Properties

Label 723.1.bc.a.194.1
Level $723$
Weight $1$
Character 723.194
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 194.1
Root \(-0.453990 + 0.891007i\) of defining polynomial
Character \(\chi\) \(=\) 723.194
Dual form 723.1.bc.a.41.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.156434 + 0.987688i) q^{3} +1.00000i q^{4} +(1.04178 - 0.0819895i) q^{7} +(-0.951057 + 0.309017i) q^{9} +O(q^{10})\) \(q+(0.156434 + 0.987688i) q^{3} +1.00000i q^{4} +(1.04178 - 0.0819895i) q^{7} +(-0.951057 + 0.309017i) q^{9} +(-0.987688 + 0.156434i) q^{12} +(1.65816 - 1.01612i) q^{13} -1.00000 q^{16} +(-1.57547 + 0.652583i) q^{19} +(0.243950 + 1.01612i) q^{21} +(-0.951057 - 0.309017i) q^{25} +(-0.453990 - 0.891007i) q^{27} +(0.0819895 + 1.04178i) q^{28} +(-0.581990 - 0.497066i) q^{31} +(-0.309017 - 0.951057i) q^{36} +(0.133795 + 0.0819895i) q^{37} +(1.26301 + 1.47879i) q^{39} +(0.101910 - 1.29489i) q^{43} +(-0.156434 - 0.987688i) q^{48} +(0.0908861 - 0.0143949i) q^{49} +(1.01612 + 1.65816i) q^{52} +(-0.891007 - 1.45399i) q^{57} +(1.87869 - 0.297556i) q^{61} +(-0.965451 + 0.399903i) q^{63} -1.00000i q^{64} +(-1.04744 + 0.533698i) q^{67} +(1.29671 + 0.794622i) q^{73} +(0.156434 - 0.987688i) q^{75} +(-0.652583 - 1.57547i) q^{76} +(1.59811 + 0.253116i) q^{79} +(0.809017 - 0.587785i) q^{81} +(-1.01612 + 0.243950i) q^{84} +(1.64412 - 1.19453i) q^{91} +(0.399903 - 0.652583i) q^{93} +(0.297556 - 0.0966818i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) 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 \(\left(\mathbb{Z}/723\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(242\)
\(\chi(n)\) \(e\left(\frac{39}{40}\right)\) \(-1\)

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 \(0.250000\pi\)
\(3\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(4\) 1.00000i 1.00000i
\(5\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(6\) 0 0
\(7\) 1.04178 0.0819895i 1.04178 0.0819895i 0.453990 0.891007i \(-0.350000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(8\) 0 0
\(9\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(10\) 0 0
\(11\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(12\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(13\) 1.65816 1.01612i 1.65816 1.01612i 0.707107 0.707107i \(-0.250000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) 0 0 −0.760406 0.649448i \(-0.775000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(18\) 0 0
\(19\) −1.57547 + 0.652583i −1.57547 + 0.652583i −0.987688 0.156434i \(-0.950000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(20\) 0 0
\(21\) 0.243950 + 1.01612i 0.243950 + 1.01612i
\(22\) 0 0
\(23\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(24\) 0 0
\(25\) −0.951057 0.309017i −0.951057 0.309017i
\(26\) 0 0
\(27\) −0.453990 0.891007i −0.453990 0.891007i
\(28\) 0.0819895 + 1.04178i 0.0819895 + 1.04178i
\(29\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(30\) 0 0
\(31\) −0.581990 0.497066i −0.581990 0.497066i 0.309017 0.951057i \(-0.400000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.309017 0.951057i −0.309017 0.951057i
\(37\) 0.133795 + 0.0819895i 0.133795 + 0.0819895i 0.587785 0.809017i \(-0.300000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(38\) 0 0
\(39\) 1.26301 + 1.47879i 1.26301 + 1.47879i
\(40\) 0 0
\(41\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(42\) 0 0
\(43\) 0.101910 1.29489i 0.101910 1.29489i −0.707107 0.707107i \(-0.750000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(48\) −0.156434 0.987688i −0.156434 0.987688i
\(49\) 0.0908861 0.0143949i 0.0908861 0.0143949i
\(50\) 0 0
\(51\) 0 0
\(52\) 1.01612 + 1.65816i 1.01612 + 1.65816i
\(53\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.891007 1.45399i −0.891007 1.45399i
\(58\) 0 0
\(59\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(60\) 0 0
\(61\) 1.87869 0.297556i 1.87869 0.297556i 0.891007 0.453990i \(-0.150000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(62\) 0 0
\(63\) −0.965451 + 0.399903i −0.965451 + 0.399903i
