Properties

Label 2576.1.cn.b.1371.1
Level $2576$
Weight $1$
Character 2576.1371
Analytic conductor $1.286$
Analytic rank $0$
Dimension $20$
Projective image $D_{44}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2576,1,Mod(83,2576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2576, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([22, 33, 22, 42]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2576.83");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2576.cn (of order \(44\), degree \(20\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.28559147254\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{44})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{44}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{44} - \cdots)\)

Embedding invariants

Embedding label 1371.1
Root \(0.755750 + 0.654861i\) of defining polynomial
Character \(\chi\) \(=\) 2576.1371
Dual form 2576.1.cn.b.419.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.755750 - 0.654861i) q^{2} +(0.142315 + 0.989821i) q^{4} +(-0.909632 - 0.415415i) q^{7} +(0.540641 - 0.841254i) q^{8} +(-0.989821 - 0.142315i) q^{9} +O(q^{10})\) \(q+(-0.755750 - 0.654861i) q^{2} +(0.142315 + 0.989821i) q^{4} +(-0.909632 - 0.415415i) q^{7} +(0.540641 - 0.841254i) q^{8} +(-0.989821 - 0.142315i) q^{9} +(0.0683785 + 0.125226i) q^{11} +(0.415415 + 0.909632i) q^{14} +(-0.959493 + 0.281733i) q^{16} +(0.654861 + 0.755750i) q^{18} +(0.0303285 - 0.139418i) q^{22} +(0.142315 + 0.989821i) q^{23} +(0.281733 + 0.959493i) q^{25} +(0.281733 - 0.959493i) q^{28} +(-1.71524 + 0.373128i) q^{29} +(0.909632 + 0.415415i) q^{32} -1.00000i q^{36} +(1.12299 + 1.50013i) q^{37} +(0.114220 - 1.59700i) q^{43} +(-0.114220 + 0.0855040i) q^{44} +(0.540641 - 0.841254i) q^{46} +(0.654861 + 0.755750i) q^{49} +(0.415415 - 0.909632i) q^{50} +(0.613435 + 1.64468i) q^{53} +(-0.841254 + 0.540641i) q^{56} +(1.54064 + 0.841254i) q^{58} +(0.841254 + 0.540641i) q^{63} +(-0.415415 - 0.909632i) q^{64} +(-0.936593 + 1.71524i) q^{67} +(-1.45027 + 0.425839i) q^{71} +(-0.654861 + 0.755750i) q^{72} +(0.133682 - 1.86912i) q^{74} +(-0.0101786 - 0.142315i) q^{77} +(-0.234072 - 0.512546i) q^{79} +(0.959493 + 0.281733i) q^{81} +(-1.13214 + 1.13214i) q^{86} +(0.142315 + 0.0101786i) q^{88} +(-0.959493 + 0.281733i) q^{92} -1.00000i q^{98} +(-0.0498610 - 0.133682i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{4} + 2 q^{11} - 2 q^{14} - 2 q^{16} + 2 q^{18} - 2 q^{22} + 2 q^{23} - 2 q^{29} - 2 q^{37} + 2 q^{43} - 2 q^{44} + 2 q^{49} - 2 q^{50} - 2 q^{53} + 2 q^{56} + 20 q^{58} - 2 q^{63} + 2 q^{64} - 2 q^{67} - 2 q^{72} - 2 q^{74} - 20 q^{77} + 2 q^{81} - 2 q^{86} + 2 q^{88} - 2 q^{92} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2576\mathbb{Z}\right)^\times\).

