Properties

Label 32.4.b.a.17.2
Level $32$
Weight $4$
Character 32.17
Analytic conductor $1.888$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [32,4,Mod(17,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 32.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.88806112018\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 17.2
Root \(0.500000 - 1.32288i\) of defining polynomial
Character \(\chi\) \(=\) 32.17
Dual form 32.4.b.a.17.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.29150i q^{3} +10.5830i q^{5} +8.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+5.29150i q^{3} +10.5830i q^{5} +8.00000 q^{7} -1.00000 q^{9} -15.8745i q^{11} -52.9150i q^{13} -56.0000 q^{15} -14.0000 q^{17} -37.0405i q^{19} +42.3320i q^{21} +152.000 q^{23} +13.0000 q^{25} +137.579i q^{27} +158.745i q^{29} -224.000 q^{31} +84.0000 q^{33} +84.6640i q^{35} -243.409i q^{37} +280.000 q^{39} -70.0000 q^{41} -439.195i q^{43} -10.5830i q^{45} -336.000 q^{47} -279.000 q^{49} -74.0810i q^{51} -31.7490i q^{53} +168.000 q^{55} +196.000 q^{57} +534.442i q^{59} -95.2470i q^{61} -8.00000 q^{63} +560.000 q^{65} +174.620i q^{67} +804.308i q^{69} +72.0000 q^{71} -294.000 q^{73} +68.7895i q^{75} -126.996i q^{77} +464.000 q^{79} -755.000 q^{81} -545.025i q^{83} -148.162i q^{85} -840.000 q^{87} +266.000 q^{89} -423.320i q^{91} -1185.30i q^{93} +392.000 q^{95} +994.000 q^{97} +15.8745i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{7} - 2 q^{9} - 112 q^{15} - 28 q^{17} + 304 q^{23} + 26 q^{25} - 448 q^{31} + 168 q^{33} + 560 q^{39} - 140 q^{41} - 672 q^{47} - 558 q^{49} + 336 q^{55} + 392 q^{57} - 16 q^{63} + 1120 q^{65} + 144 q^{71} - 588 q^{73} + 928 q^{79} - 1510 q^{81} - 1680 q^{87} + 532 q^{89} + 784 q^{95} + 1988 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 5.29150i 1.01835i 0.860663 + 0.509175i \(0.170049\pi\)
−0.860663 + 0.509175i \(0.829951\pi\)
\(4\) 0 0
\(5\) 10.5830i 0.946573i 0.880909 + 0.473286i \(0.156932\pi\)
−0.880909 + 0.473286i \(0.843068\pi\)
\(6\) 0 0
\(7\) 8.00000 0.431959 0.215980 0.976398i \(-0.430705\pi\)
0.215980 + 0.976398i \(0.430705\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.0370370
\(10\) 0 0
\(11\) − 15.8745i − 0.435122i −0.976047 0.217561i \(-0.930190\pi\)
0.976047 0.217561i \(-0.0698101\pi\)
\(12\) 0 0
\(13\) − 52.9150i − 1.12892i −0.825460 0.564461i \(-0.809084\pi\)
0.825460 0.564461i \(-0.190916\pi\)
\(14\) 0 0
\(15\) −56.0000 −0.963943
\(16\) 0 0
\(17\) −14.0000 −0.199735 −0.0998676 0.995001i \(-0.531842\pi\)
−0.0998676 + 0.995001i \(0.531842\pi\)
\(18\) 0 0
\(19\) − 37.0405i − 0.447246i −0.974676 0.223623i \(-0.928212\pi\)
0.974676 0.223623i \(-0.0717885\pi\)
\(20\) 0 0
\(21\) 42.3320i 0.439886i
\(22\) 0 0
\(23\) 152.000 1.37801 0.689004 0.724757i \(-0.258048\pi\)
0.689004 + 0.724757i \(0.258048\pi\)
\(24\) 0 0
\(25\) 13.0000 0.104000
\(26\) 0 0
\(27\) 137.579i 0.980633i
\(28\) 0 0
\(29\) 158.745i 1.01649i 0.861212 + 0.508245i \(0.169706\pi\)
−0.861212 + 0.508245i \(0.830294\pi\)
\(30\) 0 0
\(31\) −224.000 −1.29779 −0.648897 0.760877i \(-0.724769\pi\)
−0.648897 + 0.760877i \(0.724769\pi\)
\(32\) 0 0
\(33\) 84.0000 0.443107
\(34\) 0 0
\(35\) 84.6640i 0.408881i
\(36\) 0 0
\(37\) − 243.409i − 1.08152i −0.841177 0.540760i \(-0.818137\pi\)
0.841177 0.540760i \(-0.181863\pi\)
\(38\) 0 0
\(39\) 280.000 1.14964
\(40\) 0 0
\(41\) −70.0000 −0.266638 −0.133319 0.991073i \(-0.542564\pi\)
−0.133319 + 0.991073i \(0.542564\pi\)
\(42\) 0 0
\(43\) − 439.195i − 1.55759i −0.627276 0.778797i \(-0.715830\pi\)
0.627276 0.778797i \(-0.284170\pi\)
\(44\) 0 0
\(45\) − 10.5830i − 0.0350583i
\(46\) 0 0
\(47\) −336.000 −1.04278 −0.521390 0.853319i \(-0.674586\pi\)
−0.521390 + 0.853319i \(0.674586\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) 0 0
\(51\) − 74.0810i − 0.203400i
\(52\) 0 0
\(53\) − 31.7490i − 0.0822842i −0.999153 0.0411421i \(-0.986900\pi\)
0.999153 0.0411421i \(-0.0130996\pi\)
\(54\) 0 0
\(55\) 168.000 0.411875
\(56\) 0 0
\(57\) 196.000 0.455453
\(58\) 0 0
\(59\) 534.442i 1.17929i 0.807661 + 0.589647i \(0.200733\pi\)
−0.807661 + 0.589647i \(0.799267\pi\)
\(60\) 0 0
\(61\) − 95.2470i − 0.199920i −0.994991 0.0999601i \(-0.968128\pi\)
0.994991 0.0999601i \(-0.0318715\pi\)
\(62\) 0 0
\(63\) −8.00000 −0.0159985
\(64\) 0 0
\(65\) 560.000 1.06861
\(66\) 0 0
\(67\) 174.620i 0.318406i 0.987246 + 0.159203i \(0.0508924\pi\)
−0.987246 + 0.159203i \(0.949108\pi\)
\(68\) 0 0
\(69\) 804.308i 1.40329i
\(70\) 0 0
