Properties

Label 8032.2.a.g.1.2
Level $8032$
Weight $2$
Character 8032.1
Self dual yes
Analytic conductor $64.136$
Analytic rank $1$
Dimension $30$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8032,2,Mod(1,8032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8032.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8032 = 2^{5} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8032.a (trivial)

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: yes
Analytic conductor: \(64.1358429035\)
Analytic rank: \(1\)
Dimension: \(30\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Character \(\chi\) \(=\) 8032.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.13996 q^{3} -2.26457 q^{5} +2.38671 q^{7} +6.85933 q^{9} +O(q^{10})\) \(q-3.13996 q^{3} -2.26457 q^{5} +2.38671 q^{7} +6.85933 q^{9} -2.83190 q^{11} +5.60226 q^{13} +7.11065 q^{15} +1.36097 q^{17} -2.30596 q^{19} -7.49415 q^{21} +0.856792 q^{23} +0.128269 q^{25} -12.1181 q^{27} +1.86972 q^{29} -4.41452 q^{31} +8.89204 q^{33} -5.40486 q^{35} -1.21387 q^{37} -17.5908 q^{39} +0.904775 q^{41} -1.53684 q^{43} -15.5334 q^{45} -0.899086 q^{47} -1.30364 q^{49} -4.27338 q^{51} -13.1544 q^{53} +6.41303 q^{55} +7.24061 q^{57} -4.36931 q^{59} -2.15372 q^{61} +16.3712 q^{63} -12.6867 q^{65} +12.5533 q^{67} -2.69029 q^{69} -5.16981 q^{71} +10.4687 q^{73} -0.402761 q^{75} -6.75891 q^{77} +5.89045 q^{79} +17.4724 q^{81} +9.87590 q^{83} -3.08200 q^{85} -5.87085 q^{87} +13.6159 q^{89} +13.3709 q^{91} +13.8614 q^{93} +5.22200 q^{95} +4.52615 q^{97} -19.4249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9} - 31 q^{11} + 5 q^{13} - 8 q^{15} - q^{17} - 29 q^{19} + 6 q^{21} - 27 q^{23} + 34 q^{25} - 36 q^{27} + q^{29} + 9 q^{31} - 8 q^{33} - 51 q^{35} + 5 q^{37} + 8 q^{39} - 3 q^{41} - 43 q^{43} - 42 q^{47} + 41 q^{49} - 54 q^{51} - 11 q^{53} + 10 q^{55} + 2 q^{57} - 49 q^{59} + 21 q^{61} - 15 q^{63} - 14 q^{65} - 68 q^{67} - 10 q^{69} - 44 q^{71} + 25 q^{73} - 51 q^{75} - 20 q^{77} + 49 q^{79} + 6 q^{81} - 88 q^{83} - 5 q^{85} - 16 q^{87} - 16 q^{89} - 46 q^{91} - 4 q^{93} - 28 q^{95} - 4 q^{97} - 71 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −3.13996 −1.81285 −0.906427 0.422362i \(-0.861201\pi\)
−0.906427 + 0.422362i \(0.861201\pi\)
\(4\) 0 0
\(5\) −2.26457 −1.01275 −0.506373 0.862315i \(-0.669014\pi\)
−0.506373 + 0.862315i \(0.669014\pi\)
\(6\) 0 0
\(7\) 2.38671 0.902090 0.451045 0.892501i \(-0.351051\pi\)
0.451045 + 0.892501i \(0.351051\pi\)
\(8\) 0 0
\(9\) 6.85933 2.28644
\(10\) 0 0
\(11\) −2.83190 −0.853850 −0.426925 0.904287i \(-0.640403\pi\)
−0.426925 + 0.904287i \(0.640403\pi\)
\(12\) 0 0
\(13\) 5.60226 1.55379 0.776893 0.629632i \(-0.216795\pi\)
0.776893 + 0.629632i \(0.216795\pi\)
\(14\) 0 0
\(15\) 7.11065 1.83596
\(16\) 0 0
\(17\) 1.36097 0.330083 0.165042 0.986287i \(-0.447224\pi\)
0.165042 + 0.986287i \(0.447224\pi\)
\(18\) 0 0
\(19\) −2.30596 −0.529023 −0.264512 0.964383i \(-0.585211\pi\)
−0.264512 + 0.964383i \(0.585211\pi\)
\(20\) 0 0
\(21\) −7.49415 −1.63536
\(22\) 0 0
\(23\) 0.856792 0.178653 0.0893267 0.996002i \(-0.471528\pi\)
0.0893267 + 0.996002i \(0.471528\pi\)
\(24\) 0 0
\(25\) 0.128269 0.0256539
\(26\) 0 0
\(27\) −12.1181 −2.33213
\(28\) 0 0
\(29\) 1.86972 0.347199 0.173599 0.984816i \(-0.444460\pi\)
0.173599 + 0.984816i \(0.444460\pi\)
\(30\) 0 0
\(31\) −4.41452 −0.792872 −0.396436 0.918062i \(-0.629753\pi\)
−0.396436 + 0.918062i \(0.629753\pi\)
\(32\) 0 0
\(33\) 8.89204 1.54791
\(34\) 0 0
\(35\) −5.40486 −0.913588
\(36\) 0 0
\(37\) −1.21387 −0.199559 −0.0997793 0.995010i \(-0.531814\pi\)
−0.0997793 + 0.995010i \(0.531814\pi\)
\(38\) 0 0
\(39\) −17.5908 −2.81679
\(40\) 0 0
\(41\) 0.904775 0.141302 0.0706510 0.997501i \(-0.477492\pi\)
0.0706510 + 0.997501i \(0.477492\pi\)
\(42\) 0 0
\(43\) −1.53684 −0.234366 −0.117183 0.993110i \(-0.537386\pi\)
−0.117183 + 0.993110i \(0.537386\pi\)
\(44\) 0 0
\(45\) −15.5334 −2.31559
\(46\) 0 0
\(47\) −0.899086 −0.131145 −0.0655726 0.997848i \(-0.520887\pi\)
−0.0655726 + 0.997848i \(0.520887\pi\)
\(48\) 0 0
\(49\) −1.30364 −0.186234
\(50\) 0 0
\(51\) −4.27338 −0.598393
\(52\) 0 0
\(53\) −13.1544 −1.80689 −0.903447 0.428701i \(-0.858971\pi\)
−0.903447 + 0.428701i \(0.858971\pi\)
\(54\) 0 0