\(n\) \(645\) \(1473\) \(1569\) \(2255\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(e\left(\frac{21}{22}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.755750 0.654861i −0.755750 0.654861i
\(3\) 0 0 0.0713392 0.997452i \(-0.477273\pi\)
−0.0713392 + 0.997452i \(0.522727\pi\)
\(4\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(5\) 0 0 −0.800541 0.599278i \(-0.795455\pi\)
0.800541 + 0.599278i \(0.204545\pi\)
\(6\) 0 0
\(7\) −0.909632 0.415415i −0.909632 0.415415i
\(8\) 0.540641 0.841254i 0.540641 0.841254i
\(9\) −0.989821 0.142315i −0.989821 0.142315i
\(10\) 0 0
\(11\) 0.0683785 + 0.125226i 0.0683785 + 0.125226i 0.909632 0.415415i \(-0.136364\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(12\) 0 0
\(13\) 0 0 −0.936950 0.349464i \(-0.886364\pi\)
0.936950 + 0.349464i \(0.113636\pi\)
\(14\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(15\) 0 0
\(16\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(17\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(18\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(19\) 0 0 0.212565 0.977147i \(-0.431818\pi\)
−0.212565 + 0.977147i \(0.568182\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.0303285 0.139418i 0.0303285 0.139418i
\(23\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(24\) 0 0
\(25\) 0.281733 + 0.959493i 0.281733 + 0.959493i
\(26\) 0 0
\(27\) 0 0
\(28\) 0.281733 0.959493i 0.281733 0.959493i
\(29\) −1.71524 + 0.373128i −1.71524 + 0.373128i −0.959493 0.281733i \(-0.909091\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(30\) 0 0
\(31\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(32\) 0.909632 + 0.415415i 0.909632 + 0.415415i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000i 1.00000i
\(37\) 1.12299 + 1.50013i 1.12299 + 1.50013i 0.841254 + 0.540641i \(0.181818\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(42\) 0 0
\(43\) 0.114220 1.59700i 0.114220 1.59700i −0.540641 0.841254i \(-0.681818\pi\)
0.654861 0.755750i \(-0.272727\pi\)
\(44\) −0.114220 + 0.0855040i −0.114220 + 0.0855040i
\(45\) 0 0
\(46\) 0.540641 0.841254i 0.540641 0.841254i
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(50\) 0.415415 0.909632i 0.415415 0.909632i
\(51\) 0 0
\(52\) 0 0
\(53\) 0.613435 + 1.64468i 0.613435 + 1.64468i 0.755750 + 0.654861i \(0.227273\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(57\) 0 0
\(58\) 1.54064 + 0.841254i 1.54064 + 0.841254i
\(59\) 0 0 −0.936950 0.349464i \(-0.886364\pi\)
0.936950 + 0.349464i \(0.113636\pi\)
\(60\) 0 0
\(61\) 0 0 0.997452 0.0713392i \(-0.0227273\pi\)
−0.997452 + 0.0713392i \(0.977273\pi\)
\(62\) 0 0
\(63\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(64\) −0.415415 0.909632i −0.415415 0.909632i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.936593 + 1.71524i −0.936593 + 1.71524i −0.281733 + 0.959493i \(0.590909\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.45027 + 0.425839i −1.45027 + 0.425839i −0.909632 0.415415i \(-0.863636\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(72\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(73\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(74\) 0.133682 1.86912i 0.133682 1.86912i
\(75\) 0 0
\(76\) 0 0
\(77\) −0.0101786 0.142315i −0.0101786 0.142315i
\(78\) 0 0
\(79\) −0.234072 0.512546i −0.234072 0.512546i 0.755750 0.654861i \(-0.227273\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(80\) 0 0
\(81\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(82\) 0 0
\(83\) 0 0 −0.599278 0.800541i \(-0.704545\pi\)
0.599278 + 0.800541i \(0.295455\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.13214 + 1.13214i −1.13214 + 1.13214i
\(87\) 0 0
\(88\) 0.142315 + 0.0101786i 0.142315 + 0.0101786i
\(89\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(98\) 1.00000i 1.00000i
\(99\) −0.0498610 0.133682i −0.0498610 0.133682i
\(100\) −0.909632 + 0.415415i −0.909632 + 0.415415i
\(101\) 0 0 0.800541 0.599278i \(-0.204545\pi\)
−0.800541 + 0.599278i \(0.795455\pi\)
\(102\) 0 0
\(103\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.613435 1.64468i 0.613435 1.64468i
\(107\) −0.697148 + 0.0498610i −0.697148 + 0.0498610i −0.415415 0.909632i \(-0.636364\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(108\) 0 0
\(109\) 0.415415 + 1.90963i 0.415415 + 1.90963i 0.415415 + 0.909632i \(0.363636\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.989821 + 0.142315i 0.989821 + 0.142315i
