.23205 1.86603i
\(255\) 0 0
\(256\) −1.36603 + 2.36603i −1.36603 + 2.36603i
\(257\) −0.991445 1.71723i −0.991445 1.71723i −0.608761 0.793353i \(-0.708333\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1.67303 0.448288i 1.67303 0.448288i
\(262\) 2.65461 + 1.53264i 2.65461 + 1.53264i
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.991445 + 1.71723i −0.991445 + 1.71723i −0.382683 + 0.923880i \(0.625000\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(270\) 0 0
\(271\) 1.60021 0.923880i 1.60021 0.923880i 0.608761 0.793353i \(-0.291667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 1.66317 2.16748i 1.66317 2.16748i
\(277\) −0.258819 + 0.448288i −0.258819 + 0.448288i −0.965926 0.258819i \(-0.916667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(278\) −0.252157 0.436749i −0.252157 0.436749i
\(279\) −0.410670 + 1.53264i −0.410670 + 1.53264i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −1.46593 3.53906i −1.46593 3.53906i
\(283\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(284\) 2.36603 + 1.36603i 2.36603 + 1.36603i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 3.73205i 1.00000 3.73205i
\(289\) −0.500000 0.866025i −0.500000 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0.617657 0.356604i 0.617657 0.356604i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.608761 + 1.05441i 0.608761 + 1.05441i
\(300\) 0.356604 2.70868i 0.356604 2.70868i
\(301\) 0 0
\(302\) 1.00000i 1.00000i
\(303\) 0.707107 0.292893i 0.707107 0.292893i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −0.793353 + 1.37413i −0.793353 + 1.37413i 0.130526 + 0.991445i \(0.458333\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(312\) 3.23205 + 2.48004i 3.23205 + 2.48004i
\(313\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 2.73205i 2.73205i
\(325\) 1.05441 + 0.608761i 1.05441 + 0.608761i
\(326\) 1.67303 + 0.965926i 1.67303 + 0.965926i
\(327\) 0 0
\(328\) 4.07391i 4.07391i
\(329\) 0 0
\(330\) 0 0
\(331\) −0.258819 0.448288i −0.258819 0.448288i 0.707107 0.707107i \(-0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −1.28048 + 0.739288i −1.28048 + 0.739288i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −0.807007 + 0.465926i −0.807007 + 0.465926i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −3.09136 1.78480i −3.09136 1.78480i
\(347\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(348\) 1.81088 + 4.37184i 1.81088 + 4.37184i
\(349\) 1.58671i 1.58671i 0.608761 + 0.793353i \(0.291667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(350\) 0 0
\(351\) −1.12484 0.465926i −1.12484 0.465926i
\(352\) 0 0
\(353\) 0.793353 1.37413i 0.793353 1.37413i −0.130526 0.991445i \(-0.541667\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(354\) −2.17303 + 2.83195i −2.17303 + 2.83195i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 1.00000 1.00000
\(359\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(360\) 0 0
\(361\) 0.500000 0.866025i 0.500000 0.866025i
\(362\) 0 0
\(363\) 0.130526 0.991445i 0.130526 0.991445i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(368\) 3.23205 + 1.86603i 3.23205 + 1.86603i
\(369\) −0.315118 1.17604i −0.315118 1.17604i
\(370\) 0 0
\(371\) 0 0
\(372\) −4.29788 0.565826i −4.29788 0.565826i
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 5.74597 3.31744i 5.74597 3.31744i
\(377\) −2.10881 −2.10881
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −1.53264 1.17604i −1.53264 1.17604i
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 3.31744 + 0.436749i 3.31744 + 0.436749i
\(385\) 0 0
\(386\) 1.93185i 1.93185i
\(387\) 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
\(391\) 0 0
\(392\) 0 0
\(393\) 0.207107 1.57313i 0.207107 1.57313i
\(394\) −0.965926 1.67303i −0.965926 1.67303i
\(395\) 0 0
\(396\) 0 0
\(397\) −1.71723 + 0.991445i −1.71723 + 0.991445i −0.793353 + 0.608761i \(0.791667\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 3.73205 3.73205
\(401\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) 0 0
\(403\) 0.965926 1.67303i 0.965926 1.67303i
