Properties

Label 723.1.bc.a.236.1
Level $723$
Weight $1$
Character 723.236
Analytic conductor $0.361$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [723,1,Mod(5,723)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(723, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 23]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("723.5");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 723 = 3 \cdot 241 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 723.bc (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: \(0.360824004134\)
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: \(\Z[a_1, a_2, a_3]\)
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 236.1
Root \(0.987688 + 0.156434i\) of defining polynomial
Character \(\chi\) \(=\) 723.236
Dual form 723.1.bc.a.530.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.453990 + 0.891007i) q^{3} +1.00000i q^{4} +(-1.93874 - 0.465451i) q^{7} +(-0.587785 - 0.809017i) q^{9} +O(q^{10})\) \(q+(-0.453990 + 0.891007i) q^{3} +1.00000i q^{4} +(-1.93874 - 0.465451i) q^{7} +(-0.587785 - 0.809017i) q^{9} +(-0.891007 - 0.453990i) q^{12} +(-0.119322 + 1.51612i) q^{13} -1.00000 q^{16} +(0.0600500 + 0.144974i) q^{19} +(1.29489 - 1.51612i) q^{21} +(-0.587785 + 0.809017i) q^{25} +(0.987688 - 0.156434i) q^{27} +(0.465451 - 1.93874i) q^{28} +(-0.965451 - 1.57547i) q^{31} +(0.809017 - 0.587785i) q^{36} +(0.0366318 + 0.465451i) q^{37} +(-1.29671 - 0.794622i) q^{39} +(0.398090 + 1.65816i) q^{43} +(0.453990 - 0.891007i) q^{48} +(2.65108 + 1.35079i) q^{49} +(-1.51612 - 0.119322i) q^{52} +(-0.156434 - 0.0123117i) q^{57} +(1.04744 + 0.533698i) q^{61} +(0.763007 + 1.84206i) q^{63} -1.00000i q^{64} +(0.297556 + 1.87869i) q^{67} +(0.0819895 + 1.04178i) q^{73} +(-0.453990 - 0.891007i) q^{75} +(-0.144974 + 0.0600500i) q^{76} +(-0.550672 + 0.280582i) q^{79} +(-0.309017 + 0.951057i) q^{81} +(1.51612 + 1.29489i) q^{84} +(0.937016 - 2.88384i) q^{91} +(1.84206 - 0.144974i) q^{93} +(-0.533698 - 0.734572i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{16} - 4 q^{28} - 4 q^{31} + 4 q^{36} + 4 q^{39} + 4 q^{43} + 4 q^{49} - 4 q^{52} - 4 q^{63} - 4 q^{73} - 4 q^{76} + 4 q^{81} + 4 q^{84} + 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/723\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(242\)
\(\chi(n)\) \(e\left(\frac{3}{40}\right)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) −0.453990 + 0.891007i −0.453990 + 0.891007i
\(4\) 1.00000i 1.00000i
\(5\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(6\) 0 0
\(7\) −1.93874 0.465451i −1.93874 0.465451i −0.987688 0.156434i \(-0.950000\pi\)
−0.951057 0.309017i \(-0.900000\pi\)
\(8\) 0 0
\(9\) −0.587785 0.809017i −0.587785 0.809017i
\(10\) 0 0
\(11\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(12\) −0.891007 0.453990i −0.891007 0.453990i
\(13\) −0.119322 + 1.51612i −0.119322 + 1.51612i 0.587785 + 0.809017i \(0.300000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) 0 0 −0.522499 0.852640i \(-0.675000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(18\) 0 0
\(19\) 0.0600500 + 0.144974i 0.0600500 + 0.144974i 0.951057 0.309017i \(-0.100000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(20\) 0 0
\(21\) 1.29489 1.51612i 1.29489 1.51612i
\(22\) 0 0
\(23\) 0 0 −0.649448 0.760406i \(-0.725000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(24\) 0 0
\(25\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(26\) 0 0
\(27\) 0.987688 0.156434i 0.987688 0.156434i
\(28\) 0.465451 1.93874i 0.465451 1.93874i
\(29\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(30\) 0 0
\(31\) −0.965451 1.57547i −0.965451 1.57547i −0.809017 0.587785i \(-0.800000\pi\)