\(71\) 72.0000 0.120350 0.0601748 0.998188i \(-0.480834\pi\)
0.0601748 + 0.998188i \(0.480834\pi\)
\(72\) 0 0
\(73\) −294.000 −0.471371 −0.235686 0.971829i \(-0.575734\pi\)
−0.235686 + 0.971829i \(0.575734\pi\)
\(74\) 0 0
\(75\) 68.7895i 0.105908i
\(76\) 0 0
\(77\) − 126.996i − 0.187955i
\(78\) 0 0
\(79\) 464.000 0.660811 0.330406 0.943839i \(-0.392814\pi\)
0.330406 + 0.943839i \(0.392814\pi\)
\(80\) 0 0
\(81\) −755.000 −1.03567
\(82\) 0 0
\(83\) − 545.025i − 0.720774i −0.932803 0.360387i \(-0.882645\pi\)
0.932803 0.360387i \(-0.117355\pi\)
\(84\) 0 0
\(85\) − 148.162i − 0.189064i
\(86\) 0 0
\(87\) −840.000 −1.03514
\(88\) 0 0
\(89\) 266.000 0.316808 0.158404 0.987374i \(-0.449365\pi\)
0.158404 + 0.987374i \(0.449365\pi\)
\(90\) 0 0
\(91\) − 423.320i − 0.487649i
\(92\) 0 0
\(93\) − 1185.30i − 1.32161i
\(94\) 0 0
\(95\) 392.000 0.423351
\(96\) 0 0
\(97\) 994.000 1.04047 0.520234 0.854024i \(-0.325845\pi\)
0.520234 + 0.854024i \(0.325845\pi\)
\(98\) 0 0
\(99\) 15.8745i 0.0161156i
\(100\) 0 0
\(101\) − 751.393i − 0.740262i −0.928980 0.370131i \(-0.879313\pi\)
0.928980 0.370131i \(-0.120687\pi\)
\(102\) 0 0
\(103\) −1176.00 −1.12500 −0.562499 0.826798i \(-0.690160\pi\)
−0.562499 + 0.826798i \(0.690160\pi\)
\(104\) 0 0
\(105\) −448.000 −0.416384
\(106\) 0 0
\(107\) − 269.867i − 0.243822i −0.992541 0.121911i \(-0.961098\pi\)
0.992541 0.121911i \(-0.0389023\pi\)
\(108\) 0 0
\(109\) 1894.36i 1.66465i 0.554290 + 0.832324i \(0.312990\pi\)
−0.554290 + 0.832324i \(0.687010\pi\)
\(110\) 0 0
\(111\) 1288.00 1.10137
\(112\) 0 0
\(113\) −1710.00 −1.42357 −0.711784 0.702398i \(-0.752113\pi\)
−0.711784 + 0.702398i \(0.752113\pi\)
\(114\) 0 0
\(115\) 1608.62i 1.30439i
\(116\) 0 0
\(117\) 52.9150i 0.0418119i
\(118\) 0 0
\(119\) −112.000 −0.0862775
\(120\) 0 0
\(121\) 1079.00 0.810669
\(122\) 0 0
\(123\) − 370.405i − 0.271531i
\(124\) 0 0
\(125\) 1460.45i 1.04502i
\(126\) 0 0
\(127\) 1664.00 1.16265 0.581323 0.813673i \(-0.302535\pi\)
0.581323 + 0.813673i \(0.302535\pi\)
\(128\) 0 0
\(129\) 2324.00 1.58618
\(130\) 0 0
\(131\) − 672.021i − 0.448204i −0.974566 0.224102i \(-0.928055\pi\)
0.974566 0.224102i \(-0.0719449\pi\)
\(132\) 0 0
\(133\) − 296.324i − 0.193192i
\(134\) 0 0
\(135\) −1456.00 −0.928241
\(136\) 0 0
\(137\) −1062.00 −0.662283 −0.331142 0.943581i \(-0.607434\pi\)
−0.331142 + 0.943581i \(0.607434\pi\)
\(138\) 0 0
\(139\) 2693.37i 1.64352i 0.569835 + 0.821759i \(0.307007\pi\)
−0.569835 + 0.821759i \(0.692993\pi\)
\(140\) 0 0
\(141\) − 1777.94i − 1.06191i
\(142\) 0 0
\(143\) −840.000 −0.491219
\(144\) 0 0
\(145\) −1680.00 −0.962182
\(146\) 0 0
\(147\) − 1476.33i − 0.828337i
\(148\) 0 0
\(149\) − 793.725i − 0.436406i −0.975903 0.218203i \(-0.929980\pi\)
0.975903 0.218203i \(-0.0700195\pi\)
\(150\) 0 0
\(151\) −744.000 −0.400966 −0.200483 0.979697i \(-0.564251\pi\)
−0.200483 + 0.979697i \(0.564251\pi\)
\(152\) 0 0
\(153\) 14.0000 0.00739760
\(154\) 0 0
\(155\) − 2370.59i − 1.22846i
\(156\) 0 0
\(157\) − 179.911i − 0.0914552i −0.998954 0.0457276i \(-0.985439\pi\)
0.998954 0.0457276i \(-0.0145606\pi\)
\(158\) 0 0
\(159\) 168.000 0.0837941
\(160\) 0 0
\(161\) 1216.00 0.595244
\(162\) 0 0
\(163\) − 1772.65i − 0.851809i −0.904768 0.425905i \(-0.859956\pi\)
0.904768 0.425905i \(-0.140044\pi\)
\(164\) 0 0
\(165\) 888.972i 0.419433i
\(166\) 0 0
\(167\) 1960.00 0.908200 0.454100 0.890951i \(-0.349961\pi\)
0.454100 + 0.890951i \(0.349961\pi\)
\(168\) 0 0
\(169\) −603.000 −0.274465
\(170\) 0 0
\(171\) 37.0405i 0.0165647i
\(172\) 0 0
\(173\) − 2000.19i − 0.879026i −0.898236 0.439513i \(-0.855151\pi\)
0.898236 0.439513i \(-0.144849\pi\)
\(174\) 0 0
\(175\) 104.000 0.0449238
\(176\) 0 0
\(177\) −2828.00 −1.20094
\(178\) 0 0
\(179\) 3264.86i 1.36328i 0.731688 + 0.681639i \(0.238733\pi\)
−0.731688 + 0.681639i \(0.761267\pi\)
\(180\) 0 0
\(181\) 2338.84i 0.960469i 0.877140 + 0.480235i \(0.159448\pi\)
−0.877140 + 0.480235i \(0.840552\pi\)
\(182\) 0 0
\(183\) 504.000 0.203589
\(184\) 0 0
\(185\) 2576.00 1.02374
\(186\) 0 0
\(187\) 222.243i 0.0869092i
\(188\) 0 0
\(189\) 1100.63i 0.423594i
\(190\) 0 0
\(191\) −3904.00 −1.47897 −0.739486 0.673172i \(-0.764931\pi\)
−0.739486 + 0.673172i \(0.764931\pi\)
\(192\) 0 0
\(193\) 3330.00 1.24196 0.620981 0.783826i \(-0.286734\pi\)
0.620981 + 0.783826i \(0.286734\pi\)
\(194\) 0 0
\(195\) 2963.24i 1.08822i
\(196\) 0 0
\(197\) 1195.88i 0.432502i 0.976338 + 0.216251i \(0.0693829\pi\)
−0.976338 + 0.216251i \(0.930617\pi\)
\(198\) 0 0
\(199\) 1736.00 0.618401 0.309200 0.950997i \(-0.399939\pi\)