\(64\) 1.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) −1.04744 + 0.533698i −1.04744 + 0.533698i −0.891007 0.453990i \(-0.850000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.996917 0.0784591i \(-0.975000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(72\) 0 0
\(73\) 1.29671 + 0.794622i 1.29671 + 0.794622i 0.987688 0.156434i \(-0.0500000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(74\) 0 0
\(75\) 0.156434 0.987688i 0.156434 0.987688i
\(76\) −0.652583 1.57547i −0.652583 1.57547i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.59811 + 0.253116i 1.59811 + 0.253116i 0.891007 0.453990i \(-0.150000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(80\) 0 0
\(81\) 0.809017 0.587785i 0.809017 0.587785i
\(82\) 0 0
\(83\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(84\) −1.01612 + 0.243950i −1.01612 + 0.243950i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(90\) 0 0
\(91\) 1.64412 1.19453i 1.64412 1.19453i
\(92\) 0 0
\(93\) 0.399903 0.652583i 0.399903 0.652583i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.297556 0.0966818i 0.297556 0.0966818i −0.156434 0.987688i \(-0.550000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0.309017 0.951057i 0.309017 0.951057i
\(101\) 0 0 −0.996917 0.0784591i \(-0.975000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(102\) 0 0
\(103\) −0.303221 0.355026i −0.303221 0.355026i 0.587785 0.809017i \(-0.300000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(108\) 0.891007 0.453990i 0.891007 0.453990i
\(109\) −0.965451 1.57547i −0.965451 1.57547i −0.809017 0.587785i \(-0.800000\pi\)
−0.156434 0.987688i \(-0.550000\pi\)
\(110\) 0 0
\(111\) −0.0600500 + 0.144974i −0.0600500 + 0.144974i
\(112\) −1.04178 + 0.0819895i −1.04178 + 0.0819895i
\(113\) 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
\(117\) −1.26301 + 1.47879i −1.26301 + 1.47879i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0.497066 0.581990i 0.497066 0.581990i
\(125\) 0 0
\(126\) 0 0
\(127\) −0.465451 + 1.93874i −0.465451 + 1.93874i −0.156434 + 0.987688i \(0.550000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(128\) 0 0
\(129\) 1.29489 0.101910i 1.29489 0.101910i
\(130\) 0 0
\(131\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(132\) 0 0
\(133\) −1.58779 + 0.809017i −1.58779 + 0.809017i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(138\) 0 0
\(139\) −1.79671 + 0.431351i −1.79671 + 0.431351i −0.987688 0.156434i \(-0.950000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.951057 0.309017i 0.951057 0.309017i
\(145\) 0 0
\(146\) 0 0
\(147\) 0.0284354 + 0.0875153i 0.0284354 + 0.0875153i
\(148\) −0.0819895 + 0.133795i −0.0819895 + 0.133795i
\(149\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(150\) 0 0
\(151\) 0.183900 0.253116i 0.183900 0.253116i −0.707107 0.707107i \(-0.750000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −1.47879 + 1.26301i −1.47879 + 1.26301i
\(157\) −1.93874 + 0.465451i −1.93874 + 0.465451i −0.951057 + 0.309017i \(0.900000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.0366318 + 0.465451i −0.0366318 + 0.465451i 0.951057 + 0.309017i \(0.100000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.233445 0.972370i \(-0.575000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(168\) 0 0
\(169\) 1.26301 2.47879i 1.26301 2.47879i
\(170\) 0 0
\(171\) 1.29671 1.10749i 1.29671 1.10749i
\(172\) 1.29489 + 0.101910i 1.29489 + 0.101910i
\(173\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(174\) 0 0
\(175\) −1.01612 0.243950i −1.01612 0.243950i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.0784591 0.996917i \(-0.525000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(180\) 0 0
\(181\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(182\) 0 0
\(183\) 0.587785 + 1.80902i 0.587785 + 1.80902i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −0.546010 0.891007i −0.546010 0.891007i
\(190\) 0 0
\(191\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(192\) 0.987688 0.156434i 0.987688 0.156434i
\(193\) −0.309017 1.95106i −0.309017 1.95106i −0.309017 0.951057i \(-0.600000\pi\)
1.00000i \(-0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.0143949 + 0.0908861i 0.0143949 + 0.0908861i
\(197\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(198\) 0 0
