Properties

Label 820.1.bz.a.263.1
Level $820$
Weight $1$
Character 820.263
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(7,820)]
 
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, 39]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("820.7");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 820.bz (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 263.1
Root \(0.891007 + 0.453990i\) of defining polynomial
Character \(\chi\) \(=\) 820.263
Dual form 820.1.bz.a.767.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.809017 + 0.587785i) q^{2} +(0.309017 + 0.951057i) q^{4} +(0.453990 + 0.891007i) q^{5} +(-0.309017 + 0.951057i) q^{8} +(-0.707107 - 0.707107i) q^{9} +O(q^{10})\) \(q+(0.809017 + 0.587785i) q^{2} +(0.309017 + 0.951057i) q^{4} +(0.453990 + 0.891007i) q^{5} +(-0.309017 + 0.951057i) q^{8} +(-0.707107 - 0.707107i) q^{9} +(-0.156434 + 0.987688i) q^{10} +(1.10749 - 0.678671i) q^{13} +(-0.809017 + 0.587785i) q^{16} +(-1.04178 + 0.0819895i) q^{17} +(-0.156434 - 0.987688i) q^{18} +(-0.707107 + 0.707107i) q^{20} +(-0.587785 + 0.809017i) q^{25} +(1.29489 + 0.101910i) q^{26} +(-1.20002 - 1.40505i) q^{29} -1.00000 q^{32} +(-0.891007 - 0.546010i) q^{34} +(0.453990 - 0.891007i) q^{36} +(1.69480 + 0.863541i) q^{37} +(-0.987688 + 0.156434i) q^{40} +(0.587785 - 0.809017i) q^{41} +(0.309017 - 0.951057i) q^{45} +(0.891007 - 0.453990i) q^{49} +(-0.951057 + 0.309017i) q^{50} +(0.987688 + 0.843566i) q^{52} +(0.0366318 - 0.465451i) q^{53} +(-0.144974 - 1.84206i) q^{58} +(-1.76007 - 0.278768i) q^{61} +(-0.809017 - 0.587785i) q^{64} +(1.10749 + 0.678671i) q^{65} +(-0.399903 - 0.965451i) q^{68} +(0.891007 - 0.453990i) q^{72} +0.907981i q^{73} +(0.863541 + 1.69480i) q^{74} +(-0.891007 - 0.453990i) q^{80} +1.00000i q^{81} +(0.951057 - 0.309017i) q^{82} +(-0.546010 - 0.891007i) q^{85} +(-0.399903 + 0.652583i) q^{89} +(0.809017 - 0.587785i) q^{90} +(0.101910 + 0.119322i) q^{97} +(0.987688 + 0.156434i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 4 q^{4} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 4 q^{4} + 4 q^{8} - 4 q^{16} + 4 q^{29} - 16 q^{32} - 4 q^{45} - 4 q^{58} - 4 q^{61} - 4 q^{64} - 16 q^{85} + 4 q^{90} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(411\) \(621\) \(657\)
\(\chi(n)\) \(-1\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(3\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(4\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(5\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(6\) 0 0
\(7\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(8\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(9\) −0.707107 0.707107i −0.707107 0.707107i
\(10\) −0.156434 + 0.987688i −0.156434 + 0.987688i
\(11\) 0 0 −0.0784591 0.996917i \(-0.525000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(12\) 0 0
\(13\) 1.10749 0.678671i 1.10749 0.678671i 0.156434 0.987688i \(-0.450000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(17\) −1.04178 + 0.0819895i −1.04178 + 0.0819895i −0.587785 0.809017i \(-0.700000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(18\) −0.156434 0.987688i −0.156434 0.987688i
\(19\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(20\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(24\) 0 0
\(25\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(26\) 1.29489 + 0.101910i 1.29489 + 0.101910i
\(27\) 0 0
\(28\) 0 0
\(29\) −1.20002 1.40505i −1.20002 1.40505i −0.891007 0.453990i \(-0.850000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(32\) −1.00000 −1.00000
\(33\) 0 0
\(34\) −0.891007 0.546010i −0.891007 0.546010i
\(35\) 0 0
\(36\) 0.453990 0.891007i 0.453990 0.891007i
\(37\) 1.69480 + 0.863541i 1.69480 + 0.863541i 0.987688 + 0.156434i \(0.0500000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(41\) 0.587785 0.809017i 0.587785 0.809017i
\(42\) 0 0
\(43\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(44\) 0 0
\(45\) 0.309017 0.951057i 0.309017 0.951057i
\(46\) 0 0
