Properties

Label 1412.1.r.a.995.1
Level $1412$
Weight $1$
Character 1412.995
Analytic conductor $0.705$
Analytic rank $0$
Dimension $20$
Projective image $D_{44}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1412,1,Mod(35,1412)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1412, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([22, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1412.35");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1412 = 2^{2} \cdot 353 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1412.r (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: \(0.704679797838\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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 995.1
Root \(0.540641 + 0.841254i\) of defining polynomial
Character \(\chi\) \(=\) 1412.995
Dual form 1412.1.r.a.535.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.142315 + 0.989821i) q^{2} +(-0.959493 + 0.281733i) q^{4} +(-1.19550 - 1.59700i) q^{5} +(-0.415415 - 0.909632i) q^{8} +(-0.281733 + 0.959493i) q^{9} +O(q^{10})\) \(q+(0.142315 + 0.989821i) q^{2} +(-0.959493 + 0.281733i) q^{4} +(-1.19550 - 1.59700i) q^{5} +(-0.415415 - 0.909632i) q^{8} +(-0.281733 + 0.959493i) q^{9} +(1.41061 - 1.41061i) q^{10} +(-0.340335 + 1.56449i) q^{13} +(0.841254 - 0.540641i) q^{16} +(0.797176 + 1.74557i) q^{17} +(-0.989821 - 0.142315i) q^{18} +(1.59700 + 1.19550i) q^{20} +(-0.839462 + 2.85895i) q^{25} +(-1.59700 - 0.114220i) q^{26} +(-1.03748 - 0.304632i) q^{29} +(0.654861 + 0.755750i) q^{32} +(-1.61435 + 1.03748i) q^{34} -1.00000i q^{36} +(-0.244250 + 0.654861i) q^{37} +(-0.956056 + 1.75089i) q^{40} +(-0.627899 - 0.544078i) q^{41} +(1.86912 - 0.697148i) q^{45} +1.00000i q^{49} +(-2.94931 - 0.424047i) q^{50} +(-0.114220 - 1.59700i) q^{52} +(1.40524 - 1.05195i) q^{53} +(0.153882 - 1.07028i) q^{58} +(-0.909632 + 0.584585i) q^{61} +(-0.654861 + 0.755750i) q^{64} +(2.90537 - 1.32684i) q^{65} +(-1.25667 - 1.45027i) q^{68} +(0.989821 - 0.142315i) q^{72} +(-0.682956 - 0.148568i) q^{74} +(-1.86912 - 0.697148i) q^{80} +(-0.841254 - 0.540641i) q^{81} +(0.449181 - 0.698939i) q^{82} +(1.83466 - 3.35992i) q^{85} +(0.697148 + 1.86912i) q^{89} +(0.956056 + 1.75089i) q^{90} +(-0.118239 + 0.258908i) q^{97} +(-0.989821 + 0.142315i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} - 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} - 2 q^{4} - 2 q^{5} + 2 q^{8} + 2 q^{10} - 2 q^{13} - 2 q^{16} + 4 q^{17} - 2 q^{20} + 2 q^{26} + 2 q^{32} - 4 q^{34} - 20 q^{37} + 2 q^{40} + 2 q^{45} - 22 q^{50} - 2 q^{52} - 2 q^{53} - 2 q^{64} + 4 q^{68} - 2 q^{74} - 2 q^{80} + 2 q^{81} + 4 q^{85} - 2 q^{89} - 2 q^{90} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1412\mathbb{Z}\right)^\times\).