\(55\) 6.41303 0.864732
\(56\) 0 0
\(57\) 7.24061 0.959042
\(58\) 0 0
\(59\) −4.36931 −0.568836 −0.284418 0.958700i \(-0.591800\pi\)
−0.284418 + 0.958700i \(0.591800\pi\)
\(60\) 0 0
\(61\) −2.15372 −0.275756 −0.137878 0.990449i \(-0.544028\pi\)
−0.137878 + 0.990449i \(0.544028\pi\)
\(62\) 0 0
\(63\) 16.3712 2.06258
\(64\) 0 0
\(65\) −12.6867 −1.57359
\(66\) 0 0
\(67\) 12.5533 1.53363 0.766814 0.641869i \(-0.221841\pi\)
0.766814 + 0.641869i \(0.221841\pi\)
\(68\) 0 0
\(69\) −2.69029 −0.323873
\(70\) 0 0
\(71\) −5.16981 −0.613543 −0.306772 0.951783i \(-0.599249\pi\)
−0.306772 + 0.951783i \(0.599249\pi\)
\(72\) 0 0
\(73\) 10.4687 1.22526 0.612632 0.790368i \(-0.290111\pi\)
0.612632 + 0.790368i \(0.290111\pi\)
\(74\) 0 0
\(75\) −0.402761 −0.0465068
\(76\) 0 0
\(77\) −6.75891 −0.770249
\(78\) 0 0
\(79\) 5.89045 0.662728 0.331364 0.943503i \(-0.392491\pi\)
0.331364 + 0.943503i \(0.392491\pi\)
\(80\) 0 0
\(81\) 17.4724 1.94138
\(82\) 0 0
\(83\) 9.87590 1.08402 0.542010 0.840372i \(-0.317663\pi\)
0.542010 + 0.840372i \(0.317663\pi\)
\(84\) 0 0
\(85\) −3.08200 −0.334290
\(86\) 0 0
\(87\) −5.87085 −0.629421
\(88\) 0 0
\(89\) 13.6159 1.44328 0.721640 0.692268i \(-0.243388\pi\)
0.721640 + 0.692268i \(0.243388\pi\)
\(90\) 0 0
\(91\) 13.3709 1.40166
\(92\) 0 0
\(93\) 13.8614 1.43736
\(94\) 0 0
\(95\) 5.22200 0.535766
\(96\) 0 0
\(97\) 4.52615 0.459560 0.229780 0.973243i \(-0.426199\pi\)
0.229780 + 0.973243i \(0.426199\pi\)
\(98\) 0 0
\(99\) −19.4249 −1.95228
\(100\) 0 0
\(101\) −5.24580 −0.521976 −0.260988 0.965342i \(-0.584048\pi\)
−0.260988 + 0.965342i \(0.584048\pi\)
\(102\) 0 0
\(103\) −2.44635 −0.241046 −0.120523 0.992711i \(-0.538457\pi\)
−0.120523 + 0.992711i \(0.538457\pi\)
\(104\) 0 0
\(105\) 16.9710 1.65620
\(106\) 0 0
\(107\) −17.1122 −1.65430 −0.827152 0.561979i \(-0.810040\pi\)
−0.827152 + 0.561979i \(0.810040\pi\)
\(108\) 0 0
\(109\) 11.8001 1.13024 0.565121 0.825008i \(-0.308829\pi\)
0.565121 + 0.825008i \(0.308829\pi\)
\(110\) 0 0
\(111\) 3.81149 0.361771
\(112\) 0 0
\(113\) 18.1037 1.70305 0.851525 0.524315i \(-0.175678\pi\)
0.851525 + 0.524315i \(0.175678\pi\)
\(114\) 0 0
\(115\) −1.94026 −0.180931
\(116\) 0 0
\(117\) 38.4277 3.55264
\(118\) 0 0
\(119\) 3.24823 0.297765
\(120\) 0 0
\(121\) −2.98035 −0.270941
\(122\) 0 0
\(123\) −2.84095 −0.256160
\(124\) 0 0
\(125\) 11.0324 0.986765
\(126\) 0 0
\(127\) −0.996277 −0.0884053 −0.0442026 0.999023i \(-0.514075\pi\)
−0.0442026 + 0.999023i \(0.514075\pi\)
\(128\) 0 0
\(129\) 4.82562 0.424872
\(130\) 0 0
\(131\) −15.6185 −1.36460 −0.682298 0.731075i \(-0.739019\pi\)
−0.682298 + 0.731075i \(0.739019\pi\)
\(132\) 0 0
\(133\) −5.50364 −0.477226
\(134\) 0 0
\(135\) 27.4423 2.36186
\(136\) 0 0
\(137\) −8.29331 −0.708545 −0.354273 0.935142i \(-0.615271\pi\)
−0.354273 + 0.935142i \(0.615271\pi\)
\(138\) 0 0
\(139\) −12.4062 −1.05228 −0.526141 0.850398i \(-0.676361\pi\)
−0.526141 + 0.850398i \(0.676361\pi\)
\(140\) 0 0
\(141\) 2.82309 0.237747
\(142\) 0 0
\(143\) −15.8650 −1.32670
\(144\) 0 0
\(145\) −4.23412 −0.351624
\(146\) 0 0
\(147\) 4.09336 0.337614
\(148\) 0 0
\(149\) −6.14114 −0.503102 −0.251551 0.967844i \(-0.580941\pi\)
−0.251551 + 0.967844i \(0.580941\pi\)
\(150\) 0 0
\(151\) 0.353629 0.0287779 0.0143889 0.999896i \(-0.495420\pi\)
0.0143889 + 0.999896i \(0.495420\pi\)
\(152\) 0 0
\(153\) 9.33532 0.754716
\(154\) 0 0
\(155\) 9.99699 0.802977
\(156\) 0 0
\(157\) −0.473917 −0.0378227 −0.0189113 0.999821i \(-0.506020\pi\)
−0.0189113 + 0.999821i \(0.506020\pi\)
\(158\) 0 0
\(159\) 41.3042 3.27563
\(160\) 0 0
\(161\) 2.04491 0.161162
\(162\) 0 0
\(163\) 6.31884 0.494930 0.247465 0.968897i \(-0.420402\pi\)
0.247465 + 0.968897i \(0.420402\pi\)
\(164\) 0 0
\(165\) −20.1366 −1.56763
\(166\) 0 0
\(167\) −8.37454 −0.648042 −0.324021 0.946050i \(-0.605035\pi\)
−0.324021 + 0.946050i \(0.605035\pi\)
\(168\) 0 0
\(169\) 18.3853 1.41425
\(170\) 0 0
\(171\) −15.8173 −1.20958
\(172\) 0 0
\(173\) 14.8959 1.13251 0.566256 0.824230i \(-0.308391\pi\)
0.566256 + 0.824230i \(0.308391\pi\)
\(174\) 0 0
\(175\) 0.306142 0.0231421
\(176\) 0 0
\(177\) 13.7195 1.03122
\(178\) 0 0
\(179\) −2.89130 −0.216106 −0.108053 0.994145i \(-0.534462\pi\)
−0.108053 + 0.994145i \(0.534462\pi\)
\(180\) 0 0
