Properties

Label 1265.1.bi.b.1187.1
Level $1265$
Weight $1$
Character 1265.1187
Analytic conductor $0.631$
Analytic rank $0$
Dimension $20$
Projective image $D_{44}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1265,1,Mod(43,1265)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1265, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([33, 22, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1265.43");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1265 = 5 \cdot 11 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1265.bi (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: \(0.631317240981\)
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: \(\Z[a_1, \ldots, a_{4}]\)
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 1187.1
Root \(0.989821 - 0.142315i\) of defining polynomial
Character \(\chi\) \(=\) 1265.1187
Dual form 1265.1.bi.b.373.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.697148 + 0.0498610i) q^{3} +(0.540641 + 0.841254i) q^{4} +(0.909632 - 0.415415i) q^{5} +(-0.506293 - 0.0727939i) q^{9} +O(q^{10})\) \(q+(0.697148 + 0.0498610i) q^{3} +(0.540641 + 0.841254i) q^{4} +(0.909632 - 0.415415i) q^{5} +(-0.506293 - 0.0727939i) q^{9} +(-0.281733 + 0.959493i) q^{11} +(0.334961 + 0.613435i) q^{12} +(0.654861 - 0.244250i) q^{15} +(-0.415415 + 0.909632i) q^{16} +(0.841254 + 0.540641i) q^{20} +(-0.959493 - 0.281733i) q^{23} +(0.654861 - 0.755750i) q^{25} +(-1.03229 - 0.224560i) q^{27} +(1.29639 - 1.49611i) q^{31} +(-0.244250 + 0.654861i) q^{33} +(-0.212484 - 0.465276i) q^{36} +(-0.254771 - 0.340335i) q^{37} +(-0.959493 + 0.281733i) q^{44} +(-0.490780 + 0.144106i) q^{45} +(-1.13214 - 1.13214i) q^{47} +(-0.334961 + 0.613435i) q^{48} +(-0.755750 + 0.654861i) q^{49} +(1.83107 - 0.682956i) q^{53} +(0.142315 + 0.989821i) q^{55} +(-0.983568 + 0.449181i) q^{59} +(0.559521 + 0.418852i) q^{60} +(-0.989821 + 0.142315i) q^{64} +(0.898064 - 1.64468i) q^{67} +(-0.654861 - 0.244250i) q^{69} +(-1.84125 + 0.540641i) q^{71} +(0.494217 - 0.494217i) q^{75} +1.00000i q^{80} +(-0.217680 - 0.0639165i) q^{81} +(1.19136 + 1.37491i) q^{89} +(-0.281733 - 0.959493i) q^{92} +(0.978373 - 0.978373i) q^{93} +(0.959493 + 0.718267i) q^{97} +(0.212484 - 0.465276i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{3} - 2 q^{12} + 2 q^{15} + 2 q^{16} - 2 q^{20} - 2 q^{23} + 2 q^{25} - 20 q^{33} - 2 q^{36} + 2 q^{37} - 2 q^{44} + 2 q^{45} - 2 q^{47} + 2 q^{48} - 2 q^{53} + 2 q^{55} - 2 q^{60} + 2 q^{67} - 2 q^{69} - 18 q^{71} + 2 q^{75} - 2 q^{81} + 2 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1265\mathbb{Z}\right)^\times\).

\(n\) \(166\) \(507\) \(1036\)
\(\chi(n)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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 −0.877679 0.479249i \(-0.840909\pi\)
0.877679 + 0.479249i \(0.159091\pi\)
\(3\) 0.697148 + 0.0498610i 0.697148 + 0.0498610i 0.415415 0.909632i \(-0.363636\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(4\) 0.540641 + 0.841254i 0.540641 + 0.841254i
\(5\) 0.909632 0.415415i 0.909632 0.415415i
\(6\) 0 0
\(7\) 0 0 −0.349464 0.936950i \(-0.613636\pi\)
0.349464 + 0.936950i \(0.386364\pi\)
\(8\) 0 0
\(9\) −0.506293 0.0727939i −0.506293 0.0727939i
\(10\) 0 0
\(11\) −0.281733 + 0.959493i −0.281733 + 0.959493i
\(12\) 0.334961 + 0.613435i 0.334961 + 0.613435i
\(13\) 0 0 −0.936950 0.349464i \(-0.886364\pi\)
0.936950 + 0.349464i \(0.113636\pi\)
\(14\) 0 0
\(15\) 0.654861 0.244250i 0.654861 0.244250i
\(16\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(17\) 0 0 0.977147 0.212565i \(-0.0681818\pi\)
−0.977147 + 0.212565i \(0.931818\pi\)
\(18\) 0 0
\(19\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(20\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(21\) 0 0
\(22\) 0 0
\(23\) −0.959493 0.281733i −0.959493 0.281733i
\(24\) 0 0
\(25\) 0.654861 0.755750i 0.654861 0.755750i
\(26\) 0 0
\(27\) −1.03229 0.224560i −1.03229 0.224560i
\(28\) 0 0
\(29\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(30\) 0 0
\(31\) 1.29639 1.49611i 1.29639 1.49611i 0.540641 0.841254i \(-0.318182\pi\)
0.755750 0.654861i \(-0.227273\pi\)
\(32\) 0 0
\(33\) −0.244250 + 0.654861i −0.244250 + 0.654861i
\(34\) 0 0
\(35\) 0 0
\(36\) −0.212484 0.465276i −0.212484 0.465276i