0.309200 + 0.950997i \(0.399939\pi\)
\(200\) 0 0
\(201\) −924.000 −0.324248
\(202\) 0 0
\(203\) 1269.96i 0.439083i
\(204\) 0 0
\(205\) − 740.810i − 0.252392i
\(206\) 0 0
\(207\) −152.000 −0.0510373
\(208\) 0 0
\(209\) −588.000 −0.194607
\(210\) 0 0
\(211\) − 2915.62i − 0.951277i −0.879641 0.475638i \(-0.842217\pi\)
0.879641 0.475638i \(-0.157783\pi\)
\(212\) 0 0
\(213\) 380.988i 0.122558i
\(214\) 0 0
\(215\) 4648.00 1.47438
\(216\) 0 0
\(217\) −1792.00 −0.560594
\(218\) 0 0
\(219\) − 1555.70i − 0.480021i
\(220\) 0 0
\(221\) 740.810i 0.225486i
\(222\) 0 0
\(223\) −1568.00 −0.470857 −0.235428 0.971892i \(-0.575649\pi\)
−0.235428 + 0.971892i \(0.575649\pi\)
\(224\) 0 0
\(225\) −13.0000 −0.00385185
\(226\) 0 0
\(227\) − 1264.67i − 0.369775i −0.982760 0.184888i \(-0.940808\pi\)
0.982760 0.184888i \(-0.0591922\pi\)
\(228\) 0 0
\(229\) − 5153.92i − 1.48725i −0.668595 0.743626i \(-0.733104\pi\)
0.668595 0.743626i \(-0.266896\pi\)
\(230\) 0 0
\(231\) 672.000 0.191404
\(232\) 0 0
\(233\) −838.000 −0.235619 −0.117809 0.993036i \(-0.537587\pi\)
−0.117809 + 0.993036i \(0.537587\pi\)
\(234\) 0 0
\(235\) − 3555.89i − 0.987067i
\(236\) 0 0
\(237\) 2455.26i 0.672937i
\(238\) 0 0
\(239\) −6288.00 −1.70183 −0.850914 0.525305i \(-0.823951\pi\)
−0.850914 + 0.525305i \(0.823951\pi\)
\(240\) 0 0
\(241\) −2926.00 −0.782076 −0.391038 0.920375i \(-0.627884\pi\)
−0.391038 + 0.920375i \(0.627884\pi\)
\(242\) 0 0
\(243\) − 280.450i − 0.0740364i
\(244\) 0 0
\(245\) − 2952.66i − 0.769953i
\(246\) 0 0
\(247\) −1960.00 −0.504906
\(248\) 0 0
\(249\) 2884.00 0.734000
\(250\) 0 0
\(251\) 5444.96i 1.36925i 0.728894 + 0.684627i \(0.240035\pi\)
−0.728894 + 0.684627i \(0.759965\pi\)
\(252\) 0 0
\(253\) − 2412.93i − 0.599602i
\(254\) 0 0
\(255\) 784.000 0.192533
\(256\) 0 0
\(257\) 2562.00 0.621841 0.310921 0.950436i \(-0.399363\pi\)
0.310921 + 0.950436i \(0.399363\pi\)
\(258\) 0 0
\(259\) − 1947.27i − 0.467172i
\(260\) 0 0
\(261\) − 158.745i − 0.0376478i
\(262\) 0 0
\(263\) 5896.00 1.38237 0.691184 0.722679i \(-0.257089\pi\)
0.691184 + 0.722679i \(0.257089\pi\)
\(264\) 0 0
\(265\) 336.000 0.0778880
\(266\) 0 0
\(267\) 1407.54i 0.322622i
\(268\) 0 0
\(269\) 5365.58i 1.21615i 0.793878 + 0.608077i \(0.208059\pi\)
−0.793878 + 0.608077i \(0.791941\pi\)
\(270\) 0 0
\(271\) 1680.00 0.376578 0.188289 0.982114i \(-0.439706\pi\)
0.188289 + 0.982114i \(0.439706\pi\)
\(272\) 0 0
\(273\) 2240.00 0.496597
\(274\) 0 0
\(275\) − 206.369i − 0.0452527i
\(276\) 0 0
\(277\) 1576.87i 0.342039i 0.985268 + 0.171019i \(0.0547061\pi\)
−0.985268 + 0.171019i \(0.945294\pi\)
\(278\) 0 0
\(279\) 224.000 0.0480664
\(280\) 0 0
\(281\) −2742.00 −0.582114 −0.291057 0.956706i \(-0.594007\pi\)
−0.291057 + 0.956706i \(0.594007\pi\)
\(282\) 0 0
\(283\) 2989.70i 0.627983i 0.949426 + 0.313991i \(0.101666\pi\)
−0.949426 + 0.313991i \(0.898334\pi\)
\(284\) 0 0
\(285\) 2074.27i 0.431120i
\(286\) 0 0
\(287\) −560.000 −0.115177
\(288\) 0 0
\(289\) −4717.00 −0.960106
\(290\) 0 0
\(291\) 5259.75i 1.05956i
\(292\) 0 0
\(293\) 9238.96i 1.84214i 0.389401 + 0.921068i \(0.372682\pi\)
−0.389401 + 0.921068i \(0.627318\pi\)
\(294\) 0 0
\(295\) −5656.00 −1.11629
\(296\) 0 0
\(297\) 2184.00 0.426695
\(298\) 0 0
\(299\) − 8043.08i − 1.55566i
\(300\) 0 0
\(301\) − 3513.56i − 0.672818i
\(302\) 0 0
\(303\) 3976.00 0.753846
\(304\) 0 0
\(305\) 1008.00 0.189239
\(306\) 0 0
\(307\) 2587.54i 0.481039i 0.970644 + 0.240520i \(0.0773178\pi\)
−0.970644 + 0.240520i \(0.922682\pi\)
\(308\) 0 0
\(309\) − 6222.81i − 1.14564i
\(310\) 0 0
\(311\) 2744.00 0.500315 0.250157 0.968205i \(-0.419518\pi\)
0.250157 + 0.968205i \(0.419518\pi\)
\(312\) 0 0
\(313\) 2282.00 0.412097 0.206048 0.978542i \(-0.433940\pi\)
0.206048 + 0.978542i \(0.433940\pi\)
\(314\) 0 0
\(315\) − 84.6640i − 0.0151437i
\(316\) 0 0
\(317\) − 9577.62i − 1.69695i −0.529237 0.848474i \(-0.677522\pi\)
0.529237 0.848474i \(-0.322478\pi\)
\(318\) 0 0
\(319\) 2520.00 0.442298
\(320\) 0 0
\(321\) 1428.00 0.248297
\(322\) 0 0
\(323\) 518.567i 0.0893308i
\(324\) 0 0
\(325\) − 687.895i − 0.117408i
\(326\) 0 0
\(327\) −10024.0 −1.69519
\(328\) 0 0
\(329\) −2688.00 −0.450438
\(330\) 0 0
\(331\) − 4249.08i − 0.705590i −0.935701 0.352795i \(-0.885231\pi\)
0.935701 0.352795i \(-0.114769\pi\)
\(332\) 0 0
\(333\) 243.409i 0.0400563i
\(334\) 0 0
\(335\) −1848.00 −0.301394
\(336\) 0 0
\(337\) 6130.00 0.990868 0.495434 0.868646i \(-0.335009\pi\)
0.495434 + 0.868646i \(0.335009\pi\)
\(338\) 0 0