−0.156434 0.987688i \(-0.550000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.809017 0.587785i 0.809017 0.587785i
\(37\) 0.0366318 + 0.465451i 0.0366318 + 0.465451i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(38\) 0 0
\(39\) −1.29671 0.794622i −1.29671 0.794622i
\(40\) 0 0
\(41\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(42\) 0 0
\(43\) 0.398090 + 1.65816i 0.398090 + 1.65816i 0.707107 + 0.707107i \(0.250000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(48\) 0.453990 0.891007i 0.453990 0.891007i
\(49\) 2.65108 + 1.35079i 2.65108 + 1.35079i
\(50\) 0 0
\(51\) 0 0
\(52\) −1.51612 0.119322i −1.51612 0.119322i
\(53\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.156434 0.0123117i −0.156434 0.0123117i
\(58\) 0 0
\(59\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(60\) 0 0
\(61\) 1.04744 + 0.533698i 1.04744 + 0.533698i 0.891007 0.453990i \(-0.150000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(62\) 0 0
\(63\) 0.763007 + 1.84206i 0.763007 + 1.84206i
\(64\) 1.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.297556 + 1.87869i 0.297556 + 1.87869i 0.453990 + 0.891007i \(0.350000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(72\) 0 0
\(73\) 0.0819895 + 1.04178i 0.0819895 + 1.04178i 0.891007 + 0.453990i \(0.150000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(74\) 0 0
\(75\) −0.453990 0.891007i −0.453990 0.891007i
\(76\) −0.144974 + 0.0600500i −0.144974 + 0.0600500i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.550672 + 0.280582i −0.550672 + 0.280582i −0.707107 0.707107i \(-0.750000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(80\) 0 0
\(81\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(82\) 0 0
\(83\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(84\) 1.51612 + 1.29489i 1.51612 + 1.29489i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(90\) 0 0
\(91\) 0.937016 2.88384i 0.937016 2.88384i
\(92\) 0 0
\(93\) 1.84206 0.144974i 1.84206 0.144974i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.533698 0.734572i −0.533698 0.734572i 0.453990 0.891007i \(-0.350000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −0.809017 0.587785i −0.809017 0.587785i
\(101\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(102\) 0 0
\(103\) −1.10749 0.678671i −1.10749 0.678671i −0.156434 0.987688i \(-0.550000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(108\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(109\) 0.763007 + 0.0600500i 0.763007 + 0.0600500i 0.453990 0.891007i \(-0.350000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(110\) 0 0
\(111\) −0.431351 0.178671i −0.431351 0.178671i
\(112\) 1.93874 + 0.465451i 1.93874 + 0.465451i
\(113\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.29671 0.794622i 1.29671 0.794622i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.707107 0.707107i 0.707107 0.707107i
\(122\) 0 0
\(123\) 0 0
\(124\) 1.57547 0.965451i 1.57547 0.965451i
\(125\) 0 0
\(126\) 0 0
\(127\) 1.26301 + 1.47879i 1.26301 + 1.47879i 0.809017 + 0.587785i \(0.200000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(128\) 0 0
\(129\) −1.65816 0.398090i −1.65816 0.398090i
\(130\) 0 0
\(131\) 0 0 −0.0784591 0.996917i \(-0.525000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(132\) 0 0
\(133\) −0.0489435 0.309017i −0.0489435 0.309017i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.649448 0.760406i \(-0.725000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(138\) 0 0
\(139\) −0.581990 0.497066i −0.581990 0.497066i 0.309017 0.951057i \(-0.400000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(145\) 0 0
\(146\) 0 0
\(147\) −2.40713 + 1.74888i −2.40713 + 1.74888i
\(148\) −0.465451 + 0.0366318i −0.465451 + 0.0366318i