\(113\) 0.0801894 + 0.273100i 0.0801894 + 0.273100i 0.989821 0.142315i \(-0.0454545\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.613435 1.64468i −0.613435 1.64468i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.529635 0.824128i 0.529635 0.824128i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −0.281733 0.959493i −0.281733 0.959493i
\(127\) −0.557730 + 1.89945i −0.557730 + 1.89945i −0.142315 + 0.989821i \(0.545455\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(128\) −0.281733 + 0.959493i −0.281733 + 0.959493i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.936950 0.349464i \(-0.113636\pi\)
−0.936950 + 0.349464i \(0.886364\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.83107 0.682956i 1.83107 0.682956i
\(135\) 0 0
\(136\) 0 0
\(137\) −1.68251 −1.68251 −0.841254 0.540641i \(-0.818182\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(138\) 0 0
\(139\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.37491 + 0.627899i 1.37491 + 0.627899i
\(143\) 0 0
\(144\) 0.989821 0.142315i 0.989821 0.142315i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −1.32505 + 1.32505i −1.32505 + 1.32505i
\(149\) −1.40524 + 0.767317i −1.40524 + 0.767317i −0.989821 0.142315i \(-0.954545\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(150\) 0 0
\(151\) −0.797176 1.74557i −0.797176 1.74557i −0.654861 0.755750i \(-0.727273\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −0.0855040 + 0.114220i −0.0855040 + 0.114220i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.977147 0.212565i \(-0.931818\pi\)
0.977147 + 0.212565i \(0.0681818\pi\)
\(158\) −0.158746 + 0.540641i −0.158746 + 0.540641i
\(159\) 0 0
\(160\) 0 0
\(161\) 0.281733 0.959493i 0.281733 0.959493i
\(162\) −0.540641 0.841254i −0.540641 0.841254i
\(163\) 0.841254 + 0.459359i 0.841254 + 0.459359i 0.841254 0.540641i \(-0.181818\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(168\) 0 0
\(169\) 0.755750 + 0.654861i 0.755750 + 0.654861i
\(170\) 0 0
\(171\) 0 0
\(172\) 1.59700 0.114220i 1.59700 0.114220i
\(173\) 0 0 0.877679 0.479249i \(-0.159091\pi\)
−0.877679 + 0.479249i \(0.840909\pi\)
\(174\) 0 0
\(175\) 0.142315 0.989821i 0.142315 0.989821i
\(176\) −0.100889 0.100889i −0.100889 0.100889i
\(177\) 0 0
\(178\) 0 0
\(179\) 0.718267 0.959493i 0.718267 0.959493i −0.281733 0.959493i \(-0.590909\pi\)
1.00000 \(0\)
\(180\) 0 0
\(181\) 0 0 −0.997452 0.0713392i \(-0.977273\pi\)
0.997452 + 0.0713392i \(0.0227273\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0.909632 + 0.415415i 0.909632 + 0.415415i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.755750 1.65486i 0.755750 1.65486i 1.00000i \(-0.5\pi\)
0.755750 0.654861i \(-0.227273\pi\)
\(192\) 0 0
\(193\) −0.215109 + 1.49611i −0.215109 + 1.49611i 0.540641 + 0.841254i \(0.318182\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(197\) 0.418852 1.12299i 0.418852 1.12299i −0.540641 0.841254i \(-0.681818\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(198\) −0.0498610 + 0.133682i −0.0498610 + 0.133682i
\(199\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(200\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(201\) 0 0
\(202\) 0 0
\(203\) 1.71524 + 0.373128i 1.71524 + 0.373128i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000i 1.00000i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 1.94931 + 0.424047i 1.94931 + 0.424047i 0.989821 + 0.142315i \(0.0454545\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(212\) −1.54064 + 0.841254i −1.54064 + 0.841254i
\(213\) 0 0
\(214\) 0.559521 + 0.418852i 0.559521 + 0.418852i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0.936593 1.71524i 0.936593 1.71524i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(224\) −0.654861 0.755750i −0.654861 0.755750i
\(225\) −0.142315 0.989821i −0.142315 0.989821i
\(226\) 0.118239 0.258908i 0.118239 0.258908i
\(227\) 0 0 −0.997452 0.0713392i \(-0.977273\pi\)
0.997452 + 0.0713392i \(0.0227273\pi\)
\(228\) 0 0
\(229\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.613435 + 1.64468i −0.613435 + 1.64468i
\(233\) 1.37491 1.19136i 1.37491 1.19136i 0.415415 0.909632i \(-0.363636\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.80075 0.258908i −1.80075 0.258908i −0.841254 0.540641i \(-0.818182\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(240\) 0 0
\(241\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(242\) −0.939960 + 0.275997i −0.939960 + 0.275997i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.877679 0.479249i \(-0.840909\pi\)
0.877679 + 0.479249i \(0.159091\pi\)