\(47\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(48\) 0 0
\(49\) 0.891007 0.453990i 0.891007 0.453990i
\(50\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(51\) 0 0
\(52\) 0.987688 + 0.843566i 0.987688 + 0.843566i
\(53\) 0.0366318 0.465451i 0.0366318 0.465451i −0.951057 0.309017i \(-0.900000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −0.144974 1.84206i −0.144974 1.84206i
\(59\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(60\) 0 0
\(61\) −1.76007 0.278768i −1.76007 0.278768i −0.809017 0.587785i \(-0.800000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.809017 0.587785i −0.809017 0.587785i
\(65\) 1.10749 + 0.678671i 1.10749 + 0.678671i
\(66\) 0 0
\(67\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(68\) −0.399903 0.965451i −0.399903 0.965451i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.996917 0.0784591i \(-0.0250000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(72\) 0.891007 0.453990i 0.891007 0.453990i
\(73\) 0.907981i 0.907981i 0.891007 + 0.453990i \(0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(74\) 0.863541 + 1.69480i 0.863541 + 1.69480i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(80\) −0.891007 0.453990i −0.891007 0.453990i
\(81\) 1.00000i 1.00000i
\(82\) 0.951057 0.309017i 0.951057 0.309017i
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) −0.546010 0.891007i −0.546010 0.891007i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.399903 + 0.652583i −0.399903 + 0.652583i −0.987688 0.156434i \(-0.950000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(90\) 0.809017 0.587785i 0.809017 0.587785i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.101910 + 0.119322i 0.101910 + 0.119322i 0.809017 0.587785i \(-0.200000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(99\) 0 0
\(100\) −0.951057 0.309017i −0.951057 0.309017i
\(101\) −0.453990 1.89101i −0.453990 1.89101i −0.453990 0.891007i \(-0.650000\pi\)
1.00000i \(-0.5\pi\)
\(102\) 0 0
\(103\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(104\) 0.303221 + 1.26301i 0.303221 + 1.26301i
\(105\) 0 0
\(106\) 0.303221 0.355026i 0.303221 0.355026i
\(107\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(108\) 0 0
\(109\) 0.652583 1.57547i 0.652583 1.57547i −0.156434 0.987688i \(-0.550000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.280582 0.550672i −0.280582 0.550672i 0.707107 0.707107i \(-0.250000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.965451 1.57547i 0.965451 1.57547i
\(117\) −1.26301 0.303221i −1.26301 0.303221i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(122\) −1.26007 1.26007i −1.26007 1.26007i
\(123\) 0 0
\(124\) 0 0
\(125\) −0.987688 0.156434i −0.987688 0.156434i
\(126\) 0 0
\(127\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(128\) −0.309017 0.951057i −0.309017 0.951057i
\(129\) 0 0
\(130\) 0.497066 + 1.20002i 0.497066 + 1.20002i
\(131\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0.243950 1.01612i 0.243950 1.01612i
\(137\) 0.581990 + 1.40505i 0.581990 + 1.40505i 0.891007 + 0.453990i \(0.150000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(138\) 0 0
\(139\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(145\) 0.707107 1.70711i 0.707107 1.70711i
\(146\) −0.533698 + 0.734572i −0.533698 + 0.734572i
\(147\) 0 0
\(148\) −0.297556 + 1.87869i −0.297556 + 1.87869i
\(149\) −0.156434 + 1.98769i −0.156434 + 1.98769i 1.00000i \(0.5\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(150\) 0 0
\(151\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(152\) 0 0
\(153\) 0.794622 + 0.678671i 0.794622 + 0.678671i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.465451 + 1.93874i 0.465451 + 1.93874i 0.309017 + 0.951057i \(0.400000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −0.453990 0.891007i −0.453990 0.891007i
\(161\) 0 0
\(162\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(163\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(164\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(168\) 0 0