\(339\) − 9048.47i − 1.44969i
\(340\) 0 0
\(341\) 3555.89i 0.564699i
\(342\) 0 0
\(343\) −4976.00 −0.783320
\(344\) 0 0
\(345\) −8512.00 −1.32832
\(346\) 0 0
\(347\) 2481.71i 0.383935i 0.981401 + 0.191967i \(0.0614868\pi\)
−0.981401 + 0.191967i \(0.938513\pi\)
\(348\) 0 0
\(349\) 328.073i 0.0503191i 0.999683 + 0.0251595i \(0.00800937\pi\)
−0.999683 + 0.0251595i \(0.991991\pi\)
\(350\) 0 0
\(351\) 7280.00 1.10706
\(352\) 0 0
\(353\) −10206.0 −1.53884 −0.769420 0.638743i \(-0.779455\pi\)
−0.769420 + 0.638743i \(0.779455\pi\)
\(354\) 0 0
\(355\) 761.976i 0.113920i
\(356\) 0 0
\(357\) − 592.648i − 0.0878607i
\(358\) 0 0
\(359\) 3176.00 0.466916 0.233458 0.972367i \(-0.424996\pi\)
0.233458 + 0.972367i \(0.424996\pi\)
\(360\) 0 0
\(361\) 5487.00 0.799971
\(362\) 0 0
\(363\) 5709.53i 0.825545i
\(364\) 0 0
\(365\) − 3111.40i − 0.446187i
\(366\) 0 0
\(367\) 11760.0 1.67266 0.836331 0.548225i \(-0.184696\pi\)
0.836331 + 0.548225i \(0.184696\pi\)
\(368\) 0 0
\(369\) 70.0000 0.00987549
\(370\) 0 0
\(371\) − 253.992i − 0.0355434i
\(372\) 0 0
\(373\) 10974.6i 1.52344i 0.647908 + 0.761719i \(0.275644\pi\)
−0.647908 + 0.761719i \(0.724356\pi\)
\(374\) 0 0
\(375\) −7728.00 −1.06419
\(376\) 0 0
\(377\) 8400.00 1.14754
\(378\) 0 0
\(379\) 3074.36i 0.416674i 0.978057 + 0.208337i \(0.0668051\pi\)
−0.978057 + 0.208337i \(0.933195\pi\)
\(380\) 0 0
\(381\) 8805.06i 1.18398i
\(382\) 0 0
\(383\) −2688.00 −0.358617 −0.179309 0.983793i \(-0.557386\pi\)
−0.179309 + 0.983793i \(0.557386\pi\)
\(384\) 0 0
\(385\) 1344.00 0.177913
\(386\) 0 0
\(387\) 439.195i 0.0576887i
\(388\) 0 0
\(389\) − 10487.8i − 1.36697i −0.729966 0.683484i \(-0.760464\pi\)
0.729966 0.683484i \(-0.239536\pi\)
\(390\) 0 0
\(391\) −2128.00 −0.275237
\(392\) 0 0
\(393\) 3556.00 0.456429
\(394\) 0 0
\(395\) 4910.51i 0.625506i
\(396\) 0 0
\(397\) 5704.24i 0.721127i 0.932735 + 0.360564i \(0.117416\pi\)
−0.932735 + 0.360564i \(0.882584\pi\)
\(398\) 0 0
\(399\) 1568.00 0.196737
\(400\) 0 0
\(401\) 12402.0 1.54445 0.772227 0.635346i \(-0.219143\pi\)
0.772227 + 0.635346i \(0.219143\pi\)
\(402\) 0 0
\(403\) 11853.0i 1.46511i
\(404\) 0 0
\(405\) − 7990.17i − 0.980333i
\(406\) 0 0
\(407\) −3864.00 −0.470593
\(408\) 0 0
\(409\) −12278.0 −1.48437 −0.742186 0.670194i \(-0.766211\pi\)
−0.742186 + 0.670194i \(0.766211\pi\)
\(410\) 0 0
\(411\) − 5619.58i − 0.674436i
\(412\) 0 0
\(413\) 4275.53i 0.509407i
\(414\) 0 0
\(415\) 5768.00 0.682265
\(416\) 0 0
\(417\) −14252.0 −1.67368
\(418\) 0 0
\(419\) − 8207.12i − 0.956907i −0.878113 0.478454i \(-0.841198\pi\)
0.878113 0.478454i \(-0.158802\pi\)
\(420\) 0 0
\(421\) 1449.87i 0.167844i 0.996472 + 0.0839221i \(0.0267447\pi\)
−0.996472 + 0.0839221i \(0.973255\pi\)
\(422\) 0 0
\(423\) 336.000 0.0386215
\(424\) 0 0
\(425\) −182.000 −0.0207725
\(426\) 0 0
\(427\) − 761.976i − 0.0863574i
\(428\) 0 0
\(429\) − 4444.86i − 0.500233i
\(430\) 0 0
\(431\) −7632.00 −0.852948 −0.426474 0.904500i \(-0.640244\pi\)
−0.426474 + 0.904500i \(0.640244\pi\)
\(432\) 0 0
\(433\) 3794.00 0.421081 0.210540 0.977585i \(-0.432478\pi\)
0.210540 + 0.977585i \(0.432478\pi\)
\(434\) 0 0
\(435\) − 8889.72i − 0.979838i
\(436\) 0 0
\(437\) − 5630.16i − 0.616309i
\(438\) 0 0
\(439\) 1848.00 0.200912 0.100456 0.994942i \(-0.467970\pi\)
0.100456 + 0.994942i \(0.467970\pi\)
\(440\) 0 0
\(441\) 279.000 0.0301263
\(442\) 0 0
\(443\) − 12334.5i − 1.32287i −0.750004 0.661433i \(-0.769949\pi\)
0.750004 0.661433i \(-0.230051\pi\)
\(444\) 0 0
\(445\) 2815.08i 0.299882i
\(446\) 0 0
\(447\) 4200.00 0.444414
\(448\) 0 0
\(449\) −3582.00 −0.376492 −0.188246 0.982122i \(-0.560280\pi\)
−0.188246 + 0.982122i \(0.560280\pi\)
\(450\) 0 0
\(451\) 1111.22i 0.116020i
\(452\) 0 0
\(453\) − 3936.88i − 0.408324i
\(454\) 0 0
\(455\) 4480.00 0.461595
\(456\) 0 0
\(457\) 2714.00 0.277802 0.138901 0.990306i \(-0.455643\pi\)
0.138901 + 0.990306i \(0.455643\pi\)
\(458\) 0 0
\(459\) − 1926.11i − 0.195867i
\(460\) 0 0
\(461\) − 8349.99i − 0.843596i −0.906690 0.421798i \(-0.861399\pi\)
0.906690 0.421798i \(-0.138601\pi\)
\(462\) 0 0
\(463\) −2224.00 −0.223236 −0.111618 0.993751i \(-0.535603\pi\)
−0.111618 + 0.993751i \(0.535603\pi\)
\(464\) 0 0
\(465\) 12544.0 1.25100
\(466\) 0 0
\(467\) 10292.0i 1.01982i 0.860228 + 0.509910i \(0.170321\pi\)
−0.860228 + 0.509910i \(0.829679\pi\)
\(468\) 0 0
\(469\) 1396.96i 0.137538i
\(470\) 0 0
\(471\) 952.000 0.0931334
\(472\) 0 0
\(473\) −6972.00 −0.677744
\(474\) 0 0
\(475\) − 481.527i − 0.0465136i
\(476\) 0 0
