Properties

Label 5520.2.be.d.1471.17
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5520,2,Mod(1471,5520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5520.1471");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.be (of order \(2\), degree \(1\), 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: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.17
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.d.1471.18

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.00000i q^{5} +0.143131 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +1.00000i q^{5} +0.143131 q^{7} -1.00000 q^{9} +4.80659 q^{11} +4.78151 q^{13} +1.00000 q^{15} -4.86743i q^{17} +0.379340 q^{19} -0.143131i q^{21} +(1.15365 - 4.65501i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +8.39933 q^{29} +2.55983i q^{31} -4.80659i q^{33} +0.143131i q^{35} +5.56282i q^{37} -4.78151i q^{39} -5.17202 q^{41} +1.03505 q^{43} -1.00000i q^{45} -8.93486i q^{47} -6.97951 q^{49} -4.86743 q^{51} +13.3413i q^{53} +4.80659i q^{55} -0.379340i q^{57} -6.57340i q^{59} -7.49837i q^{61} -0.143131 q^{63} +4.78151i q^{65} +4.96888 q^{67} +(-4.65501 - 1.15365i) q^{69} +6.77385i q^{71} -8.04529 q^{73} +1.00000i q^{75} +0.687974 q^{77} -1.53132 q^{79} +1.00000 q^{81} +14.3973 q^{83} +4.86743 q^{85} -8.39933i q^{87} +12.6001i q^{89} +0.684383 q^{91} +2.55983 q^{93} +0.379340i q^{95} -5.67311i q^{97} -4.80659 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 8 q^{7} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 8 q^{7} - 32 q^{9} - 8 q^{11} - 8 q^{13} + 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} + 4 q^{51} - 8 q^{63} - 32 q^{67} - 40 q^{73} - 24 q^{77} - 32 q^{79} + 32 q^{81} - 4 q^{85} + 48 q^{91} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1841\) \(4417\) \(4831\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 0.143131 0.0540986 0.0270493 0.999634i \(-0.491389\pi\)
0.0270493 + 0.999634i \(0.491389\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.80659 1.44924 0.724621 0.689148i \(-0.242015\pi\)
0.724621 + 0.689148i \(0.242015\pi\)
\(12\) 0 0
\(13\) 4.78151 1.32615 0.663075 0.748553i \(-0.269251\pi\)
0.663075 + 0.748553i \(0.269251\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 4.86743i 1.18053i −0.807211 0.590263i \(-0.799024\pi\)
0.807211 0.590263i \(-0.200976\pi\)
\(18\) 0 0
\(19\) 0.379340 0.0870266 0.0435133 0.999053i \(-0.486145\pi\)
0.0435133 + 0.999053i \(0.486145\pi\)
\(20\) 0 0
\(21\) 0.143131i 0.0312338i
\(22\) 0 0
\(23\) 1.15365 4.65501i 0.240552 0.970636i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 8.39933 1.55972 0.779858 0.625956i \(-0.215291\pi\)
0.779858 + 0.625956i \(0.215291\pi\)
\(30\) 0 0
\(31\) 2.55983i 0.459759i 0.973219 + 0.229879i \(0.0738332\pi\)
−0.973219 + 0.229879i \(0.926167\pi\)