\(37\) −0.254771 0.340335i −0.254771 0.340335i 0.654861 0.755750i \(-0.272727\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(42\) 0 0
\(43\) 0 0 0.0713392 0.997452i \(-0.477273\pi\)
−0.0713392 + 0.997452i \(0.522727\pi\)
\(44\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(45\) −0.490780 + 0.144106i −0.490780 + 0.144106i
\(46\) 0 0
\(47\) −1.13214 1.13214i −1.13214 1.13214i −0.989821 0.142315i \(-0.954545\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(48\) −0.334961 + 0.613435i −0.334961 + 0.613435i
\(49\) −0.755750 + 0.654861i −0.755750 + 0.654861i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.83107 0.682956i 1.83107 0.682956i 0.841254 0.540641i \(-0.181818\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(54\) 0 0
\(55\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.983568 + 0.449181i −0.983568 + 0.449181i −0.841254 0.540641i \(-0.818182\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(60\) 0.559521 + 0.418852i 0.559521 + 0.418852i
\(61\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.989821 + 0.142315i −0.989821 + 0.142315i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.898064 1.64468i 0.898064 1.64468i 0.142315 0.989821i \(-0.454545\pi\)
0.755750 0.654861i \(-0.227273\pi\)
\(68\) 0 0
\(69\) −0.654861 0.244250i −0.654861 0.244250i
\(70\) 0 0
\(71\) −1.84125 + 0.540641i −1.84125 + 0.540641i −0.841254 + 0.540641i \(0.818182\pi\)
−1.00000 \(1.00000\pi\)
\(72\) 0 0
\(73\) 0 0 0.212565 0.977147i \(-0.431818\pi\)
−0.212565 + 0.977147i \(0.568182\pi\)
\(74\) 0 0
\(75\) 0.494217 0.494217i 0.494217 0.494217i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(80\) 1.00000i 1.00000i
\(81\) −0.217680 0.0639165i −0.217680 0.0639165i
\(82\) 0 0
\(83\) 0 0 0.800541 0.599278i \(-0.204545\pi\)
−0.800541 + 0.599278i \(0.795455\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.19136 + 1.37491i 1.19136 + 1.37491i 0.909632 + 0.415415i \(0.136364\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.281733 0.959493i −0.281733 0.959493i
\(93\) 0.978373 0.978373i 0.978373 0.978373i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.959493 + 0.718267i 0.959493 + 0.718267i 0.959493 0.281733i \(-0.0909091\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0.212484 0.465276i 0.212484 0.465276i
\(100\) 0.989821 + 0.142315i 0.989821 + 0.142315i
\(101\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(102\) 0 0
\(103\) 0.956056 + 1.75089i 0.956056 + 1.75089i 0.540641 + 0.841254i \(0.318182\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.0713392 0.997452i \(-0.522727\pi\)
0.0713392 + 0.997452i \(0.477273\pi\)
\(108\) −0.369184 0.989821i −0.369184 0.989821i
\(109\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(110\) 0 0
\(111\) −0.160644 0.249967i −0.160644 0.249967i
\(112\) 0 0
\(113\) −1.40524 0.767317i −1.40524 0.767317i −0.415415 0.909632i \(-0.636364\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(114\) 0 0
\(115\) −0.989821 + 0.142315i −0.989821 + 0.142315i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.841254 0.540641i −0.841254 0.540641i
\(122\) 0 0
\(123\) 0 0
\(124\) 1.95949 + 0.281733i 1.95949 + 0.281733i
\(125\) 0.281733 0.959493i 0.281733 0.959493i
\(126\) 0 0
\(127\) 0 0 −0.479249 0.877679i \(-0.659091\pi\)
0.479249 + 0.877679i \(0.340909\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(132\) −0.682956 + 0.148568i −0.682956 + 0.148568i
\(133\) 0 0
\(134\) 0 0
\(135\) −1.03229 + 0.224560i −1.03229 + 0.224560i
\(136\) 0 0
\(137\) 0.100889 0.100889i 0.100889 0.100889i −0.654861 0.755750i \(-0.727273\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −0.732817 0.845715i −0.732817 0.845715i
\(142\) 0 0
\(143\) 0 0
\(144\) 0.276537 0.430300i 0.276537 0.430300i
\(145\) 0 0
\(146\) 0 0
\(147\) −0.559521 + 0.418852i −0.559521 + 0.418852i
\(148\) 0.148568 0.398326i 0.148568 0.398326i
\(149\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(150\) 0 0
\(151\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.557730 1.89945i 0.557730 1.89945i
\(156\) 0 0
\(157\) 0.300613 1.38189i 0.300613 1.38189i −0.540641 0.841254i \(-0.681818\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(158\) 0 0
