Properties

Label 820.1.by.a.423.1
Level $820$
Weight $1$
Character 820.423
Analytic conductor $0.409$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [820,1,Mod(47,820)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(820, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 10, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("820.47");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 820.by (of order \(40\), degree \(16\), 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: \(0.409233310359\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} - \cdots)\)

Embedding invariants

Embedding label 423.1
Root \(0.453990 - 0.891007i\) of defining polynomial
Character \(\chi\) \(=\) 820.423
Dual form 820.1.by.a.347.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.951057 + 0.309017i) q^{2} +(0.809017 - 0.587785i) q^{4} +(0.453990 - 0.891007i) q^{5} +(-0.587785 + 0.809017i) q^{8} +(0.707107 - 0.707107i) q^{9} +O(q^{10})\) \(q+(-0.951057 + 0.309017i) q^{2} +(0.809017 - 0.587785i) q^{4} +(0.453990 - 0.891007i) q^{5} +(-0.587785 + 0.809017i) q^{8} +(0.707107 - 0.707107i) q^{9} +(-0.156434 + 0.987688i) q^{10} +(-1.26301 - 1.47879i) q^{13} +(0.309017 - 0.951057i) q^{16} +(0.678671 + 1.10749i) q^{17} +(-0.453990 + 0.891007i) q^{18} +(-0.156434 - 0.987688i) q^{20} +(-0.587785 - 0.809017i) q^{25} +(1.65816 + 1.01612i) q^{26} +(-1.79671 + 0.431351i) q^{29} +1.00000i q^{32} +(-0.987688 - 0.843566i) q^{34} +(0.156434 - 0.987688i) q^{36} +(0.183900 - 1.16110i) q^{37} +(0.453990 + 0.891007i) q^{40} +(0.951057 - 0.309017i) q^{41} +(-0.309017 - 0.951057i) q^{45} +(0.987688 - 0.156434i) q^{49} +(0.809017 + 0.587785i) q^{50} +(-1.89101 - 0.453990i) q^{52} +(1.70002 + 1.04178i) q^{53} +(1.57547 - 0.965451i) q^{58} +(0.896802 + 1.76007i) q^{61} +(-0.309017 - 0.951057i) q^{64} +(-1.89101 + 0.453990i) q^{65} +(1.20002 + 0.497066i) q^{68} +(0.156434 + 0.987688i) q^{72} +0.312869 q^{73} +(0.183900 + 1.16110i) q^{74} +(-0.707107 - 0.707107i) q^{80} -1.00000i q^{81} +(-0.809017 + 0.587785i) q^{82} +(1.29489 - 0.101910i) q^{85} +(-0.497066 + 0.581990i) q^{89} +(0.587785 + 0.809017i) q^{90} +(0.243950 + 1.01612i) q^{97} +(-0.891007 + 0.453990i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} - 4 q^{13} - 4 q^{16} + 4 q^{17} - 4 q^{29} + 4 q^{45} + 4 q^{50} - 16 q^{52} + 4 q^{53} - 4 q^{61} + 4 q^{64} - 16 q^{65} - 4 q^{68} - 4 q^{82}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/820\mathbb{Z}\right)^\times\).