\(252\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(253\) −0.114220 + 0.0855040i −0.114220 + 0.0855040i
\(254\) 1.66538 1.07028i 1.66538 1.07028i
\(255\) 0 0
\(256\) 0.841254 0.540641i 0.841254 0.540641i
\(257\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(258\) 0 0
\(259\) −0.398326 1.83107i −0.398326 1.83107i
\(260\) 0 0
\(261\) 1.75089 0.125226i 1.75089 0.125226i
\(262\) 0 0
\(263\) −1.53046 + 0.698939i −1.53046 + 0.698939i −0.989821 0.142315i \(-0.954545\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.83107 0.682956i −1.83107 0.682956i
\(269\) 0 0 −0.349464 0.936950i \(-0.613636\pi\)
0.349464 + 0.936950i \(0.386364\pi\)
\(270\) 0 0
\(271\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 1.27155 + 1.10181i 1.27155 + 1.10181i
\(275\) −0.100889 + 0.100889i −0.100889 + 0.100889i
\(276\) 0 0
\(277\) 1.32505 + 1.32505i 1.32505 + 1.32505i 0.909632 + 0.415415i \(0.136364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.273100 1.89945i −0.273100 1.89945i −0.415415 0.909632i \(-0.636364\pi\)
0.142315 0.989821i \(-0.454545\pi\)
\(282\) 0 0
\(283\) 0 0 0.936950 0.349464i \(-0.113636\pi\)
−0.936950 + 0.349464i \(0.886364\pi\)
\(284\) −0.627899 1.37491i −0.627899 1.37491i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.841254 0.540641i −0.841254 0.540641i
\(289\) −0.415415 0.909632i −0.415415 0.909632i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.977147 0.212565i \(-0.0681818\pi\)
−0.977147 + 0.212565i \(0.931818\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.86912 0.133682i 1.86912 0.133682i
\(297\) 0 0
\(298\) 1.56449 + 0.340335i 1.56449 + 0.340335i
\(299\) 0 0
\(300\) 0 0
\(301\) −0.767317 + 1.40524i −0.767317 + 1.40524i
\(302\) −0.540641 + 1.84125i −0.540641 + 1.84125i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.997452 0.0713392i \(-0.0227273\pi\)
−0.997452 + 0.0713392i \(0.977273\pi\)
\(308\) 0.139418 0.0303285i 0.139418 0.0303285i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(312\) 0 0
\(313\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0.474017 0.304632i 0.474017 0.304632i
\(317\) 0.559521 + 0.418852i 0.559521 + 0.418852i 0.841254 0.540641i \(-0.181818\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(318\) 0 0
\(319\) −0.164011 0.189279i −0.164011 0.189279i
\(320\) 0 0
\(321\) 0 0
\(322\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(323\) 0 0
\(324\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(325\) 0 0
\(326\) −0.334961 0.898064i −0.334961 0.898064i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.959493 + 1.28173i 0.959493 + 1.28173i 0.959493 + 0.281733i \(0.0909091\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) −0.898064 1.64468i −0.898064 1.64468i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.989821 0.857685i −0.989821 0.857685i 1.00000i \(-0.5\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(338\) −0.142315 0.989821i −0.142315 0.989821i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.281733 0.959493i −0.281733 0.959493i
\(344\) −1.28173 0.959493i −1.28173 0.959493i
\(345\) 0 0
\(346\) 0 0
\(347\) 0.334961 0.613435i 0.334961 0.613435i −0.654861 0.755750i \(-0.727273\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(348\) 0 0
\(349\) 0 0 0.212565 0.977147i \(-0.431818\pi\)
−0.212565 + 0.977147i \(0.568182\pi\)
\(350\) −0.755750 + 0.654861i −0.755750 + 0.654861i
\(351\) 0 0
\(352\) 0.0101786 + 0.142315i 0.0101786 + 0.142315i
\(353\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −1.17116 + 0.254771i −1.17116 + 0.254771i
\(359\) −1.95949 0.281733i −1.95949 0.281733i −0.959493 0.281733i \(-0.909091\pi\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −0.909632 0.415415i −0.909632 0.415415i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −0.415415 0.909632i −0.415415 0.909632i
\(369\) 0 0
\(370\) 0 0
\(371\) 0.125226 1.75089i 0.125226 1.75089i
\(372\) 0 0
\(373\) −0.114220 0.0855040i −0.114220 0.0855040i 0.540641 0.841254i \(-0.318182\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.459359 0.841254i −0.459359 0.841254i 0.540641 0.841254i \(-0.318182\pi\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.65486 + 0.755750i −1.65486 + 0.755750i
\(383\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.14231 0.989821i 1.14231 0.989821i
\(387\) −0.340335 + 1.56449i −0.340335 + 1.56449i
\(388\) 0 0
\(389\) −0.956056 + 1.75089i −0.956056 + 1.75089i −0.415415 + 0.909632i \(0.636364\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.989821 0.142315i 0.989821 0.142315i