\(159\) 1.31058 0.384822i 1.31058 0.384822i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.956056 + 1.75089i −0.956056 + 1.75089i −0.415415 + 0.909632i \(0.636364\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(164\) 0 0
\(165\) 0.0498610 + 0.697148i 0.0498610 + 0.697148i
\(166\) 0 0
\(167\) 0 0 −0.212565 0.977147i \(-0.568182\pi\)
0.212565 + 0.977147i \(0.431818\pi\)
\(168\) 0 0
\(169\) 0.755750 + 0.654861i 0.755750 + 0.654861i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.877679 0.479249i \(-0.159091\pi\)
−0.877679 + 0.479249i \(0.840909\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.755750 0.654861i −0.755750 0.654861i
\(177\) −0.708089 + 0.264103i −0.708089 + 0.264103i
\(178\) 0 0
\(179\) −1.29639 + 0.186393i −1.29639 + 0.186393i −0.755750 0.654861i \(-0.772727\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(180\) −0.386565 0.334961i −0.386565 0.334961i
\(181\) −0.989821 + 0.857685i −0.989821 + 0.857685i −0.989821 0.142315i \(-0.954545\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.373128 0.203743i −0.373128 0.203743i
\(186\) 0 0
\(187\) 0 0
\(188\) 0.340335 1.56449i 0.340335 1.56449i
\(189\) 0 0
\(190\) 0 0
\(191\) −0.755750 0.345139i −0.755750 0.345139i 1.00000i \(-0.5\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(192\) −0.697148 + 0.0498610i −0.697148 + 0.0498610i
\(193\) 0 0 −0.599278 0.800541i \(-0.704545\pi\)
0.599278 + 0.800541i \(0.295455\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.959493 0.281733i −0.959493 0.281733i
\(197\) 0 0 0.349464 0.936950i \(-0.386364\pi\)
−0.349464 + 0.936950i \(0.613636\pi\)
\(198\) 0 0
\(199\) −0.544078 + 0.627899i −0.544078 + 0.627899i −0.959493 0.281733i \(-0.909091\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(200\) 0 0
\(201\) 0.708089 1.10181i 0.708089 1.10181i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.465276 + 0.212484i 0.465276 + 0.212484i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(212\) 1.56449 + 1.17116i 1.56449 + 1.17116i
\(213\) −1.31058 + 0.285100i −1.31058 + 0.285100i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −0.755750 + 0.654861i −0.755750 + 0.654861i
\(221\) 0 0
\(222\) 0 0
\(223\) 0.0498610 + 0.133682i 0.0498610 + 0.133682i 0.959493 0.281733i \(-0.0909091\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(224\) 0 0
\(225\) −0.386565 + 0.334961i −0.386565 + 0.334961i
\(226\) 0 0
\(227\) 0 0 −0.997452 0.0713392i \(-0.977273\pi\)
0.997452 + 0.0713392i \(0.0227273\pi\)
\(228\) 0 0
\(229\) 0.284630 0.284630 0.142315 0.989821i \(-0.454545\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.997452 0.0713392i \(-0.977273\pi\)
0.997452 + 0.0713392i \(0.0227273\pi\)
\(234\) 0 0
\(235\) −1.50013 0.559521i −1.50013 0.559521i
\(236\) −0.909632 0.584585i −0.909632 0.584585i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(240\) −0.0498610 + 0.697148i −0.0498610 + 0.697148i
\(241\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(242\) 0 0
\(243\) 0.841254 + 0.313771i 0.841254 + 0.313771i
\(244\) 0 0
\(245\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.304632 + 1.03748i 0.304632 + 1.03748i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(252\) 0 0
\(253\) 0.540641 0.841254i 0.540641 0.841254i
\(254\) 0 0
\(255\) 0 0
\(256\) −0.654861 0.755750i −0.654861 0.755750i
\(257\) −1.90963 0.415415i −1.90963 0.415415i −0.909632 0.415415i \(-0.863636\pi\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.349464 0.936950i \(-0.386364\pi\)
−0.349464 + 0.936950i \(0.613636\pi\)
\(264\) 0 0
\(265\) 1.38189 1.38189i 1.38189 1.38189i
\(266\) 0 0
\(267\) 0.762003 + 1.01792i 0.762003 + 1.01792i
\(268\) 1.86912 0.133682i 1.86912 0.133682i
\(269\) 0.512546 + 0.234072i 0.512546 + 0.234072i 0.654861 0.755750i \(-0.272727\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(270\) 0 0
\(271\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.540641 + 0.841254i 0.540641 + 0.841254i
\(276\) −0.148568 0.682956i −0.148568 0.682956i
\(277\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(278\) 0 0
\(279\) −0.765261 + 0.663103i −0.765261 + 0.663103i
\(280\) 0 0
\(281\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(282\) 0 0
\(283\) 0 0 0.936950 0.349464i \(-0.113636\pi\)
−0.936950 + 0.349464i \(0.886364\pi\)