\(169\) 0.311951 0.612238i 0.311951 0.612238i
\(170\) 0.0819895 1.04178i 0.0819895 1.04178i
\(171\) 0 0
\(172\) 0 0
\(173\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(179\) 0 0 −0.996917 0.0784591i \(-0.975000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(180\) 1.00000 1.00000
\(181\) −1.15643 0.987688i −1.15643 0.987688i −0.156434 0.987688i \(-0.550000\pi\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.90211i 1.90211i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(192\) 0 0
\(193\) 1.70002 + 0.133795i 1.70002 + 0.133795i 0.891007 0.453990i \(-0.150000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(194\) 0.0123117 + 0.156434i 0.0123117 + 0.156434i
\(195\) 0 0
\(196\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(197\) −0.363271 + 1.11803i −0.363271 + 1.11803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(198\) 0 0
\(199\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(200\) −0.587785 0.809017i −0.587785 0.809017i
\(201\) 0 0
\(202\) 0.744220 1.79671i 0.744220 1.79671i
\(203\) 0 0
\(204\) 0 0
\(205\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(206\) 0 0
\(207\) 0 0
\(208\) −0.497066 + 1.20002i −0.497066 + 1.20002i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(212\) 0.453990 0.108993i 0.453990 0.108993i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 1.45399 0.891007i 1.45399 0.891007i
\(219\) 0 0
\(220\) 0 0
\(221\) −1.09811 + 0.797826i −1.09811 + 0.797826i
\(222\) 0 0
\(223\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(224\) 0 0
\(225\) 0.987688 0.156434i 0.987688 0.156434i
\(226\) 0.0966818 0.610425i 0.0966818 0.610425i
\(227\) 0 0 0.522499 0.852640i \(-0.325000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(228\) 0 0
\(229\) −0.119322 0.101910i −0.119322 0.101910i 0.587785 0.809017i \(-0.300000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.70711 0.707107i 1.70711 0.707107i
\(233\) 1.01612 + 1.65816i 1.01612 + 1.65816i 0.707107 + 0.707107i \(0.250000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(234\) −0.843566 0.987688i −0.843566 0.987688i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(240\) 0 0
\(241\) −0.896802 + 1.76007i −0.896802 + 1.76007i −0.309017 + 0.951057i \(0.600000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(242\) −0.891007 0.453990i −0.891007 0.453990i
\(243\) 0 0
\(244\) −0.278768 1.76007i −0.278768 1.76007i
\(245\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.707107 0.707107i −0.707107 0.707107i
\(251\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.309017 0.951057i 0.309017 0.951057i
\(257\) −0.581990 0.497066i −0.581990 0.497066i 0.309017 0.951057i \(-0.400000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −0.303221 + 1.26301i −0.303221 + 1.26301i
\(261\) −0.144974 + 1.84206i −0.144974 + 1.84206i
\(262\) 0 0
\(263\) 0 0 −0.0784591 0.996917i \(-0.525000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(264\) 0 0
\(265\) 0.431351 0.178671i 0.431351 0.178671i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(270\) 0 0
\(271\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(272\) 0.794622 0.678671i 0.794622 0.678671i
\(273\) 0 0
\(274\) −0.355026 + 1.47879i −0.355026 + 1.47879i
\(275\) 0 0
\(276\) 0 0
\(277\) −1.04744 + 0.533698i −1.04744 + 0.533698i −0.891007 0.453990i \(-0.850000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.794622 + 1.29671i 0.794622 + 1.29671i 0.951057 + 0.309017i \(0.100000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(282\) 0 0
\(283\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(289\) 0.0908861 0.0143949i 0.0908861 0.0143949i
\(290\) 1.57547 0.965451i 1.57547 0.965451i
\(291\) 0 0
\(292\) −0.863541 + 0.280582i −0.863541 + 0.280582i
\(293\) −1.89101 0.453990i −1.89101 0.453990i −0.891007 0.453990i \(-0.850000\pi\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.34500 + 1.34500i −1.34500 + 1.34500i
\(297\) 0 0