\(n\) \(411\) \(621\) \(657\)
\(\chi(n)\) \(-1\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(3\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(4\) 0.809017 0.587785i 0.809017 0.587785i
\(5\) 0.453990 0.891007i 0.453990 0.891007i
\(6\) 0 0
\(7\) 0 0 0.996917 0.0784591i \(-0.0250000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(8\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(9\) 0.707107 0.707107i 0.707107 0.707107i
\(10\) −0.156434 + 0.987688i −0.156434 + 0.987688i
\(11\) 0 0 −0.522499 0.852640i \(-0.675000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(12\) 0 0
\(13\) −1.26301 1.47879i −1.26301 1.47879i −0.809017 0.587785i \(-0.800000\pi\)
−0.453990 0.891007i \(-0.650000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.309017 0.951057i 0.309017 0.951057i
\(17\) 0.678671 + 1.10749i 0.678671 + 1.10749i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(18\) −0.453990 + 0.891007i −0.453990 + 0.891007i
\(19\) 0 0 −0.0784591 0.996917i \(-0.525000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(20\) −0.156434 0.987688i −0.156434 0.987688i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(24\) 0 0
\(25\) −0.587785 0.809017i −0.587785 0.809017i
\(26\) 1.65816 + 1.01612i 1.65816 + 1.01612i
\(27\) 0 0
\(28\) 0 0
\(29\) −1.79671 + 0.431351i −1.79671 + 0.431351i −0.987688 0.156434i \(-0.950000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(30\) 0 0
\(31\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(32\) 1.00000i 1.00000i
\(33\) 0 0
\(34\) −0.987688 0.843566i −0.987688 0.843566i
\(35\) 0 0
\(36\) 0.156434 0.987688i 0.156434 0.987688i
\(37\) 0.183900 1.16110i 0.183900 1.16110i −0.707107 0.707107i \(-0.750000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(41\) 0.951057 0.309017i 0.951057 0.309017i
\(42\) 0 0
\(43\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(44\) 0 0
\(45\) −0.309017 0.951057i −0.309017 0.951057i
\(46\) 0 0
\(47\) 0 0 0.0784591 0.996917i \(-0.475000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(48\) 0 0
\(49\) 0.987688 0.156434i 0.987688 0.156434i
\(50\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(51\) 0 0
\(52\) −1.89101 0.453990i −1.89101 0.453990i
\(53\) 1.70002 + 1.04178i 1.70002 + 1.04178i 0.891007 + 0.453990i \(0.150000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 1.57547 0.965451i 1.57547 0.965451i
\(59\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(60\) 0 0
\(61\) 0.896802 + 1.76007i 0.896802 + 1.76007i 0.587785 + 0.809017i \(0.300000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.309017 0.951057i −0.309017 0.951057i
\(65\) −1.89101 + 0.453990i −1.89101 + 0.453990i
\(66\) 0 0
\(67\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(68\) 1.20002 + 0.497066i 1.20002 + 0.497066i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(72\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(73\) 0.312869 0.312869 0.156434 0.987688i \(-0.450000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(74\) 0.183900 + 1.16110i 0.183900 + 1.16110i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(80\) −0.707107 0.707107i −0.707107 0.707107i
\(81\) 1.00000i 1.00000i
\(82\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) 1.29489 0.101910i 1.29489 0.101910i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.497066 + 0.581990i −0.497066 + 0.581990i −0.951057 0.309017i \(-0.900000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(90\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.243950 + 1.01612i 0.243950 + 1.01612i 0.951057 + 0.309017i \(0.100000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) −0.891007 + 0.453990i −0.891007 + 0.453990i
\(99\) 0 0
\(100\) −0.951057 0.309017i −0.951057 0.309017i
\(101\) −0.156434 + 0.0123117i −0.156434 + 0.0123117i −0.156434 0.987688i \(-0.550000\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(104\) 1.93874 0.152583i 1.93874 0.152583i
\(105\) 0 0
\(106\) −1.93874 0.465451i −1.93874 0.465451i
\(107\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(108\) 0 0
\(109\) −0.581990 1.40505i −0.581990 1.40505i −0.891007 0.453990i \(-0.850000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.59811 + 0.253116i −1.59811 + 0.253116i −0.891007 0.453990i \(-0.850000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.20002 + 1.40505i −1.20002 + 1.40505i