\(284\) −1.45027 1.25667i −1.45027 1.25667i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.909632 0.415415i 0.909632 0.415415i
\(290\) 0 0
\(291\) 0.633095 + 0.548580i 0.633095 + 0.548580i
\(292\) 0 0
\(293\) 0 0 −0.212565 0.977147i \(-0.568182\pi\)
0.212565 + 0.977147i \(0.431818\pi\)
\(294\) 0 0
\(295\) −0.708089 + 0.817178i −0.708089 + 0.817178i
\(296\) 0 0
\(297\) 0.506293 0.927206i 0.506293 0.927206i
\(298\) 0 0
\(299\) 0 0
\(300\) 0.682956 + 0.148568i 0.682956 + 0.148568i
\(301\) 0 0
\(302\) 0 0
\(303\) 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 0
\(309\) 0.579211 + 1.26830i 0.579211 + 1.26830i
\(310\) 0 0
\(311\) 1.61435 + 0.474017i 1.61435 + 0.474017i 0.959493 0.281733i \(-0.0909091\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(312\) 0 0
\(313\) −0.767317 + 0.574406i −0.767317 + 0.574406i −0.909632 0.415415i \(-0.863636\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.718267 0.959493i 0.718267 0.959493i −0.281733 0.959493i \(-0.590909\pi\)
1.00000 \(0\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.0639165 0.217680i −0.0639165 0.217680i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.0801894 0.557730i 0.0801894 0.557730i −0.909632 0.415415i \(-0.863636\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(332\) 0 0
\(333\) 0.104215 + 0.190855i 0.104215 + 0.190855i
\(334\) 0 0
\(335\) 0.133682 1.86912i 0.133682 1.86912i
\(336\) 0 0
\(337\) 0 0 −0.0713392 0.997452i \(-0.522727\pi\)
0.0713392 + 0.997452i \(0.477273\pi\)
\(338\) 0 0
\(339\) −0.941398 0.605000i −0.941398 0.605000i
\(340\) 0 0
\(341\) 1.07028 + 1.66538i 1.07028 + 1.66538i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −0.697148 + 0.0498610i −0.697148 + 0.0498610i
\(346\) 0 0
\(347\) 0 0 −0.877679 0.479249i \(-0.840909\pi\)
0.877679 + 0.479249i \(0.159091\pi\)
\(348\) 0 0
\(349\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.142315 1.98982i −0.142315 1.98982i −0.142315 0.989821i \(-0.545455\pi\)
1.00000i \(-0.5\pi\)
\(354\) 0 0
\(355\) −1.45027 + 1.25667i −1.45027 + 1.25667i
\(356\) −0.512546 + 1.74557i −0.512546 + 1.74557i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(360\) 0 0
\(361\) 0.415415 0.909632i 0.415415 0.909632i
\(362\) 0 0
\(363\) −0.559521 0.418852i −0.559521 0.418852i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.24123 + 1.24123i −1.24123 + 1.24123i −0.281733 + 0.959493i \(0.590909\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(368\) 0.654861 0.755750i 0.654861 0.755750i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 1.35201 + 0.294111i 1.35201 + 0.294111i
\(373\) 0 0 0.599278 0.800541i \(-0.295455\pi\)
−0.599278 + 0.800541i \(0.704545\pi\)
\(374\) 0 0
\(375\) 0.244250 0.654861i 0.244250 0.654861i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.61435 + 0.474017i 1.61435 + 0.474017i 0.959493 0.281733i \(-0.0909091\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.98982 0.142315i 1.98982 0.142315i 0.989821 0.142315i \(-0.0454545\pi\)
1.00000 \(0\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −0.0855040 + 1.19550i −0.0855040 + 1.19550i
\(389\) 1.25667 0.368991i 1.25667 0.368991i 0.415415 0.909632i \(-0.363636\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0.506293 0.0727939i 0.506293 0.0727939i
\(397\) 0.203743 + 0.936593i 0.203743 + 0.936593i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(401\) −0.258908 + 0.118239i −0.258908 + 0.118239i −0.540641 0.841254i \(-0.681818\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.224560 + 0.0322869i −0.224560 + 0.0322869i
\(406\) 0 0
\(407\) 0.398326 0.148568i 0.398326 0.148568i
\(408\) 0 0
\(409\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(410\) 0 0
\(411\) 0.0753648 0.0653040i 0.0753648 0.0653040i
\(412\) −0.956056 + 1.75089i −0.956056 + 1.75089i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.215109 1.49611i −0.215109 1.49611i −0.755750 0.654861i \(-0.772727\pi\)
0.540641 0.841254i \(-0.318182\pi\)
\(420\) 0 0
\(421\) 1.74557 + 0.797176i 1.74557 + 0.797176i 0.989821 + 0.142315i \(0.0454545\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(422\) 0 0
\(423\) 0.490780 + 0.655605i 0.490780 + 0.655605i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(432\) 0.633095 0.845715i 0.633095 0.845715i