\(477\) 31.7490i 0.00304756i
\(478\) 0 0
\(479\) −17696.0 −1.68800 −0.843999 0.536345i \(-0.819805\pi\)
−0.843999 + 0.536345i \(0.819805\pi\)
\(480\) 0 0
\(481\) −12880.0 −1.22095
\(482\) 0 0
\(483\) 6434.47i 0.606166i
\(484\) 0 0
\(485\) 10519.5i 0.984879i
\(486\) 0 0
\(487\) −1304.00 −0.121334 −0.0606672 0.998158i \(-0.519323\pi\)
−0.0606672 + 0.998158i \(0.519323\pi\)
\(488\) 0 0
\(489\) 9380.00 0.867440
\(490\) 0 0
\(491\) 16662.9i 1.53154i 0.643112 + 0.765772i \(0.277643\pi\)
−0.643112 + 0.765772i \(0.722357\pi\)
\(492\) 0 0
\(493\) − 2222.43i − 0.203029i
\(494\) 0 0
\(495\) −168.000 −0.0152546
\(496\) 0 0
\(497\) 576.000 0.0519862
\(498\) 0 0
\(499\) 3095.53i 0.277705i 0.990313 + 0.138853i \(0.0443414\pi\)
−0.990313 + 0.138853i \(0.955659\pi\)
\(500\) 0 0
\(501\) 10371.3i 0.924865i
\(502\) 0 0
\(503\) 19320.0 1.71260 0.856298 0.516481i \(-0.172758\pi\)
0.856298 + 0.516481i \(0.172758\pi\)
\(504\) 0 0
\(505\) 7952.00 0.700712
\(506\) 0 0
\(507\) − 3190.78i − 0.279502i
\(508\) 0 0
\(509\) 4476.61i 0.389828i 0.980820 + 0.194914i \(0.0624427\pi\)
−0.980820 + 0.194914i \(0.937557\pi\)
\(510\) 0 0
\(511\) −2352.00 −0.203613
\(512\) 0 0
\(513\) 5096.00 0.438585
\(514\) 0 0
\(515\) − 12445.6i − 1.06489i
\(516\) 0 0
\(517\) 5333.83i 0.453737i
\(518\) 0 0
\(519\) 10584.0 0.895156
\(520\) 0 0
\(521\) −2982.00 −0.250756 −0.125378 0.992109i \(-0.540014\pi\)
−0.125378 + 0.992109i \(0.540014\pi\)
\(522\) 0 0
\(523\) 2016.06i 0.168559i 0.996442 + 0.0842794i \(0.0268588\pi\)
−0.996442 + 0.0842794i \(0.973141\pi\)
\(524\) 0 0
\(525\) 550.316i 0.0457481i
\(526\) 0 0
\(527\) 3136.00 0.259215
\(528\) 0 0
\(529\) 10937.0 0.898907
\(530\) 0 0
\(531\) − 534.442i − 0.0436776i
\(532\) 0 0
\(533\) 3704.05i 0.301014i
\(534\) 0 0
\(535\) 2856.00 0.230796
\(536\) 0 0
\(537\) −17276.0 −1.38830
\(538\) 0 0
\(539\) 4428.99i 0.353933i
\(540\) 0 0
\(541\) − 15419.4i − 1.22539i −0.790321 0.612693i \(-0.790086\pi\)
0.790321 0.612693i \(-0.209914\pi\)
\(542\) 0 0
\(543\) −12376.0 −0.978094
\(544\) 0 0
\(545\) −20048.0 −1.57571
\(546\) 0 0
\(547\) − 12609.7i − 0.985649i −0.870129 0.492824i \(-0.835965\pi\)
0.870129 0.492824i \(-0.164035\pi\)
\(548\) 0 0
\(549\) 95.2470i 0.00740445i
\(550\) 0 0
\(551\) 5880.00 0.454621
\(552\) 0 0
\(553\) 3712.00 0.285444
\(554\) 0 0
\(555\) 13630.9i 1.04252i
\(556\) 0 0
\(557\) 7143.53i 0.543413i 0.962380 + 0.271706i \(0.0875880\pi\)
−0.962380 + 0.271706i \(0.912412\pi\)
\(558\) 0 0
\(559\) −23240.0 −1.75840
\(560\) 0 0
\(561\) −1176.00 −0.0885040
\(562\) 0 0
\(563\) − 7572.14i − 0.566834i −0.958997 0.283417i \(-0.908532\pi\)
0.958997 0.283417i \(-0.0914681\pi\)
\(564\) 0 0
\(565\) − 18096.9i − 1.34751i
\(566\) 0 0
\(567\) −6040.00 −0.447365
\(568\) 0 0
\(569\) 15594.0 1.14892 0.574459 0.818533i \(-0.305212\pi\)
0.574459 + 0.818533i \(0.305212\pi\)
\(570\) 0 0
\(571\) − 16737.0i − 1.22666i −0.789827 0.613330i \(-0.789830\pi\)
0.789827 0.613330i \(-0.210170\pi\)
\(572\) 0 0
\(573\) − 20658.0i − 1.50611i
\(574\) 0 0
\(575\) 1976.00 0.143313
\(576\) 0 0
\(577\) 6594.00 0.475757 0.237879 0.971295i \(-0.423548\pi\)
0.237879 + 0.971295i \(0.423548\pi\)
\(578\) 0 0
\(579\) 17620.7i 1.26475i
\(580\) 0 0
\(581\) − 4360.20i − 0.311345i
\(582\) 0 0
\(583\) −504.000 −0.0358037
\(584\) 0 0
\(585\) −560.000 −0.0395780
\(586\) 0 0
\(587\) − 23213.8i − 1.63226i −0.577868 0.816130i \(-0.696115\pi\)
0.577868 0.816130i \(-0.303885\pi\)
\(588\) 0 0
\(589\) 8297.08i 0.580433i
\(590\) 0 0
\(591\) −6328.00 −0.440438
\(592\) 0 0
\(593\) 14322.0 0.991794 0.495897 0.868381i \(-0.334839\pi\)
0.495897 + 0.868381i \(0.334839\pi\)
\(594\) 0 0
\(595\) − 1185.30i − 0.0816679i
\(596\) 0 0
\(597\) 9186.05i 0.629749i
\(598\) 0 0
\(599\) 16088.0 1.09739 0.548696 0.836022i \(-0.315124\pi\)
0.548696 + 0.836022i \(0.315124\pi\)
\(600\) 0 0
\(601\) −21238.0 −1.44146 −0.720729 0.693217i \(-0.756193\pi\)
−0.720729 + 0.693217i \(0.756193\pi\)
\(602\) 0 0
\(603\) − 174.620i − 0.0117928i
\(604\) 0 0
\(605\) 11419.1i 0.767357i
\(606\) 0 0
\(607\) 13664.0 0.913681 0.456841 0.889549i \(-0.348981\pi\)
0.456841 + 0.889549i \(0.348981\pi\)
\(608\) 0 0
\(609\) −6720.00 −0.447140
\(610\) 0 0
\(611\) 17779.4i 1.17722i
\(612\) 0 0
\(613\) − 20393.5i − 1.34369i −0.740690 0.671846i \(-0.765501\pi\)
0.740690 0.671846i \(-0.234499\pi\)
\(614\) 0 0
\(615\) 3920.00 0.257024
\(616\) 0 0
\(617\) −3782.00 −0.246771 −0.123385 0.992359i \(-0.539375\pi\)
−0.123385 + 0.992359i \(0.539375\pi\)
\(618\) 0 0