\(199\) −0.794622 + 1.29671i −0.794622 + 1.29671i 0.156434 + 0.987688i \(0.450000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(200\) 0 0
\(201\) −0.690983 0.951057i −0.690983 0.951057i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.65816 + 1.01612i −1.65816 + 1.01612i
\(209\) 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
\(217\) −0.647057 0.470114i −0.647057 0.470114i
\(218\) 0 0
\(219\) −0.581990 + 1.40505i −0.581990 + 1.40505i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.76007 + 0.896802i 1.76007 + 0.896802i 0.951057 + 0.309017i \(0.100000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(224\) 0 0
\(225\) 1.00000 1.00000
\(226\) 0 0
\(227\) 0 0 −0.760406 0.649448i \(-0.775000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(228\) 1.45399 0.891007i 1.45399 0.891007i
\(229\) 1.59811 0.253116i 1.59811 0.253116i 0.707107 0.707107i \(-0.250000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(230\) 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
\(237\) 1.61803i 1.61803i
\(238\) 0 0
\(239\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(240\) 0 0
\(241\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(242\) 0 0
\(243\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(244\) 0.297556 + 1.87869i 0.297556 + 1.87869i
\(245\) 0 0
\(246\) 0 0
\(247\) −1.94929 + 2.68296i −1.94929 + 2.68296i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(252\) −0.399903 0.965451i −0.399903 0.965451i
\(253\) 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
\(259\) 0.146106 + 0.0744449i 0.146106 + 0.0744449i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −0.533698 1.04744i −0.533698 1.04744i
\(269\) 0 0 −0.0784591 0.996917i \(-0.525000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(270\) 0 0
\(271\) −1.39680 1.39680i −1.39680 1.39680i −0.809017 0.587785i \(-0.800000\pi\)
−0.587785 0.809017i \(-0.700000\pi\)
\(272\) 0 0
\(273\) 1.43702 + 1.43702i 1.43702 + 1.43702i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −0.610425 1.87869i −0.610425 1.87869i −0.453990 0.891007i \(-0.650000\pi\)
−0.156434 0.987688i \(-0.550000\pi\)
\(278\) 0 0
\(279\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(280\) 0 0
\(281\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(282\) 0 0
\(283\) −1.04178 + 1.70002i −1.04178 + 1.70002i −0.453990 + 0.891007i \(0.650000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(290\) 0 0
\(291\) 0.142040 + 0.278768i 0.142040 + 0.278768i
\(292\) −0.794622 + 1.29671i −0.794622 + 1.29671i
\(293\) 0 0 −0.522499 0.852640i \(-0.675000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(301\) 1.35734i 1.35734i
\(302\) 0 0
\(303\) 0 0
\(304\) 1.57547 0.652583i 1.57547 0.652583i
\(305\) 0 0
\(306\) 0 0
\(307\) 0.453990 + 0.108993i 0.453990 + 0.108993i 0.453990 0.891007i \(-0.350000\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0.303221 0.355026i 0.303221 0.355026i
\(310\) 0 0
\(311\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(312\) 0 0
\(313\) −0.412215 + 0.809017i −0.412215 + 0.809017i 0.587785 + 0.809017i \(0.300000\pi\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −0.253116 + 1.59811i −0.253116 + 1.59811i
\(317\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(325\) −1.89101 + 0.453990i −1.89101 + 0.453990i
\(326\) 0 0
\(327\) 1.40505 1.20002i 1.40505 1.20002i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.04744 1.44168i 1.04744 1.44168i 0.156434 0.987688i \(-0.450000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(332\) 0 0
\(333\) −0.152583 0.0366318i −0.152583 0.0366318i
\(334\) 0 0
\(335\) 0 0
\(336\) −0.243950 1.01612i −0.243950 1.01612i
\(337\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.922621 + 0.221502i −0.922621 + 0.221502i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(348\) 0 0
\(349\) −1.69480 + 0.863541i −1.69480 + 0.863541i −0.707107 + 0.707107i \(0.750000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(350\) 0 0
\(351\) −1.65816 1.01612i −1.65816 1.01612i
\(352\) 0 0
\(353\) 0 0 0.996917 0.0784591i \(-0.0250000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(360\) 0 0
\(361\) 1.34915 1.34915i 1.34915 1.34915i