\(433\) −0.139418 0.0303285i −0.139418 0.0303285i 0.142315 0.989821i \(-0.454545\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(440\) 0 0
\(441\) 0.430300 0.276537i 0.430300 0.276537i
\(442\) 0 0
\(443\) −0.936593 + 0.203743i −0.936593 + 0.203743i −0.654861 0.755750i \(-0.727273\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(444\) 0.123435 0.270284i 0.123435 0.270284i
\(445\) 1.65486 + 0.755750i 1.65486 + 0.755750i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.0801894 0.273100i 0.0801894 0.273100i −0.909632 0.415415i \(-0.863636\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −0.114220 1.59700i −0.114220 1.59700i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.997452 0.0713392i \(-0.977273\pi\)
0.997452 + 0.0713392i \(0.0227273\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −0.654861 0.755750i −0.654861 0.755750i
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.19550 0.0855040i −1.19550 0.0855040i −0.540641 0.841254i \(-0.681818\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(464\) 0 0
\(465\) 0.483529 1.29639i 0.483529 1.29639i
\(466\) 0 0
\(467\) 0.418852 + 1.12299i 0.418852 + 1.12299i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0.278474 0.948395i 0.278474 0.948395i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.976775 + 0.212484i −0.976775 + 0.212484i
\(478\) 0 0
\(479\) 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
\(484\) 1.00000i 1.00000i
\(485\) 1.17116 + 0.254771i 1.17116 + 0.254771i
\(486\) 0 0
\(487\) 1.83107 + 0.398326i 1.83107 + 0.398326i 0.989821 0.142315i \(-0.0454545\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(488\) 0 0
\(489\) −0.753813 + 1.17296i −0.753813 + 1.17296i
\(490\) 0 0
\(491\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0.511499i 0.511499i
\(496\) 0.822373 + 1.80075i 0.822373 + 1.80075i
\(497\) 0 0
\(498\) 0 0
\(499\) −1.65486 0.755750i −1.65486 0.755750i −0.654861 0.755750i \(-0.727273\pi\)
−1.00000 \(\pi\)
\(500\) 0.959493 0.281733i 0.959493 0.281733i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.0713392 0.997452i \(-0.477273\pi\)
−0.0713392 + 0.997452i \(0.522727\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.494217 + 0.494217i 0.494217 + 0.494217i
\(508\) 0 0
\(509\) −1.37491 + 1.19136i −1.37491 + 1.19136i −0.415415 + 0.909632i \(0.636364\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.59700 + 1.19550i 1.59700 + 1.19550i
\(516\) 0 0
\(517\) 1.40524 0.767317i 1.40524 0.767317i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.27155 + 1.10181i 1.27155 + 1.10181i 0.989821 + 0.142315i \(0.0454545\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(522\) 0 0
\(523\) 0 0 −0.212565 0.977147i \(-0.568182\pi\)
0.212565 + 0.977147i \(0.431818\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −0.494217 0.494217i −0.494217 0.494217i
\(529\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(530\) 0 0
\(531\) 0.530671 0.155819i 0.530671 0.155819i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −0.913069 + 0.0653040i −0.913069 + 0.0653040i
\(538\) 0 0
\(539\) −0.415415 0.909632i −0.415415 0.909632i
\(540\) −0.747009 0.747009i −0.747009 0.747009i
\(541\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(542\) 0 0
\(543\) −0.732817 + 0.548580i −0.732817 + 0.548580i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.599278 0.800541i \(-0.295455\pi\)
−0.599278 + 0.800541i \(0.704545\pi\)
\(548\) 0.139418 + 0.0303285i 0.139418 + 0.0303285i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −0.249967 0.160644i −0.249967 0.160644i
\(556\) 0 0
\(557\) 0 0 −0.800541 0.599278i \(-0.795455\pi\)
0.800541 + 0.599278i \(0.204545\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.479249 0.877679i \(-0.659091\pi\)
0.479249 + 0.877679i \(0.340909\pi\)
\(564\) 0.315271 1.07371i 0.315271 1.07371i
\(565\) −1.59700 0.114220i −1.59700 0.114220i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(570\) 0 0
\(571\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(572\) 0 0
\(573\) −0.509660 0.278295i −0.509660 0.278295i
\(574\) 0 0
\(575\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(576\) 0.511499 0.511499