\(32\) 0 0
\(33\) 4.80659i 0.836720i
\(34\) 0 0
\(35\) 0.143131i 0.0241936i
\(36\) 0 0
\(37\) 5.56282i 0.914521i 0.889333 + 0.457261i \(0.151169\pi\)
−0.889333 + 0.457261i \(0.848831\pi\)
\(38\) 0 0
\(39\) 4.78151i 0.765654i
\(40\) 0 0
\(41\) −5.17202 −0.807734 −0.403867 0.914818i \(-0.632334\pi\)
−0.403867 + 0.914818i \(0.632334\pi\)
\(42\) 0 0
\(43\) 1.03505 0.157843 0.0789215 0.996881i \(-0.474852\pi\)
0.0789215 + 0.996881i \(0.474852\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 8.93486i 1.30328i −0.758527 0.651642i \(-0.774081\pi\)
0.758527 0.651642i \(-0.225919\pi\)
\(48\) 0 0
\(49\) −6.97951 −0.997073
\(50\) 0 0
\(51\) −4.86743 −0.681577
\(52\) 0 0
\(53\) 13.3413i 1.83256i 0.400534 + 0.916282i \(0.368825\pi\)
−0.400534 + 0.916282i \(0.631175\pi\)
\(54\) 0 0
\(55\) 4.80659i 0.648121i
\(56\) 0 0
\(57\) 0.379340i 0.0502448i
\(58\) 0 0
\(59\) 6.57340i 0.855783i −0.903830 0.427892i \(-0.859257\pi\)
0.903830 0.427892i \(-0.140743\pi\)
\(60\) 0 0
\(61\) 7.49837i 0.960068i −0.877250 0.480034i \(-0.840624\pi\)
0.877250 0.480034i \(-0.159376\pi\)
\(62\) 0 0
\(63\) −0.143131 −0.0180329
\(64\) 0 0
\(65\) 4.78151i 0.593073i
\(66\) 0 0
\(67\) 4.96888 0.607046 0.303523 0.952824i \(-0.401837\pi\)
0.303523 + 0.952824i \(0.401837\pi\)
\(68\) 0 0
\(69\) −4.65501 1.15365i −0.560397 0.138883i
\(70\) 0 0
\(71\) 6.77385i 0.803908i 0.915660 + 0.401954i \(0.131669\pi\)
−0.915660 + 0.401954i \(0.868331\pi\)
\(72\) 0 0
\(73\) −8.04529 −0.941631 −0.470815 0.882232i \(-0.656040\pi\)
−0.470815 + 0.882232i \(0.656040\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 0.687974 0.0784019
\(78\) 0 0
\(79\) −1.53132 −0.172287 −0.0861433 0.996283i \(-0.527454\pi\)
−0.0861433 + 0.996283i \(0.527454\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.3973 1.58031 0.790157 0.612905i \(-0.209999\pi\)
0.790157 + 0.612905i \(0.209999\pi\)
\(84\) 0 0
\(85\) 4.86743 0.527947
\(86\) 0 0
\(87\) 8.39933i 0.900503i
\(88\) 0 0
\(89\) 12.6001i 1.33561i 0.744337 + 0.667804i \(0.232766\pi\)
−0.744337 + 0.667804i \(0.767234\pi\)
\(90\) 0 0
\(91\) 0.684383 0.0717429
\(92\) 0 0
\(93\) 2.55983 0.265442
\(94\) 0 0
\(95\) 0.379340i 0.0389195i
\(96\) 0 0
\(97\) 5.67311i 0.576017i −0.957628 0.288009i \(-0.907007\pi\)
0.957628 0.288009i \(-0.0929932\pi\)
\(98\) 0 0
\(99\) −4.80659 −0.483081
\(100\) 0 0
\(101\) −17.3244 −1.72384 −0.861921 0.507043i \(-0.830739\pi\)
−0.861921 + 0.507043i \(0.830739\pi\)
\(102\) 0 0
\(103\) 12.2942 1.21138 0.605691 0.795700i \(-0.292897\pi\)
0.605691 + 0.795700i \(0.292897\pi\)
\(104\) 0 0
\(105\) 0.143131 0.0139682
\(106\) 0 0
\(107\) 1.02774 0.0993557 0.0496779 0.998765i \(-0.484181\pi\)
0.0496779 + 0.998765i \(0.484181\pi\)
\(108\) 0 0
\(109\) 16.9971i 1.62802i −0.580849 0.814011i \(-0.697279\pi\)
0.580849 0.814011i \(-0.302721\pi\)