\(362\) 0 0
\(363\) −0.809017 0.587785i −0.809017 0.587785i
\(364\) 1.19453 + 1.64412i 1.19453 + 1.64412i
\(365\) 0 0
\(366\) 0 0
\(367\) 1.20002 + 0.497066i 1.20002 + 0.497066i 0.891007 0.453990i \(-0.150000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0.652583 + 0.399903i 0.652583 + 0.399903i
\(373\) −0.678671 1.10749i −0.678671 1.10749i −0.987688 0.156434i \(-0.950000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.20002 1.40505i −1.20002 1.40505i −0.891007 0.453990i \(-0.850000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(380\) 0 0
\(381\) −1.98769 0.156434i −1.98769 0.156434i
\(382\) 0 0
\(383\) 0 0 −0.649448 0.760406i \(-0.725000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.303221 + 1.26301i 0.303221 + 1.26301i
\(388\) 0.0966818 + 0.297556i 0.0966818 + 0.297556i
\(389\) 0 0 0.522499 0.852640i \(-0.325000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.119322 0.101910i 0.119322 0.101910i −0.587785 0.809017i \(-0.700000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) 0 0
\(399\) −1.04744 1.44168i −1.04744 1.44168i
\(400\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(401\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(402\) 0 0
\(403\) −1.47011 0.232843i −1.47011 0.232843i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.29671 + 0.794622i 1.29671 + 0.794622i 0.987688 0.156434i \(-0.0500000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.355026 0.303221i 0.355026 0.303221i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −0.707107 1.70711i −0.707107 1.70711i
\(418\) 0 0
\(419\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(420\) 0 0
\(421\) −1.76007 + 0.278768i −1.76007 + 0.278768i −0.951057 0.309017i \(-0.900000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.93278 0.464020i 1.93278 0.464020i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.522499 0.852640i \(-0.325000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(432\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(433\) −1.16110 + 0.183900i −1.16110 + 0.183900i −0.707107 0.707107i \(-0.750000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.57547 0.965451i 1.57547 0.965451i
\(437\) 0 0
\(438\) 0 0
\(439\) −0.144974 + 1.84206i −0.144974 + 1.84206i 0.309017 + 0.951057i \(0.400000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(440\) 0 0
\(441\) −0.0819895 + 0.0417758i −0.0819895 + 0.0417758i
\(442\) 0 0
\(443\) 0 0 −0.649448 0.760406i \(-0.725000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(444\) −0.144974 0.0600500i −0.144974 0.0600500i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.0819895 1.04178i −0.0819895 1.04178i
\(449\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0.278768 + 0.142040i 0.278768 + 0.142040i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.69480 + 0.550672i 1.69480 + 0.550672i 0.987688 0.156434i \(-0.0500000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.233445 0.972370i \(-0.575000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(462\) 0 0
\(463\) 0.144974 0.0600500i 0.144974 0.0600500i −0.309017 0.951057i \(-0.600000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −1.47879 1.26301i −1.47879 1.26301i
\(469\) −1.04744 + 0.641873i −1.04744 + 0.641873i
\(470\) 0 0
\(471\) −0.763007 1.84206i −0.763007 1.84206i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.70002 0.133795i 1.70002 0.133795i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(480\) 0 0
\(481\) 0.305165 0.305165
\(482\) 0 0
\(483\) 0 0
\(484\) −0.707107 0.707107i −0.707107 0.707107i
\(485\) 0 0
\(486\) 0 0
\(487\) 0.221232 1.39680i 0.221232 1.39680i −0.587785 0.809017i \(-0.700000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(488\) 0 0
\(489\) −0.465451 + 0.0366318i −0.465451 + 0.0366318i
\(490\) 0 0
\(491\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0.581990 + 0.497066i 0.581990 + 0.497066i
\(497\) 0 0
\(498\) 0 0
\(499\) −1.51612 1.29489i −1.51612 1.29489i −0.809017 0.587785i \(-0.800000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.233445 0.972370i \(-0.575000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.64585 + 0.859689i 2.64585 + 0.859689i