\(298\) −1.29489 + 1.51612i −1.29489 + 1.51612i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.550672 1.69480i −0.550672 1.69480i
\(306\) 0.243950 + 1.01612i 0.243950 + 1.01612i
\(307\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(312\) 0 0
\(313\) −0.497066 0.581990i −0.497066 0.581990i 0.453990 0.891007i \(-0.350000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(314\) −0.763007 + 1.84206i −0.763007 + 1.84206i
\(315\) 0 0
\(316\) 0 0
\(317\) 1.29489 1.51612i 1.29489 1.51612i 0.587785 0.809017i \(-0.300000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.156434 0.987688i 0.156434 0.987688i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(325\) −0.101910 + 1.29489i −0.101910 + 1.29489i
\(326\) 0 0
\(327\) 0 0
\(328\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(332\) 0 0
\(333\) −0.587785 1.80902i −0.587785 1.80902i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.97538i 1.97538i −0.156434 0.987688i \(-0.550000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(338\) 0.612238 0.311951i 0.612238 0.311951i
\(339\) 0 0
\(340\) 0.678671 0.794622i 0.678671 0.794622i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −1.30902 0.951057i −1.30902 0.951057i
\(347\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(348\) 0 0
\(349\) 1.16110 + 0.183900i 1.16110 + 0.183900i 0.707107 0.707107i \(-0.250000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.253116 + 1.59811i −0.253116 + 1.59811i 0.453990 + 0.891007i \(0.350000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −0.744220 0.178671i −0.744220 0.178671i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(360\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(361\) 0.891007 0.453990i 0.891007 0.453990i
\(362\) −0.355026 1.47879i −0.355026 1.47879i
\(363\) 0 0
\(364\) 0 0
\(365\) −0.809017 + 0.412215i −0.809017 + 0.412215i
\(366\) 0 0
\(367\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(368\) 0 0
\(369\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(370\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(371\) 0 0
\(372\) 0 0
\(373\) 1.58779 + 0.809017i 1.58779 + 0.809017i 1.00000 \(0\)
0.587785 + 0.809017i \(0.300000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.28258 0.741655i −2.28258 0.741655i
\(378\) 0 0
\(379\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.29671 + 1.10749i 1.29671 + 1.10749i
\(387\) 0 0
\(388\) −0.0819895 + 0.133795i −0.0819895 + 0.133795i
\(389\) −0.0489435 + 0.309017i −0.0489435 + 0.309017i 0.951057 + 0.309017i \(0.100000\pi\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(393\) 0 0
\(394\) −0.951057 + 0.690983i −0.951057 + 0.690983i
\(395\) 0 0
\(396\) 0 0
\(397\) 1.45399 0.891007i 1.45399 0.891007i 0.453990 0.891007i \(-0.350000\pi\)
1.00000 \(0\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000i 1.00000i
\(401\) 1.14412 + 1.14412i 1.14412 + 1.14412i 0.987688 + 0.156434i \(0.0500000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1.65816 1.01612i 1.65816 1.01612i
\(405\) −0.891007 + 0.453990i −0.891007 + 0.453990i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.907981 −0.907981 −0.453990 0.891007i \(-0.650000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(410\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −1.10749 + 0.678671i −1.10749 + 0.678671i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(420\) 0 0
\(421\) −0.0366318 0.465451i −0.0366318 0.465451i −0.987688 0.156434i \(-0.950000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0.431351 + 0.178671i 0.431351 + 0.178671i
\(425\) 0.546010 0.891007i 0.546010 0.891007i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(432\) 0 0
\(433\) −0.309017 + 0.0489435i −0.309017 + 0.0489435i −0.309017 0.951057i \(-0.600000\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.70002 + 0.133795i 1.70002 + 0.133795i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.649448 0.760406i \(-0.725000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(440\) 0 0
\(441\) −0.951057 0.309017i −0.951057 0.309017i