\(181\) −10.3437 −0.768840 −0.384420 0.923158i \(-0.625598\pi\)
−0.384420 + 0.923158i \(0.625598\pi\)
\(182\) 0 0
\(183\) 6.76260 0.499906
\(184\) 0 0
\(185\) 2.74889 0.202102
\(186\) 0 0
\(187\) −3.85412 −0.281841
\(188\) 0 0
\(189\) −28.9224 −2.10380
\(190\) 0 0
\(191\) 0.103580 0.00749476 0.00374738 0.999993i \(-0.498807\pi\)
0.00374738 + 0.999993i \(0.498807\pi\)
\(192\) 0 0
\(193\) −0.978935 −0.0704653 −0.0352326 0.999379i \(-0.511217\pi\)
−0.0352326 + 0.999379i \(0.511217\pi\)
\(194\) 0 0
\(195\) 39.8357 2.85269
\(196\) 0 0
\(197\) 5.41931 0.386110 0.193055 0.981188i \(-0.438160\pi\)
0.193055 + 0.981188i \(0.438160\pi\)
\(198\) 0 0
\(199\) −25.7543 −1.82567 −0.912836 0.408326i \(-0.866113\pi\)
−0.912836 + 0.408326i \(0.866113\pi\)
\(200\) 0 0
\(201\) −39.4168 −2.78025
\(202\) 0 0
\(203\) 4.46248 0.313205
\(204\) 0 0
\(205\) −2.04892 −0.143103
\(206\) 0 0
\(207\) 5.87702 0.408481
\(208\) 0 0
\(209\) 6.53024 0.451706
\(210\) 0 0
\(211\) 17.0671 1.17495 0.587475 0.809242i \(-0.300122\pi\)
0.587475 + 0.809242i \(0.300122\pi\)
\(212\) 0 0
\(213\) 16.2330 1.11227
\(214\) 0 0
\(215\) 3.48028 0.237353
\(216\) 0 0
\(217\) −10.5362 −0.715242
\(218\) 0 0
\(219\) −32.8712 −2.22123
\(220\) 0 0
\(221\) 7.62449 0.512879
\(222\) 0 0
\(223\) −0.756263 −0.0506431 −0.0253216 0.999679i \(-0.508061\pi\)
−0.0253216 + 0.999679i \(0.508061\pi\)
\(224\) 0 0
\(225\) 0.879843 0.0586562
\(226\) 0 0
\(227\) −7.41771 −0.492331 −0.246165 0.969228i \(-0.579171\pi\)
−0.246165 + 0.969228i \(0.579171\pi\)
\(228\) 0 0
\(229\) 18.5585 1.22638 0.613190 0.789935i \(-0.289886\pi\)
0.613190 + 0.789935i \(0.289886\pi\)
\(230\) 0 0
\(231\) 21.2227 1.39635
\(232\) 0 0
\(233\) 26.5430 1.73889 0.869444 0.494032i \(-0.164477\pi\)
0.869444 + 0.494032i \(0.164477\pi\)
\(234\) 0 0
\(235\) 2.03604 0.132817
\(236\) 0 0
\(237\) −18.4958 −1.20143
\(238\) 0 0
\(239\) 25.6986 1.66230 0.831152 0.556045i \(-0.187682\pi\)
0.831152 + 0.556045i \(0.187682\pi\)
\(240\) 0 0
\(241\) 1.17731 0.0758371 0.0379186 0.999281i \(-0.487927\pi\)
0.0379186 + 0.999281i \(0.487927\pi\)
\(242\) 0 0
\(243\) −18.5082 −1.18730
\(244\) 0 0
\(245\) 2.95217 0.188607
\(246\) 0 0
\(247\) −12.9186 −0.821989
\(248\) 0 0
\(249\) −31.0099 −1.96517
\(250\) 0 0
\(251\) 1.00000 0.0631194
\(252\) 0 0
\(253\) −2.42635 −0.152543
\(254\) 0 0
\(255\) 9.67736 0.606020
\(256\) 0 0
\(257\) −5.98720 −0.373471 −0.186736 0.982410i \(-0.559791\pi\)
−0.186736 + 0.982410i \(0.559791\pi\)
\(258\) 0 0
\(259\) −2.89715 −0.180020
\(260\) 0 0
\(261\) 12.8250 0.793850
\(262\) 0 0
\(263\) −19.6964 −1.21453 −0.607265 0.794499i \(-0.707733\pi\)
−0.607265 + 0.794499i \(0.707733\pi\)
\(264\) 0 0
\(265\) 29.7890 1.82992
\(266\) 0 0
\(267\) −42.7533 −2.61646
\(268\) 0 0
\(269\) −11.2348 −0.684997 −0.342499 0.939518i \(-0.611273\pi\)
−0.342499 + 0.939518i \(0.611273\pi\)
\(270\) 0 0
\(271\) 12.9484 0.786557 0.393279 0.919419i \(-0.371341\pi\)
0.393279 + 0.919419i \(0.371341\pi\)
\(272\) 0 0
\(273\) −41.9842 −2.54100
\(274\) 0 0
\(275\) −0.363246 −0.0219046
\(276\) 0 0
\(277\) 21.7827 1.30879 0.654396 0.756152i \(-0.272923\pi\)
0.654396 + 0.756152i \(0.272923\pi\)
\(278\) 0 0
\(279\) −30.2807 −1.81286
\(280\) 0 0
\(281\) −23.4725 −1.40025 −0.700127 0.714019i \(-0.746873\pi\)
−0.700127 + 0.714019i \(0.746873\pi\)
\(282\) 0 0
\(283\) 10.1297 0.602148 0.301074 0.953601i \(-0.402655\pi\)
0.301074 + 0.953601i \(0.402655\pi\)
\(284\) 0 0
\(285\) −16.3969 −0.971266
\(286\) 0 0
\(287\) 2.15943 0.127467
\(288\) 0 0
\(289\) −15.1478 −0.891045
\(290\) 0 0
\(291\) −14.2119 −0.833116
\(292\) 0 0
\(293\) 6.40062 0.373928 0.186964 0.982367i \(-0.440135\pi\)
0.186964 + 0.982367i \(0.440135\pi\)
\(294\) 0 0
\(295\) 9.89461 0.576086
\(296\) 0 0
\(297\) 34.3173 1.99129
\(298\) 0 0
\(299\) 4.79997 0.277589
\(300\) 0 0
\(301\) −3.66799 −0.211419
\(302\) 0 0
\(303\) 16.4716 0.946268
\(304\) 0 0
\(305\) 4.87725 0.279271
\(306\) 0 0
\(307\) 1.10123 0.0628504 0.0314252 0.999506i \(-0.489995\pi\)
0.0314252 + 0.999506i \(0.489995\pi\)
\(308\) 0 0
\(309\) 7.68142 0.436981
\(310\) 0 0
\(311\) 12.0589 0.683798 0.341899 0.939737i \(-0.388930\pi\)
0.341899 + 0.939737i \(0.388930\pi\)
\(312\) 0 0
\(313\) 2.16495 0.122370 0.0611851 0.998126i \(-0.480512\pi\)