\(n\) \(707\) \(709\)
\(\chi(n)\) \(-1\) \(e\left(\frac{13}{44}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(3\) 0 0 −0.599278 0.800541i \(-0.704545\pi\)
0.599278 + 0.800541i \(0.295455\pi\)
\(4\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(5\) −1.19550 1.59700i −1.19550 1.59700i −0.654861 0.755750i \(-0.727273\pi\)
−0.540641 0.841254i \(-0.681818\pi\)
\(6\) 0 0
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) −0.415415 0.909632i −0.415415 0.909632i
\(9\) −0.281733 + 0.959493i −0.281733 + 0.959493i
\(10\) 1.41061 1.41061i 1.41061 1.41061i
\(11\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(12\) 0 0
\(13\) −0.340335 + 1.56449i −0.340335 + 1.56449i 0.415415 + 0.909632i \(0.363636\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.841254 0.540641i 0.841254 0.540641i
\(17\) 0.797176 + 1.74557i 0.797176 + 1.74557i 0.654861 + 0.755750i \(0.272727\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(18\) −0.989821 0.142315i −0.989821 0.142315i
\(19\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(20\) 1.59700 + 1.19550i 1.59700 + 1.19550i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(24\) 0 0
\(25\) −0.839462 + 2.85895i −0.839462 + 2.85895i
\(26\) −1.59700 0.114220i −1.59700 0.114220i
\(27\) 0 0
\(28\) 0 0
\(29\) −1.03748 0.304632i −1.03748 0.304632i −0.281733 0.959493i \(-0.590909\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(30\) 0 0
\(31\) 0 0 −0.0713392 0.997452i \(-0.522727\pi\)
0.0713392 + 0.997452i \(0.477273\pi\)
\(32\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(33\) 0 0
\(34\) −1.61435 + 1.03748i −1.61435 + 1.03748i
\(35\) 0 0
\(36\) 1.00000i 1.00000i
\(37\) −0.244250 + 0.654861i −0.244250 + 0.654861i 0.755750 + 0.654861i \(0.227273\pi\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.956056 + 1.75089i −0.956056 + 1.75089i
\(41\) −0.627899 0.544078i −0.627899 0.544078i 0.281733 0.959493i \(-0.409091\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(42\) 0 0
\(43\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(44\) 0 0
\(45\) 1.86912 0.697148i 1.86912 0.697148i
\(46\) 0 0
\(47\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(48\) 0 0
\(49\) 1.00000i 1.00000i
\(50\) −2.94931 0.424047i −2.94931 0.424047i
\(51\) 0 0
\(52\) −0.114220 1.59700i −0.114220 1.59700i
\(53\) 1.40524 1.05195i 1.40524 1.05195i 0.415415 0.909632i \(-0.363636\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0.153882 1.07028i 0.153882 1.07028i
\(59\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(60\) 0 0
\(61\) −0.909632 + 0.584585i −0.909632 + 0.584585i −0.909632 0.415415i \(-0.863636\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(65\) 2.90537 1.32684i 2.90537 1.32684i
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) −1.25667 1.45027i −1.25667 1.45027i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.877679 0.479249i \(-0.840909\pi\)
0.877679 + 0.479249i \(0.159091\pi\)
\(72\) 0.989821 0.142315i 0.989821 0.142315i
\(73\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(74\) −0.682956 0.148568i −0.682956 0.148568i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.599278 0.800541i \(-0.295455\pi\)
−0.599278 + 0.800541i \(0.704545\pi\)
\(80\) −1.86912 0.697148i −1.86912 0.697148i
\(81\) −0.841254 0.540641i −0.841254 0.540641i
\(82\) 0.449181 0.698939i 0.449181 0.698939i
\(83\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(84\) 0 0
\(85\) 1.83466 3.35992i 1.83466 3.35992i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.697148 + 1.86912i 0.697148 + 1.86912i 0.415415 + 0.909632i \(0.363636\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(90\) 0.956056 + 1.75089i 0.956056 + 1.75089i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.118239 + 0.258908i −0.118239 + 0.258908i −0.959493 0.281733i \(-0.909091\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(98\) −0.989821 + 0.142315i −0.989821 + 0.142315i
\(99\) 0 0
\(100\) 2.97964i 2.97964i
\(101\) −1.13214 + 1.13214i −1.13214 + 1.13214i −0.142315 + 0.989821i \(0.545455\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(102\) 0 0
\(103\) 0 0 −0.479249 0.877679i \(-0.659091\pi\)
0.479249 + 0.877679i \(0.340909\pi\)
\(104\) 1.56449 0.340335i 1.56449 0.340335i
\(105\) 0 0
\(106\) 1.24123 + 1.24123i 1.24123 + 1.24123i
\(107\) 0 0 0.997452 0.0713392i \(-0.0227273\pi\)
−0.997452 + 0.0713392i \(0.977273\pi\)
\(108\) 0 0
\(109\) 0.857685 0.989821i 0.857685 0.989821i −0.142315 0.989821i \(-0.545455\pi\)
1.00000 \(0\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.540641 1.84125i −0.540641 1.84125i −0.540641 0.841254i \(-0.681818\pi\)