\(404\) 1.04551 + 1.81088i 1.04551 + 1.81088i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.71723 + 0.991445i 1.71723 + 0.991445i 0.923880 + 0.382683i \(0.125000\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −1.86603 0.500000i −1.86603 0.500000i
\(415\) 0 0
\(416\) −2.35207 + 4.07391i −2.35207 + 4.07391i
\(417\) −0.158919 + 0.207107i −0.158919 + 0.207107i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 2.36603 1.36603i 2.36603 1.36603i
\(423\) −1.40211 + 1.40211i −1.40211 + 1.40211i
\(424\) 0 0
\(425\) 0 0
\(426\) 0.252157 1.91532i 0.252157 1.91532i
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(432\) −3.70012 + 0.487130i −3.70012 + 0.487130i
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −0.400100 0.307007i −0.400100 0.307007i
\(439\) 0.226078 0.130526i 0.226078 0.130526i −0.382683 0.923880i \(-0.625000\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −0.739288 1.28048i −0.739288 1.28048i
\(447\) 0 0
\(448\) 0 0
\(449\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(450\) −1.86603 + 0.500000i −1.86603 + 0.500000i
\(451\) 0 0
\(452\) 0 0
\(453\) −0.478235 + 0.198092i −0.478235 + 0.198092i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.261052 0.261052 0.130526 0.991445i \(-0.458333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(462\) 0 0
\(463\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(464\) −5.59808 + 3.23205i −5.59808 + 3.23205i
\(465\) 0 0
\(466\) −0.965926 + 1.67303i −0.965926 + 1.67303i
\(467\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(468\) 0.860919 3.21299i 0.860919 3.21299i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −5.35439 3.09136i −5.35439 3.09136i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.73205 2.73205
\(485\) 0 0
\(486\) 1.78480 0.739288i 1.78480 0.739288i
\(487\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0.130526 0.991445i 0.130526 0.991445i
\(490\) 0 0
\(491\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) 3.07313 1.27293i 3.07313 1.27293i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 5.92167i 5.92167i
\(497\) 0 0
\(498\) 0 0
\(499\) −0.965926 1.67303i −0.965926 1.67303i −0.707107 0.707107i \(-0.750000\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(500\) 0 0
\(501\) 0.607206 + 0.465926i 0.607206 + 0.465926i
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.382683 + 0.293643i 0.382683 + 0.293643i
\(508\) 2.63896 4.57081i 2.63896 4.57081i
\(509\) −0.130526 0.226078i −0.130526 0.226078i 0.793353 0.608761i \(-0.208333\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.93185i 1.93185i
\(513\) 0 0
\(514\) −3.31744 1.91532i −3.31744 1.91532i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −0.241181 + 1.83195i −0.241181 + 1.83195i
\(520\) 0 0
\(521\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 2.36603 2.36603i 2.36603 2.36603i
\(523\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(524\) 4.33496 4.33496
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.500000 0.866025i 0.500000 0.866025i
\(530\) 0 0
\(531\) 1.78480 + 0.478235i 1.78480 + 0.478235i
\(532\) 0 0
\(533\) 1.48236i 1.48236i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −0.198092 0.478235i −0.198092 0.478235i
\(538\) 3.83065i 3.83065i
\(539\) 0 0
\(540\) 0 0
\(541\) 0.965926 + 1.67303i 0.965926 + 1.67303i 0.707107 + 0.707107i \(0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(542\) 1.78480 3.09136i 1.78480 3.09136i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.517638 −0.517638 −0.258819 0.965926i \(-0.583333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0.436749 3.31744i 0.436749 3.31744i
\(553\) 0 0
\(554\) 1.00000i 1.00000i
\(555\) 0 0
\(556\) −0.617657 0.356604i −0.617657 0.356604i
\(557\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(558\) 0.793353 + 2.96083i 0.793353 + 2.96083i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) −4.29788 3.29788i −4.29788 3.29788i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 3.34607 3.34607
\(569\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(570\) 0 0