\(149\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(150\) 0 0
\(151\) 0.863541 0.280582i 0.863541 0.280582i 0.156434 0.987688i \(-0.450000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0.794622 1.29671i 0.794622 1.29671i
\(157\) −1.47879 1.26301i −1.47879 1.26301i −0.891007 0.453990i \(-0.850000\pi\)
−0.587785 0.809017i \(-0.700000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.303221 1.26301i −0.303221 1.26301i −0.891007 0.453990i \(-0.850000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(168\) 0 0
\(169\) −1.29671 0.205378i −1.29671 0.205378i
\(170\) 0 0
\(171\) 0.0819895 0.133795i 0.0819895 0.133795i
\(172\) −1.65816 + 0.398090i −1.65816 + 0.398090i
\(173\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(174\) 0 0
\(175\) 1.51612 1.29489i 1.51612 1.29489i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(180\) 0 0
\(181\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(182\) 0 0
\(183\) −0.951057 + 0.690983i −0.951057 + 0.690983i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.98769 0.156434i −1.98769 0.156434i
\(190\) 0 0
\(191\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(192\) 0.891007 + 0.453990i 0.891007 + 0.453990i
\(193\) 0.809017 1.58779i 0.809017 1.58779i 1.00000i \(-0.5\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.35079 + 2.65108i −1.35079 + 2.65108i
\(197\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(198\) 0 0
\(199\) −1.04178 + 0.0819895i −1.04178 + 0.0819895i −0.587785 0.809017i \(-0.700000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(200\) 0 0
\(201\) −1.80902 0.587785i −1.80902 0.587785i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.119322 1.51612i 0.119322 1.51612i
\(209\) 0 0
\(210\) 0 0
\(211\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.13846 + 3.50381i 1.13846 + 3.50381i
\(218\) 0 0
\(219\) −0.965451 0.399903i −0.965451 0.399903i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.278768 1.76007i 0.278768 1.76007i −0.309017 0.951057i \(-0.600000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(224\) 0 0
\(225\) 1.00000 1.00000
\(226\) 0 0
\(227\) 0 0 −0.522499 0.852640i \(-0.675000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(228\) 0.0123117 0.156434i 0.0123117 0.156434i
\(229\) −0.550672 0.280582i −0.550672 0.280582i 0.156434 0.987688i \(-0.450000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.618034i 0.618034i
\(238\) 0 0
\(239\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(240\) 0 0
\(241\) −0.453990 + 0.891007i −0.453990 + 0.891007i
\(242\) 0 0
\(243\) −0.707107 0.707107i −0.707107 0.707107i
\(244\) −0.533698 + 1.04744i −0.533698 + 1.04744i
\(245\) 0 0
\(246\) 0 0
\(247\) −0.226963 + 0.0737448i −0.226963 + 0.0737448i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(252\) −1.84206 + 0.763007i −1.84206 + 0.763007i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0.145625 0.919442i 0.145625 0.919442i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.87869 + 0.297556i −1.87869 + 0.297556i
\(269\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(270\) 0 0
\(271\) 1.26007 + 1.26007i 1.26007 + 1.26007i 0.951057 + 0.309017i \(0.100000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(272\) 0 0
\(273\) 2.14412 + 2.14412i 2.14412 + 2.14412i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.44168 1.04744i 1.44168 1.04744i 0.453990 0.891007i \(-0.350000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(278\) 0 0
\(279\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(280\) 0 0
\(281\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(282\) 0 0