1.00000i \(-0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.08128 1.08128
\(117\) −1.40524 0.767317i −1.40524 0.767317i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.142315 0.989821i −0.142315 0.989821i
\(122\) −0.708089 0.817178i −0.708089 0.817178i
\(123\) 0 0
\(124\) 0 0
\(125\) 3.70020 1.38010i 3.70020 1.38010i
\(126\) 0 0
\(127\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(128\) −0.841254 0.540641i −0.841254 0.540641i
\(129\) 0 0
\(130\) 1.72681 + 2.68697i 1.72681 + 2.68697i
\(131\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 1.25667 1.45027i 1.25667 1.45027i
\(137\) −0.300613 0.300613i −0.300613 0.300613i 0.540641 0.841254i \(-0.318182\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(138\) 0 0
\(139\) 0 0 −0.212565 0.977147i \(-0.568182\pi\)
0.212565 + 0.977147i \(0.431818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.281733 + 0.959493i 0.281733 + 0.959493i
\(145\) 0.753813 + 2.02105i 0.753813 + 2.02105i
\(146\) 0 0
\(147\) 0 0
\(148\) 0.0498610 0.697148i 0.0498610 0.697148i
\(149\) −0.0903680 + 0.415415i −0.0903680 + 0.415415i 0.909632 + 0.415415i \(0.136364\pi\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 −0.800541 0.599278i \(-0.795455\pi\)
0.800541 + 0.599278i \(0.204545\pi\)
\(152\) 0 0
\(153\) −1.89945 + 0.273100i −1.89945 + 0.273100i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.368991 + 1.25667i 0.368991 + 1.25667i 0.909632 + 0.415415i \(0.136364\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.424047 1.94931i 0.424047 1.94931i
\(161\) 0 0
\(162\) 0.415415 0.909632i 0.415415 0.909632i
\(163\) 0 0 −0.212565 0.977147i \(-0.568182\pi\)
0.212565 + 0.977147i \(0.431818\pi\)
\(164\) 0.755750 + 0.345139i 0.755750 + 0.345139i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(168\) 0 0
\(169\) −1.42218 0.649487i −1.42218 0.649487i
\(170\) 3.58682 + 1.33782i 3.58682 + 1.33782i
\(171\) 0 0
\(172\) 0 0
\(173\) −0.114220 + 0.0855040i −0.114220 + 0.0855040i −0.654861 0.755750i \(-0.727273\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −1.75089 + 0.956056i −1.75089 + 0.956056i
\(179\) 0 0 0.877679 0.479249i \(-0.159091\pi\)
−0.877679 + 0.479249i \(0.840909\pi\)
\(180\) −1.59700 + 1.19550i −1.59700 + 1.19550i
\(181\) −1.80075 0.822373i −1.80075 0.822373i −0.959493 0.281733i \(-0.909091\pi\)
−0.841254 0.540641i \(-0.818182\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.33782 0.392818i 1.33782 0.392818i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(192\) 0 0
\(193\) −0.300613 + 1.38189i −0.300613 + 1.38189i 0.540641 + 0.841254i \(0.318182\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(194\) −0.273100 0.0801894i −0.273100 0.0801894i
\(195\) 0 0
\(196\) −0.281733 0.959493i −0.281733 0.959493i
\(197\) −0.215109 + 0.186393i −0.215109 + 0.186393i −0.755750 0.654861i \(-0.772727\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(198\) 0 0
\(199\) 0 0 −0.212565 0.977147i \(-0.568182\pi\)
0.212565 + 0.977147i \(0.431818\pi\)
\(200\) 2.94931 0.424047i 2.94931 0.424047i
\(201\) 0 0
\(202\) −1.28173 0.959493i −1.28173 0.959493i
\(203\) 0 0
\(204\) 0 0
\(205\) −0.118239 + 1.65320i −0.118239 + 1.65320i
\(206\) 0 0
\(207\) 0 0
\(208\) 0.559521 + 1.50013i 0.559521 + 1.50013i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.349464 0.936950i \(-0.386364\pi\)
−0.349464 + 0.936950i \(0.613636\pi\)
\(212\) −1.05195 + 1.40524i −1.05195 + 1.40524i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 1.10181 + 0.708089i 1.10181 + 0.708089i
\(219\) 0 0
\(220\) 0 0
\(221\) −3.00224 + 0.653097i −3.00224 + 0.653097i
\(222\) 0 0
\(223\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(224\) 0 0
\(225\) −2.50664 1.61092i −2.50664 1.61092i
\(226\) 1.74557 0.797176i 1.74557 0.797176i
\(227\) 0 0 −0.599278 0.800541i \(-0.704545\pi\)
0.599278 + 0.800541i \(0.295455\pi\)
\(228\) 0 0
\(229\) 0.494217 + 1.32505i 0.494217 + 1.32505i 0.909632 + 0.415415i \(0.136364\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.153882 + 1.07028i 0.153882 + 1.07028i
\(233\) 1.80075 0.258908i 1.80075 0.258908i 0.841254 0.540641i \(-0.181818\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(234\) 0.559521 1.50013i 0.559521 1.50013i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.212565 0.977147i \(-0.568182\pi\)
0.212565 + 0.977147i \(0.431818\pi\)
\(240\) 0 0
\(241\) 1.71524 + 0.373128i 1.71524 + 0.373128i 0.959493 0.281733i \(-0.0909091\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(242\) 0.959493 0.281733i 0.959493 0.281733i