\(508\) −1.93874 0.465451i −1.93874 0.465451i
\(509\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(510\) 0 0
\(511\) 1.41603 + 0.721502i 1.41603 + 0.721502i
\(512\) 0 0
\(513\) 1.29671 + 1.10749i 1.29671 + 1.10749i
\(514\) 0 0
\(515\) 0 0
\(516\) 0.101910 + 1.29489i 0.101910 + 1.29489i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.649448 0.760406i \(-0.725000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(522\) 0 0
\(523\) −1.26007 + 0.642040i −1.26007 + 0.642040i −0.951057 0.309017i \(-0.900000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(524\) 0 0
\(525\) 0.0819895 1.04178i 0.0819895 1.04178i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.891007 0.453990i −0.891007 0.453990i
\(530\) 0 0
\(531\) 0 0
\(532\) −0.809017 1.58779i −0.809017 1.58779i
\(533\) 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 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(542\) 0 0
\(543\) 0.610425 0.0966818i 0.610425 0.0966818i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.292893 0.707107i −0.292893 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) −1.69480 + 0.863541i −1.69480 + 0.863541i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.68563 + 0.132662i 1.68563 + 0.132662i
\(554\) 0 0
\(555\) 0 0
\(556\) −0.431351 1.79671i −0.431351 1.79671i
\(557\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(558\) 0 0
\(559\) −1.14679 2.25070i −1.14679 2.25070i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.794622 0.678671i 0.794622 0.678671i
\(568\) 0 0
\(569\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(570\) 0 0
\(571\) 0.178671 0.431351i 0.178671 0.431351i −0.809017 0.587785i \(-0.800000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(577\) 0.178671 + 0.744220i 0.178671 + 0.744220i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(578\) 0 0
\(579\) 1.87869 0.610425i 1.87869 0.610425i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.996917 0.0784591i \(-0.0250000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(588\) −0.0875153 + 0.0284354i −0.0875153 + 0.0284354i
\(589\) 1.24129 + 0.403318i 1.24129 + 0.403318i
\(590\) 0 0
\(591\) 0 0
\(592\) −0.133795 0.0819895i −0.133795 0.0819895i
\(593\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.40505 0.581990i −1.40505 0.581990i
\(598\) 0 0
\(599\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(600\) 0 0
\(601\) 0.500000 + 0.363271i 0.500000 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(602\) 0 0
\(603\) 0.831254 0.831254i 0.831254 0.831254i
\(604\) 0.253116 + 0.183900i 0.253116 + 0.183900i
\(605\) 0 0
\(606\) 0 0
\(607\) 0.280582 0.550672i 0.280582 0.550672i −0.707107 0.707107i \(-0.750000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.57547 0.965451i −1.57547 0.965451i −0.987688 0.156434i \(-0.950000\pi\)
−0.587785 0.809017i \(-0.700000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(618\) 0 0
\(619\) 0.303221 1.26301i 0.303221 1.26301i −0.587785 0.809017i \(-0.700000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −1.26301 1.47879i −1.26301 1.47879i
\(625\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(626\) 0 0
\(627\) 0 0
\(628\) −0.465451 1.93874i −0.465451 1.93874i
\(629\) 0 0
\(630\) 0 0
\(631\) 0.453990 + 0.108993i 0.453990 + 0.108993i 0.453990 0.891007i \(-0.350000\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) −0.831254 + 1.14412i −0.831254 + 1.14412i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.136077 0.116221i 0.136077 0.116221i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(642\) 0 0
\(643\) −0.221232 + 1.39680i −0.221232 + 1.39680i 0.587785 + 0.809017i \(0.300000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.363104 0.712633i 0.363104 0.712633i
\(652\) −0.465451 0.0366318i −0.465451 0.0366318i
\(653\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.47879 0.355026i −1.47879 0.355026i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0.119322 + 1.51612i 0.119322 + 1.51612i 0.707107 + 0.707107i \(0.250000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −0.610425 + 1.87869i −0.610425 + 1.87869i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.734572 + 1.44168i 0.734572 + 1.44168i 0.891007 + 0.453990i \(0.150000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(674\) 0 0