\(442\) −1.35734 −1.35734
\(443\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(444\) 0 0
\(445\) −0.763007 0.0600500i −0.763007 0.0600500i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.0489435 + 0.309017i 0.0489435 + 0.309017i 1.00000 \(0\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(450\) 0.891007 + 0.453990i 0.891007 + 0.453990i
\(451\) 0 0
\(452\) 0.437016 0.437016i 0.437016 0.437016i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.0366318 + 0.152583i −0.0366318 + 0.152583i −0.987688 0.156434i \(-0.950000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(458\) −0.0366318 0.152583i −0.0366318 0.152583i
\(459\) 0 0
\(460\) 0 0
\(461\) 0.280582 0.863541i 0.280582 0.863541i −0.707107 0.707107i \(-0.750000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(462\) 0 0
\(463\) 0 0 0.0784591 0.996917i \(-0.475000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(464\) 1.79671 + 0.431351i 1.79671 + 0.431351i
\(465\) 0 0
\(466\) −0.152583 + 1.93874i −0.152583 + 1.93874i
\(467\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(468\) −0.101910 1.29489i −0.101910 1.29489i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.355026 + 0.303221i −0.355026 + 0.303221i
\(478\) 0 0
\(479\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(480\) 0 0
\(481\) 2.46303 0.193845i 2.46303 0.193845i
\(482\) −1.76007 + 0.896802i −1.76007 + 0.896802i
\(483\) 0 0
\(484\) −0.453990 0.891007i −0.453990 0.891007i
\(485\) −0.0600500 + 0.144974i −0.0600500 + 0.144974i
\(486\) 0 0
\(487\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(488\) 0.809017 1.58779i 0.809017 1.58779i
\(489\) 0 0
\(490\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 1.36535 + 1.36535i 1.36535 + 1.36535i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.522499 0.852640i \(-0.325000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(500\) −0.156434 0.987688i −0.156434 0.987688i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(504\) 0 0
\(505\) 1.47879 1.26301i 1.47879 1.26301i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.10749 1.29671i 1.10749 1.29671i 0.156434 0.987688i \(-0.450000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.809017 0.587785i 0.809017 0.587785i
\(513\) 0 0
\(514\) −0.178671 0.744220i −0.178671 0.744220i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −0.987688 + 0.843566i −0.987688 + 0.843566i
\(521\) 0.119322 0.101910i 0.119322 0.101910i −0.587785 0.809017i \(-0.700000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(522\) −1.20002 + 1.40505i −1.20002 + 1.40505i
\(523\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.951057 0.309017i 0.951057 0.309017i
\(530\) 0.453990 + 0.108993i 0.453990 + 0.108993i
\(531\) 0 0
\(532\) 0 0
\(533\) 0.101910 1.29489i 0.101910 1.29489i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −0.190983 0.587785i −0.190983 0.587785i
\(539\) 0 0
\(540\) 0 0
\(541\) −0.734572 1.44168i −0.734572 1.44168i −0.891007 0.453990i \(-0.850000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.04178 0.0819895i 1.04178 0.0819895i
\(545\) 1.70002 0.133795i 1.70002 0.133795i
\(546\) 0 0
\(547\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(548\) −1.15643 + 0.987688i −1.15643 + 0.987688i
\(549\) 1.04744 + 1.44168i 1.04744 + 1.44168i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −1.16110 0.183900i −1.16110 0.183900i
\(555\) 0 0
\(556\) 0 0
\(557\) −0.0600500 0.763007i −0.0600500 0.763007i −0.951057 0.309017i \(-0.900000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −0.119322 + 1.51612i −0.119322 + 1.51612i
\(563\) 0 0 −0.760406 0.649448i \(-0.775000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(564\) 0 0
\(565\) 0.363271 0.500000i 0.363271 0.500000i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.69480 + 0.863541i 1.69480 + 0.863541i 0.987688 + 0.156434i \(0.0500000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(570\) 0 0