\(243\) 0 0
\(244\) 0.708089 0.817178i 0.708089 0.817178i
\(245\) 1.59700 1.19550i 1.59700 1.19550i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 1.89265 + 3.46613i 1.89265 + 3.46613i
\(251\) 0 0 −0.936950 0.349464i \(-0.886364\pi\)
0.936950 + 0.349464i \(0.113636\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.415415 0.909632i 0.415415 0.909632i
\(257\) 0.0303285 0.139418i 0.0303285 0.139418i −0.959493 0.281733i \(-0.909091\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.41387 + 2.09163i −2.41387 + 2.09163i
\(261\) 0.584585 0.909632i 0.584585 0.909632i
\(262\) 0 0
\(263\) 0 0 −0.479249 0.877679i \(-0.659091\pi\)
0.479249 + 0.877679i \(0.340909\pi\)
\(264\) 0 0
\(265\) −3.35992 0.986563i −3.35992 0.986563i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.215109 + 1.49611i 0.215109 + 1.49611i 0.755750 + 0.654861i \(0.227273\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(270\) 0 0
\(271\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(272\) 1.61435 + 1.03748i 1.61435 + 1.03748i
\(273\) 0 0
\(274\) 0.254771 0.340335i 0.254771 0.340335i
\(275\) 0 0
\(276\) 0 0
\(277\) 0.512546 1.74557i 0.512546 1.74557i −0.142315 0.989821i \(-0.545455\pi\)
0.654861 0.755750i \(-0.272727\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.49611 0.215109i 1.49611 0.215109i 0.654861 0.755750i \(-0.272727\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.909632 + 0.415415i −0.909632 + 0.415415i
\(289\) −1.75667 + 2.02730i −1.75667 + 2.02730i
\(290\) −1.89320 + 1.03377i −1.89320 + 1.03377i
\(291\) 0 0
\(292\) 0 0
\(293\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.697148 0.0498610i 0.697148 0.0498610i
\(297\) 0 0
\(298\) −0.424047 0.0303285i −0.424047 0.0303285i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.02105 + 0.753813i 2.02105 + 0.753813i
\(306\) −0.540641 1.84125i −0.540641 1.84125i
\(307\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −0.334961 + 0.613435i −0.334961 + 0.613435i −0.989821 0.142315i \(-0.954545\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(314\) −1.19136 + 0.544078i −1.19136 + 0.544078i
\(315\) 0 0
\(316\) 0 0
\(317\) 0.284630i 0.284630i −0.989821 0.142315i \(-0.954545\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.98982 + 0.142315i 1.98982 + 0.142315i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(325\) −4.18710 2.28633i −4.18710 2.28633i
\(326\) 0 0
\(327\) 0 0
\(328\) −0.234072 + 0.797176i −0.234072 + 0.797176i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(332\) 0 0
\(333\) −0.559521 0.418852i −0.559521 0.418852i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.698939 + 0.449181i −0.698939 + 0.449181i −0.841254 0.540641i \(-0.818182\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(338\) 0.440479 1.50013i 0.440479 1.50013i
\(339\) 0 0
\(340\) −0.813741 + 3.74071i −0.813741 + 3.74071i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −0.100889 0.100889i −0.100889 0.100889i
\(347\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0 0
\(349\) 1.74557 0.512546i 1.74557 0.512546i 0.755750 0.654861i \(-0.227273\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.415415 0.909632i 0.415415 0.909632i
\(354\) 0 0
\(355\) 0 0
\(356\) −1.19550 1.59700i −1.19550 1.59700i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(360\) −1.41061 1.41061i −1.41061 1.41061i
\(361\) −0.415415 0.909632i −0.415415 0.909632i
\(362\) 0.557730 1.89945i 0.557730 1.89945i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.599278 0.800541i \(-0.295455\pi\)
−0.599278 + 0.800541i \(0.704545\pi\)
\(368\) 0 0
\(369\) 0.698939 0.449181i 0.698939 0.449181i
\(370\) 0.579211 + 1.26830i 0.579211 + 1.26830i
\(371\) 0 0
\(372\) 0 0
\(373\) 1.13214 + 0.847507i 1.13214 + 0.847507i 0.989821 0.142315i \(-0.0454545\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.829686 1.51946i 0.829686 1.51946i
\(378\) 0 0
\(379\) 0 0 −0.997452 0.0713392i \(-0.977273\pi\)
0.997452 + 0.0713392i \(0.0227273\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.41061 0.100889i −1.41061 0.100889i
\(387\) 0 0
\(388\) 0.0405070 0.281733i 0.0405070 0.281733i
\(389\) 0.563465i 0.563465i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.909632 0.415415i 0.909632 0.415415i
\(393\) 0 0
\(394\) −0.215109 0.186393i −0.215109 0.186393i
\(395\) 0 0
\(396\) 0 0
\(397\) 0.544078 1.19136i 0.544078 1.19136i −0.415415 0.909632i \(-0.636364\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.839462 + 2.85895i 0.839462 + 2.85895i