\(283\) 1.93874 0.152583i 1.93874 0.152583i 0.951057 0.309017i \(-0.100000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.453990 + 0.891007i −0.453990 + 0.891007i
\(290\) 0 0
\(291\) 0.896802 0.142040i 0.896802 0.142040i
\(292\) −1.04178 + 0.0819895i −1.04178 + 0.0819895i
\(293\) 0 0 −0.996917 0.0784591i \(-0.975000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0.891007 0.453990i 0.891007 0.453990i
\(301\) 3.40005i 3.40005i
\(302\) 0 0
\(303\) 0 0
\(304\) −0.0600500 0.144974i −0.0600500 0.144974i
\(305\) 0 0
\(306\) 0 0
\(307\) −0.987688 + 0.843566i −0.987688 + 0.843566i −0.987688 0.156434i \(-0.950000\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 1.10749 0.678671i 1.10749 0.678671i
\(310\) 0 0
\(311\) 0 0 0.522499 0.852640i \(-0.325000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(312\) 0 0
\(313\) −1.95106 0.309017i −1.95106 0.309017i −0.951057 0.309017i \(-0.900000\pi\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −0.280582 0.550672i −0.280582 0.550672i
\(317\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.951057 0.309017i −0.951057 0.309017i
\(325\) −1.15643 0.987688i −1.15643 0.987688i
\(326\) 0 0
\(327\) −0.399903 + 0.652583i −0.399903 + 0.652583i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.297556 + 0.0966818i −0.297556 + 0.0966818i −0.453990 0.891007i \(-0.650000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(332\) 0 0
\(333\) 0.355026 0.303221i 0.355026 0.303221i
\(334\) 0 0
\(335\) 0 0
\(336\) −1.29489 + 1.51612i −1.29489 + 1.51612i
\(337\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −2.99492 2.55790i −2.99492 2.55790i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(348\) 0 0
\(349\) −0.183900 1.16110i −0.183900 1.16110i −0.891007 0.453990i \(-0.850000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(350\) 0 0
\(351\) 0.119322 + 1.51612i 0.119322 + 1.51612i
\(352\) 0 0
\(353\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(360\) 0 0
\(361\) 0.689695 0.689695i 0.689695 0.689695i
\(362\) 0 0
\(363\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(364\) 2.88384 + 0.937016i 2.88384 + 0.937016i
\(365\) 0 0
\(366\) 0 0
\(367\) −0.652583 + 1.57547i −0.652583 + 1.57547i 0.156434 + 0.987688i \(0.450000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0.144974 + 1.84206i 0.144974 + 1.84206i
\(373\) −1.70002 0.133795i −1.70002 0.133795i −0.809017 0.587785i \(-0.800000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.652583 + 0.399903i 0.652583 + 0.399903i 0.809017 0.587785i \(-0.200000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(380\) 0 0
\(381\) −1.89101 + 0.453990i −1.89101 + 0.453990i
\(382\) 0 0
\(383\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.10749 1.29671i 1.10749 1.29671i
\(388\) 0.734572 0.533698i 0.734572 0.533698i
\(389\) 0 0 0.996917 0.0784591i \(-0.0250000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.243950 0.398090i 0.243950 0.398090i −0.707107 0.707107i \(-0.750000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(398\) 0 0
\(399\) 0.297556 + 0.0966818i 0.297556 + 0.0966818i
\(400\) 0.587785 0.809017i 0.587785 0.809017i
\(401\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(402\) 0 0
\(403\) 2.50381 1.27576i 2.50381 1.27576i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.0819895 + 1.04178i 0.0819895 + 1.04178i 0.891007 + 0.453990i \(0.150000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.678671 1.10749i 0.678671 1.10749i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.707107 0.292893i 0.707107 0.292893i
\(418\) 0 0
\(419\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(420\) 0 0