\(675\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(676\) 2.47879 + 1.26301i 2.47879 + 1.26301i
\(677\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(678\) 0 0
\(679\) 0.302060 0.125117i 0.302060 0.125117i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(684\) 1.10749 + 1.29671i 1.10749 + 1.29671i
\(685\) 0 0
\(686\) 0 0
\(687\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(688\) −0.101910 + 1.29489i −0.101910 + 1.29489i
\(689\) 0 0
\(690\) 0 0
\(691\) −0.437016 0.437016i −0.437016 0.437016i 0.453990 0.891007i \(-0.350000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.243950 1.01612i 0.243950 1.01612i
\(701\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(702\) 0 0
\(703\) −0.264295 0.0418602i −0.264295 0.0418602i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.40505 + 1.20002i 1.40505 + 1.20002i 0.951057 + 0.309017i \(0.100000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(710\) 0 0
\(711\) −1.59811 + 0.253116i −1.59811 + 0.253116i
\(712\) 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
\(721\) −0.344997 0.344997i −0.344997 0.344997i
\(722\) 0 0
\(723\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(724\) 0.618034 0.618034
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(730\) 0 0
\(731\) 0 0
\(732\) −1.80902 + 0.587785i −1.80902 + 0.587785i
\(733\) −0.951057 + 0.690983i −0.951057 + 0.690983i −0.951057 0.309017i \(-0.900000\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(740\) 0 0
\(741\) −2.95487 1.50558i −2.95487 1.50558i
\(742\) 0 0
\(743\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −0.0123117 0.156434i −0.0123117 0.156434i 0.987688 0.156434i \(-0.0500000\pi\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0.891007 0.546010i 0.891007 0.546010i
\(757\) −0.0366318 0.465451i −0.0366318 0.465451i −0.987688 0.156434i \(-0.950000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(762\) 0 0
\(763\) −1.13496 1.56213i −1.13496 1.56213i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(769\) 1.45399 0.891007i 1.45399 0.891007i 0.453990 0.891007i \(-0.350000\pi\)
1.00000 \(0\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.95106 0.309017i 1.95106 0.309017i
\(773\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(774\) 0 0
\(775\) 0.399903 + 0.652583i 0.399903 + 0.652583i
\(776\) 0 0
\(777\) −0.0506723 + 0.155953i −0.0506723 + 0.155953i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.0908861 + 0.0143949i −0.0908861 + 0.0143949i
\(785\) 0 0
\(786\) 0 0
\(787\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.81283 2.40238i 2.81283 2.40238i
\(794\) 0 0
\(795\) 0 0
\(796\) −1.29671 0.794622i −1.29671 0.794622i
\(797\) 0 0 −0.233445 0.972370i \(-0.575000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0.951057 0.690983i 0.951057 0.690983i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(810\) 0 0
\(811\) −0.707107 0.292893i −0.707107 0.292893i 1.00000i \(-0.5\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(812\) 0 0
\(813\) 1.16110 1.59811i 1.16110 1.59811i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.684467 + 2.10657i 0.684467 + 2.10657i
\(818\) 0 0
\(819\) −1.19453 + 1.64412i −1.19453 + 1.64412i
\(820\) 0 0
\(821\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(822\) 0 0
\(823\) −0.587785 + 1.80902i −0.587785 + 1.80902i 1.00000i \(0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(828\) 0 0
\(829\) 1.34500 0.437016i 1.34500 0.437016i 0.453990 0.891007i \(-0.350000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(830\) 0 0
\(831\) 1.76007 0.896802i 1.76007 0.896802i
\(832\) −1.01612 1.65816i −1.01612 1.65816i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −0.178671 + 0.744220i −0.178671 + 0.744220i
\(838\) 0 0
\(839\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(840\) 0 0
\(841\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(842\) 0 0
\(843\) 0 0
\(844\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(845\) 0 0
\(846\) 0 0
\(847\) −0.678671 + 0.794622i −0.678671 + 0.794622i