0.0611851 + 0.998126i \(0.480512\pi\)
\(314\) 0 0
\(315\) −37.0737 −2.08887
\(316\) 0 0
\(317\) −21.6775 −1.21753 −0.608764 0.793351i \(-0.708334\pi\)
−0.608764 + 0.793351i \(0.708334\pi\)
\(318\) 0 0
\(319\) −5.29487 −0.296456
\(320\) 0 0
\(321\) 53.7317 2.99901
\(322\) 0 0
\(323\) −3.13833 −0.174622
\(324\) 0 0
\(325\) 0.718599 0.0398607
\(326\) 0 0
\(327\) −37.0517 −2.04897
\(328\) 0 0
\(329\) −2.14585 −0.118305
\(330\) 0 0
\(331\) −15.0529 −0.827381 −0.413691 0.910417i \(-0.635761\pi\)
−0.413691 + 0.910417i \(0.635761\pi\)
\(332\) 0 0
\(333\) −8.32632 −0.456279
\(334\) 0 0
\(335\) −28.4278 −1.55318
\(336\) 0 0
\(337\) −3.16475 −0.172395 −0.0861975 0.996278i \(-0.527472\pi\)
−0.0861975 + 0.996278i \(0.527472\pi\)
\(338\) 0 0
\(339\) −56.8447 −3.08738
\(340\) 0 0
\(341\) 12.5015 0.676993
\(342\) 0 0
\(343\) −19.8183 −1.07009
\(344\) 0 0
\(345\) 6.09234 0.328001
\(346\) 0 0
\(347\) −13.2042 −0.708839 −0.354419 0.935087i \(-0.615321\pi\)
−0.354419 + 0.935087i \(0.615321\pi\)
\(348\) 0 0
\(349\) 10.6989 0.572700 0.286350 0.958125i \(-0.407558\pi\)
0.286350 + 0.958125i \(0.407558\pi\)
\(350\) 0 0
\(351\) −67.8888 −3.62364
\(352\) 0 0
\(353\) −23.9918 −1.27696 −0.638478 0.769640i \(-0.720436\pi\)
−0.638478 + 0.769640i \(0.720436\pi\)
\(354\) 0 0
\(355\) 11.7074 0.621364
\(356\) 0 0
\(357\) −10.1993 −0.539804
\(358\) 0 0
\(359\) −26.1374 −1.37948 −0.689741 0.724056i \(-0.742276\pi\)
−0.689741 + 0.724056i \(0.742276\pi\)
\(360\) 0 0
\(361\) −13.6826 −0.720135
\(362\) 0 0
\(363\) 9.35817 0.491177
\(364\) 0 0
\(365\) −23.7070 −1.24088
\(366\) 0 0
\(367\) 14.1155 0.736825 0.368412 0.929662i \(-0.379901\pi\)
0.368412 + 0.929662i \(0.379901\pi\)
\(368\) 0 0
\(369\) 6.20615 0.323079
\(370\) 0 0
\(371\) −31.3956 −1.62998
\(372\) 0 0
\(373\) −35.3420 −1.82994 −0.914968 0.403525i \(-0.867785\pi\)
−0.914968 + 0.403525i \(0.867785\pi\)
\(374\) 0 0
\(375\) −34.6412 −1.78886
\(376\) 0 0
\(377\) 10.4747 0.539473
\(378\) 0 0
\(379\) 34.1121 1.75222 0.876111 0.482110i \(-0.160129\pi\)
0.876111 + 0.482110i \(0.160129\pi\)
\(380\) 0 0
\(381\) 3.12827 0.160266
\(382\) 0 0
\(383\) −29.8940 −1.52751 −0.763757 0.645504i \(-0.776647\pi\)
−0.763757 + 0.645504i \(0.776647\pi\)
\(384\) 0 0
\(385\) 15.3060 0.780067
\(386\) 0 0
\(387\) −10.5417 −0.535865
\(388\) 0 0
\(389\) −2.32922 −0.118096 −0.0590480 0.998255i \(-0.518807\pi\)
−0.0590480 + 0.998255i \(0.518807\pi\)
\(390\) 0 0
\(391\) 1.16607 0.0589705
\(392\) 0 0
\(393\) 49.0414 2.47381
\(394\) 0 0
\(395\) −13.3393 −0.671175
\(396\) 0 0
\(397\) 18.3689 0.921907 0.460953 0.887424i \(-0.347508\pi\)
0.460953 + 0.887424i \(0.347508\pi\)
\(398\) 0 0
\(399\) 17.2812 0.865142
\(400\) 0 0
\(401\) −38.3880 −1.91701 −0.958503 0.285082i \(-0.907979\pi\)
−0.958503 + 0.285082i \(0.907979\pi\)
\(402\) 0 0
\(403\) −24.7313 −1.23195
\(404\) 0 0
\(405\) −39.5674 −1.96612
\(406\) 0 0
\(407\) 3.43755 0.170393
\(408\) 0 0
\(409\) −7.56170 −0.373902 −0.186951 0.982369i \(-0.559861\pi\)
−0.186951 + 0.982369i \(0.559861\pi\)
\(410\) 0 0
\(411\) 26.0406 1.28449
\(412\) 0 0
\(413\) −10.4283 −0.513142
\(414\) 0 0
\(415\) −22.3646 −1.09784
\(416\) 0 0
\(417\) 38.9550 1.90763
\(418\) 0 0
\(419\) −0.448297 −0.0219007 −0.0109504 0.999940i \(-0.503486\pi\)
−0.0109504 + 0.999940i \(0.503486\pi\)
\(420\) 0 0
\(421\) −16.0003 −0.779806 −0.389903 0.920856i \(-0.627492\pi\)
−0.389903 + 0.920856i \(0.627492\pi\)
\(422\) 0 0
\(423\) −6.16713 −0.299856
\(424\) 0 0
\(425\) 0.174571 0.00846792
\(426\) 0 0
\(427\) −5.14030 −0.248757
\(428\) 0 0
\(429\) 49.8155 2.40511
\(430\) 0 0
\(431\) 14.9477 0.720006 0.360003 0.932951i \(-0.382776\pi\)
0.360003 + 0.932951i \(0.382776\pi\)
\(432\) 0 0
\(433\) −11.0750 −0.532231 −0.266116 0.963941i \(-0.585740\pi\)
−0.266116 + 0.963941i \(0.585740\pi\)
\(434\) 0 0
\(435\) 13.2949 0.637444
\(436\) 0 0
\(437\) −1.97573 −0.0945118
\(438\) 0 0
\(439\) −0.213196 −0.0101753 −0.00508765 0.999987i \(-0.501619\pi\)
−0.00508765 + 0.999987i \(0.501619\pi\)
\(440\) 0 0
\(441\) −8.94206 −0.425812
\(442\) 0 0
\(443\) −10.3439 −0.491455 −0.245727 0.969339i \(-0.579027\pi\)
−0.245727 + 0.969339i \(0.579027\pi\)
\(444\) 0 0
\(445\) −30.8341 −1.46168
\(446\) 0 0
\(447\) 19.2829 0.912051
\(448\) 0 0