\(577\) −0.373128 0.203743i −0.373128 0.203743i 0.281733 0.959493i \(-0.409091\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.139418 + 1.94931i 0.139418 + 1.94931i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.898064 + 1.64468i 0.898064 + 1.64468i 0.755750 + 0.654861i \(0.227273\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(588\) −0.654861 0.244250i −0.654861 0.244250i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.415415 0.0903680i 0.415415 0.0903680i
\(593\) 0 0 −0.800541 0.599278i \(-0.795455\pi\)
0.800541 + 0.599278i \(0.204545\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.410610 + 0.410610i −0.410610 + 0.410610i
\(598\) 0 0
\(599\) 0.830830i 0.830830i 0.909632 + 0.415415i \(0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(600\) 0 0
\(601\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(602\) 0 0
\(603\) −0.574406 + 0.767317i −0.574406 + 0.767317i
\(604\) 0 0
\(605\) −0.989821 0.142315i −0.989821 0.142315i
\(606\) 0 0
\(607\) 0 0 0.800541 0.599278i \(-0.204545\pi\)
−0.800541 + 0.599278i \(0.795455\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.997452 0.0713392i \(-0.0227273\pi\)
−0.997452 + 0.0713392i \(0.977273\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.398326 + 1.83107i −0.398326 + 1.83107i 0.142315 + 0.989821i \(0.454545\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(618\) 0 0
\(619\) −0.540641 + 0.158746i −0.540641 + 0.158746i −0.540641 0.841254i \(-0.681818\pi\)
1.00000i \(0.5\pi\)
\(620\) 1.89945 0.557730i 1.89945 0.557730i
\(621\) 0.927206 + 0.506293i 0.927206 + 0.506293i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.142315 0.989821i −0.142315 0.989821i
\(626\) 0 0
\(627\) 0 0
\(628\) 1.32505 0.494217i 1.32505 0.494217i
\(629\) 0 0
\(630\) 0 0
\(631\) 1.80075 0.822373i 1.80075 0.822373i 0.841254 0.540641i \(-0.181818\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 1.03229 + 0.894482i 1.03229 + 0.894482i
\(637\) 0 0
\(638\) 0 0
\(639\) 0.971569 0.139691i 0.971569 0.139691i
\(640\) 0 0
\(641\) −1.27155 + 1.10181i −1.27155 + 1.10181i −0.281733 + 0.959493i \(0.590909\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(642\) 0 0
\(643\) −1.13214 1.13214i −1.13214 1.13214i −0.989821 0.142315i \(-0.954545\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.0101786 + 0.142315i −0.0101786 + 0.142315i 0.989821 + 0.142315i \(0.0454545\pi\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −0.153882 1.07028i −0.153882 1.07028i
\(650\) 0 0
\(651\) 0 0
\(652\) −1.98982 + 0.142315i −1.98982 + 0.142315i
\(653\) 0.0855040 + 0.114220i 0.0855040 + 0.114220i 0.841254 0.540641i \(-0.181818\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(660\) −0.559521 + 0.418852i −0.559521 + 0.418852i
\(661\) −0.817178 + 1.27155i −0.817178 + 1.27155i 0.142315 + 0.989821i \(0.454545\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0.0280949 + 0.0956825i 0.0280949 + 0.0956825i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.977147 0.212565i \(-0.0681818\pi\)
−0.977147 + 0.212565i \(0.931818\pi\)
\(674\) 0 0
\(675\) −0.845715 + 0.633095i −0.845715 + 0.633095i
\(676\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(677\) 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
\(683\) −0.0498610 0.133682i −0.0498610 0.133682i 0.909632 0.415415i \(-0.136364\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(684\) 0 0
\(685\) 0.0498610 0.133682i 0.0498610 0.133682i
\(686\) 0 0
\(687\) 0.198429 + 0.0141919i 0.198429 + 0.0141919i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.30972 1.30972 0.654861 0.755750i \(-0.272727\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(692\) 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 0
\(701\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.142315 0.989821i 0.142315 0.989821i
\(705\) −1.01792 0.464867i −1.01792 0.464867i
\(706\) 0 0
\(707\) 0 0
\(708\) −0.605000 0.452897i −0.605000 0.452897i
\(709\) 1.53046 0.983568i 1.53046 0.983568i 0.540641 0.841254i \(-0.318182\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.66538 + 1.07028i −1.66538 + 1.07028i
\(714\) 0 0
\(715\) 0 0
\(716\) −0.857685 0.989821i −0.857685 0.989821i
\(717\) 0 0
\(718\) 0 0
\(719\) 0.449181 0.698939i 0.449181 0.698939i −0.540641 0.841254i \(-0.681818\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(720\) 0.0727939 0.506293i 0.0727939 0.506293i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −1.25667 0.368991i −1.25667 0.368991i