\(117\) −1.93874 0.152583i −1.93874 0.152583i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.453990 + 0.891007i −0.453990 + 0.891007i
\(122\) −1.39680 1.39680i −1.39680 1.39680i
\(123\) 0 0
\(124\) 0 0
\(125\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(126\) 0 0
\(127\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(128\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(129\) 0 0
\(130\) 1.65816 1.01612i 1.65816 1.01612i
\(131\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −1.29489 0.101910i −1.29489 0.101910i
\(137\) 0.431351 + 0.178671i 0.431351 + 0.178671i 0.587785 0.809017i \(-0.300000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(138\) 0 0
\(139\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.453990 0.891007i −0.453990 0.891007i
\(145\) −0.431351 + 1.79671i −0.431351 + 1.79671i
\(146\) −0.297556 + 0.0966818i −0.297556 + 0.0966818i
\(147\) 0 0
\(148\) −0.533698 1.04744i −0.533698 1.04744i
\(149\) −0.891007 + 1.45399i −0.891007 + 1.45399i 1.00000i \(0.5\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(150\) 0 0
\(151\) 0 0 0.0784591 0.996917i \(-0.475000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(152\) 0 0
\(153\) 1.26301 + 0.303221i 1.26301 + 0.303221i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.133795 + 1.70002i 0.133795 + 1.70002i 0.587785 + 0.809017i \(0.300000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.891007 + 0.453990i 0.891007 + 0.453990i
\(161\) 0 0
\(162\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(163\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(164\) 0.587785 0.809017i 0.587785 0.809017i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(168\) 0 0
\(169\) −0.435203 + 2.74776i −0.435203 + 2.74776i
\(170\) −1.20002 + 0.497066i −1.20002 + 0.497066i
\(171\) 0 0
\(172\) 0 0
\(173\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0.292893 0.707107i 0.292893 0.707107i
\(179\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(180\) −0.809017 0.587785i −0.809017 0.587785i
\(181\) −0.108993 + 0.453990i −0.108993 + 0.453990i 0.891007 + 0.453990i \(0.150000\pi\)
−1.00000 \(1.00000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.951057 0.690983i −0.951057 0.690983i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(192\) 0 0
\(193\) 0.794622 1.29671i 0.794622 1.29671i −0.156434 0.987688i \(-0.550000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(194\) −0.546010 0.891007i −0.546010 0.891007i
\(195\) 0 0
\(196\) 0.707107 0.707107i 0.707107 0.707107i
\(197\) 1.11803 1.53884i 1.11803 1.53884i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(198\) 0 0
\(199\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(200\) 1.00000 1.00000
\(201\) 0 0
\(202\) 0.144974 0.0600500i 0.144974 0.0600500i
\(203\) 0 0
\(204\) 0 0
\(205\) 0.156434 0.987688i 0.156434 0.987688i
\(206\) 0 0
\(207\) 0 0
\(208\) −1.79671 + 0.744220i −1.79671 + 0.744220i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(212\) 1.98769 0.156434i 1.98769 0.156434i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0.987688 + 1.15643i 0.987688 + 1.15643i
\(219\) 0 0
\(220\) 0 0
\(221\) 0.780582 2.40238i 0.780582 2.40238i
\(222\) 0 0
\(223\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(224\) 0 0
\(225\) −0.987688 0.156434i −0.987688 0.156434i
\(226\) 1.44168 0.734572i 1.44168 0.734572i
\(227\) 0 0 −0.760406 0.649448i \(-0.775000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(228\) 0 0
\(229\) −0.243950 + 1.01612i −0.243950 + 1.01612i 0.707107 + 0.707107i \(0.250000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.707107 1.70711i 0.707107 1.70711i
\(233\) 0.119322 0.101910i 0.119322 0.101910i −0.587785 0.809017i \(-0.700000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(234\) 1.89101 0.453990i 1.89101 0.453990i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.760406 0.649448i \(-0.775000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(240\) 0 0
\(241\) −0.142040 + 0.896802i −0.142040 + 0.896802i 0.809017 + 0.587785i \(0.200000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(242\) 0.156434 0.987688i 0.156434 0.987688i
\(243\) 0 0
\(244\) 1.76007 + 0.896802i 1.76007 + 0.896802i