\(449\) 7.79773 0.367998 0.183999 0.982926i \(-0.441096\pi\)
0.183999 + 0.982926i \(0.441096\pi\)
\(450\) 0 0
\(451\) −2.56223 −0.120651
\(452\) 0 0
\(453\) −1.11038 −0.0521701
\(454\) 0 0
\(455\) −30.2794 −1.41952
\(456\) 0 0
\(457\) 17.3806 0.813030 0.406515 0.913644i \(-0.366744\pi\)
0.406515 + 0.913644i \(0.366744\pi\)
\(458\) 0 0
\(459\) −16.4924 −0.769798
\(460\) 0 0
\(461\) 1.34676 0.0627249 0.0313625 0.999508i \(-0.490015\pi\)
0.0313625 + 0.999508i \(0.490015\pi\)
\(462\) 0 0
\(463\) 3.66132 0.170156 0.0850781 0.996374i \(-0.472886\pi\)
0.0850781 + 0.996374i \(0.472886\pi\)
\(464\) 0 0
\(465\) −31.3901 −1.45568
\(466\) 0 0
\(467\) −23.1204 −1.06988 −0.534942 0.844889i \(-0.679667\pi\)
−0.534942 + 0.844889i \(0.679667\pi\)
\(468\) 0 0
\(469\) 29.9610 1.38347
\(470\) 0 0
\(471\) 1.48808 0.0685670
\(472\) 0 0
\(473\) 4.35218 0.200114
\(474\) 0 0
\(475\) −0.295784 −0.0135715
\(476\) 0 0
\(477\) −90.2302 −4.13136
\(478\) 0 0
\(479\) −19.0089 −0.868540 −0.434270 0.900783i \(-0.642994\pi\)
−0.434270 + 0.900783i \(0.642994\pi\)
\(480\) 0 0
\(481\) −6.80040 −0.310071
\(482\) 0 0
\(483\) −6.42093 −0.292162
\(484\) 0 0
\(485\) −10.2498 −0.465418
\(486\) 0 0
\(487\) −29.7339 −1.34737 −0.673686 0.739017i \(-0.735290\pi\)
−0.673686 + 0.739017i \(0.735290\pi\)
\(488\) 0 0
\(489\) −19.8409 −0.897236
\(490\) 0 0
\(491\) −7.90998 −0.356973 −0.178486 0.983942i \(-0.557120\pi\)
−0.178486 + 0.983942i \(0.557120\pi\)
\(492\) 0 0
\(493\) 2.54463 0.114604
\(494\) 0 0
\(495\) 43.9891 1.97716
\(496\) 0 0
\(497\) −12.3388 −0.553471
\(498\) 0 0
\(499\) 15.8519 0.709628 0.354814 0.934937i \(-0.384544\pi\)
0.354814 + 0.934937i \(0.384544\pi\)
\(500\) 0 0
\(501\) 26.2957 1.17481
\(502\) 0 0
\(503\) 22.6578 1.01026 0.505132 0.863042i \(-0.331444\pi\)
0.505132 + 0.863042i \(0.331444\pi\)
\(504\) 0 0
\(505\) 11.8795 0.528629
\(506\) 0 0
\(507\) −57.7290 −2.56383
\(508\) 0 0
\(509\) −10.8726 −0.481919 −0.240959 0.970535i \(-0.577462\pi\)
−0.240959 + 0.970535i \(0.577462\pi\)
\(510\) 0 0
\(511\) 24.9856 1.10530
\(512\) 0 0
\(513\) 27.9439 1.23375
\(514\) 0 0
\(515\) 5.53992 0.244118
\(516\) 0 0
\(517\) 2.54612 0.111978
\(518\) 0 0
\(519\) −46.7724 −2.05308
\(520\) 0 0
\(521\) 2.93700 0.128672 0.0643362 0.997928i \(-0.479507\pi\)
0.0643362 + 0.997928i \(0.479507\pi\)
\(522\) 0 0
\(523\) 16.7036 0.730397 0.365198 0.930930i \(-0.381001\pi\)
0.365198 + 0.930930i \(0.381001\pi\)
\(524\) 0 0
\(525\) −0.961271 −0.0419533
\(526\) 0 0
\(527\) −6.00802 −0.261714
\(528\) 0 0
\(529\) −22.2659 −0.968083
\(530\) 0 0
\(531\) −29.9706 −1.30061
\(532\) 0 0
\(533\) 5.06878 0.219553
\(534\) 0 0
\(535\) 38.7518 1.67539
\(536\) 0 0
\(537\) 9.07855 0.391768
\(538\) 0 0
\(539\) 3.69176 0.159015
\(540\) 0 0
\(541\) 34.6446 1.48949 0.744745 0.667350i \(-0.232571\pi\)
0.744745 + 0.667350i \(0.232571\pi\)
\(542\) 0 0
\(543\) 32.4787 1.39379
\(544\) 0 0
\(545\) −26.7221 −1.14465
\(546\) 0 0
\(547\) 41.8744 1.79042 0.895210 0.445644i \(-0.147025\pi\)
0.895210 + 0.445644i \(0.147025\pi\)
\(548\) 0 0
\(549\) −14.7731 −0.630500
\(550\) 0 0
\(551\) −4.31150 −0.183676
\(552\) 0 0
\(553\) 14.0588 0.597840
\(554\) 0 0
\(555\) −8.63139 −0.366382
\(556\) 0 0
\(557\) −11.1774 −0.473604 −0.236802 0.971558i \(-0.576099\pi\)
−0.236802 + 0.971558i \(0.576099\pi\)
\(558\) 0 0
\(559\) −8.60978 −0.364155
\(560\) 0 0
\(561\) 12.1018 0.510937
\(562\) 0 0
\(563\) 32.5168 1.37042 0.685210 0.728346i \(-0.259711\pi\)
0.685210 + 0.728346i \(0.259711\pi\)
\(564\) 0 0
\(565\) −40.9970 −1.72476
\(566\) 0 0
\(567\) 41.7015 1.75130
\(568\) 0 0
\(569\) −28.9106 −1.21200 −0.605999 0.795466i \(-0.707226\pi\)
−0.605999 + 0.795466i \(0.707226\pi\)
\(570\) 0 0
\(571\) 15.0723 0.630755 0.315378 0.948966i \(-0.397869\pi\)
0.315378 + 0.948966i \(0.397869\pi\)
\(572\) 0 0
\(573\) −0.325236 −0.0135869
\(574\) 0 0
\(575\) 0.109900 0.00458316
\(576\) 0 0
\(577\) −12.8922 −0.536709 −0.268355 0.963320i \(-0.586480\pi\)
−0.268355 + 0.963320i \(0.586480\pi\)
\(578\) 0 0
\(579\) 3.07381 0.127743
\(580\) 0 0
\(581\) 23.5709 0.977884
\(582\) 0 0
\(583\) 37.2519 1.54281
\(584\) 0 0
\(585\) −87.0222 −3.59792
\(586\) 0 0
\(587\) −9.44592 −0.389875 −0.194937 0.980816i \(-0.562450\pi\)
−0.194937 + 0.980816i \(0.562450\pi\)