\(725\) 0 0
\(726\) 0 0
\(727\) −1.17116 1.56449i −1.17116 1.56449i −0.755750 0.654861i \(-0.772727\pi\)
−0.415415 0.909632i \(-0.636364\pi\)
\(728\) 0 0
\(729\) 0.777201 + 0.354936i 0.777201 + 0.354936i
\(730\) 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
\(735\) −0.334961 + 0.613435i −0.334961 + 0.613435i
\(736\) 0 0
\(737\) 1.32505 + 1.32505i 1.32505 + 1.32505i
\(738\) 0 0
\(739\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(740\) −0.0303285 0.424047i −0.0303285 0.424047i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.936950 0.349464i \(-0.113636\pi\)
−0.936950 + 0.349464i \(0.886364\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(752\) 1.50013 0.559521i 1.50013 0.559521i
\(753\) 0.160644 + 0.738467i 0.160644 + 0.738467i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.0683785 0.125226i 0.0683785 0.125226i −0.841254 0.540641i \(-0.818182\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(758\) 0 0
\(759\) 0.418852 0.559521i 0.418852 0.559521i
\(760\) 0 0
\(761\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −0.118239 0.822373i −0.118239 0.822373i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.418852 0.559521i −0.418852 0.559521i
\(769\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(770\) 0 0
\(771\) −1.31058 0.384822i −1.31058 0.384822i
\(772\) 0 0
\(773\) −0.959493 + 0.718267i −0.959493 + 0.718267i −0.959493 0.281733i \(-0.909091\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −0.281733 1.95949i −0.281733 1.95949i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 1.91899i 1.91899i
\(782\) 0 0
\(783\) 0 0
\(784\) −0.281733 0.959493i −0.281733 0.959493i
\(785\) −0.300613 1.38189i −0.300613 1.38189i
\(786\) 0 0
\(787\) 0 0 −0.800541 0.599278i \(-0.795455\pi\)
0.800541 + 0.599278i \(0.204545\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 1.03229 0.894482i 1.03229 0.894482i
\(796\) −0.822373 0.118239i −0.822373 0.118239i
\(797\) −0.133682 1.86912i −0.133682 1.86912i −0.415415 0.909632i \(-0.636364\pi\)
0.281733 0.959493i \(-0.409091\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −0.503094 0.782830i −0.503094 0.782830i
\(802\) 0 0
\(803\) 0 0
\(804\) 1.30972 1.30972
\(805\) 0 0
\(806\) 0 0
\(807\) 0.345649 + 0.188739i 0.345649 + 0.188739i
\(808\) 0 0
\(809\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(810\) 0 0
\(811\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.142315 + 1.98982i −0.142315 + 1.98982i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(822\) 0 0
\(823\) 1.56449 + 1.17116i 1.56449 + 1.17116i 0.909632 + 0.415415i \(0.136364\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(824\) 0 0
\(825\) 0.334961 + 0.613435i 0.334961 + 0.613435i
\(826\) 0 0
\(827\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(828\) 0.0727939 + 0.506293i 0.0727939 + 0.506293i
\(829\) 1.30972i 1.30972i −0.755750 0.654861i \(-0.772727\pi\)
0.755750 0.654861i \(-0.227273\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.67421 + 1.25330i −1.67421 + 1.25330i
\(838\) 0 0
\(839\) −0.273100 0.0801894i −0.273100 0.0801894i 0.142315 0.989821i \(-0.454545\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(840\) 0 0
\(841\) −0.415415 0.909632i −0.415415 0.909632i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(846\) 0 0
\(847\) 0 0
\(848\) −0.139418 + 1.94931i −0.139418 + 1.94931i
\(849\) 0 0
\(850\) 0 0
\(851\) 0.148568 + 0.398326i 0.148568 + 0.398326i
\(852\) −0.948395 0.948395i −0.948395 0.948395i
\(853\) 0 0 0.479249 0.877679i \(-0.340909\pi\)
−0.479249 + 0.877679i \(0.659091\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.212565 0.977147i \(-0.568182\pi\)
0.212565 + 0.977147i \(0.431818\pi\)
\(858\) 0 0
\(859\) −0.627899 0.544078i −0.627899 0.544078i 0.281733 0.959493i \(-0.409091\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −0.841254 + 0.459359i −0.841254 + 0.459359i −0.841254 0.540641i \(-0.818182\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.654861 0.244250i 0.654861 0.244250i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −0.433499 0.433499i −0.433499 0.433499i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.0713392 0.997452i \(-0.477273\pi\)