\(571\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000i 1.00000i
\(576\) −0.965926 3.60488i −0.965926 3.60488i
\(577\) −1.37413 0.793353i −1.37413 0.793353i −0.382683 0.923880i \(-0.625000\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(578\) −1.67303 0.965926i −1.67303 0.965926i
\(579\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0.436749 0.756472i 0.436749 0.756472i
\(585\) 0 0
\(586\) 0 0
\(587\) 1.21752 1.21752 0.608761 0.793353i \(-0.291667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −0.608761 + 0.793353i −0.608761 + 0.793353i
\(592\) 0 0
\(593\) 0.382683 + 0.662827i 0.382683 + 0.662827i 0.991445 0.130526i \(-0.0416667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 2.03696 + 1.17604i 2.03696 + 1.17604i
\(599\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(600\) −1.28048 3.09136i −1.28048 3.09136i
\(601\) 1.98289i 1.98289i 0.130526 + 0.991445i \(0.458333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −0.707107 1.22474i −0.707107 1.22474i
\(605\) 0 0
\(606\) 0.900100 1.17303i 0.900100 1.17303i
\(607\) −1.60021 + 0.923880i −1.60021 + 0.923880i −0.608761 + 0.793353i \(0.708333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.09077 1.20711i 2.09077 1.20711i
\(612\) 0 0
\(613\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) 0.739288 + 1.28048i 0.739288 + 1.28048i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(620\) 0 0
\(621\) 0.130526 + 0.991445i 0.130526 + 0.991445i
\(622\) 3.06528i 3.06528i
\(623\) 0 0
\(624\) 4.50498 + 0.593092i 4.50498 + 0.593092i
\(625\) −0.500000 0.866025i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −1.12197 0.860919i −1.12197 0.860919i
\(634\) −1.36603 + 2.36603i −1.36603 + 2.36603i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(640\) 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 0
\(647\) −0.608761 + 1.05441i −0.608761 + 1.05441i 0.382683 + 0.923880i \(0.375000\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(648\) 1.67303 + 2.89778i 1.67303 + 2.89778i
\(649\) 0 0
\(650\) 2.35207 2.35207
\(651\) 0 0
\(652\) 2.73205 2.73205
\(653\) −0.448288 + 0.258819i −0.448288 + 0.258819i −0.707107 0.707107i \(-0.750000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.27193 + 3.93510i 2.27193 + 3.93510i
\(657\) −0.0675653 + 0.252157i −0.0675653 + 0.252157i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) −0.866025 0.500000i −0.866025 0.500000i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(668\) −1.04551 + 1.81088i −1.04551 + 1.81088i
\(669\) −0.465926 + 0.607206i −0.465926 + 0.607206i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(674\) 0 0
\(675\) 0.608761 + 0.793353i 0.608761 + 0.793353i
\(676\) −0.658919 + 1.14128i −0.658919 + 1.14128i
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.60021 + 0.923880i −1.60021 + 0.923880i −0.608761 + 0.793353i \(0.708333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(692\) −5.04817 −5.04817
\(693\) 0 0
\(694\) −2.73205 −2.73205
\(695\) 0 0
\(696\) 4.59792 + 3.52811i 4.59792 + 3.52811i
\(697\) 0 0
\(698\) 1.53264 + 2.65461i 1.53264 + 2.65461i
\(699\) 0.991445 + 0.130526i 0.991445 + 0.130526i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) −2.33195 + 0.307007i −2.33195 + 0.307007i
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 3.06528i 3.06528i
\(707\) 0 0
\(708\) −0.658919 + 5.00498i −0.658919 + 5.00498i
\(709\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.58671 −1.58671
\(714\) 0 0
\(715\) 0 0
\(716\) 1.22474 0.707107i 1.22474 0.707107i
\(717\) 0.315118 0.410670i 0.315118 0.410670i
\(718\) 0 0
\(719\) 0.923880 + 1.60021i 0.923880 + 1.60021i 0.793353 + 0.608761i \(0.208333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.93185i 1.93185i
\(723\) 0 0
\(724\) 0 0
\(725\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(726\) −0.739288 1.78480i −0.739288 1.78480i
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −0.707107 0.707107i −0.707107 0.707107i
\(730\) 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
\(736\) 3.86370 3.86370
\(737\) 0 0