\(401\) 1.32505 + 0.494217i 1.32505 + 0.494217i 0.909632 0.415415i \(-0.136364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0.767317 1.40524i 0.767317 1.40524i
\(405\) 0.142315 + 1.98982i 0.142315 + 1.98982i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.654861 + 1.75575i −0.654861 + 1.75575i 1.00000i \(0.5\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(410\) −1.65320 + 0.118239i −1.65320 + 0.118239i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −1.40524 + 0.767317i −1.40524 + 0.767317i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.0713392 0.997452i \(-0.522727\pi\)
0.0713392 + 0.997452i \(0.477273\pi\)
\(420\) 0 0
\(421\) 0.368991 + 0.425839i 0.368991 + 0.425839i 0.909632 0.415415i \(-0.136364\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −1.54064 0.841254i −1.54064 0.841254i
\(425\) −5.65969 + 0.813741i −5.65969 + 0.813741i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(432\) 0 0
\(433\) 1.12299 + 0.418852i 1.12299 + 0.418852i 0.841254 0.540641i \(-0.181818\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.544078 + 1.19136i −0.544078 + 1.19136i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(440\) 0 0
\(441\) −0.959493 0.281733i −0.959493 0.281733i
\(442\) −1.07371 2.87874i −1.07371 2.87874i
\(443\) 0 0 −0.479249 0.877679i \(-0.659091\pi\)
0.479249 + 0.877679i \(0.340909\pi\)
\(444\) 0 0
\(445\) 2.15156 3.34789i 2.15156 3.34789i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.373128 1.71524i 0.373128 1.71524i −0.281733 0.959493i \(-0.590909\pi\)
0.654861 0.755750i \(-0.272727\pi\)
\(450\) 1.23779 2.71038i 1.23779 2.71038i
\(451\) 0 0
\(452\) 1.03748 + 1.61435i 1.03748 + 1.61435i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.936593 0.203743i 0.936593 0.203743i 0.281733 0.959493i \(-0.409091\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(458\) −1.24123 + 0.677760i −1.24123 + 0.677760i
\(459\) 0 0
\(460\) 0 0
\(461\) 0.340335 0.254771i 0.340335 0.254771i −0.415415 0.909632i \(-0.636364\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(462\) 0 0
\(463\) 0 0 −0.0713392 0.997452i \(-0.522727\pi\)
0.0713392 + 0.997452i \(0.477273\pi\)
\(464\) −1.03748 + 0.304632i −1.03748 + 0.304632i
\(465\) 0 0
\(466\) 0.512546 + 1.74557i 0.512546 + 1.74557i
\(467\) 0 0 −0.212565 0.977147i \(-0.568182\pi\)
0.212565 + 0.977147i \(0.431818\pi\)
\(468\) 1.56449 + 0.340335i 1.56449 + 0.340335i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.613435 + 1.64468i 0.613435 + 1.64468i
\(478\) 0 0
\(479\) 0 0 −0.599278 0.800541i \(-0.704545\pi\)
0.599278 + 0.800541i \(0.295455\pi\)
\(480\) 0 0
\(481\) −0.941398 0.605000i −0.941398 0.605000i
\(482\) −0.125226 + 1.75089i −0.125226 + 1.75089i
\(483\) 0 0
\(484\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(485\) 0.554833 0.120696i 0.554833 0.120696i
\(486\) 0 0
\(487\) 0 0 −0.800541 0.599278i \(-0.795455\pi\)
0.800541 + 0.599278i \(0.204545\pi\)
\(488\) 0.909632 + 0.584585i 0.909632 + 0.584585i
\(489\) 0 0
\(490\) 1.41061 + 1.41061i 1.41061 + 1.41061i
\(491\) 0 0 0.0713392 0.997452i \(-0.477273\pi\)
−0.0713392 + 0.997452i \(0.522727\pi\)
\(492\) 0 0
\(493\) −0.295298 2.05384i −0.295298 2.05384i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(500\) −3.16150 + 2.36667i −3.16150 + 2.36667i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.479249 0.877679i \(-0.659091\pi\)
0.479249 + 0.877679i \(0.340909\pi\)
\(504\) 0 0
\(505\) 3.16150 + 0.454554i 3.16150 + 0.454554i
\(506\) 0 0
\(507\) 0 0
\(508\) 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
\(512\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(513\) 0 0
\(514\) 0.142315 + 0.0101786i 0.142315 + 0.0101786i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −2.41387 2.09163i −2.41387 2.09163i
\(521\) −1.74557 + 0.512546i −1.74557 + 0.512546i −0.989821 0.142315i \(-0.954545\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(522\) 0.983568 + 0.449181i 0.983568 + 0.449181i
\(523\) 0 0 −0.936950 0.349464i \(-0.886364\pi\)
0.936950 + 0.349464i \(0.113636\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.142315 0.989821i 0.142315 0.989821i
\(530\) 0.498354 3.46613i 0.498354 3.46613i
\(531\) 0 0
\(532\) 0 0
\(533\) 1.06490 0.797176i 1.06490 0.797176i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −1.45027 + 0.425839i −1.45027 + 0.425839i
\(539\) 0 0
\(540\) 0 0
\(541\) 0.983568 + 1.53046i 0.983568 + 1.53046i 0.841254 + 0.540641i \(0.181818\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.797176 + 1.74557i −0.797176 + 1.74557i