\(245\) 0.309017 0.951057i 0.309017 0.951057i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0.891007 0.453990i 0.891007 0.453990i
\(251\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.809017 0.587785i −0.809017 0.587785i
\(257\) −0.744220 0.178671i −0.744220 0.178671i −0.156434 0.987688i \(-0.550000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.26301 + 1.47879i −1.26301 + 1.47879i
\(261\) −0.965451 + 1.57547i −0.965451 + 1.57547i
\(262\) 0 0
\(263\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(264\) 0 0
\(265\) 1.70002 1.04178i 1.70002 1.04178i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.500000 + 1.53884i 0.500000 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(270\) 0 0
\(271\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(272\) 1.26301 0.303221i 1.26301 0.303221i
\(273\) 0 0
\(274\) −0.465451 0.0366318i −0.465451 0.0366318i
\(275\) 0 0
\(276\) 0 0
\(277\) −0.297556 1.87869i −0.297556 1.87869i −0.453990 0.891007i \(-0.650000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.303221 + 0.355026i 0.303221 + 0.355026i 0.891007 0.453990i \(-0.150000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(282\) 0 0
\(283\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(289\) −0.311951 + 0.612238i −0.311951 + 0.612238i
\(290\) −0.144974 1.84206i −0.144974 1.84206i
\(291\) 0 0
\(292\) 0.253116 0.183900i 0.253116 0.183900i
\(293\) −0.156434 0.0123117i −0.156434 0.0123117i 1.00000i \(-0.5\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.831254 + 0.831254i 0.831254 + 0.831254i
\(297\) 0 0
\(298\) 0.398090 1.65816i 0.398090 1.65816i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.97538 1.97538
\(306\) −1.29489 + 0.101910i −1.29489 + 0.101910i
\(307\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(312\) 0 0
\(313\) 0.178671 + 0.744220i 0.178671 + 0.744220i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(314\) −0.652583 1.57547i −0.652583 1.57547i
\(315\) 0 0
\(316\) 0 0
\(317\) 0.398090 1.65816i 0.398090 1.65816i −0.309017 0.951057i \(-0.600000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.987688 0.156434i −0.987688 0.156434i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.587785 0.809017i −0.587785 0.809017i
\(325\) −0.453990 + 1.89101i −0.453990 + 1.89101i
\(326\) 0 0
\(327\) 0 0
\(328\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(332\) 0 0
\(333\) −0.690983 0.951057i −0.690983 0.951057i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.907981 0.907981 0.453990 0.891007i \(-0.350000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(338\) −0.435203 2.74776i −0.435203 2.74776i
\(339\) 0 0
\(340\) 0.987688 0.843566i 0.987688 0.843566i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −0.190983 0.587785i −0.190983 0.587785i
\(347\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(348\) 0 0
\(349\) −0.863541 1.69480i −0.863541 1.69480i −0.707107 0.707107i \(-0.750000\pi\)
−0.156434 0.987688i \(-0.550000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.280582 0.550672i −0.280582 0.550672i 0.707107 0.707107i \(-0.250000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −0.0600500 + 0.763007i −0.0600500 + 0.763007i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(360\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(361\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(362\) −0.0366318 0.465451i −0.0366318 0.465451i
\(363\) 0 0
\(364\) 0 0
\(365\) 0.142040 0.278768i 0.142040 0.278768i
\(366\) 0 0
\(367\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(368\) 0 0
\(369\) 0.453990 0.891007i 0.453990 0.891007i
\(370\) 1.11803 + 0.363271i 1.11803 + 0.363271i
\(371\) 0 0
\(372\) 0 0
\(373\) −0.309017 + 1.95106i −0.309017 + 1.95106i 1.00000i \(0.5\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.90713 + 2.11215i 2.90713 + 2.11215i
\(378\) 0 0
\(379\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.355026 + 1.47879i −0.355026 + 1.47879i
\(387\) 0 0
\(388\) 0.794622 + 0.678671i 0.794622 + 0.678671i
\(389\) 1.58779 0.809017i 1.58779 0.809017i 0.587785 0.809017i \(-0.300000\pi\)
1.00000 \(0\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.453990 + 0.891007i −0.453990 + 0.891007i
\(393\) 0 0
\(394\) −0.587785 + 1.80902i −0.587785 + 1.80902i
\(395\) 0 0
\(396\) 0 0