\(738\) −1.66317 1.66317i −1.66317 1.66317i
\(739\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −4.90508 + 2.03175i −4.90508 + 2.03175i
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) 3.70012 6.40880i 3.70012 6.40880i
\(753\) 0 0
\(754\) −3.52811 + 2.03696i −3.52811 + 2.03696i
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.130526 + 0.226078i 0.130526 + 0.226078i 0.923880 0.382683i \(-0.125000\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(762\) −3.70012 0.487130i −3.70012 0.487130i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.94829 1.12484i −1.94829 1.12484i
\(768\) 2.52409 1.04551i 2.52409 1.04551i
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) −0.258819 + 1.96593i −0.258819 + 1.96593i
\(772\) −1.36603 2.36603i −1.36603 2.36603i
\(773\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(774\) 0 0
\(775\) −1.37413 + 0.793353i −1.37413 + 0.793353i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1.60021 0.662827i −1.60021 0.662827i
\(784\) 0 0
\(785\) 0 0
\(786\) −1.17303 2.83195i −1.17303 2.83195i
\(787\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(788\) −2.36603 1.36603i −2.36603 1.36603i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −1.91532 + 3.31744i −1.91532 + 3.31744i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 3.34607 1.93185i 3.34607 1.93185i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 3.73205i 3.73205i
\(807\) 1.83195 0.758819i 1.83195 0.758819i
\(808\) 2.21786 + 1.28048i 2.21786 + 1.28048i
\(809\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(810\) 0 0
\(811\) 1.21752i 1.21752i 0.793353 + 0.608761i \(0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(812\) 0 0
\(813\) −1.83195 0.241181i −1.83195 0.241181i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 3.83065 3.83065
\(819\) 0 0
\(820\) 0 0
\(821\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(822\) 0 0
\(823\) 0.866025 1.50000i 0.866025 1.50000i 1.00000i \(-0.5\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) −2.63896 + 0.707107i −2.63896 + 0.707107i
\(829\) 0.662827 + 0.382683i 0.662827 + 0.382683i 0.793353 0.608761i \(-0.208333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(830\) 0 0
\(831\) 0.478235 0.198092i 0.478235 0.198092i
\(832\) 4.54386i 4.54386i
\(833\) 0 0
\(834\) −0.0658262 + 0.500000i −0.0658262 + 0.500000i
\(835\) 0 0
\(836\) 0 0
\(837\) 1.25882 0.965926i 1.25882 0.965926i
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −2.00000 −2.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 1.93185 3.34607i 1.93185 3.34607i
\(845\) 0 0
\(846\) −0.991445 + 3.70012i −0.991445 + 3.70012i
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) −1.04551 2.52409i −1.04551 2.52409i
\(853\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.130526 + 0.226078i −0.130526 + 0.226078i −0.923880 0.382683i \(-0.875000\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(858\) 0 0
\(859\) −1.05441 + 0.608761i −1.05441 + 0.608761i −0.923880 0.382683i \(-0.875000\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(864\) −3.06528 + 2.35207i −3.06528 + 2.35207i
\(865\) 0 0
\(866\) 0 0
\(867\) −0.130526 + 0.991445i −0.130526 + 0.991445i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) −0.707107 0.0930924i −0.707107 0.0930924i
\(877\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(878\) 0.252157 0.436749i 0.252157 0.436749i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1.67303 + 2.89778i −1.67303 + 2.89778i
\(887\) −0.793353 1.37413i −0.793353 1.37413i −0.923880 0.382683i \(-0.875000\pi\)
0.130526 0.991445i \(-0.458333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −1.81088 1.04551i −1.81088 1.04551i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0.158919 1.20711i 0.158919 1.20711i
\(898\) 1.36603 + 2.36603i 1.36603 + 2.36603i
\(899\) 1.37413 2.38006i 1.37413 2.38006i
\(900\) −1.93185 + 1.93185i −1.93185 + 1.93185i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) −0.608761 + 0.793353i −0.608761 + 0.793353i