\(393\) 0 0
\(394\) −1.05195 + 0.574406i −1.05195 + 0.574406i
\(395\) 0 0
\(396\) 0.125226 0.0683785i 0.125226 0.0683785i
\(397\) 0 0 0.977147 0.212565i \(-0.0681818\pi\)
−0.977147 + 0.212565i \(0.931818\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.540641 0.841254i −0.540641 0.841254i
\(401\) −0.755750 + 0.345139i −0.755750 + 0.345139i −0.755750 0.654861i \(-0.772727\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −1.05195 1.40524i −1.05195 1.40524i
\(407\) −0.111067 + 0.243204i −0.111067 + 0.243204i
\(408\) 0 0
\(409\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.800541 0.599278i \(-0.795455\pi\)
0.800541 + 0.599278i \(0.204545\pi\)
\(420\) 0 0
\(421\) −0.418852 1.12299i −0.418852 1.12299i −0.959493 0.281733i \(-0.909091\pi\)
0.540641 0.841254i \(-0.318182\pi\)
\(422\) −1.19550 1.59700i −1.19550 1.59700i
\(423\) 0 0
\(424\) 1.71524 + 0.373128i 1.71524 + 0.373128i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −0.148568 0.682956i −0.148568 0.682956i
\(429\) 0 0
\(430\) 0 0
\(431\) −0.239446 0.153882i −0.239446 0.153882i 0.415415 0.909632i \(-0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(432\) 0 0
\(433\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.83107 + 0.682956i −1.83107 + 0.682956i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(440\) 0 0
\(441\) −0.540641 0.841254i −0.540641 0.841254i
\(442\) 0 0
\(443\) −0.139418 + 0.0303285i −0.139418 + 0.0303285i −0.281733 0.959493i \(-0.590909\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 1.00000i 1.00000i
\(449\) −0.797176 0.234072i −0.797176 0.234072i −0.142315 0.989821i \(-0.545455\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(450\) −0.540641 + 0.841254i −0.540641 + 0.841254i
\(451\) 0 0
\(452\) −0.258908 + 0.118239i −0.258908 + 0.118239i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.368991 0.425839i −0.368991 0.425839i 0.540641 0.841254i \(-0.318182\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) 0 0
\(463\) 1.27155 1.10181i 1.27155 1.10181i 0.281733 0.959493i \(-0.409091\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(464\) 1.54064 0.841254i 1.54064 0.841254i
\(465\) 0 0
\(466\) −1.81926 −1.81926
\(467\) 0 0 −0.349464 0.936950i \(-0.613636\pi\)
0.349464 + 0.936950i \(0.386364\pi\)
\(468\) 0 0
\(469\) 1.56449 1.17116i 1.56449 1.17116i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.207796 0.0948973i 0.207796 0.0948973i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.373128 1.71524i −0.373128 1.71524i
\(478\) 1.19136 + 1.37491i 1.19136 + 1.37491i
\(479\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0.891115 + 0.406958i 0.891115 + 0.406958i
\(485\) 0 0
\(486\) 0 0
\(487\) −0.239446 + 0.153882i −0.239446 + 0.153882i −0.654861 0.755750i \(-0.727273\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0.139418 + 1.94931i 0.139418 + 1.94931i 0.281733 + 0.959493i \(0.409091\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.49611 + 0.215109i 1.49611 + 0.215109i
\(498\) 0 0
\(499\) 0.398326 0.148568i 0.398326 0.148568i −0.142315 0.989821i \(-0.545455\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(504\) 0.909632 0.415415i 0.909632 0.415415i
\(505\) 0 0
\(506\) 0.142315 + 0.0101786i 0.142315 + 0.0101786i
\(507\) 0 0
\(508\) −1.95949 0.281733i −1.95949 0.281733i
\(509\) 0 0 −0.997452 0.0713392i \(-0.977273\pi\)
0.997452 + 0.0713392i \(0.0227273\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.989821 0.142315i −0.989821 0.142315i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −0.898064 + 1.64468i −0.898064 + 1.64468i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(522\) −1.40524 1.05195i −1.40524 1.05195i
\(523\) 0 0 −0.212565 0.977147i \(-0.568182\pi\)
0.212565 + 0.977147i \(0.431818\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 1.61435 + 0.474017i 1.61435 + 0.474017i
\(527\) 0 0
\(528\) 0 0
\(529\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0.936593 + 1.71524i 0.936593 + 1.71524i
\(537\) 0 0
\(538\) 0 0
\(539\) −0.0498610 + 0.133682i −0.0498610 + 0.133682i
\(540\) 0 0
\(541\) 0.373128 0.203743i 0.373128 0.203743i −0.281733 0.959493i \(-0.590909\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.0855040 + 0.114220i −0.0855040 + 0.114220i −0.841254 0.540641i \(-0.818182\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(548\) −0.239446 1.66538i −0.239446 1.66538i
\(549\) 0 0
\(550\) 0.142315 0.0101786i 0.142315 0.0101786i
\(551\) 0 0