\(619\) 5825.94i 0.378295i 0.981949 + 0.189147i \(0.0605724\pi\)
−0.981949 + 0.189147i \(0.939428\pi\)
\(620\) 0 0
\(621\) 20912.0i 1.35132i
\(622\) 0 0
\(623\) 2128.00 0.136848
\(624\) 0 0
\(625\) −13831.0 −0.885184
\(626\) 0 0
\(627\) − 3111.40i − 0.198178i
\(628\) 0 0
\(629\) 3407.73i 0.216017i
\(630\) 0 0
\(631\) −2056.00 −0.129712 −0.0648558 0.997895i \(-0.520659\pi\)
−0.0648558 + 0.997895i \(0.520659\pi\)
\(632\) 0 0
\(633\) 15428.0 0.968733
\(634\) 0 0
\(635\) 17610.1i 1.10053i
\(636\) 0 0
\(637\) 14763.3i 0.918278i
\(638\) 0 0
\(639\) −72.0000 −0.00445740
\(640\) 0 0
\(641\) 11842.0 0.729689 0.364845 0.931068i \(-0.381122\pi\)
0.364845 + 0.931068i \(0.381122\pi\)
\(642\) 0 0
\(643\) − 16250.2i − 0.996649i −0.866991 0.498325i \(-0.833949\pi\)
0.866991 0.498325i \(-0.166051\pi\)
\(644\) 0 0
\(645\) 24594.9i 1.50143i
\(646\) 0 0
\(647\) −19320.0 −1.17395 −0.586976 0.809604i \(-0.699682\pi\)
−0.586976 + 0.809604i \(0.699682\pi\)
\(648\) 0 0
\(649\) 8484.00 0.513137
\(650\) 0 0
\(651\) − 9482.37i − 0.570881i
\(652\) 0 0
\(653\) 2317.68i 0.138894i 0.997586 + 0.0694470i \(0.0221235\pi\)
−0.997586 + 0.0694470i \(0.977877\pi\)
\(654\) 0 0
\(655\) 7112.00 0.424258
\(656\) 0 0
\(657\) 294.000 0.0174582
\(658\) 0 0
\(659\) 27732.8i 1.63932i 0.572847 + 0.819662i \(0.305839\pi\)
−0.572847 + 0.819662i \(0.694161\pi\)
\(660\) 0 0
\(661\) − 22467.7i − 1.32208i −0.750352 0.661039i \(-0.770116\pi\)
0.750352 0.661039i \(-0.229884\pi\)
\(662\) 0 0
\(663\) −3920.00 −0.229623
\(664\) 0 0
\(665\) 3136.00 0.182870
\(666\) 0 0
\(667\) 24129.3i 1.40073i
\(668\) 0 0
\(669\) − 8297.08i − 0.479497i
\(670\) 0 0
\(671\) −1512.00 −0.0869897
\(672\) 0 0
\(673\) −10078.0 −0.577234 −0.288617 0.957445i \(-0.593195\pi\)
−0.288617 + 0.957445i \(0.593195\pi\)
\(674\) 0 0
\(675\) 1788.53i 0.101986i
\(676\) 0 0
\(677\) − 16160.2i − 0.917413i −0.888588 0.458707i \(-0.848313\pi\)
0.888588 0.458707i \(-0.151687\pi\)
\(678\) 0 0
\(679\) 7952.00 0.449440
\(680\) 0 0
\(681\) 6692.00 0.376561
\(682\) 0 0
\(683\) − 16356.0i − 0.916320i −0.888870 0.458160i \(-0.848509\pi\)
0.888870 0.458160i \(-0.151491\pi\)
\(684\) 0 0
\(685\) − 11239.2i − 0.626899i
\(686\) 0 0
\(687\) 27272.0 1.51454
\(688\) 0 0
\(689\) −1680.00 −0.0928925
\(690\) 0 0
\(691\) − 29246.1i − 1.61009i −0.593211 0.805047i \(-0.702140\pi\)
0.593211 0.805047i \(-0.297860\pi\)
\(692\) 0 0
\(693\) 126.996i 0.00696130i
\(694\) 0 0
\(695\) −28504.0 −1.55571
\(696\) 0 0
\(697\) 980.000 0.0532570
\(698\) 0 0
\(699\) − 4434.28i − 0.239943i
\(700\) 0 0
\(701\) − 2465.84i − 0.132858i −0.997791 0.0664290i \(-0.978839\pi\)
0.997791 0.0664290i \(-0.0211606\pi\)
\(702\) 0 0
\(703\) −9016.00 −0.483705
\(704\) 0 0
\(705\) 18816.0 1.00518
\(706\) 0 0
\(707\) − 6011.15i − 0.319763i
\(708\) 0 0
\(709\) 31674.9i 1.67782i 0.544267 + 0.838912i \(0.316808\pi\)
−0.544267 + 0.838912i \(0.683192\pi\)
\(710\) 0 0
\(711\) −464.000 −0.0244745
\(712\) 0 0
\(713\) −34048.0 −1.78837
\(714\) 0 0
\(715\) − 8889.72i − 0.464975i
\(716\) 0 0
\(717\) − 33273.0i − 1.73306i
\(718\) 0 0
\(719\) 9296.00 0.482173 0.241086 0.970504i \(-0.422496\pi\)
0.241086 + 0.970504i \(0.422496\pi\)
\(720\) 0 0
\(721\) −9408.00 −0.485953
\(722\) 0 0
\(723\) − 15482.9i − 0.796427i
\(724\) 0 0
\(725\) 2063.69i 0.105715i
\(726\) 0 0
\(727\) −21672.0 −1.10560 −0.552799 0.833315i \(-0.686440\pi\)
−0.552799 + 0.833315i \(0.686440\pi\)
\(728\) 0 0
\(729\) −18901.0 −0.960270
\(730\) 0 0
\(731\) 6148.73i 0.311106i
\(732\) 0 0
\(733\) 9471.79i 0.477283i 0.971108 + 0.238642i \(0.0767021\pi\)
−0.971108 + 0.238642i \(0.923298\pi\)
\(734\) 0 0
\(735\) 15624.0 0.784082
\(736\) 0 0
\(737\) 2772.00 0.138545
\(738\) 0 0
\(739\) 6863.08i 0.341627i 0.985303 + 0.170814i \(0.0546396\pi\)
−0.985303 + 0.170814i \(0.945360\pi\)
\(740\) 0 0
\(741\) − 10371.3i − 0.514171i
\(742\) 0 0
\(743\) −17432.0 −0.860724 −0.430362 0.902656i \(-0.641614\pi\)
−0.430362 + 0.902656i \(0.641614\pi\)
\(744\) 0 0
\(745\) 8400.00 0.413090
\(746\) 0 0
\(747\) 545.025i 0.0266953i
\(748\) 0 0
\(749\) − 2158.93i − 0.105321i
\(750\) 0 0
\(751\) 11632.0 0.565190 0.282595 0.959239i \(-0.408805\pi\)
0.282595 + 0.959239i \(0.408805\pi\)
\(752\) 0 0
\(753\) −28812.0 −1.39438
\(754\) 0 0
\(755\) − 7873.76i − 0.379543i
\(756\) 0 0
\(757\) 16731.7i 0.803336i 0.915785 + 0.401668i \(0.131569\pi\)
−0.915785 + 0.401668i \(0.868431\pi\)
\(758\) 0 0
\(759\) 12768.0 0.610605
\(760\) 0 0
\(761\) 39466.0 1.87995 0.939975 0.341244i \(-0.110848\pi\)
0.939975 + 0.341244i \(0.110848\pi\)