\(397\) −0.987688 1.15643i −0.987688 1.15643i −0.987688 0.156434i \(-0.950000\pi\)
1.00000i \(-0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(401\) −0.437016 + 0.437016i −0.437016 + 0.437016i −0.891007 0.453990i \(-0.850000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.119322 + 0.101910i −0.119322 + 0.101910i
\(405\) −0.891007 0.453990i −0.891007 0.453990i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.312869 0.312869 0.156434 0.987688i \(-0.450000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(410\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.47879 1.26301i 1.47879 1.26301i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(420\) 0 0
\(421\) −1.04178 1.70002i −1.04178 1.70002i −0.587785 0.809017i \(-0.700000\pi\)
−0.453990 0.891007i \(-0.650000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −1.84206 + 0.763007i −1.84206 + 0.763007i
\(425\) 0.497066 1.20002i 0.497066 1.20002i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(432\) 0 0
\(433\) 1.58779 + 0.809017i 1.58779 + 0.809017i 1.00000 \(0\)
0.587785 + 0.809017i \(0.300000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.29671 0.794622i −1.29671 0.794622i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(440\) 0 0
\(441\) 0.587785 0.809017i 0.587785 0.809017i
\(442\) 2.52601i 2.52601i
\(443\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(444\) 0 0
\(445\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.58779 0.809017i −1.58779 0.809017i −0.587785 0.809017i \(-0.700000\pi\)
−1.00000 \(\pi\)
\(450\) 0.987688 0.156434i 0.987688 0.156434i
\(451\) 0 0
\(452\) −1.14412 + 1.14412i −1.14412 + 1.14412i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.0819895 + 1.04178i −0.0819895 + 1.04178i 0.809017 + 0.587785i \(0.200000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(458\) −0.0819895 1.04178i −0.0819895 1.04178i
\(459\) 0 0
\(460\) 0 0
\(461\) −0.253116 0.183900i −0.253116 0.183900i 0.453990 0.891007i \(-0.350000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) 0 0
\(463\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(464\) −0.144974 + 1.84206i −0.144974 + 1.84206i
\(465\) 0 0
\(466\) −0.0819895 + 0.133795i −0.0819895 + 0.133795i
\(467\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(468\) −1.65816 + 1.01612i −1.65816 + 1.01612i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.93874 0.465451i 1.93874 0.465451i
\(478\) 0 0
\(479\) 0 0 −0.996917 0.0784591i \(-0.975000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(480\) 0 0
\(481\) −1.94929 + 1.19453i −1.94929 + 1.19453i
\(482\) −0.142040 0.896802i −0.142040 0.896802i
\(483\) 0 0
\(484\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(485\) 1.01612 + 0.243950i 1.01612 + 0.243950i
\(486\) 0 0
\(487\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(488\) −1.95106 0.309017i −1.95106 0.309017i
\(489\) 0 0
\(490\) 1.00000i 1.00000i
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −1.69709 1.69709i −1.69709 1.69709i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(500\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(504\) 0 0
\(505\) −0.0600500 + 0.144974i −0.0600500 + 0.144974i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.47879 + 0.355026i 1.47879 + 0.355026i 0.891007 0.453990i \(-0.150000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(513\) 0 0
\(514\) 0.763007 0.0600500i 0.763007 0.0600500i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0.744220 1.79671i 0.744220 1.79671i
\(521\) −0.243950 1.01612i −0.243950 1.01612i −0.951057 0.309017i \(-0.900000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(522\) 0.431351 1.79671i 0.431351 1.79671i
\(523\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(530\) −1.29489 + 1.51612i −1.29489 + 1.51612i
\(531\) 0 0
\(532\) 0 0
\(533\) −1.65816 1.01612i −1.65816 1.01612i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −0.951057 1.30902i −0.951057 1.30902i
\(539\) 0 0
\(540\) 0 0
\(541\) 0.0966818 + 0.610425i 0.0966818 + 0.610425i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −1.10749 + 0.678671i −1.10749 + 0.678671i
\(545\) −1.51612 0.119322i −1.51612 0.119322i
\(546\) 0 0
\(547\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(548\) 0.453990 0.108993i 0.453990 0.108993i