\(552\) 0 0
\(553\) 0.563465i 0.563465i
\(554\) −0.133682 1.86912i −0.133682 1.86912i
\(555\) 0 0
\(556\) 0 0
\(557\) −0.847507 + 1.13214i −0.847507 + 1.13214i 0.142315 + 0.989821i \(0.454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −1.03748 + 1.61435i −1.03748 + 1.61435i
\(563\) 0 0 0.877679 0.479249i \(-0.159091\pi\)
−0.877679 + 0.479249i \(0.840909\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.755750 0.654861i −0.755750 0.654861i
\(568\) −0.425839 + 1.45027i −0.425839 + 1.45027i
\(569\) 1.53046 + 0.983568i 1.53046 + 0.983568i 0.989821 + 0.142315i \(0.0454545\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(570\) 0 0
\(571\) −1.83107 0.398326i −1.83107 0.398326i −0.841254 0.540641i \(-0.818182\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.909632 + 0.415415i −0.909632 + 0.415415i
\(576\) 0.281733 + 0.959493i 0.281733 + 0.959493i
\(577\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(578\) −0.281733 + 0.959493i −0.281733 + 0.959493i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.164011 + 0.189279i −0.164011 + 0.189279i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.877679 0.479249i \(-0.159091\pi\)
−0.877679 + 0.479249i \(0.840909\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.50013 1.12299i −1.50013 1.12299i
\(593\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.959493 1.28173i −0.959493 1.28173i
\(597\) 0 0
\(598\) 0 0
\(599\) 1.08128 1.08128 0.540641 0.841254i \(-0.318182\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(600\) 0 0
\(601\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(602\) 1.50013 0.559521i 1.50013 0.559521i
\(603\) 1.17116 1.56449i 1.17116 1.56449i
\(604\) 1.61435 1.03748i 1.61435 1.03748i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0.0303285 + 0.424047i 0.0303285 + 0.424047i 0.989821 + 0.142315i \(0.0454545\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −0.125226 0.0683785i −0.125226 0.0683785i
\(617\) 0.909632 0.584585i 0.909632 0.584585i 1.00000i \(-0.5\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(618\) 0 0
\(619\) 0 0 −0.877679 0.479249i \(-0.840909\pi\)
0.877679 + 0.479249i \(0.159091\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0.258908 0.118239i 0.258908 0.118239i −0.281733 0.959493i \(-0.590909\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(632\) −0.557730 0.0801894i −0.557730 0.0801894i
\(633\) 0 0
\(634\) −0.148568 0.682956i −0.148568 0.682956i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0.250452i 0.250452i
\(639\) 1.49611 0.215109i 1.49611 0.215109i
\(640\) 0 0
\(641\) 1.51150 1.30972i 1.51150 1.30972i 0.755750 0.654861i \(-0.227273\pi\)
0.755750 0.654861i \(-0.227273\pi\)
\(642\) 0 0
\(643\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 0.989821 + 0.142315i 0.989821 + 0.142315i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(648\) 0.755750 0.654861i 0.755750 0.654861i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −0.334961 + 0.898064i −0.334961 + 0.898064i
\(653\) 0.254771 + 0.340335i 0.254771 + 0.340335i 0.909632 0.415415i \(-0.136364\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0.0683785 + 0.956056i 0.0683785 + 0.956056i 0.909632 + 0.415415i \(0.136364\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(660\) 0 0
\(661\) 0 0 0.977147 0.212565i \(-0.0681818\pi\)
−0.977147 + 0.212565i \(0.931818\pi\)
\(662\) 0.114220 1.59700i 0.114220 1.59700i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −0.398326 + 1.83107i −0.398326 + 1.83107i
\(667\) −0.613435 1.64468i −0.613435 1.64468i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.41542 + 0.909632i 1.41542 + 0.909632i 1.00000 \(0\)
0.415415 + 0.909632i \(0.363636\pi\)
\(674\) 0.186393 + 1.29639i 0.186393 + 1.29639i
\(675\) 0 0
\(676\) −0.540641 + 0.841254i −0.540641 + 0.841254i
\(677\) 0 0 −0.936950 0.349464i \(-0.886364\pi\)
0.936950 + 0.349464i \(0.113636\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.244250 + 0.654861i 0.244250 + 0.654861i 1.00000 \(0\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(687\) 0 0
\(688\) 0.340335 + 1.56449i 0.340335 + 1.56449i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(692\) 0 0
\(693\) −0.0101786 + 0.142315i −0.0101786 + 0.142315i
\(694\) −0.654861 + 0.244250i −0.654861 + 0.244250i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.00000 1.00000