\(421\) −0.278768 0.142040i −0.278768 0.142040i 0.309017 0.951057i \(-0.400000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.78231 1.52224i −1.78231 1.52224i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.996917 0.0784591i \(-0.0250000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(432\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(433\) 1.69480 + 0.863541i 1.69480 + 0.863541i 0.987688 + 0.156434i \(0.0500000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.0600500 + 0.763007i −0.0600500 + 0.763007i
\(437\) 0 0
\(438\) 0 0
\(439\) 0.178671 + 0.744220i 0.178671 + 0.744220i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(440\) 0 0
\(441\) −0.465451 2.93874i −0.465451 2.93874i
\(442\) 0 0
\(443\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(444\) 0.178671 0.431351i 0.178671 0.431351i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.465451 + 1.93874i −0.465451 + 1.93874i
\(449\) 0 0 0.0784591 0.996917i \(-0.475000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −0.142040 + 0.896802i −0.142040 + 0.896802i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.183900 0.253116i 0.183900 0.253116i −0.707107 0.707107i \(-0.750000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(462\) 0 0
\(463\) −0.178671 0.431351i −0.178671 0.431351i 0.809017 0.587785i \(-0.200000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0.794622 + 1.29671i 0.794622 + 1.29671i
\(469\) 0.297556 3.78081i 0.297556 3.78081i
\(470\) 0 0
\(471\) 1.79671 0.744220i 1.79671 0.744220i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −0.152583 0.0366318i −0.152583 0.0366318i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(480\) 0 0
\(481\) −0.710053 −0.710053
\(482\) 0 0
\(483\) 0 0
\(484\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(485\) 0 0
\(486\) 0 0
\(487\) 0.642040 + 1.26007i 0.642040 + 1.26007i 0.951057 + 0.309017i \(0.100000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(488\) 0 0
\(489\) 1.26301 + 0.303221i 1.26301 + 0.303221i
\(490\) 0 0
\(491\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0.965451 + 1.57547i 0.965451 + 1.57547i
\(497\) 0 0
\(498\) 0 0
\(499\) 1.01612 + 1.65816i 1.01612 + 1.65816i 0.707107 + 0.707107i \(0.250000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(500\) 0 0
\(501\) 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
\(505\) 0 0
\(506\) 0 0
\(507\) 0.771685 1.06213i 0.771685 1.06213i
\(508\) −1.47879 + 1.26301i −1.47879 + 1.26301i
\(509\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(510\) 0 0
\(511\) 0.325939 2.05790i 0.325939 2.05790i
\(512\) 0 0
\(513\) 0.0819895 + 0.133795i 0.0819895 + 0.133795i
\(514\) 0 0
\(515\) 0 0
\(516\) 0.398090 1.65816i 0.398090 1.65816i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(522\) 0 0
\(523\) 0.221232 + 1.39680i 0.221232 + 1.39680i 0.809017 + 0.587785i \(0.200000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(524\) 0 0
\(525\) 0.465451 + 1.93874i 0.465451 + 1.93874i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.156434 + 0.987688i −0.156434 + 0.987688i
\(530\) 0 0
\(531\) 0 0
\(532\) 0.309017 0.0489435i 0.309017 0.0489435i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(542\) 0 0
\(543\) −1.44168 0.734572i −1.44168 0.734572i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.70711 + 0.707107i −1.70711 + 0.707107i −0.707107 + 0.707107i \(0.750000\pi\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) −0.183900 1.16110i −0.183900 1.16110i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.19821 0.287665i 1.19821 0.287665i
\(554\) 0 0
\(555\) 0 0
\(556\) 0.497066 0.581990i 0.497066 0.581990i
\(557\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(558\) 0 0
\(559\) −2.56148 + 0.405699i −2.56148 + 0.405699i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.04178 1.70002i 1.04178 1.70002i