\(588\) 0 0
\(589\) 10.1797 0.419447
\(590\) 0 0
\(591\) −17.0164 −0.699962
\(592\) 0 0
\(593\) −33.3718 −1.37041 −0.685207 0.728349i \(-0.740288\pi\)
−0.685207 + 0.728349i \(0.740288\pi\)
\(594\) 0 0
\(595\) −7.35584 −0.301560
\(596\) 0 0
\(597\) 80.8673 3.30968
\(598\) 0 0
\(599\) −28.2779 −1.15540 −0.577702 0.816248i \(-0.696051\pi\)
−0.577702 + 0.816248i \(0.696051\pi\)
\(600\) 0 0
\(601\) 6.30320 0.257113 0.128556 0.991702i \(-0.458966\pi\)
0.128556 + 0.991702i \(0.458966\pi\)
\(602\) 0 0
\(603\) 86.1071 3.50655
\(604\) 0 0
\(605\) 6.74921 0.274394
\(606\) 0 0
\(607\) −25.8813 −1.05049 −0.525245 0.850951i \(-0.676026\pi\)
−0.525245 + 0.850951i \(0.676026\pi\)
\(608\) 0 0
\(609\) −14.0120 −0.567795
\(610\) 0 0
\(611\) −5.03691 −0.203772
\(612\) 0 0
\(613\) 0.402225 0.0162457 0.00812285 0.999967i \(-0.497414\pi\)
0.00812285 + 0.999967i \(0.497414\pi\)
\(614\) 0 0
\(615\) 6.43353 0.259425
\(616\) 0 0
\(617\) −15.8402 −0.637703 −0.318851 0.947805i \(-0.603297\pi\)
−0.318851 + 0.947805i \(0.603297\pi\)
\(618\) 0 0
\(619\) 43.6388 1.75399 0.876995 0.480499i \(-0.159545\pi\)
0.876995 + 0.480499i \(0.159545\pi\)
\(620\) 0 0
\(621\) −10.3827 −0.416644
\(622\) 0 0
\(623\) 32.4971 1.30197
\(624\) 0 0
\(625\) −25.6249 −1.02500
\(626\) 0 0
\(627\) −20.5047 −0.818878
\(628\) 0 0
\(629\) −1.65203 −0.0658709
\(630\) 0 0
\(631\) −15.4687 −0.615798 −0.307899 0.951419i \(-0.599626\pi\)
−0.307899 + 0.951419i \(0.599626\pi\)
\(632\) 0 0
\(633\) −53.5901 −2.13001
\(634\) 0 0
\(635\) 2.25614 0.0895321
\(636\) 0 0
\(637\) −7.30330 −0.289367
\(638\) 0 0
\(639\) −35.4614 −1.40283
\(640\) 0 0
\(641\) −8.69808 −0.343554 −0.171777 0.985136i \(-0.554951\pi\)
−0.171777 + 0.985136i \(0.554951\pi\)
\(642\) 0 0
\(643\) −11.8322 −0.466618 −0.233309 0.972403i \(-0.574955\pi\)
−0.233309 + 0.972403i \(0.574955\pi\)
\(644\) 0 0
\(645\) −10.9279 −0.430287
\(646\) 0 0
\(647\) −8.08234 −0.317750 −0.158875 0.987299i \(-0.550787\pi\)
−0.158875 + 0.987299i \(0.550787\pi\)
\(648\) 0 0
\(649\) 12.3735 0.485701
\(650\) 0 0
\(651\) 33.0831 1.29663
\(652\) 0 0
\(653\) −19.6037 −0.767154 −0.383577 0.923509i \(-0.625308\pi\)
−0.383577 + 0.923509i \(0.625308\pi\)
\(654\) 0 0
\(655\) 35.3692 1.38199
\(656\) 0 0
\(657\) 71.8080 2.80150
\(658\) 0 0
\(659\) −23.1836 −0.903105 −0.451552 0.892245i \(-0.649130\pi\)
−0.451552 + 0.892245i \(0.649130\pi\)
\(660\) 0 0
\(661\) 19.7025 0.766339 0.383169 0.923678i \(-0.374833\pi\)
0.383169 + 0.923678i \(0.374833\pi\)
\(662\) 0 0
\(663\) −23.9406 −0.929774
\(664\) 0 0
\(665\) 12.4634 0.483309
\(666\) 0 0
\(667\) 1.60196 0.0620283
\(668\) 0 0
\(669\) 2.37463 0.0918086
\(670\) 0 0
\(671\) 6.09912 0.235454
\(672\) 0 0
\(673\) −48.8593 −1.88339 −0.941695 0.336469i \(-0.890767\pi\)
−0.941695 + 0.336469i \(0.890767\pi\)
\(674\) 0 0
\(675\) −1.55439 −0.0598283
\(676\) 0 0
\(677\) −41.5110 −1.59540 −0.797699 0.603056i \(-0.793950\pi\)
−0.797699 + 0.603056i \(0.793950\pi\)
\(678\) 0 0
\(679\) 10.8026 0.414565
\(680\) 0 0
\(681\) 23.2913 0.892524
\(682\) 0 0
\(683\) −29.7708 −1.13915 −0.569574 0.821940i \(-0.692892\pi\)
−0.569574 + 0.821940i \(0.692892\pi\)
\(684\) 0 0
\(685\) 18.7808 0.717576
\(686\) 0 0
\(687\) −58.2729 −2.22325
\(688\) 0 0
\(689\) −73.6942 −2.80753
\(690\) 0 0
\(691\) 2.22104 0.0844924 0.0422462 0.999107i \(-0.486549\pi\)
0.0422462 + 0.999107i \(0.486549\pi\)
\(692\) 0 0
\(693\) −46.3616 −1.76113
\(694\) 0 0
\(695\) 28.0947 1.06569
\(696\) 0 0
\(697\) 1.23137 0.0466414
\(698\) 0 0
\(699\) −83.3438 −3.15235
\(700\) 0 0
\(701\) −29.8962 −1.12916 −0.564582 0.825377i \(-0.690963\pi\)
−0.564582 + 0.825377i \(0.690963\pi\)
\(702\) 0 0
\(703\) 2.79913 0.105571
\(704\) 0 0
\(705\) −6.39309 −0.240778
\(706\) 0 0
\(707\) −12.5202 −0.470870
\(708\) 0 0
\(709\) 0.489981 0.0184016 0.00920081 0.999958i \(-0.497071\pi\)
0.00920081 + 0.999958i \(0.497071\pi\)
\(710\) 0 0
\(711\) 40.4046 1.51529
\(712\) 0 0
\(713\) −3.78233 −0.141649
\(714\) 0 0
\(715\) 35.9274 1.34361
\(716\) 0 0
\(717\) −80.6925 −3.01352
\(718\) 0 0
\(719\) −6.84580 −0.255305 −0.127653 0.991819i \(-0.540744\pi\)
−0.127653 + 0.991819i \(0.540744\pi\)
\(720\) 0 0
\(721\) −5.83871 −0.217445
\(722\) 0 0
\(723\) −3.69670 −0.137482
\(724\) 0 0
\(725\) 0.239828 0.00890700