\(545\) −2.60611 0.186393i −2.60611 0.186393i
\(546\) 0 0
\(547\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(548\) 0.373128 + 0.203743i 0.373128 + 0.203743i
\(549\) −0.304632 1.03748i −0.304632 1.03748i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 1.80075 + 0.258908i 1.80075 + 0.258908i
\(555\) 0 0
\(556\) 0 0
\(557\) −0.415415 + 1.90963i −0.415415 + 1.90963i 1.00000i \(0.5\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.425839 + 1.45027i 0.425839 + 1.45027i
\(563\) 0 0 −0.599278 0.800541i \(-0.704545\pi\)
0.599278 + 0.800541i \(0.295455\pi\)
\(564\) 0 0
\(565\) −2.29415 + 3.06463i −2.29415 + 3.06463i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.677760 0.677760i −0.677760 0.677760i 0.281733 0.959493i \(-0.409091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(570\) 0 0
\(571\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.540641 0.841254i −0.540641 0.841254i
\(577\) −0.0303285 + 0.424047i −0.0303285 + 0.424047i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(578\) −2.25667 1.45027i −2.25667 1.45027i
\(579\) 0 0
\(580\) −1.29267 1.72681i −1.29267 1.72681i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0.454554 + 3.16150i 0.454554 + 3.16150i
\(586\) 1.97964 0.284630i 1.97964 0.284630i
\(587\) 0 0 0.349464 0.936950i \(-0.386364\pi\)
−0.349464 + 0.936950i \(0.613636\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.148568 + 0.682956i 0.148568 + 0.682956i
\(593\) −0.540641 1.84125i −0.540641 1.84125i −0.540641 0.841254i \(-0.681818\pi\)
1.00000i \(-0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.0303285 0.424047i −0.0303285 0.424047i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.997452 0.0713392i \(-0.0227273\pi\)
−0.997452 + 0.0713392i \(0.977273\pi\)
\(600\) 0 0
\(601\) −0.613435 + 0.334961i −0.613435 + 0.334961i −0.755750 0.654861i \(-0.772727\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.41061 + 1.41061i −1.41061 + 1.41061i
\(606\) 0 0
\(607\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −0.458515 + 2.10776i −0.458515 + 2.10776i
\(611\) 0 0
\(612\) 1.74557 0.797176i 1.74557 0.797176i
\(613\) 0.989821 0.857685i 0.989821 0.857685i 1.00000i \(-0.5\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.559521 1.50013i −0.559521 1.50013i −0.841254 0.540641i \(-0.818182\pi\)
0.281733 0.959493i \(-0.409091\pi\)
\(618\) 0 0
\(619\) 0 0 −0.349464 0.936950i \(-0.613636\pi\)
0.349464 + 0.936950i \(0.386364\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −4.12099 2.64840i −4.12099 2.64840i
\(626\) −0.654861 0.244250i −0.654861 0.244250i
\(627\) 0 0
\(628\) −0.708089 1.10181i −0.708089 1.10181i
\(629\) −1.33782 + 0.0956825i −1.33782 + 0.0956825i
\(630\) 0 0
\(631\) 0 0 0.936950 0.349464i \(-0.113636\pi\)
−0.936950 + 0.349464i \(0.886364\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0.281733 0.0405070i 0.281733 0.0405070i
\(635\) 0 0
\(636\) 0 0
\(637\) −1.56449 0.340335i −1.56449 0.340335i
\(638\) 0 0
\(639\) 0 0
\(640\) 0.142315 + 1.98982i 0.142315 + 1.98982i
\(641\) −1.19136 + 0.544078i −1.19136 + 0.544078i −0.909632 0.415415i \(-0.863636\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(642\) 0 0
\(643\) 0 0 0.877679 0.479249i \(-0.159091\pi\)
−0.877679 + 0.479249i \(0.840909\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(648\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(649\) 0 0
\(650\) 1.66717 4.46986i 1.66717 4.46986i
\(651\) 0 0
\(652\) 0 0
\(653\) −1.13214 + 0.847507i −1.13214 + 0.847507i −0.989821 0.142315i \(-0.954545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.822373 0.118239i −0.822373 0.118239i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(660\) 0 0
\(661\) −1.12299 + 0.418852i −1.12299 + 0.418852i −0.841254 0.540641i \(-0.818182\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.334961 0.613435i 0.334961 0.613435i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.956056 + 0.0683785i 0.956056 + 0.0683785i 0.540641 0.841254i \(-0.318182\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(674\) −0.544078 0.627899i −0.544078 0.627899i
\(675\) 0 0
\(676\) 1.54755 + 0.222504i 1.54755 + 0.222504i
\(677\) 0.273100 + 0.0801894i 0.273100 + 0.0801894i 0.415415 0.909632i \(-0.363636\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −3.81844 0.273100i −3.81844 0.273100i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(684\) 0 0
\(685\) −0.120696 + 0.839462i −0.120696 + 0.839462i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.16751 + 2.55650i 1.16751 + 2.55650i
\(690\) 0 0