\(571\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(577\) −1.79671 0.744220i −1.79671 0.744220i −0.987688 0.156434i \(-0.950000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(578\) 0.0819895 + 0.0417758i 0.0819895 + 0.0417758i
\(579\) 0 0
\(580\) 1.84206 + 0.144974i 1.84206 + 0.144974i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.863541 0.280582i −0.863541 0.280582i
\(585\) −0.303221 1.26301i −0.303221 1.26301i
\(586\) −1.26301 1.47879i −1.26301 1.47879i
\(587\) 0 0 −0.522499 0.852640i \(-0.675000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.87869 + 0.297556i −1.87869 + 0.297556i
\(593\) −1.04178 + 1.70002i −1.04178 + 1.70002i −0.453990 + 0.891007i \(0.650000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.93874 + 0.465451i −1.93874 + 0.465451i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(600\) 0 0
\(601\) −1.84206 0.763007i −1.84206 0.763007i −0.951057 0.309017i \(-0.900000\pi\)
−0.891007 0.453990i \(-0.850000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.587785 0.809017i −0.587785 0.809017i
\(606\) 0 0
\(607\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0.550672 1.69480i 0.550672 1.69480i
\(611\) 0 0
\(612\) −0.399903 + 0.965451i −0.399903 + 0.965451i
\(613\) −1.44168 1.04744i −1.44168 1.04744i −0.987688 0.156434i \(-0.950000\pi\)
−0.453990 0.891007i \(-0.650000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.951057 + 0.690983i 0.951057 + 0.690983i 0.951057 0.309017i \(-0.100000\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.309017 0.951057i −0.309017 0.951057i
\(626\) −0.0600500 0.763007i −0.0600500 0.763007i
\(627\) 0 0
\(628\) −1.70002 + 1.04178i −1.70002 + 1.04178i
\(629\) −1.83640 0.760661i −1.83640 0.760661i
\(630\) 0 0
\(631\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 1.93874 0.465451i 1.93874 0.465451i
\(635\) 0 0
\(636\) 0 0
\(637\) 0.678671 1.10749i 0.678671 1.10749i
\(638\) 0 0
\(639\) 0 0
\(640\) 0.707107 0.707107i 0.707107 0.707107i
\(641\) 0.465451 + 0.0366318i 0.465451 + 0.0366318i 0.309017 0.951057i \(-0.400000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(642\) 0 0
\(643\) 0 0 −0.522499 0.852640i \(-0.675000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −0.951057 0.309017i −0.951057 0.309017i
\(649\) 0 0
\(650\) −0.843566 + 0.987688i −0.843566 + 0.987688i
\(651\) 0 0
\(652\) 0 0
\(653\) −0.144974 0.0600500i −0.144974 0.0600500i 0.309017 0.951057i \(-0.400000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.00000i 1.00000i
\(657\) 0.642040 0.642040i 0.642040 0.642040i
\(658\) 0 0
\(659\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(660\) 0 0
\(661\) 1.69480 + 0.863541i 1.69480 + 0.863541i 0.987688 + 0.156434i \(0.0500000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.587785 1.80902i 0.587785 1.80902i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.119322 1.51612i −0.119322 1.51612i −0.707107 0.707107i \(-0.750000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(674\) 1.16110 1.59811i 1.16110 1.59811i
\(675\) 0 0
\(676\) 0.678671 + 0.107491i 0.678671 + 0.107491i
\(677\) −0.533698 0.734572i −0.533698 0.734572i 0.453990 0.891007i \(-0.350000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1.01612 0.243950i 1.01612 0.243950i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(684\) 0 0
\(685\) −0.987688 + 1.15643i −0.987688 + 1.15643i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.275319 0.540344i −0.275319 0.540344i
\(690\) 0 0
\(691\) 0 0 −0.522499 0.852640i \(-0.675000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(692\) −0.500000 1.53884i −0.500000 1.53884i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −0.546010 + 0.891007i −0.546010 + 0.891007i
\(698\) 0.831254 + 0.831254i 0.831254 + 0.831254i
\(699\) 0 0
\(700\) 0 0
\(701\) 1.87869 0.610425i 1.87869 0.610425i 0.891007 0.453990i \(-0.150000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −1.14412 + 1.14412i −1.14412 + 1.14412i
\(707\) 0 0
\(708\) 0 0