\(549\) 1.87869 + 0.610425i 1.87869 + 0.610425i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0.863541 + 1.69480i 0.863541 + 1.69480i
\(555\) 0 0
\(556\) 0 0
\(557\) −0.652583 + 0.399903i −0.652583 + 0.399903i −0.809017 0.587785i \(-0.800000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −0.398090 0.243950i −0.398090 0.243950i
\(563\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(564\) 0 0
\(565\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.16110 0.183900i −1.16110 0.183900i −0.453990 0.891007i \(-0.650000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(570\) 0 0
\(571\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.891007 0.453990i −0.891007 0.453990i
\(577\) 0.0600500 + 0.144974i 0.0600500 + 0.144974i 0.951057 0.309017i \(-0.100000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(578\) 0.107491 0.678671i 0.107491 0.678671i
\(579\) 0 0
\(580\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.183900 + 0.253116i −0.183900 + 0.253116i
\(585\) −1.01612 + 1.65816i −1.01612 + 1.65816i
\(586\) 0.152583 0.0366318i 0.152583 0.0366318i
\(587\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.04744 0.533698i −1.04744 0.533698i
\(593\) −1.29671 1.10749i −1.29671 1.10749i −0.987688 0.156434i \(-0.950000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.133795 + 1.70002i 0.133795 + 1.70002i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(600\) 0 0
\(601\) 1.57547 0.652583i 1.57547 0.652583i 0.587785 0.809017i \(-0.300000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(606\) 0 0
\(607\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −1.87869 + 0.610425i −1.87869 + 0.610425i
\(611\) 0 0
\(612\) 1.20002 0.497066i 1.20002 0.497066i
\(613\) 1.87869 0.610425i 1.87869 0.610425i 0.891007 0.453990i \(-0.150000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.80902 + 0.587785i −1.80902 + 0.587785i −0.809017 + 0.587785i \(0.800000\pi\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(626\) −0.399903 0.652583i −0.399903 0.652583i
\(627\) 0 0
\(628\) 1.10749 + 1.29671i 1.10749 + 1.29671i
\(629\) 1.41071 0.584336i 1.41071 0.584336i
\(630\) 0 0
\(631\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0.133795 + 1.70002i 0.133795 + 1.70002i
\(635\) 0 0
\(636\) 0 0
\(637\) −1.47879 1.26301i −1.47879 1.26301i
\(638\) 0 0
\(639\) 0 0
\(640\) 0.987688 0.156434i 0.987688 0.156434i
\(641\) −1.70002 1.04178i −1.70002 1.04178i −0.891007 0.453990i \(-0.850000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(642\) 0 0
\(643\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(649\) 0 0
\(650\) −0.152583 1.93874i −0.152583 1.93874i
\(651\) 0 0
\(652\) 0 0
\(653\) −0.399903 0.965451i −0.399903 0.965451i −0.987688 0.156434i \(-0.950000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.00000i 1.00000i
\(657\) 0.221232 0.221232i 0.221232 0.221232i
\(658\) 0 0
\(659\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(660\) 0 0
\(661\) 1.16110 + 0.183900i 1.16110 + 0.183900i 0.707107 0.707107i \(-0.250000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.951057 + 0.690983i 0.951057 + 0.690983i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.398090 + 0.243950i −0.398090 + 0.243950i −0.707107 0.707107i \(-0.750000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(674\) −0.863541 + 0.280582i −0.863541 + 0.280582i
\(675\) 0 0
\(676\) 1.26301 + 2.47879i 1.26301 + 2.47879i
\(677\) −0.0966818 + 0.297556i −0.0966818 + 0.297556i −0.987688 0.156434i \(-0.950000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −0.678671 + 1.10749i −0.678671 + 1.10749i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(684\) 0 0
\(685\) 0.355026 0.303221i 0.355026 0.303221i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.606573 3.82975i −0.606573 3.82975i
\(690\) 0 0
\(691\) 0 0 −0.649448 0.760406i \(-0.725000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(692\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0.987688 + 0.843566i 0.987688 + 0.843566i
\(698\) 1.34500 + 1.34500i 1.34500 + 1.34500i
\(699\) 0 0
\(700\) 0 0
\(701\) −0.533698 0.734572i −0.533698 0.734572i 0.453990 0.891007i \(-0.350000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0.437016 + 0.437016i 0.437016 + 0.437016i
\(707\) 0 0
\(708\) 0 0