\(691\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(692\) 0.0855040 0.114220i 0.0855040 0.114220i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0.449181 1.52977i 0.449181 1.52977i
\(698\) 0.755750 + 1.65486i 0.755750 + 1.65486i
\(699\) 0 0
\(700\) 0 0
\(701\) 1.19550 + 1.59700i 1.19550 + 1.59700i 0.654861 + 0.755750i \(0.272727\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(707\) 0 0
\(708\) 0 0
\(709\) 0.418852 + 0.559521i 0.418852 + 0.559521i 0.959493 0.281733i \(-0.0909091\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.41061 1.41061i 1.41061 1.41061i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.212565 0.977147i \(-0.431818\pi\)
−0.212565 + 0.977147i \(0.568182\pi\)
\(720\) 1.19550 1.59700i 1.19550 1.59700i
\(721\) 0 0
\(722\) 0.841254 0.540641i 0.841254 0.540641i
\(723\) 0 0
\(724\) 1.95949 + 0.281733i 1.95949 + 0.281733i
\(725\) 1.74185 2.71038i 1.74185 2.71038i
\(726\) 0 0
\(727\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(728\) 0 0
\(729\) 0.755750 0.654861i 0.755750 0.654861i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.83107 0.682956i 1.83107 0.682956i 0.841254 0.540641i \(-0.181818\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0.544078 + 0.627899i 0.544078 + 0.627899i
\(739\) 0 0 −0.997452 0.0713392i \(-0.977273\pi\)
0.997452 + 0.0713392i \(0.0227273\pi\)
\(740\) −1.17296 + 0.753813i −1.17296 + 0.753813i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.349464 0.936950i \(-0.386364\pi\)
−0.349464 + 0.936950i \(0.613636\pi\)
\(744\) 0 0
\(745\) 0.771454 0.352311i 0.771454 0.352311i
\(746\) −0.677760 + 1.24123i −0.677760 + 1.24123i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.936950 0.349464i \(-0.113636\pi\)
−0.936950 + 0.349464i \(0.886364\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 1.62207 + 0.605000i 1.62207 + 0.605000i
\(755\) 0 0
\(756\) 0 0
\(757\) 0.459359 0.841254i 0.459359 0.841254i −0.540641 0.841254i \(-0.681818\pi\)
1.00000 \(0\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.86912 + 0.133682i 1.86912 + 0.133682i 0.959493 0.281733i \(-0.0909091\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.70694 + 2.70694i 2.70694 + 2.70694i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1.24123 + 0.677760i −1.24123 + 0.677760i −0.959493 0.281733i \(-0.909091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.100889 1.41061i −0.100889 1.41061i
\(773\) 0.677760 0.677760i 0.677760 0.677760i −0.281733 0.959493i \(-0.590909\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0.284630 0.284630
\(777\) 0 0
\(778\) −0.557730 + 0.0801894i −0.557730 + 0.0801894i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.540641 + 0.841254i 0.540641 + 0.841254i
\(785\) 1.56577 2.09163i 1.56577 2.09163i
\(786\) 0 0
\(787\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(788\) 0.153882 0.239446i 0.153882 0.239446i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.605000 1.62207i −0.605000 1.62207i
\(794\) 1.25667 + 0.368991i 1.25667 + 0.368991i
\(795\) 0 0
\(796\) 0 0
\(797\) −0.239446 + 0.153882i −0.239446 + 0.153882i −0.654861 0.755750i \(-0.727273\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −2.71038 + 1.23779i −2.71038 + 1.23779i
\(801\) −1.98982 + 0.142315i −1.98982 + 0.142315i
\(802\) −0.300613 + 1.38189i −0.300613 + 1.38189i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 1.50013 + 0.559521i 1.50013 + 0.559521i
\(809\) 0.767317 + 1.40524i 0.767317 + 1.40524i 0.909632 + 0.415415i \(0.136364\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(810\) −1.94931 + 0.424047i −1.94931 + 0.424047i
\(811\) 0 0 0.877679 0.479249i \(-0.159091\pi\)
−0.877679 + 0.479249i \(0.840909\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −1.83107 0.398326i −1.83107 0.398326i
\(819\) 0 0
\(820\) −0.352311 1.61955i −0.352311 1.61955i
\(821\) −0.139418 0.0303285i −0.139418 0.0303285i 0.142315 0.989821i \(-0.454545\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(822\) 0 0
\(823\) 0 0 −0.877679 0.479249i \(-0.840909\pi\)
0.877679 + 0.479249i \(0.159091\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(828\) 0 0
\(829\) 0.114220 1.59700i 0.114220 1.59700i −0.540641 0.841254i \(-0.681818\pi\)
0.654861 0.755750i \(-0.272727\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −0.959493 1.28173i −0.959493 1.28173i
\(833\) −1.74557 + 0.797176i −1.74557 + 0.797176i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.977147 0.212565i \(-0.0681818\pi\)
−0.977147 + 0.212565i \(0.931818\pi\)
\(840\) 0 0
\(841\) 0.142315 + 0.0914602i 0.142315 + 0.0914602i