−0.0713392 + 0.997452i \(0.522727\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −0.959493 0.281733i −0.959493 0.281733i
\(881\) 1.37491 + 0.627899i 1.37491 + 0.627899i 0.959493 0.281733i \(-0.0909091\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(882\) 0 0
\(883\) −0.718267 0.959493i −0.718267 0.959493i 0.281733 0.959493i \(-0.409091\pi\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) −0.534388 + 0.534388i −0.534388 + 0.534388i
\(886\) 0 0
\(887\) 0 0 0.349464 0.936950i \(-0.386364\pi\)
−0.349464 + 0.936950i \(0.613636\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.122655 0.190855i 0.122655 0.190855i
\(892\) −0.0855040 + 0.114220i −0.0855040 + 0.114220i
\(893\) 0 0
\(894\) 0 0
\(895\) −1.10181 + 0.708089i −1.10181 + 0.708089i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.490780 0.144106i −0.490780 0.144106i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −0.544078 + 1.19136i −0.544078 + 1.19136i
\(906\) 0 0
\(907\) −1.64468 0.613435i −1.64468 0.613435i −0.654861 0.755750i \(-0.727273\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.29639 0.186393i −1.29639 0.186393i −0.540641 0.841254i \(-0.681818\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0.153882 + 0.239446i 0.153882 + 0.239446i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −0.424047 0.0303285i −0.424047 0.0303285i
\(926\) 0 0
\(927\) −0.356590 0.956056i −0.356590 0.956056i
\(928\) 0 0
\(929\) 1.80075 + 0.258908i 1.80075 + 0.258908i 0.959493 0.281733i \(-0.0909091\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.10181 + 0.410953i 1.10181 + 0.410953i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.977147 0.212565i \(-0.0681818\pi\)
−0.977147 + 0.212565i \(0.931818\pi\)
\(938\) 0 0
\(939\) −0.563574 + 0.362187i −0.563574 + 0.362187i
\(940\) −0.340335 1.56449i −0.340335 1.56449i
\(941\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.08128i 1.08128i
\(945\) 0 0
\(946\) 0 0
\(947\) −1.94931 0.424047i −1.94931 0.424047i −0.989821 0.142315i \(-0.954545\pi\)
−0.959493 0.281733i \(-0.909091\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0.548580 0.633095i 0.548580 0.633095i
\(952\) 0 0
\(953\) 0 0 0.349464 0.936950i \(-0.386364\pi\)
−0.349464 + 0.936950i \(0.613636\pi\)
\(954\) 0 0
\(955\) −0.830830 −0.830830
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −0.613435 + 0.334961i −0.613435 + 0.334961i
\(961\) −0.415415 2.88927i −0.415415 2.88927i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1.29639 + 0.186393i −1.29639 + 0.186393i −0.755750 0.654861i \(-0.772727\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(972\) 0.190855 + 0.877345i 0.190855 + 0.877345i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.75089 0.956056i 1.75089 0.956056i 0.841254 0.540641i \(-0.181818\pi\)
0.909632 0.415415i \(-0.136364\pi\)
\(978\) 0 0
\(979\) −1.65486 + 0.755750i −1.65486 + 0.755750i
\(980\) −0.989821 + 0.142315i −0.989821 + 0.142315i
\(981\) 0 0
\(982\) 0 0
\(983\) 0.0903680 + 0.415415i 0.0903680 + 0.415415i 1.00000 \(0\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0.540641 0.158746i 0.540641 0.158746i 1.00000i \(-0.5\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(992\) 0 0
\(993\) 0.0837128 0.384822i 0.0837128 0.384822i
\(994\) 0 0
\(995\) −0.234072 + 0.797176i −0.234072 + 0.797176i
\(996\) 0 0
\(997\) 0 0 0.997452 0.0713392i \(-0.0227273\pi\)
−0.997452 + 0.0713392i \(0.977273\pi\)
\(998\) 0 0
\(999\) 0.186571 + 0.408534i 0.186571 + 0.408534i
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 1265.1.bi.b.1187.1 yes 20
5.3 odd 4 1265.1.bi.a.428.1 20
11.10 odd 2 CM 1265.1.bi.b.1187.1 yes 20
23.5 odd 22 1265.1.bi.a.1132.1 yes 20
55.43 even 4 1265.1.bi.a.428.1 20
115.28 even 44 inner 1265.1.bi.b.373.1 yes 20
253.120 even 22 1265.1.bi.a.1132.1 yes 20
1265.373 odd 44 inner 1265.1.bi.b.373.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1265.1.bi.a.428.1 20 5.3 odd 4
1265.1.bi.a.428.1 20 55.43 even 4
1265.1.bi.a.1132.1 yes 20 23.5 odd 22
1265.1.bi.a.1132.1 yes 20 253.120 even 22
1265.1.bi.b.373.1 yes 20 115.28 even 44 inner
1265.1.bi.b.373.1 yes 20 1265.373 odd 44 inner
1265.1.bi.b.1187.1 yes 20 1.1 even 1 trivial
1265.1.bi.b.1187.1 yes 20 11.10 odd 2 CM