\(726\) 0 0
\(727\) −37.0644 −1.37464 −0.687322 0.726353i \(-0.741214\pi\)
−0.687322 + 0.726353i \(0.741214\pi\)
\(728\) 0 0
\(729\) 5.69779 0.211029
\(730\) 0 0
\(731\) −2.09159 −0.0773603
\(732\) 0 0
\(733\) −38.8813 −1.43611 −0.718057 0.695985i \(-0.754968\pi\)
−0.718057 + 0.695985i \(0.754968\pi\)
\(734\) 0 0
\(735\) −9.26969 −0.341918
\(736\) 0 0
\(737\) −35.5496 −1.30949
\(738\) 0 0
\(739\) 1.53607 0.0565051 0.0282525 0.999601i \(-0.491006\pi\)
0.0282525 + 0.999601i \(0.491006\pi\)
\(740\) 0 0
\(741\) 40.5637 1.49015
\(742\) 0 0
\(743\) −15.8411 −0.581155 −0.290578 0.956851i \(-0.593847\pi\)
−0.290578 + 0.956851i \(0.593847\pi\)
\(744\) 0 0
\(745\) 13.9070 0.509514
\(746\) 0 0
\(747\) 67.7420 2.47855
\(748\) 0 0
\(749\) −40.8419 −1.49233
\(750\) 0 0
\(751\) 50.0654 1.82691 0.913456 0.406937i \(-0.133403\pi\)
0.913456 + 0.406937i \(0.133403\pi\)
\(752\) 0 0
\(753\) −3.13996 −0.114426
\(754\) 0 0
\(755\) −0.800816 −0.0291447
\(756\) 0 0
\(757\) −33.3030 −1.21042 −0.605210 0.796066i \(-0.706911\pi\)
−0.605210 + 0.796066i \(0.706911\pi\)
\(758\) 0 0
\(759\) 7.61863 0.276539
\(760\) 0 0
\(761\) 32.7116 1.18580 0.592898 0.805278i \(-0.297984\pi\)
0.592898 + 0.805278i \(0.297984\pi\)
\(762\) 0 0
\(763\) 28.1633 1.01958
\(764\) 0 0
\(765\) −21.1405 −0.764336
\(766\) 0 0
\(767\) −24.4780 −0.883850
\(768\) 0 0
\(769\) −5.39161 −0.194426 −0.0972132 0.995264i \(-0.530993\pi\)
−0.0972132 + 0.995264i \(0.530993\pi\)
\(770\) 0 0
\(771\) 18.7996 0.677049
\(772\) 0 0
\(773\) 49.7123 1.78803 0.894013 0.448041i \(-0.147878\pi\)
0.894013 + 0.448041i \(0.147878\pi\)
\(774\) 0 0
\(775\) −0.566249 −0.0203402
\(776\) 0 0
\(777\) 9.09691 0.326350
\(778\) 0 0
\(779\) −2.08637 −0.0747520
\(780\) 0 0
\(781\) 14.6404 0.523874
\(782\) 0 0
\(783\) −22.6575 −0.809714
\(784\) 0 0
\(785\) 1.07322 0.0383047
\(786\) 0 0
\(787\) 2.23424 0.0796420 0.0398210 0.999207i \(-0.487321\pi\)
0.0398210 + 0.999207i \(0.487321\pi\)
\(788\) 0 0
\(789\) 61.8458 2.20177
\(790\) 0 0
\(791\) 43.2081 1.53630
\(792\) 0 0
\(793\) −12.0657 −0.428466
\(794\) 0 0
\(795\) −93.5361 −3.31739
\(796\) 0 0
\(797\) 40.6850 1.44113 0.720567 0.693385i \(-0.243881\pi\)
0.720567 + 0.693385i \(0.243881\pi\)
\(798\) 0 0
\(799\) −1.22363 −0.0432888
\(800\) 0 0
\(801\) 93.3958 3.29998
\(802\) 0 0
\(803\) −29.6462 −1.04619
\(804\) 0 0
\(805\) −4.63084 −0.163216
\(806\) 0 0
\(807\) 35.2768 1.24180
\(808\) 0 0
\(809\) −41.7414 −1.46755 −0.733775 0.679392i \(-0.762243\pi\)
−0.733775 + 0.679392i \(0.762243\pi\)
\(810\) 0 0
\(811\) 9.29843 0.326512 0.163256 0.986584i \(-0.447800\pi\)
0.163256 + 0.986584i \(0.447800\pi\)
\(812\) 0 0
\(813\) −40.6573 −1.42591
\(814\) 0 0
\(815\) −14.3095 −0.501238
\(816\) 0 0
\(817\) 3.54389 0.123985
\(818\) 0 0
\(819\) 91.7157 3.20480
\(820\) 0 0
\(821\) 39.0610 1.36324 0.681619 0.731707i \(-0.261276\pi\)
0.681619 + 0.731707i \(0.261276\pi\)
\(822\) 0 0
\(823\) −13.1337 −0.457813 −0.228907 0.973448i \(-0.573515\pi\)
−0.228907 + 0.973448i \(0.573515\pi\)
\(824\) 0 0
\(825\) 1.14058 0.0397098
\(826\) 0 0
\(827\) −29.1382 −1.01323 −0.506617 0.862171i \(-0.669104\pi\)
−0.506617 + 0.862171i \(0.669104\pi\)
\(828\) 0 0
\(829\) −30.7340 −1.06744 −0.533718 0.845663i \(-0.679206\pi\)
−0.533718 + 0.845663i \(0.679206\pi\)
\(830\) 0 0
\(831\) −68.3966 −2.37265
\(832\) 0 0
\(833\) −1.77421 −0.0614726
\(834\) 0 0
\(835\) 18.9647 0.656301
\(836\) 0 0
\(837\) 53.4957 1.84908
\(838\) 0 0
\(839\) 10.2263 0.353049 0.176525 0.984296i \(-0.443514\pi\)
0.176525 + 0.984296i \(0.443514\pi\)
\(840\) 0 0
\(841\) −25.5041 −0.879453
\(842\) 0 0
\(843\) 73.7027 2.53846
\(844\) 0 0
\(845\) −41.6347 −1.43228
\(846\) 0 0
\(847\) −7.11322 −0.244413
\(848\) 0 0
\(849\) −31.8068 −1.09161
\(850\) 0 0
\(851\) −1.04003 −0.0356518
\(852\) 0 0
\(853\) 31.1549 1.06672 0.533361 0.845888i \(-0.320929\pi\)
0.533361 + 0.845888i \(0.320929\pi\)
\(854\) 0 0
\(855\) 35.8194 1.22500
\(856\) 0 0
\(857\) 6.13967 0.209727 0.104864 0.994487i \(-0.466559\pi\)
0.104864 + 0.994487i \(0.466559\pi\)
\(858\) 0 0
\(859\) −25.7512 −0.878621 −0.439310 0.898335i \(-0.644777\pi\)
−0.439310 + 0.898335i \(0.644777\pi\)
\(860\) 0 0
\(861\) −6.78052 −0.231079
\(862\) 0 0
\(863\) 48.3413 1.64556 0.822778 0.568363i \(-0.192423\pi\)