\(568\) 0 0
\(569\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(570\) 0 0
\(571\) 1.20002 + 0.497066i 1.20002 + 0.497066i 0.891007 0.453990i \(-0.150000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(577\) 1.20002 1.40505i 1.20002 1.40505i 0.309017 0.951057i \(-0.400000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(578\) 0 0
\(579\) 1.04744 + 1.44168i 1.04744 + 1.44168i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(588\) −1.74888 2.40713i −1.74888 2.40713i
\(589\) 0.170427 0.234572i 0.170427 0.234572i
\(590\) 0 0
\(591\) 0 0
\(592\) −0.0366318 0.465451i −0.0366318 0.465451i
\(593\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.399903 0.965451i 0.399903 0.965451i
\(598\) 0 0
\(599\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(600\) 0 0
\(601\) 0.500000 + 1.53884i 0.500000 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(602\) 0 0
\(603\) 1.34500 1.34500i 1.34500 1.34500i
\(604\) 0.280582 + 0.863541i 0.280582 + 0.863541i
\(605\) 0 0
\(606\) 0 0
\(607\) 1.59811 + 0.253116i 1.59811 + 0.253116i 0.891007 0.453990i \(-0.150000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0.0600500 + 0.763007i 0.0600500 + 0.763007i 0.951057 + 0.309017i \(0.100000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(618\) 0 0
\(619\) 1.10749 + 1.29671i 1.10749 + 1.29671i 0.951057 + 0.309017i \(0.100000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 1.29671 + 0.794622i 1.29671 + 0.794622i
\(625\) −0.309017 0.951057i −0.309017 0.951057i
\(626\) 0 0
\(627\) 0 0
\(628\) 1.26301 1.47879i 1.26301 1.47879i
\(629\) 0 0
\(630\) 0 0
\(631\) −0.987688 + 0.843566i −0.987688 + 0.843566i −0.987688 0.156434i \(-0.950000\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) −1.34500 + 0.437016i −1.34500 + 0.437016i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.36430 + 3.85819i −2.36430 + 3.85819i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(642\) 0 0
\(643\) −0.642040 1.26007i −0.642040 1.26007i −0.951057 0.309017i \(-0.900000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −3.63877 0.576324i −3.63877 0.576324i
\(652\) 1.26301 0.303221i 1.26301 0.303221i
\(653\) 0 0 0.522499 0.852640i \(-0.325000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.794622 0.678671i 0.794622 0.678671i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0.243950 1.01612i 0.243950 1.01612i −0.707107 0.707107i \(-0.750000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1.44168 + 1.04744i 1.44168 + 1.04744i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.610425 0.0966818i 0.610425 0.0966818i 0.156434 0.987688i \(-0.450000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(674\) 0 0
\(675\) −0.453990 + 0.891007i −0.453990 + 0.891007i
\(676\) 0.205378 1.29671i 0.205378 1.29671i
\(677\) 0 0 0.0784591 0.996917i \(-0.475000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(678\) 0 0
\(679\) 0.692796 + 1.67256i 0.692796 + 1.67256i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(684\) 0.133795 + 0.0819895i 0.133795 + 0.0819895i
\(685\) 0 0
\(686\) 0 0
\(687\) 0.500000 0.363271i 0.500000 0.363271i
\(688\) −0.398090 1.65816i −0.398090 1.65816i
\(689\) 0 0
\(690\) 0 0
\(691\) −1.14412 1.14412i −1.14412 1.14412i −0.987688 0.156434i \(-0.950000\pi\)
−0.156434 0.987688i \(-0.550000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.29489 + 1.51612i 1.29489 + 1.51612i
\(701\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(702\) 0 0
\(703\) −0.0652784 + 0.0332610i −0.0652784 + 0.0332610i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.399903 0.652583i −0.399903 0.652583i 0.587785 0.809017i \(-0.300000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(710\) 0 0