\(842\) −0.368991 + 0.425839i −0.368991 + 0.425839i
\(843\) 0 0
\(844\) 0 0
\(845\) 0.662983 + 3.04769i 0.662983 + 3.04769i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.613435 1.64468i 0.613435 1.64468i
\(849\) 0 0
\(850\) −1.61092 5.48628i −1.61092 5.48628i
\(851\) 0 0
\(852\) 0 0
\(853\) 1.59700 1.19550i 1.59700 1.19550i 0.755750 0.654861i \(-0.227273\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.767317 0.574406i −0.767317 0.574406i 0.142315 0.989821i \(-0.454545\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(858\) 0 0
\(859\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(864\) 0 0
\(865\) 0.273100 + 0.0801894i 0.273100 + 0.0801894i
\(866\) −0.254771 + 1.17116i −0.254771 + 1.17116i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −1.25667 0.368991i −1.25667 0.368991i
\(873\) −0.215109 0.186393i −0.215109 0.186393i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.544078 + 0.627899i 0.544078 + 0.627899i 0.959493 0.281733i \(-0.0909091\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.841254 + 0.459359i −0.841254 + 0.459359i −0.841254 0.540641i \(-0.818182\pi\)
1.00000i \(0.5\pi\)
\(882\) 0.142315 0.989821i 0.142315 0.989821i
\(883\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(884\) 2.69663 1.47247i 2.69663 1.47247i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 3.62001 + 1.65320i 3.62001 + 1.65320i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 1.75089 + 0.125226i 1.75089 + 0.125226i
\(899\) 0 0
\(900\) 2.85895 + 0.839462i 2.85895 + 0.839462i
\(901\) 2.95647 + 1.61435i 2.95647 + 1.61435i
\(902\) 0 0
\(903\) 0 0
\(904\) −1.45027 + 1.25667i −1.45027 + 1.25667i
\(905\) 0.839462 + 3.85895i 0.839462 + 3.85895i
\(906\) 0 0
\(907\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(908\) 0 0
\(909\) −0.767317 1.40524i −0.767317 1.40524i
\(910\) 0 0
\(911\) 0 0 0.0713392 0.997452i \(-0.477273\pi\)
−0.0713392 + 0.997452i \(0.522727\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0.334961 + 0.898064i 0.334961 + 0.898064i
\(915\) 0 0
\(916\) −0.847507 1.13214i −0.847507 1.13214i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.300613 + 0.300613i 0.300613 + 0.300613i
\(923\) 0 0
\(924\) 0 0
\(925\) −1.66717 1.24803i −1.66717 1.24803i
\(926\) 0 0
\(927\) 0 0
\(928\) −0.449181 0.983568i −0.449181 0.983568i
\(929\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.65486 + 0.755750i −1.65486 + 0.755750i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −0.114220 + 1.59700i −0.114220 + 1.59700i
\(937\) 0.857685 + 0.989821i 0.857685 + 0.989821i 1.00000 \(0\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −0.254771 + 0.340335i −0.254771 + 0.340335i −0.909632 0.415415i \(-0.863636\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.977147 0.212565i \(-0.931818\pi\)
0.977147 + 0.212565i \(0.0681818\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.494217 + 0.494217i 0.494217 + 0.494217i 0.909632 0.415415i \(-0.136364\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(954\) −1.54064 + 0.841254i −1.54064 + 0.841254i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.989821 + 0.142315i −0.989821 + 0.142315i
\(962\) 0.464867 1.01792i 0.464867 1.01792i
\(963\) 0 0
\(964\) −1.75089 + 0.125226i −1.75089 + 0.125226i
\(965\) 2.56627 1.17198i 2.56627 1.17198i
\(966\) 0 0
\(967\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(968\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(969\) 0 0
\(970\) 0.198429 + 0.532008i 0.198429 + 0.532008i
\(971\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −0.449181 + 0.983568i −0.449181 + 0.983568i
\(977\) −0.909632 + 1.41542i −0.909632 + 1.41542i 1.00000i \(0.5\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.19550 + 1.59700i −1.19550 + 1.59700i
\(981\) 0.708089 + 1.10181i 0.708089 + 1.10181i
\(982\) 0 0
\(983\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(984\) 0 0
\(985\) 0.554833 + 0.120696i 0.554833 + 0.120696i
\(986\) 1.99091 0.584585i 1.99091 0.584585i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.90963 0.415415i 1.90963 0.415415i 0.909632 0.415415i \(-0.136364\pi\)
1.00000 \(0\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1412.1.r.a.995.1 yes 20
4.3 odd 2 CM 1412.1.r.a.995.1 yes 20
353.182 even 44 inner 1412.1.r.a.535.1 20
1412.535 odd 44 inner 1412.1.r.a.535.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1412.1.r.a.535.1 20 353.182 even 44 inner
1412.1.r.a.535.1 20 1412.535 odd 44 inner
1412.1.r.a.995.1 yes 20 1.1 even 1 trivial
1412.1.r.a.995.1 yes 20 4.3 odd 2 CM