\(110\) 0 0
\(111\) 5.56282 0.527999
\(112\) 0 0
\(113\) 0.637571i 0.0599776i −0.999550 0.0299888i \(-0.990453\pi\)
0.999550 0.0299888i \(-0.00954716\pi\)
\(114\) 0 0
\(115\) 4.65501 + 1.15365i 0.434082 + 0.107578i
\(116\) 0 0
\(117\) −4.78151 −0.442050
\(118\) 0 0
\(119\) 0.696682i 0.0638648i
\(120\) 0 0
\(121\) 12.1033 1.10030
\(122\) 0 0
\(123\) 5.17202i 0.466345i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 3.74112i 0.331970i −0.986128 0.165985i \(-0.946920\pi\)
0.986128 0.165985i \(-0.0530804\pi\)
\(128\) 0 0
\(129\) 1.03505i 0.0911308i
\(130\) 0 0
\(131\) 11.9550i 1.04451i 0.852790 + 0.522255i \(0.174909\pi\)
−0.852790 + 0.522255i \(0.825091\pi\)
\(132\) 0 0
\(133\) 0.0542955 0.00470802
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 8.16863i 0.697893i −0.937143 0.348947i \(-0.886539\pi\)
0.937143 0.348947i \(-0.113461\pi\)
\(138\) 0 0
\(139\) 0.870322i 0.0738198i 0.999319 + 0.0369099i \(0.0117514\pi\)
−0.999319 + 0.0369099i \(0.988249\pi\)
\(140\) 0 0
\(141\) −8.93486 −0.752451
\(142\) 0 0
\(143\) 22.9827 1.92191
\(144\) 0 0
\(145\) 8.39933i 0.697526i
\(146\) 0 0
\(147\) 6.97951i 0.575661i
\(148\) 0 0
\(149\) 11.4050i 0.934337i 0.884168 + 0.467168i \(0.154726\pi\)
−0.884168 + 0.467168i \(0.845274\pi\)
\(150\) 0 0
\(151\) 5.66516i 0.461024i −0.973069 0.230512i \(-0.925960\pi\)
0.973069 0.230512i \(-0.0740401\pi\)
\(152\) 0 0
\(153\) 4.86743i 0.393509i
\(154\) 0 0
\(155\) −2.55983 −0.205610
\(156\) 0 0
\(157\) 5.94964i 0.474833i −0.971408 0.237417i \(-0.923699\pi\)
0.971408 0.237417i \(-0.0763006\pi\)
\(158\) 0 0
\(159\) 13.3413 1.05803
\(160\) 0 0
\(161\) 0.165123 0.666278i 0.0130135 0.0525100i
\(162\) 0 0
\(163\) 12.0905i 0.946997i 0.880795 + 0.473499i \(0.157009\pi\)
−0.880795 + 0.473499i \(0.842991\pi\)
\(164\) 0 0
\(165\) 4.80659 0.374193
\(166\) 0 0
\(167\) 14.1105i 1.09191i −0.837816 0.545953i \(-0.816168\pi\)
0.837816 0.545953i \(-0.183832\pi\)
\(168\) 0 0
\(169\) 9.86279 0.758676
\(170\) 0 0
\(171\) −0.379340 −0.0290089
\(172\) 0 0
\(173\) 3.68070 0.279838 0.139919 0.990163i \(-0.455316\pi\)
0.139919 + 0.990163i \(0.455316\pi\)
\(174\) 0 0
\(175\) −0.143131 −0.0108197
\(176\) 0 0
\(177\) −6.57340 −0.494087
\(178\) 0 0
\(179\) 9.27393i 0.693166i 0.938019 + 0.346583i \(0.112658\pi\)
−0.938019 + 0.346583i \(0.887342\pi\)
\(180\) 0 0
\(181\) 5.10162i 0.379200i −0.981861 0.189600i \(-0.939281\pi\)
0.981861 0.189600i \(-0.0607191\pi\)
\(182\) 0 0
\(183\) −7.49837 −0.554295
\(184\) 0 0
\(185\) −5.56282 −0.408986
\(186\) 0 0
\(187\) 23.3958i 1.71087i
\(188\) 0 0
\(189\) 0.143131i 0.0104113i
\(190\) 0 0
\(191\) 20.1633 1.45897 0.729484 0.683997i \(-0.239760\pi\)
0.729484 + 0.683997i \(0.239760\pi\)
\(192\) 0 0
\(193\) 1.06748 0.0768392 0.0384196 0.999262i \(-0.487768\pi\)
0.0384196 + 0.999262i \(0.487768\pi\)