\(711\) 0.550672 + 0.280582i 0.550672 + 0.280582i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 1.83125 + 1.83125i 1.83125 + 1.83125i
\(722\) 0 0
\(723\) −0.587785 0.809017i −0.587785 0.809017i
\(724\) −1.61803 −1.61803
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0.951057 0.309017i 0.951057 0.309017i
\(730\) 0 0
\(731\) 0 0
\(732\) −0.690983 0.951057i −0.690983 0.951057i
\(733\) −0.587785 + 1.80902i −0.587785 + 1.80902i 1.00000i \(0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(740\) 0 0
\(741\) 0.0373320 0.235705i 0.0373320 0.235705i
\(742\) 0 0
\(743\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −0.108993 + 0.453990i −0.108993 + 0.453990i 0.891007 + 0.453990i \(0.150000\pi\)
−1.00000 \(1.00000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0.156434 1.98769i 0.156434 1.98769i
\(757\) −0.303221 + 1.26301i −0.303221 + 1.26301i 0.587785 + 0.809017i \(0.300000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(762\) 0 0
\(763\) −1.45133 0.471565i −1.45133 0.471565i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.453990 + 0.891007i −0.453990 + 0.891007i
\(769\) 0.0123117 0.156434i 0.0123117 0.156434i −0.987688 0.156434i \(-0.950000\pi\)
1.00000 \(0\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.58779 + 0.809017i 1.58779 + 0.809017i
\(773\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(774\) 0 0
\(775\) 1.84206 + 0.144974i 1.84206 + 0.144974i
\(776\) 0 0
\(777\) 0.753116 + 0.547171i 0.753116 + 0.547171i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −2.65108 1.35079i −2.65108 1.35079i
\(785\) 0 0
\(786\) 0 0
\(787\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.934134 + 1.52437i −0.934134 + 1.52437i
\(794\) 0 0
\(795\) 0 0
\(796\) −0.0819895 1.04178i −0.0819895 1.04178i
\(797\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0.587785 1.80902i 0.587785 1.80902i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.522499 0.852640i \(-0.325000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(810\) 0 0
\(811\) 0.707107 1.70711i 0.707107 1.70711i 1.00000i \(-0.5\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(812\) 0 0
\(813\) −1.69480 + 0.550672i −1.69480 + 0.550672i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −0.216484 + 0.157285i −0.216484 + 0.157285i
\(818\) 0 0
\(819\) −2.88384 + 0.937016i −2.88384 + 0.937016i
\(820\) 0 0
\(821\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(822\) 0 0
\(823\) 0.951057 + 0.690983i 0.951057 + 0.690983i 0.951057 0.309017i \(-0.100000\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.649448 0.760406i \(-0.725000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(828\) 0 0
\(829\) −0.831254 1.14412i −0.831254 1.14412i −0.987688 0.156434i \(-0.950000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(830\) 0 0
\(831\) 0.278768 + 1.76007i 0.278768 + 1.76007i
\(832\) 1.51612 + 0.119322i 1.51612 + 0.119322i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.20002 1.40505i −1.20002 1.40505i
\(838\) 0 0
\(839\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(840\) 0 0
\(841\) −0.951057 0.309017i −0.951057 0.309017i
\(842\) 0 0
\(843\) 0 0
\(844\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(845\) 0 0
\(846\) 0 0
\(847\) −1.70002 + 1.04178i −1.70002 + 1.04178i
\(848\) 0 0
\(849\) −0.744220 + 1.79671i −0.744220 + 1.79671i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.79671 0.744220i −1.79671 0.744220i −0.987688 0.156434i \(-0.950000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(858\) 0 0
\(859\) 0.152583 + 0.0366318i 0.152583 + 0.0366318i 0.309017 0.951057i \(-0.400000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.587785 0.809017i −0.587785 0.809017i