\(709\) −1.47879 + 1.26301i −1.47879 + 1.26301i −0.587785 + 0.809017i \(0.700000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −0.497066 0.581990i −0.497066 0.581990i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.233445 0.972370i \(-0.575000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(720\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(721\) 0 0
\(722\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(723\) 0 0
\(724\) 0.581990 1.40505i 0.581990 1.40505i
\(725\) 1.84206 0.144974i 1.84206 0.144974i
\(726\) 0 0
\(727\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(728\) 0 0
\(729\) 0.707107 0.707107i 0.707107 0.707107i
\(730\) −0.896802 0.142040i −0.896802 0.142040i
\(731\) 0 0
\(732\) 0 0
\(733\) 0.587785 0.190983i 0.587785 0.190983i 1.00000i \(-0.5\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −0.891007 0.453990i −0.891007 0.453990i
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) −1.80902 + 0.587785i −1.80902 + 0.587785i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(744\) 0 0
\(745\) −1.84206 + 0.763007i −1.84206 + 0.763007i
\(746\) 0.809017 + 1.58779i 0.809017 + 1.58779i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −1.41071 1.94168i −1.41071 1.94168i
\(755\) 0 0
\(756\) 0 0
\(757\) −1.65816 1.01612i −1.65816 1.01612i −0.951057 0.309017i \(-0.900000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.16110 1.59811i 1.16110 1.59811i 0.453990 0.891007i \(-0.350000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −0.243950 + 1.01612i −0.243950 + 1.01612i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.398090 + 1.65816i 0.398090 + 1.65816i
\(773\) 0.431351 1.79671i 0.431351 1.79671i −0.156434 0.987688i \(-0.550000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.144974 + 0.0600500i −0.144974 + 0.0600500i
\(777\) 0 0
\(778\) −0.221232 + 0.221232i −0.221232 + 0.221232i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.453990 + 0.891007i −0.453990 + 0.891007i
\(785\) −1.51612 + 1.29489i −1.51612 + 1.29489i
\(786\) 0 0
\(787\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(788\) −1.17557 −1.17557
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.13846 + 0.885778i −2.13846 + 0.885778i
\(794\) 1.70002 + 0.133795i 1.70002 + 0.133795i
\(795\) 0 0
\(796\) 0 0
\(797\) 1.95106 0.309017i 1.95106 0.309017i 0.951057 0.309017i \(-0.100000\pi\)
1.00000 \(0\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.587785 0.809017i 0.587785 0.809017i
\(801\) 0.744220 0.178671i 0.744220 0.178671i
\(802\) 0.253116 + 1.59811i 0.253116 + 1.59811i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 1.93874 + 0.152583i 1.93874 + 0.152583i
\(809\) −0.0123117 0.156434i −0.0123117 0.156434i 0.987688 0.156434i \(-0.0500000\pi\)
−1.00000 \(\pi\)
\(810\) −0.987688 0.156434i −0.987688 0.156434i
\(811\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −0.734572 0.533698i −0.734572 0.533698i
\(819\) 0 0
\(820\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(821\) 0.907981 0.907981 0.453990 0.891007i \(-0.350000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(822\) 0 0
\(823\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(828\) 0 0
\(829\) 1.34500 + 1.34500i 1.34500 + 1.34500i 0.891007 + 0.453990i \(0.150000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.29489 0.101910i −1.29489 0.101910i
\(833\) −0.891007 + 0.546010i −0.891007 + 0.546010i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(840\) 0 0
\(841\) −0.377666 + 2.38449i −0.377666 + 2.38449i
\(842\) 0.243950 0.398090i 0.243950 0.398090i
\(843\) 0 0
\(844\) 0 0
\(845\) 0.687131 0.687131
\(846\) 0 0
\(847\) 0 0
\(848\) 0.243950 + 0.398090i 0.243950 + 0.398090i
\(849\) 0 0
\(850\) 0.965451 0.399903i 0.965451 0.399903i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.04744 0.533698i −1.04744 0.533698i −0.156434 0.987688i \(-0.550000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(858\) 0 0
\(859\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(864\) 0 0
\(865\) −0.734572 1.44168i −0.734572 1.44168i