\(701\) 0.203743 + 0.373128i 0.203743 + 0.373128i 0.959493 0.281733i \(-0.0909091\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.0855040 0.114220i 0.0855040 0.114220i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.424047 1.94931i 0.424047 1.94931i 0.142315 0.989821i \(-0.454545\pi\)
0.281733 0.959493i \(-0.409091\pi\)
\(710\) 0 0
\(711\) 0.158746 + 0.540641i 0.158746 + 0.540641i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.05195 + 0.574406i 1.05195 + 0.574406i
\(717\) 0 0
\(718\) 1.29639 + 1.49611i 1.29639 + 1.49611i
\(719\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(723\) 0 0
\(724\) 0 0
\(725\) −0.841254 1.54064i −0.841254 1.54064i
\(726\) 0 0
\(727\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(728\) 0 0
\(729\) −0.909632 0.415415i −0.909632 0.415415i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.0713392 0.997452i \(-0.477273\pi\)
−0.0713392 + 0.997452i \(0.522727\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −0.281733 + 0.959493i −0.281733 + 0.959493i
\(737\) −0.278835 −0.278835
\(738\) 0 0
\(739\) 0.0303285 0.424047i 0.0303285 0.424047i −0.959493 0.281733i \(-0.909091\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.24123 + 1.24123i −1.24123 + 1.24123i
\(743\) 0.983568 + 0.449181i 0.983568 + 0.449181i 0.841254 0.540641i \(-0.181818\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.0303285 + 0.139418i 0.0303285 + 0.139418i
\(747\) 0 0
\(748\) 0 0
\(749\) 0.654861 + 0.244250i 0.654861 + 0.244250i
\(750\) 0 0
\(751\) 1.25667 1.45027i 1.25667 1.45027i 0.415415 0.909632i \(-0.363636\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.0683785 + 0.125226i −0.0683785 + 0.125226i −0.909632 0.415415i \(-0.863636\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(758\) −0.203743 + 0.936593i −0.203743 + 0.936593i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(762\) 0 0
\(763\) 0.415415 1.90963i 0.415415 1.90963i
\(764\) 1.74557 + 0.512546i 1.74557 + 0.512546i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.51150 −1.51150
\(773\) 0 0 −0.599278 0.800541i \(-0.704545\pi\)
0.599278 + 0.800541i \(0.295455\pi\)
\(774\) 1.28173 0.959493i 1.28173 0.959493i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 1.86912 0.697148i 1.86912 0.697148i
\(779\) 0 0
\(780\) 0 0
\(781\) −0.152493 0.152493i −0.152493 0.152493i
\(782\) 0 0
\(783\) 0 0
\(784\) −0.841254 0.540641i −0.841254 0.540641i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.800541 0.599278i \(-0.795455\pi\)
0.800541 + 0.599278i \(0.204545\pi\)
\(788\) 1.17116 + 0.254771i 1.17116 + 0.254771i
\(789\) 0 0
\(790\) 0 0
\(791\) 0.0405070 0.281733i 0.0405070 0.281733i
\(792\) −0.139418 0.0303285i −0.139418 0.0303285i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.997452 0.0713392i \(-0.0227273\pi\)
−0.997452 + 0.0713392i \(0.977273\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(801\) 0 0
\(802\) 0.797176 + 0.234072i 0.797176 + 0.234072i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.03748 1.61435i −1.03748 1.61435i −0.755750 0.654861i \(-0.772727\pi\)
−0.281733 0.959493i \(-0.590909\pi\)
\(810\) 0 0
\(811\) 0 0 0.977147 0.212565i \(-0.0681818\pi\)
−0.977147 + 0.212565i \(0.931818\pi\)
\(812\) −0.125226 + 1.75089i −0.125226 + 1.75089i
\(813\) 0 0
\(814\) 0.243204 0.111067i 0.243204 0.111067i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.86912 + 0.697148i −1.86912 + 0.697148i −0.909632 + 0.415415i \(0.863636\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(822\) 0 0
\(823\) 0.0801894 + 0.557730i 0.0801894 + 0.557730i 0.989821 + 0.142315i \(0.0454545\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.32505 + 1.32505i 1.32505 + 1.32505i 0.909632 + 0.415415i \(0.136364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(828\) 0.989821 0.142315i 0.989821 0.142315i
\(829\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(840\) 0 0
\(841\) 1.89320 0.864596i 1.89320 0.864596i
\(842\) −0.418852 + 1.12299i −0.418852 + 1.12299i
\(843\) 0 0
\(844\) −0.142315 + 1.98982i −0.142315 + 1.98982i
\(845\) 0 0
\(846\) 0 0
\(847\) −0.824128 + 0.529635i −0.824128 + 0.529635i
\(848\) −1.05195 1.40524i −1.05195 1.40524i
\(849\) 0 0
\(850\) 0 0
\(851\) −1.32505 + 1.32505i −1.32505 + 1.32505i
\(852\) 0 0
\(853\) 0 0 −0.877679 0.479249i \(-0.840909\pi\)
0.877679 + 0.479249i \(0.159091\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −0.334961 + 0.613435i −0.334961 + 0.613435i
\(857\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(858\) 0 0
\(859\) 0 0 −0.0713392 0.997452i \(-0.522727\pi\)
0.0713392 + 0.997452i \(0.477273\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0.0801894 + 0.273100i 0.0801894 + 0.273100i