0.822778 + 0.568363i \(0.192423\pi\)
\(864\) 0 0
\(865\) −33.7327 −1.14695
\(866\) 0 0
\(867\) 47.5633 1.61534
\(868\) 0 0
\(869\) −16.6812 −0.565870
\(870\) 0 0
\(871\) 70.3267 2.38293
\(872\) 0 0
\(873\) 31.0463 1.05076
\(874\) 0 0
\(875\) 26.3310 0.890151
\(876\) 0 0
\(877\) 25.9748 0.877107 0.438554 0.898705i \(-0.355491\pi\)
0.438554 + 0.898705i \(0.355491\pi\)
\(878\) 0 0
\(879\) −20.0977 −0.677878
\(880\) 0 0
\(881\) −13.7848 −0.464422 −0.232211 0.972665i \(-0.574596\pi\)
−0.232211 + 0.972665i \(0.574596\pi\)
\(882\) 0 0
\(883\) −14.3520 −0.482983 −0.241492 0.970403i \(-0.577637\pi\)
−0.241492 + 0.970403i \(0.577637\pi\)
\(884\) 0 0
\(885\) −31.0686 −1.04436
\(886\) 0 0
\(887\) 51.2285 1.72008 0.860042 0.510224i \(-0.170437\pi\)
0.860042 + 0.510224i \(0.170437\pi\)
\(888\) 0 0
\(889\) −2.37782 −0.0797495
\(890\) 0 0
\(891\) −49.4801 −1.65764
\(892\) 0 0
\(893\) 2.07326 0.0693789
\(894\) 0 0
\(895\) 6.54754 0.218860
\(896\) 0 0
\(897\) −15.0717 −0.503229
\(898\) 0 0
\(899\) −8.25393 −0.275284
\(900\) 0 0
\(901\) −17.9027 −0.596425
\(902\) 0 0
\(903\) 11.5173 0.383273
\(904\) 0 0
\(905\) 23.4240 0.778639
\(906\) 0 0
\(907\) −37.0873 −1.23146 −0.615732 0.787956i \(-0.711140\pi\)
−0.615732 + 0.787956i \(0.711140\pi\)
\(908\) 0 0
\(909\) −35.9827 −1.19347
\(910\) 0 0
\(911\) 56.9130 1.88561 0.942807 0.333340i \(-0.108176\pi\)
0.942807 + 0.333340i \(0.108176\pi\)
\(912\) 0 0
\(913\) −27.9675 −0.925590
\(914\) 0 0
\(915\) −15.3144 −0.506277
\(916\) 0 0
\(917\) −37.2768 −1.23099
\(918\) 0 0
\(919\) 54.1891 1.78753 0.893766 0.448533i \(-0.148053\pi\)
0.893766 + 0.448533i \(0.148053\pi\)
\(920\) 0 0
\(921\) −3.45781 −0.113939
\(922\) 0 0
\(923\) −28.9626 −0.953315
\(924\) 0 0
\(925\) −0.155702 −0.00511946
\(926\) 0 0
\(927\) −16.7803 −0.551137
\(928\) 0 0
\(929\) 39.9664 1.31125 0.655627 0.755085i \(-0.272404\pi\)
0.655627 + 0.755085i \(0.272404\pi\)
\(930\) 0 0
\(931\) 3.00613 0.0985219
\(932\) 0 0
\(933\) −37.8645 −1.23963
\(934\) 0 0
\(935\) 8.72792 0.285434
\(936\) 0 0
\(937\) 28.7969 0.940752 0.470376 0.882466i \(-0.344118\pi\)
0.470376 + 0.882466i \(0.344118\pi\)
\(938\) 0 0
\(939\) −6.79785 −0.221839
\(940\) 0 0
\(941\) 6.77139 0.220741 0.110371 0.993891i \(-0.464796\pi\)
0.110371 + 0.993891i \(0.464796\pi\)
\(942\) 0 0
\(943\) 0.775204 0.0252441
\(944\) 0 0
\(945\) 65.4967 2.13061
\(946\) 0 0
\(947\) 33.7660 1.09725 0.548624 0.836069i \(-0.315152\pi\)
0.548624 + 0.836069i \(0.315152\pi\)
\(948\) 0 0
\(949\) 58.6482 1.90380
\(950\) 0 0
\(951\) 68.0663 2.20720
\(952\) 0 0
\(953\) −40.2508 −1.30385 −0.651925 0.758283i \(-0.726038\pi\)
−0.651925 + 0.758283i \(0.726038\pi\)
\(954\) 0 0
\(955\) −0.234563 −0.00759028
\(956\) 0 0
\(957\) 16.6256 0.537431
\(958\) 0 0
\(959\) −19.7937 −0.639171
\(960\) 0 0
\(961\) −11.5120 −0.371354
\(962\) 0 0
\(963\) −117.379 −3.78247
\(964\) 0 0
\(965\) 2.21686 0.0713634
\(966\) 0 0
\(967\) −20.9363 −0.673265 −0.336633 0.941636i \(-0.609288\pi\)
−0.336633 + 0.941636i \(0.609288\pi\)
\(968\) 0 0
\(969\) 9.85423 0.316564
\(970\) 0 0
\(971\) −14.1992 −0.455674 −0.227837 0.973699i \(-0.573165\pi\)
−0.227837 + 0.973699i \(0.573165\pi\)
\(972\) 0 0
\(973\) −29.6100 −0.949252
\(974\) 0 0
\(975\) −2.25637 −0.0722616
\(976\) 0 0
\(977\) −56.3368 −1.80238 −0.901188 0.433429i \(-0.857303\pi\)
−0.901188 + 0.433429i \(0.857303\pi\)
\(978\) 0 0
\(979\) −38.5588 −1.23234
\(980\) 0 0
\(981\) 80.9406 2.58424
\(982\) 0 0
\(983\) 42.2487 1.34753 0.673763 0.738948i \(-0.264677\pi\)
0.673763 + 0.738948i \(0.264677\pi\)
\(984\) 0 0
\(985\) −12.2724 −0.391031
\(986\) 0 0
\(987\) 6.73789 0.214469
\(988\) 0 0
\(989\) −1.31675 −0.0418703
\(990\) 0 0
\(991\) −6.08450 −0.193280 −0.0966402 0.995319i \(-0.530810\pi\)
−0.0966402 + 0.995319i \(0.530810\pi\)
\(992\) 0 0
\(993\) 47.2654 1.49992
\(994\) 0 0
\(995\) 58.3223 1.84894
\(996\) 0 0
\(997\) −44.5460 −1.41079 −0.705393 0.708816i \(-0.749230\pi\)
−0.705393 + 0.708816i \(0.749230\pi\)
\(998\) 0 0
\(999\) 14.7098 0.465398
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 8032.2.a.g.1.2 30
4.3 odd 2 8032.2.a.j.1.29 yes 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8032.2.a.g.1.2 30 1.1 even 1 trivial
8032.2.a.j.1.29 yes 30 4.3 odd 2