\(709\) −0.0366318 0.152583i −0.0366318 0.152583i 0.951057 0.309017i \(-0.100000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −0.178671 0.744220i −0.178671 0.744220i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.996917 0.0784591i \(-0.0250000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(720\) −1.00000 −1.00000
\(721\) 0 0
\(722\) 0.891007 0.453990i 0.891007 0.453990i
\(723\) 0 0
\(724\) 0.178671 + 0.431351i 0.178671 + 0.431351i
\(725\) 1.40505 + 1.20002i 1.40505 + 1.20002i
\(726\) 0 0
\(727\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(728\) 0 0
\(729\) −0.707107 0.707107i −0.707107 0.707107i
\(730\) −0.0489435 + 0.309017i −0.0489435 + 0.309017i
\(731\) 0 0
\(732\) 0 0
\(733\) −1.30902 + 0.951057i −1.30902 + 0.951057i −0.309017 + 0.951057i \(0.600000\pi\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −0.156434 + 0.987688i −0.156434 + 0.987688i
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) −1.17557 −1.17557
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(744\) 0 0
\(745\) 0.891007 + 1.45399i 0.891007 + 1.45399i
\(746\) −0.309017 1.95106i −0.309017 1.95106i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.996917 0.0784591i \(-0.975000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −3.41754 1.11042i −3.41754 1.11042i
\(755\) 0 0
\(756\) 0 0
\(757\) −0.101910 + 0.119322i −0.101910 + 0.119322i −0.809017 0.587785i \(-0.800000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.863541 0.280582i 0.863541 0.280582i 0.156434 0.987688i \(-0.450000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.843566 0.987688i 0.843566 0.987688i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1.30902 0.951057i −1.30902 0.951057i −0.309017 0.951057i \(-0.600000\pi\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.119322 1.51612i −0.119322 1.51612i
\(773\) −0.144974 + 1.84206i −0.144974 + 1.84206i 0.309017 + 0.951057i \(0.400000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.965451 0.399903i −0.965451 0.399903i
\(777\) 0 0
\(778\) −1.26007 + 1.26007i −1.26007 + 1.26007i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.156434 0.987688i 0.156434 0.987688i
\(785\) 1.57547 + 0.652583i 1.57547 + 0.652583i
\(786\) 0 0
\(787\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(788\) 1.90211i 1.90211i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.47011 3.54917i 1.47011 3.54917i
\(794\) 1.29671 + 0.794622i 1.29671 + 0.794622i
\(795\) 0 0
\(796\) 0 0
\(797\) −0.809017 0.412215i −0.809017 0.412215i 1.00000i \(-0.5\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.809017 0.587785i 0.809017 0.587785i
\(801\) 0.0600500 + 0.763007i 0.0600500 + 0.763007i
\(802\) 0.280582 0.550672i 0.280582 0.550672i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0.0819895 0.133795i 0.0819895 0.133795i
\(809\) 0.546010 + 0.891007i 0.546010 + 0.891007i 1.00000 \(0\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(810\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(811\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −0.297556 + 0.0966818i −0.297556 + 0.0966818i
\(819\) 0 0
\(820\) −0.453990 0.891007i −0.453990 0.891007i
\(821\) 0.312869 0.312869 0.156434 0.987688i \(-0.450000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(822\) 0 0
\(823\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.996917 0.0784591i \(-0.0250000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(828\) 0 0
\(829\) 0.831254 0.831254i 0.831254 0.831254i −0.156434 0.987688i \(-0.550000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.01612 + 1.65816i −1.01612 + 1.65816i
\(833\) 0.843566 + 0.987688i 0.843566 + 0.987688i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.0784591 0.996917i \(-0.525000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(840\) 0 0
\(841\) 2.15108 1.09603i 2.15108 1.09603i
\(842\) 1.51612 + 1.29489i 1.51612 + 1.29489i
\(843\) 0 0
\(844\) 0 0
\(845\) 2.25070 + 1.63523i 2.25070 + 1.63523i
\(846\) 0 0
\(847\) 0 0
\(848\) 1.51612 1.29489i 1.51612 1.29489i
\(849\) 0 0
\(850\) −0.101910 + 1.29489i −0.101910 + 1.29489i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.297556 1.87869i 0.297556 1.87869i −0.156434 0.987688i \(-0.550000\pi\)
0.453990 0.891007i \(-0.350000\pi\)
\(858\) 0 0