\(907\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(908\) 0 0
\(909\) −0.739288 0.198092i −0.739288 0.198092i
\(910\) 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 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(920\) 0 0
\(921\) 0.465926 0.607206i 0.465926 0.607206i
\(922\) 0.436749 0.252157i 0.436749 0.252157i
\(923\) 1.21752 1.21752
\(924\) 0 0
\(925\) 0 0
\(926\) −2.36603 + 1.36603i −2.36603 + 1.36603i
\(927\) 0 0
\(928\) −3.34607 + 5.79555i −3.34607 + 5.79555i
\(929\) −0.793353 1.37413i −0.793353 1.37413i −0.923880 0.382683i \(-0.875000\pi\)
0.130526 0.991445i \(-0.458333\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.73205i 2.73205i
\(933\) 1.46593 0.607206i 1.46593 0.607206i
\(934\) 0 0
\(935\) 0 0
\(936\) −1.05441 3.93510i −1.05441 3.93510i
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(942\) 0 0
\(943\) 1.05441 0.608761i 1.05441 0.608761i
\(944\) −6.89593 −6.89593
\(945\) 0 0
\(946\) 0 0
\(947\) 0.448288 0.258819i 0.448288 0.258819i −0.258819 0.965926i \(-0.583333\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(948\) 0 0
\(949\) 0.158919 0.275255i 0.158919 0.275255i
\(950\) 0 0
\(951\) 1.40211 + 0.184592i 1.40211 + 0.184592i
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.758819 + 1.31431i 0.758819 + 1.31431i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(968\) 2.89778 1.67303i 2.89778 1.67303i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(972\) 1.66317 2.16748i 1.66317 2.16748i
\(973\) 0 0
\(974\) 1.93185i 1.93185i
\(975\) −0.465926 1.12484i −0.465926 1.12484i
\(976\) 0 0
\(977\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(978\) −0.739288 1.78480i −0.739288 1.78480i
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 1.67303 + 2.89778i 1.67303 + 2.89778i
\(983\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(984\) 2.48004 3.23205i 2.48004 3.23205i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(992\) −3.06528 5.30922i −3.06528 5.30922i
\(993\) −0.0675653 + 0.513210i −0.0675653 + 0.513210i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −0.662827 0.382683i −0.662827 0.382683i 0.130526 0.991445i \(-0.458333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(998\) −3.23205 1.86603i −3.23205 1.86603i
\(999\) 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 3381.1.o.b.2138.7 16
3.2 odd 2 3381.1.o.a.2138.1 16
7.2 even 3 3381.1.o.a.68.2 16
7.3 odd 6 3381.1.c.b.3380.15 yes 16
7.4 even 3 3381.1.c.b.3380.14 yes 16
7.5 odd 6 3381.1.o.a.68.1 16
7.6 odd 2 inner 3381.1.o.b.2138.8 16
21.2 odd 6 inner 3381.1.o.b.68.8 16
21.5 even 6 inner 3381.1.o.b.68.7 16
21.11 odd 6 3381.1.c.b.3380.3 yes 16
21.17 even 6 3381.1.c.b.3380.2 16
21.20 even 2 3381.1.o.a.2138.2 16
23.22 odd 2 CM 3381.1.o.b.2138.7 16
69.68 even 2 3381.1.o.a.2138.1 16
161.45 even 6 3381.1.c.b.3380.15 yes 16
161.68 even 6 3381.1.o.a.68.1 16
161.114 odd 6 3381.1.o.a.68.2 16
161.137 odd 6 3381.1.c.b.3380.14 yes 16
161.160 even 2 inner 3381.1.o.b.2138.8 16
483.68 odd 6 inner 3381.1.o.b.68.7 16
483.137 even 6 3381.1.c.b.3380.3 yes 16
483.206 odd 6 3381.1.c.b.3380.2 16
483.275 even 6 inner 3381.1.o.b.68.8 16
483.482 odd 2 3381.1.o.a.2138.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3381.1.c.b.3380.2 16 21.17 even 6
3381.1.c.b.3380.2 16 483.206 odd 6
3381.1.c.b.3380.3 yes 16 21.11 odd 6
3381.1.c.b.3380.3 yes 16 483.137 even 6
3381.1.c.b.3380.14 yes 16 7.4 even 3
3381.1.c.b.3380.14 yes 16 161.137 odd 6
3381.1.c.b.3380.15 yes 16 7.3 odd 6
3381.1.c.b.3380.15 yes 16 161.45 even 6
3381.1.o.a.68.1 16 7.5 odd 6
3381.1.o.a.68.1 16 161.68 even 6
3381.1.o.a.68.2 16 7.2 even 3
3381.1.o.a.68.2 16 161.114 odd 6
3381.1.o.a.2138.1 16 3.2 odd 2
3381.1.o.a.2138.1 16 69.68 even 2
3381.1.o.a.2138.2 16 21.20 even 2
3381.1.o.a.2138.2 16 483.482 odd 2
3381.1.o.b.68.7 16 21.5 even 6 inner
3381.1.o.b.68.7 16 483.68 odd 6 inner
3381.1.o.b.68.8 16 21.2 odd 6 inner
3381.1.o.b.68.8 16 483.275 even 6 inner
3381.1.o.b.2138.7 16 1.1 even 1 trivial
3381.1.o.b.2138.7 16 23.22 odd 2 CM
3381.1.o.b.2138.8 16 7.6 odd 2 inner
3381.1.o.b.2138.8 16 161.160 even 2 inner