\(762\) 0 0
\(763\) 15154.9i 0.719060i
\(764\) 0 0
\(765\) 148.162i 0.00700237i
\(766\) 0 0
\(767\) 28280.0 1.33133
\(768\) 0 0
\(769\) 35266.0 1.65374 0.826869 0.562395i \(-0.190120\pi\)
0.826869 + 0.562395i \(0.190120\pi\)
\(770\) 0 0
\(771\) 13556.8i 0.633252i
\(772\) 0 0
\(773\) − 16244.9i − 0.755872i −0.925832 0.377936i \(-0.876634\pi\)
0.925832 0.377936i \(-0.123366\pi\)
\(774\) 0 0
\(775\) −2912.00 −0.134970
\(776\) 0 0
\(777\) 10304.0 0.475745
\(778\) 0 0
\(779\) 2592.84i 0.119253i
\(780\) 0 0
\(781\) − 1142.96i − 0.0523668i
\(782\) 0 0
\(783\) −21840.0 −0.996805
\(784\) 0 0
\(785\) 1904.00 0.0865690
\(786\) 0 0
\(787\) 34844.5i 1.57824i 0.614240 + 0.789119i \(0.289463\pi\)
−0.614240 + 0.789119i \(0.710537\pi\)
\(788\) 0 0
\(789\) 31198.7i 1.40774i
\(790\) 0 0
\(791\) −13680.0 −0.614924
\(792\) 0 0
\(793\) −5040.00 −0.225694
\(794\) 0 0
\(795\) 1777.94i 0.0793172i
\(796\) 0 0
\(797\) − 2550.50i − 0.113354i −0.998393 0.0566772i \(-0.981949\pi\)
0.998393 0.0566772i \(-0.0180506\pi\)
\(798\) 0 0
\(799\) 4704.00 0.208280
\(800\) 0 0
\(801\) −266.000 −0.0117336
\(802\) 0 0
\(803\) 4667.11i 0.205104i
\(804\) 0 0
\(805\) 12868.9i 0.563441i
\(806\) 0 0
\(807\) −28392.0 −1.23847
\(808\) 0 0
\(809\) −24390.0 −1.05996 −0.529979 0.848010i \(-0.677800\pi\)
−0.529979 + 0.848010i \(0.677800\pi\)
\(810\) 0 0
\(811\) − 9582.91i − 0.414922i −0.978243 0.207461i \(-0.933480\pi\)
0.978243 0.207461i \(-0.0665200\pi\)
\(812\) 0 0
\(813\) 8889.72i 0.383489i
\(814\) 0 0
\(815\) 18760.0 0.806300
\(816\) 0 0
\(817\) −16268.0 −0.696628
\(818\) 0 0
\(819\) 423.320i 0.0180611i
\(820\) 0 0
\(821\) 8773.31i 0.372948i 0.982460 + 0.186474i \(0.0597061\pi\)
−0.982460 + 0.186474i \(0.940294\pi\)
\(822\) 0 0
\(823\) 21688.0 0.918586 0.459293 0.888285i \(-0.348103\pi\)
0.459293 + 0.888285i \(0.348103\pi\)
\(824\) 0 0
\(825\) 1092.00 0.0460831
\(826\) 0 0
\(827\) − 19446.3i − 0.817670i −0.912608 0.408835i \(-0.865935\pi\)
0.912608 0.408835i \(-0.134065\pi\)
\(828\) 0 0
\(829\) 19546.8i 0.818925i 0.912327 + 0.409462i \(0.134284\pi\)
−0.912327 + 0.409462i \(0.865716\pi\)
\(830\) 0 0
\(831\) −8344.00 −0.348315
\(832\) 0 0
\(833\) 3906.00 0.162467
\(834\) 0 0
\(835\) 20742.7i 0.859677i
\(836\) 0 0
\(837\) − 30817.7i − 1.27266i
\(838\) 0 0
\(839\) 18760.0 0.771951 0.385976 0.922509i \(-0.373865\pi\)
0.385976 + 0.922509i \(0.373865\pi\)
\(840\) 0 0
\(841\) −811.000 −0.0332527
\(842\) 0 0
\(843\) − 14509.3i − 0.592796i
\(844\) 0 0
\(845\) − 6381.55i − 0.259801i
\(846\) 0 0
\(847\) 8632.00 0.350176
\(848\) 0 0
\(849\) −15820.0 −0.639506
\(850\) 0 0
\(851\) − 36998.2i − 1.49034i
\(852\) 0 0
\(853\) − 28732.9i − 1.15333i −0.816979 0.576667i \(-0.804353\pi\)
0.816979 0.576667i \(-0.195647\pi\)
\(854\) 0 0
\(855\) −392.000 −0.0156797
\(856\) 0 0
\(857\) 8778.00 0.349884 0.174942 0.984579i \(-0.444026\pi\)
0.174942 + 0.984579i \(0.444026\pi\)
\(858\) 0 0
\(859\) − 5646.03i − 0.224261i −0.993693 0.112130i \(-0.964233\pi\)
0.993693 0.112130i \(-0.0357675\pi\)
\(860\) 0 0
\(861\) − 2963.24i − 0.117290i
\(862\) 0 0
\(863\) 9312.00 0.367305 0.183652 0.982991i \(-0.441208\pi\)
0.183652 + 0.982991i \(0.441208\pi\)
\(864\) 0 0
\(865\) 21168.0 0.832062
\(866\) 0 0
\(867\) − 24960.0i − 0.977724i
\(868\) 0 0
\(869\) − 7365.77i − 0.287534i
\(870\) 0 0
\(871\) 9240.00 0.359455
\(872\) 0 0
\(873\) −994.000 −0.0385359
\(874\) 0 0
\(875\) 11683.6i 0.451405i
\(876\) 0 0
\(877\) − 137.579i − 0.00529728i −0.999996 0.00264864i \(-0.999157\pi\)
0.999996 0.00264864i \(-0.000843089\pi\)
\(878\) 0 0
\(879\) −48888.0 −1.87594
\(880\) 0 0
\(881\) −31150.0 −1.19123 −0.595613 0.803272i \(-0.703091\pi\)
−0.595613 + 0.803272i \(0.703091\pi\)
\(882\) 0 0
\(883\) 12577.9i 0.479366i 0.970851 + 0.239683i \(0.0770435\pi\)
−0.970851 + 0.239683i \(0.922957\pi\)
\(884\) 0 0
\(885\) − 29928.7i − 1.13677i
\(886\) 0 0
\(887\) −37128.0 −1.40545 −0.702726 0.711460i \(-0.748034\pi\)
−0.702726 + 0.711460i \(0.748034\pi\)
\(888\) 0 0
\(889\) 13312.0 0.502216
\(890\) 0 0
\(891\) 11985.3i 0.450641i
\(892\) 0 0
\(893\) 12445.6i 0.466379i
\(894\) 0 0
\(895\) −34552.0 −1.29044
\(896\) 0 0
\(897\) 42560.0 1.58421
\(898\) 0 0
\(899\) − 35558.9i − 1.31919i
\(900\) 0 0
\(901\) 444.486i 0.0164351i
\(902\) 0 0
\(903\) 18592.0 0.685164
\(904\) 0 0
\(905\) −24752.0 −0.909154
\(906\) 0 0
\(907\) 35204.4i 1.28880i 0.764688 + 0.644400i \(0.222893\pi\)
−0.764688 + 0.644400i \(0.777107\pi\)
\(908\) 0 0
\(909\) 751.393i 0.0274171i
\(910\) 0 0