\(848\) 0 0
\(849\) −1.84206 0.763007i −1.84206 0.763007i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0.763007 1.84206i 0.763007 1.84206i 0.309017 0.951057i \(-0.400000\pi\)
0.453990 0.891007i \(-0.350000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(858\) 0 0
\(859\) −1.70002 + 0.133795i −1.70002 + 0.133795i −0.891007 0.453990i \(-0.850000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.996917 0.0784591i \(-0.975000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(868\) 0.470114 0.647057i 0.470114 0.647057i
\(869\) 0 0
\(870\) 0 0
\(871\) −1.19453 + 1.94929i −1.19453 + 1.94929i
\(872\) 0 0
\(873\) −0.253116 + 0.183900i −0.253116 + 0.183900i
\(874\) 0 0
\(875\) 0 0
\(876\) −1.40505 0.581990i −1.40505 0.581990i
\(877\) −0.550672 + 1.69480i −0.550672 + 1.69480i 0.156434 + 0.987688i \(0.450000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(882\) 0 0
\(883\) 1.53884 1.11803i 1.53884 1.11803i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(888\) 0 0
\(889\) −0.325939 + 2.05790i −0.325939 + 2.05790i
\(890\) 0 0
\(891\) 0 0
\(892\) −0.896802 + 1.76007i −0.896802 + 1.76007i
\(893\) 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
\(903\) 1.34063 0.212335i 1.34063 0.212335i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0.794622 + 1.29671i 0.794622 + 1.29671i 0.951057 + 0.309017i \(0.100000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(912\) 0.891007 + 1.45399i 0.891007 + 1.45399i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0.253116 + 1.59811i 0.253116 + 1.59811i
\(917\) 0 0
\(918\) 0 0
\(919\) −0.0489435 0.309017i −0.0489435 0.309017i 0.951057 0.309017i \(-0.100000\pi\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −0.0366318 + 0.465451i −0.0366318 + 0.465451i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −0.101910 0.119322i −0.101910 0.119322i
\(926\) 0 0
\(927\) 0.398090 + 0.243950i 0.398090 + 0.243950i
\(928\) 0 0
\(929\) 0 0 0.0784591 0.996917i \(-0.475000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(930\) 0 0
\(931\) −0.133795 + 0.0819895i −0.133795 + 0.0819895i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.412215 + 0.809017i 0.412215 + 0.809017i 1.00000 \(0\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(938\) 0 0
\(939\) −0.863541 0.280582i −0.863541 0.280582i
\(940\) 0 0
\(941\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.760406 0.649448i \(-0.775000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(948\) −1.61803 −1.61803
\(949\) 2.95758 2.95758
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.0647973 0.409114i −0.0647973 0.409114i
\(962\) 0 0
\(963\) 0 0
\(964\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(965\) 0 0
\(966\) 0 0
\(967\) −0.0966818 0.610425i −0.0966818 0.610425i −0.987688 0.156434i \(-0.950000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.996917 0.0784591i \(-0.0250000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(972\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(973\) −1.83640 + 0.596682i −1.83640 + 0.596682i
\(974\) 0 0
\(975\) −0.744220 1.79671i −0.744220 1.79671i
\(976\) −1.87869 + 0.297556i −1.87869 + 0.297556i
\(977\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.40505 + 1.20002i 1.40505 + 1.20002i
\(982\) 0 0
\(983\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −2.68296 1.94929i −2.68296 1.94929i
\(989\) 0 0
\(990\) 0 0
\(991\) 0.533698 + 1.04744i 0.533698 + 1.04744i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(992\) 0 0
\(993\) 1.58779 + 0.809017i 1.58779 + 0.809017i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.29671 0.794622i 1.29671 0.794622i 0.309017 0.951057i \(-0.400000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(998\) 0 0
\(999\) 0.0123117 0.156434i 0.0123117 0.156434i
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.194.1 yes 16
3.2 odd 2 CM 723.1.bc.a.194.1 yes 16
241.41 even 40 inner 723.1.bc.a.41.1 16
723.41 odd 40 inner 723.1.bc.a.41.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
723.1.bc.a.41.1 16 241.41 even 40 inner
723.1.bc.a.41.1 16 723.41 odd 40 inner
723.1.bc.a.194.1 yes 16 1.1 even 1 trivial
723.1.bc.a.194.1 yes 16 3.2 odd 2 CM