\(863\) 0.368991 1.25667i 0.368991 1.25667i −0.540641 0.841254i \(-0.681818\pi\)
0.909632 0.415415i \(-0.136364\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0.0481785 0.0643589i 0.0481785 0.0643589i
\(870\) 0 0
\(871\) 0 0
\(872\) 1.83107 + 0.682956i 1.83107 + 0.682956i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.98982 + 0.142315i 1.98982 + 0.142315i 1.00000 \(0\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(882\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(883\) −1.40524 + 1.05195i −1.40524 + 1.05195i −0.415415 + 0.909632i \(0.636364\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.125226 + 0.0683785i 0.125226 + 0.0683785i
\(887\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(888\) 0 0
\(889\) 1.29639 1.49611i 1.29639 1.49611i
\(890\) 0 0
\(891\) 0.0303285 + 0.139418i 0.0303285 + 0.139418i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0.654861 0.755750i 0.654861 0.755750i
\(897\) 0 0
\(898\) 0.449181 + 0.698939i 0.449181 + 0.698939i
\(899\) 0 0
\(900\) 0.959493 0.281733i 0.959493 0.281733i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0.273100 + 0.0801894i 0.273100 + 0.0801894i
\(905\) 0 0
\(906\) 0 0
\(907\) −0.148568 + 0.398326i −0.148568 + 0.398326i −0.989821 0.142315i \(-0.954545\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −0.239446 + 1.66538i −0.239446 + 1.66538i 0.415415 + 0.909632i \(0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0.563465i 0.563465i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0.830830i 0.830830i 0.909632 + 0.415415i \(0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.12299 + 1.50013i −1.12299 + 1.50013i
\(926\) −1.68251 −1.68251
\(927\) 0 0
\(928\) −1.71524 0.373128i −1.71524 0.373128i
\(929\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.37491 + 1.19136i 1.37491 + 1.19136i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(938\) −1.94931 0.139418i −1.94931 0.139418i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.877679 0.479249i \(-0.840909\pi\)
0.877679 + 0.479249i \(0.159091\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −0.219186 0.0643589i −0.219186 0.0643589i
\(947\) 1.83107 + 0.398326i 1.83107 + 0.398326i 0.989821 0.142315i \(-0.0454545\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.345139 + 0.755750i 0.345139 + 0.755750i 1.00000 \(0\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(954\) −0.841254 + 1.54064i −0.841254 + 1.54064i
\(955\) 0 0
\(956\) 1.81926i 1.81926i
\(957\) 0 0
\(958\) 0 0
\(959\) 1.53046 + 0.698939i 1.53046 + 0.698939i
\(960\) 0 0
\(961\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(962\) 0 0
\(963\) 0.697148 + 0.0498610i 0.697148 + 0.0498610i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.30972 1.30972 0.654861 0.755750i \(-0.272727\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(968\) −0.406958 0.891115i −0.406958 0.891115i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.599278 0.800541i \(-0.295455\pi\)
−0.599278 + 0.800541i \(0.704545\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0.281733 + 0.0405070i 0.281733 + 0.0405070i
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −0.139418 1.94931i −0.139418 1.94931i
\(982\) 1.17116 1.56449i 1.17116 1.56449i
\(983\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.59700 0.114220i 1.59700 0.114220i
\(990\) 0 0
\(991\) 0.474017 + 1.61435i 0.474017 + 1.61435i 0.755750 + 0.654861i \(0.227273\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −0.989821 1.14231i −0.989821 1.14231i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.997452 0.0713392i \(-0.0227273\pi\)
−0.997452 + 0.0713392i \(0.977273\pi\)
\(998\) −0.398326 0.148568i −0.398326 0.148568i
\(999\) 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 2576.1.cn.b.1371.1 yes 20
7.6 odd 2 CM 2576.1.cn.b.1371.1 yes 20
16.3 odd 4 2576.1.cn.a.83.1 20
23.5 odd 22 2576.1.cn.a.1707.1 yes 20
112.83 even 4 2576.1.cn.a.83.1 20
161.97 even 22 2576.1.cn.a.1707.1 yes 20
368.51 even 44 inner 2576.1.cn.b.419.1 yes 20
2576.419 odd 44 inner 2576.1.cn.b.419.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2576.1.cn.a.83.1 20 16.3 odd 4
2576.1.cn.a.83.1 20 112.83 even 4
2576.1.cn.a.1707.1 yes 20 23.5 odd 22
2576.1.cn.a.1707.1 yes 20 161.97 even 22
2576.1.cn.b.419.1 yes 20 368.51 even 44 inner
2576.1.cn.b.419.1 yes 20 2576.419 odd 44 inner
2576.1.cn.b.1371.1 yes 20 1.1 even 1 trivial
2576.1.cn.b.1371.1 yes 20 7.6 odd 2 CM