\(866\) −0.278768 0.142040i −0.278768 0.142040i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 1.29671 + 1.10749i 1.29671 + 1.10749i
\(873\) 0.0123117 0.156434i 0.0123117 0.156434i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.309017 1.95106i 0.309017 1.95106i 1.00000i \(-0.5\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.39680 0.221232i −1.39680 0.221232i −0.587785 0.809017i \(-0.700000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(882\) −0.587785 0.809017i −0.587785 0.809017i
\(883\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(884\) −1.09811 0.797826i −1.09811 0.797826i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −0.581990 0.497066i −0.581990 0.497066i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −0.142040 + 0.278768i −0.142040 + 0.278768i
\(899\) 0 0
\(900\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(901\) 0.487899i 0.487899i
\(902\) 0 0
\(903\) 0 0
\(904\) 0.610425 0.0966818i 0.610425 0.0966818i
\(905\) 0.355026 1.47879i 0.355026 1.47879i
\(906\) 0 0
\(907\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(908\) 0 0
\(909\) −1.01612 + 1.65816i −1.01612 + 1.65816i
\(910\) 0 0
\(911\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.119322 + 0.101910i −0.119322 + 0.101910i
\(915\) 0 0
\(916\) 0.0600500 0.144974i 0.0600500 0.144974i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.734572 0.533698i 0.734572 0.533698i
\(923\) 0 0
\(924\) 0 0
\(925\) −1.69480 + 0.863541i −1.69480 + 0.863541i
\(926\) 0 0
\(927\) 0 0
\(928\) 1.20002 + 1.40505i 1.20002 + 1.40505i
\(929\) −0.581990 + 1.40505i −0.581990 + 1.40505i 0.309017 + 0.951057i \(0.400000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.26301 + 1.47879i −1.26301 + 1.47879i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0.678671 1.10749i 0.678671 1.10749i
\(937\) 1.79671 + 0.431351i 1.79671 + 0.431351i 0.987688 0.156434i \(-0.0500000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.76007 0.278768i 1.76007 0.278768i 0.809017 0.587785i \(-0.200000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(948\) 0 0
\(949\) 0.616221 + 1.00558i 0.616221 + 1.00558i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(954\) −0.465451 + 0.0366318i −0.465451 + 0.0366318i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(962\) 2.10657 + 1.29091i 2.10657 + 1.29091i
\(963\) 0 0
\(964\) −1.95106 0.309017i −1.95106 0.309017i
\(965\) 0.652583 + 1.57547i 0.652583 + 1.57547i
\(966\) 0 0
\(967\) 0 0 −0.0784591 0.996917i \(-0.525000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(968\) 0.156434 0.987688i 0.156434 0.987688i
\(969\) 0 0
\(970\) −0.133795 + 0.0819895i −0.133795 + 0.0819895i
\(971\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.58779 0.809017i 1.58779 0.809017i
\(977\) 0.431351 + 1.79671i 0.431351 + 1.79671i 0.587785 + 0.809017i \(0.300000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(981\) −1.57547 + 0.652583i −1.57547 + 0.652583i
\(982\) 0 0
\(983\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(984\) 0 0
\(985\) −1.16110 + 0.183900i −1.16110 + 0.183900i
\(986\) 0.302060 + 1.90713i 0.302060 + 1.90713i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −0.243950 0.398090i −0.243950 0.398090i 0.707107 0.707107i \(-0.250000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(998\) 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.bz.a.263.1 yes 16
4.3 odd 2 CM 820.1.bz.a.263.1 yes 16
5.2 odd 4 820.1.by.a.427.1 16
20.7 even 4 820.1.by.a.427.1 16
41.29 odd 40 820.1.by.a.603.1 yes 16
164.111 even 40 820.1.by.a.603.1 yes 16
205.152 even 40 inner 820.1.bz.a.767.1 yes 16
820.767 odd 40 inner 820.1.bz.a.767.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
820.1.by.a.427.1 16 5.2 odd 4
820.1.by.a.427.1 16 20.7 even 4
820.1.by.a.603.1 yes 16 41.29 odd 40
820.1.by.a.603.1 yes 16 164.111 even 40
820.1.bz.a.263.1 yes 16 1.1 even 1 trivial
820.1.bz.a.263.1 yes 16 4.3 odd 2 CM
820.1.bz.a.767.1 yes 16 205.152 even 40 inner
820.1.bz.a.767.1 yes 16 820.767 odd 40 inner