\(868\) −3.50381 + 1.13846i −3.50381 + 1.13846i
\(869\) 0 0
\(870\) 0 0
\(871\) −2.88384 + 0.226963i −2.88384 + 0.226963i
\(872\) 0 0
\(873\) −0.280582 + 0.863541i −0.280582 + 0.863541i
\(874\) 0 0
\(875\) 0 0
\(876\) 0.399903 0.965451i 0.399903 0.965451i
\(877\) 0.253116 + 0.183900i 0.253116 + 0.183900i 0.707107 0.707107i \(-0.250000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(882\) 0 0
\(883\) −0.363271 + 1.11803i −0.363271 + 1.11803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(888\) 0 0
\(889\) −1.76034 3.45487i −1.76034 3.45487i
\(890\) 0 0
\(891\) 0 0
\(892\) 1.76007 + 0.278768i 1.76007 + 0.278768i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 1.00000i 1.00000i
\(901\) 0 0
\(902\) 0 0
\(903\) 3.02946 + 1.54359i 3.02946 + 1.54359i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.04178 + 0.0819895i 1.04178 + 0.0819895i 0.587785 0.809017i \(-0.300000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(912\) 0.156434 + 0.0123117i 0.156434 + 0.0123117i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0.280582 0.550672i 0.280582 0.550672i
\(917\) 0 0
\(918\) 0 0
\(919\) −0.412215 + 0.809017i −0.412215 + 0.809017i 0.587785 + 0.809017i \(0.300000\pi\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −0.303221 1.26301i −0.303221 1.26301i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −0.398090 0.243950i −0.398090 0.243950i
\(926\) 0 0
\(927\) 0.101910 + 1.29489i 0.101910 + 1.29489i
\(928\) 0 0
\(929\) 0 0 −0.233445 0.972370i \(-0.575000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(930\) 0 0
\(931\) −0.0366318 + 0.465451i −0.0366318 + 0.465451i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.95106 0.309017i 1.95106 0.309017i 0.951057 0.309017i \(-0.100000\pi\)
1.00000 \(0\)
\(938\) 0 0
\(939\) 1.16110 1.59811i 1.16110 1.59811i
\(940\) 0 0
\(941\) 0 0 −0.649448 0.760406i \(-0.725000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.522499 0.852640i \(-0.675000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(948\) 0.618034 0.618034
\(949\) −1.58924 −1.58924
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −1.09603 + 2.15108i −1.09603 + 2.15108i
\(962\) 0 0
\(963\) 0 0
\(964\) −0.891007 0.453990i −0.891007 0.453990i
\(965\) 0 0
\(966\) 0 0
\(967\) −0.734572 + 1.44168i −0.734572 + 1.44168i 0.156434 + 0.987688i \(0.450000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(972\) 0.707107 0.707107i 0.707107 0.707107i
\(973\) 0.896969 + 1.23457i 0.896969 + 1.23457i
\(974\) 0 0
\(975\) 1.40505 0.581990i 1.40505 0.581990i
\(976\) −1.04744 0.533698i −1.04744 0.533698i
\(977\) 0 0 0.0784591 0.996917i \(-0.475000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −0.399903 0.652583i −0.399903 0.652583i
\(982\) 0 0
\(983\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −0.0737448 0.226963i −0.0737448 0.226963i
\(989\) 0 0
\(990\) 0 0
\(991\) 1.87869 0.297556i 1.87869 0.297556i 0.891007 0.453990i \(-0.150000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(992\) 0 0
\(993\) 0.0489435 0.309017i 0.0489435 0.309017i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.0819895 1.04178i 0.0819895 1.04178i −0.809017 0.587785i \(-0.800000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(998\) 0 0
\(999\) 0.108993 + 0.453990i 0.108993 + 0.453990i
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 723.1.bc.a.236.1 16
3.2 odd 2 CM 723.1.bc.a.236.1 16
241.48 even 40 inner 723.1.bc.a.530.1 yes 16
723.530 odd 40 inner 723.1.bc.a.530.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
723.1.bc.a.236.1 16 1.1 even 1 trivial
723.1.bc.a.236.1 16 3.2 odd 2 CM
723.1.bc.a.530.1 yes 16 241.48 even 40 inner
723.1.bc.a.530.1 yes 16 723.530 odd 40 inner