\(911\) 10512.0 0.382303 0.191152 0.981561i \(-0.438778\pi\)
0.191152 + 0.981561i \(0.438778\pi\)
\(912\) 0 0
\(913\) −8652.00 −0.313625
\(914\) 0 0
\(915\) 5333.83i 0.192712i
\(916\) 0 0
\(917\) − 5376.17i − 0.193606i
\(918\) 0 0
\(919\) 46104.0 1.65488 0.827438 0.561557i \(-0.189798\pi\)
0.827438 + 0.561557i \(0.189798\pi\)
\(920\) 0 0
\(921\) −13692.0 −0.489866
\(922\) 0 0
\(923\) − 3809.88i − 0.135865i
\(924\) 0 0
\(925\) − 3164.32i − 0.112478i
\(926\) 0 0
\(927\) 1176.00 0.0416666
\(928\) 0 0
\(929\) −5726.00 −0.202222 −0.101111 0.994875i \(-0.532240\pi\)
−0.101111 + 0.994875i \(0.532240\pi\)
\(930\) 0 0
\(931\) 10334.3i 0.363795i
\(932\) 0 0
\(933\) 14519.9i 0.509496i
\(934\) 0 0
\(935\) −2352.00 −0.0822659
\(936\) 0 0
\(937\) 1274.00 0.0444181 0.0222091 0.999753i \(-0.492930\pi\)
0.0222091 + 0.999753i \(0.492930\pi\)
\(938\) 0 0
\(939\) 12075.2i 0.419659i
\(940\) 0 0
\(941\) 26446.9i 0.916201i 0.888900 + 0.458101i \(0.151470\pi\)
−0.888900 + 0.458101i \(0.848530\pi\)
\(942\) 0 0
\(943\) −10640.0 −0.367430
\(944\) 0 0
\(945\) −11648.0 −0.400962
\(946\) 0 0
\(947\) 23922.9i 0.820897i 0.911884 + 0.410448i \(0.134628\pi\)
−0.911884 + 0.410448i \(0.865372\pi\)
\(948\) 0 0
\(949\) 15557.0i 0.532141i
\(950\) 0 0
\(951\) 50680.0 1.72809
\(952\) 0 0
\(953\) 38250.0 1.30015 0.650073 0.759872i \(-0.274738\pi\)
0.650073 + 0.759872i \(0.274738\pi\)
\(954\) 0 0
\(955\) − 41316.1i − 1.39995i
\(956\) 0 0
\(957\) 13334.6i 0.450414i
\(958\) 0 0
\(959\) −8496.00 −0.286079
\(960\) 0 0
\(961\) 20385.0 0.684267
\(962\) 0 0
\(963\) 269.867i 0.00903046i
\(964\) 0 0
\(965\) 35241.4i 1.17561i
\(966\) 0 0
\(967\) −4664.00 −0.155103 −0.0775513 0.996988i \(-0.524710\pi\)
−0.0775513 + 0.996988i \(0.524710\pi\)
\(968\) 0 0
\(969\) −2744.00 −0.0909701
\(970\) 0 0
\(971\) 30971.2i 1.02360i 0.859106 + 0.511798i \(0.171020\pi\)
−0.859106 + 0.511798i \(0.828980\pi\)
\(972\) 0 0
\(973\) 21547.0i 0.709933i
\(974\) 0 0
\(975\) 3640.00 0.119562
\(976\) 0 0
\(977\) −4814.00 −0.157639 −0.0788196 0.996889i \(-0.525115\pi\)
−0.0788196 + 0.996889i \(0.525115\pi\)
\(978\) 0 0
\(979\) − 4222.62i − 0.137850i
\(980\) 0 0
\(981\) − 1894.36i − 0.0616536i
\(982\) 0 0
\(983\) 12376.0 0.401560 0.200780 0.979636i \(-0.435652\pi\)
0.200780 + 0.979636i \(0.435652\pi\)
\(984\) 0 0
\(985\) −12656.0 −0.409395
\(986\) 0 0
\(987\) − 14223.6i − 0.458704i
\(988\) 0 0
\(989\) − 66757.6i − 2.14638i
\(990\) 0 0
\(991\) −45344.0 −1.45348 −0.726740 0.686912i \(-0.758966\pi\)
−0.726740 + 0.686912i \(0.758966\pi\)
\(992\) 0 0
\(993\) 22484.0 0.718538
\(994\) 0 0
\(995\) 18372.1i 0.585361i
\(996\) 0 0
\(997\) 26002.4i 0.825984i 0.910735 + 0.412992i \(0.135516\pi\)
−0.910735 + 0.412992i \(0.864484\pi\)
\(998\) 0 0
\(999\) 33488.0 1.06057
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 32.4.b.a.17.2 2
3.2 odd 2 288.4.d.a.145.1 2
4.3 odd 2 8.4.b.a.5.2 yes 2
5.2 odd 4 800.4.f.a.49.4 4
5.3 odd 4 800.4.f.a.49.1 4
5.4 even 2 800.4.d.a.401.1 2
8.3 odd 2 8.4.b.a.5.1 2
8.5 even 2 inner 32.4.b.a.17.1 2
12.11 even 2 72.4.d.b.37.1 2
16.3 odd 4 256.4.a.l.1.2 2
16.5 even 4 256.4.a.j.1.2 2
16.11 odd 4 256.4.a.l.1.1 2
16.13 even 4 256.4.a.j.1.1 2
20.3 even 4 200.4.f.a.149.4 4
20.7 even 4 200.4.f.a.149.1 4
20.19 odd 2 200.4.d.a.101.1 2
24.5 odd 2 288.4.d.a.145.2 2
24.11 even 2 72.4.d.b.37.2 2
40.3 even 4 200.4.f.a.149.2 4
40.13 odd 4 800.4.f.a.49.3 4
40.19 odd 2 200.4.d.a.101.2 2
40.27 even 4 200.4.f.a.149.3 4
40.29 even 2 800.4.d.a.401.2 2
40.37 odd 4 800.4.f.a.49.2 4
48.5 odd 4 2304.4.a.v.1.2 2
48.11 even 4 2304.4.a.bn.1.2 2
48.29 odd 4 2304.4.a.v.1.1 2
48.35 even 4 2304.4.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.4.b.a.5.1 2 8.3 odd 2
8.4.b.a.5.2 yes 2 4.3 odd 2
32.4.b.a.17.1 2 8.5 even 2 inner
32.4.b.a.17.2 2 1.1 even 1 trivial
72.4.d.b.37.1 2 12.11 even 2
72.4.d.b.37.2 2 24.11 even 2
200.4.d.a.101.1 2 20.19 odd 2
200.4.d.a.101.2 2 40.19 odd 2
200.4.f.a.149.1 4 20.7 even 4
200.4.f.a.149.2 4 40.3 even 4
200.4.f.a.149.3 4 40.27 even 4
200.4.f.a.149.4 4 20.3 even 4
256.4.a.j.1.1 2 16.13 even 4
256.4.a.j.1.2 2 16.5 even 4
256.4.a.l.1.1 2 16.11 odd 4
256.4.a.l.1.2 2 16.3 odd 4
288.4.d.a.145.1 2 3.2 odd 2
288.4.d.a.145.2 2 24.5 odd 2
800.4.d.a.401.1 2 5.4 even 2
800.4.d.a.401.2 2 40.29 even 2
800.4.f.a.49.1 4 5.3 odd 4
800.4.f.a.49.2 4 40.37 odd 4
800.4.f.a.49.3 4 40.13 odd 4
800.4.f.a.49.4 4 5.2 odd 4
2304.4.a.v.1.1 2 48.29 odd 4
2304.4.a.v.1.2 2 48.5 odd 4
2304.4.a.bn.1.1 2 48.35 even 4
2304.4.a.bn.1.2 2 48.11 even 4