\(859\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(864\) 0 0
\(865\) 0.550672 + 0.280582i 0.550672 + 0.280582i
\(866\) −1.76007 0.278768i −1.76007 0.278768i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 1.47879 + 0.355026i 1.47879 + 0.355026i
\(873\) 0.891007 + 0.546010i 0.891007 + 0.546010i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.412215 + 0.809017i 0.412215 + 0.809017i 1.00000 \(0\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.642040 1.26007i −0.642040 1.26007i −0.951057 0.309017i \(-0.900000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(882\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(883\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(884\) −0.780582 2.40238i −0.780582 2.40238i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −0.497066 0.581990i −0.497066 0.581990i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 1.76007 + 0.278768i 1.76007 + 0.278768i
\(899\) 0 0
\(900\) −0.891007 + 0.453990i −0.891007 + 0.453990i
\(901\) 2.58978i 2.58978i
\(902\) 0 0
\(903\) 0 0
\(904\) 0.734572 1.44168i 0.734572 1.44168i
\(905\) 0.355026 + 0.303221i 0.355026 + 0.303221i
\(906\) 0 0
\(907\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(908\) 0 0
\(909\) −0.101910 + 0.119322i −0.101910 + 0.119322i
\(910\) 0 0
\(911\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.243950 1.01612i −0.243950 1.01612i
\(915\) 0 0
\(916\) 0.399903 + 0.965451i 0.399903 + 0.965451i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.297556 + 0.0966818i 0.297556 + 0.0966818i
\(923\) 0 0
\(924\) 0 0
\(925\) −1.04744 + 0.533698i −1.04744 + 0.533698i
\(926\) 0 0
\(927\) 0 0
\(928\) −0.431351 1.79671i −0.431351 1.79671i
\(929\) −0.178671 0.431351i −0.178671 0.431351i 0.809017 0.587785i \(-0.200000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0.0366318 0.152583i 0.0366318 0.152583i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 1.26301 1.47879i 1.26301 1.47879i
\(937\) −1.84206 0.144974i −1.84206 0.144974i −0.891007 0.453990i \(-0.850000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −0.896802 + 1.76007i −0.896802 + 1.76007i −0.309017 + 0.951057i \(0.600000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(948\) 0 0
\(949\) −0.395156 0.462668i −0.395156 0.462668i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(954\) −1.70002 + 1.04178i −1.70002 + 1.04178i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.309017 0.951057i −0.309017 0.951057i
\(962\) 1.48475 1.73842i 1.48475 1.73842i
\(963\) 0 0
\(964\) 0.412215 + 0.809017i 0.412215 + 0.809017i
\(965\) −0.794622 1.29671i −0.794622 1.29671i
\(966\) 0 0
\(967\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(968\) −0.453990 0.891007i −0.453990 0.891007i
\(969\) 0 0
\(970\) −1.04178 + 0.0819895i −1.04178 + 0.0819895i
\(971\) 0 0 0.0784591 0.996917i \(-0.475000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.95106 0.309017i 1.95106 0.309017i
\(977\) −0.144974 1.84206i −0.144974 1.84206i −0.453990 0.891007i \(-0.650000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.309017 0.951057i −0.309017 0.951057i
\(981\) −1.40505 0.581990i −1.40505 0.581990i
\(982\) 0 0
\(983\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(984\) 0 0
\(985\) −0.863541 1.69480i −0.863541 1.69480i
\(986\) 2.13846 + 1.08960i 2.13846 + 1.08960i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.760406 0.649448i \(-0.775000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.51612 1.29489i 1.51612 1.29489i 0.707107 0.707107i \(-0.250000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(998\) 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 820.1.by.a.423.1 yes 16
4.3 odd 2 CM 820.1.by.a.423.1 yes 16
5.2 odd 4 820.1.bz.a.587.1 yes 16
20.7 even 4 820.1.bz.a.587.1 yes 16
41.19 odd 40 820.1.bz.a.183.1 yes 16
164.19 even 40 820.1.bz.a.183.1 yes 16
205.142 even 40 inner 820.1.by.a.347.1 16
820.347 odd 40 inner 820.1.by.a.347.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
820.1.by.a.347.1 16 205.142 even 40 inner
820.1.by.a.347.1 16 820.347 odd 40 inner
820.1.by.a.423.1 yes 16 1.1 even 1 trivial
820.1.by.a.423.1 yes 16 4.3 odd 2 CM
820.1.bz.a.183.1 yes 16 41.19 odd 40
820.1.bz.a.183.1 yes 16 164.19 even 40
820.1.bz.a.587.1 yes 16 5.2 odd 4
820.1.bz.a.587.1 yes 16 20.7 even 4