\(194\) 0 0
\(195\) 4.78151 0.342411
\(196\) 0 0
\(197\) 4.31277 0.307272 0.153636 0.988128i \(-0.450902\pi\)
0.153636 + 0.988128i \(0.450902\pi\)
\(198\) 0 0
\(199\) 9.35032 0.662827 0.331413 0.943486i \(-0.392475\pi\)
0.331413 + 0.943486i \(0.392475\pi\)
\(200\) 0 0
\(201\) 4.96888i 0.350478i
\(202\) 0 0
\(203\) 1.20221 0.0843784
\(204\) 0 0
\(205\) 5.17202i 0.361230i
\(206\) 0 0
\(207\) −1.15365 + 4.65501i −0.0801840 + 0.323545i
\(208\) 0 0
\(209\) 1.82333 0.126123
\(210\) 0 0
\(211\) 19.8715i 1.36801i −0.729478 0.684004i \(-0.760237\pi\)
0.729478 0.684004i \(-0.239763\pi\)
\(212\) 0 0
\(213\) 6.77385 0.464136
\(214\) 0 0
\(215\) 1.03505i 0.0705896i
\(216\) 0 0
\(217\) 0.366392i 0.0248723i
\(218\) 0 0
\(219\) 8.04529i 0.543651i
\(220\) 0 0
\(221\) 23.2737i 1.56556i
\(222\) 0 0
\(223\) 13.4715i 0.902116i 0.892495 + 0.451058i \(0.148953\pi\)
−0.892495 + 0.451058i \(0.851047\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 3.56731 0.236771 0.118385 0.992968i \(-0.462228\pi\)
0.118385 + 0.992968i \(0.462228\pi\)
\(228\) 0 0
\(229\) 9.73343i 0.643203i −0.946875 0.321602i \(-0.895779\pi\)
0.946875 0.321602i \(-0.104221\pi\)
\(230\) 0 0
\(231\) 0.687974i 0.0452654i
\(232\) 0 0
\(233\) 19.7008 1.29065 0.645323 0.763910i \(-0.276723\pi\)
0.645323 + 0.763910i \(0.276723\pi\)
\(234\) 0 0
\(235\) 8.93486 0.582846
\(236\) 0 0
\(237\) 1.53132i 0.0994697i
\(238\) 0 0
\(239\) 9.71151i 0.628185i 0.949392 + 0.314093i \(0.101700\pi\)
−0.949392 + 0.314093i \(0.898300\pi\)
\(240\) 0 0
\(241\) 11.3349i 0.730144i −0.930979 0.365072i \(-0.881044\pi\)
0.930979 0.365072i \(-0.118956\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 6.97951i 0.445905i
\(246\) 0 0
\(247\) 1.81382 0.115410
\(248\) 0 0
\(249\) 14.3973i 0.912394i
\(250\) 0 0
\(251\) 12.4895 0.788329 0.394164 0.919040i \(-0.371034\pi\)
0.394164 + 0.919040i \(0.371034\pi\)
\(252\) 0 0
\(253\) 5.54511 22.3747i 0.348618 1.40669i
\(254\) 0 0
\(255\) 4.86743i 0.304810i
\(256\) 0 0
\(257\) −4.82467 −0.300954 −0.150477 0.988613i \(-0.548081\pi\)
−0.150477 + 0.988613i \(0.548081\pi\)
\(258\) 0 0
\(259\) 0.796214i 0.0494743i
\(260\) 0 0
\(261\) −8.39933 −0.519905
\(262\) 0 0
\(263\) 6.75865 0.416756 0.208378 0.978048i \(-0.433182\pi\)
0.208378 + 0.978048i \(0.433182\pi\)
\(264\) 0 0
\(265\) −13.3413 −0.819547
\(266\) 0 0
\(267\) 12.6001 0.771114
\(268\) 0 0
\(269\) 9.49855 0.579137 0.289568 0.957157i \(-0.406488\pi\)
0.289568 + 0.957157i \(0.406488\pi\)
\(270\) 0 0
\(271\) 17.6288i 1.07087i −0.844576 0.535435i \(-0.820148\pi\)
0.844576 0.535435i \(-0.179852\pi\)
\(272\) 0 0
\(273\) 0.684383i 0.0414208i
\(274\) 0 0
\(275\) −4.80659 −0.289848
\(276\) 0 0
\(277\) 8.47253 0.509065 0.254533 0.967064i \(-0.418078\pi\)
0.254533 + 0.967064i \(0.418078\pi\)
\(278\) 0 0
\(279\) 2.55983i 0.153253i
\(280\) 0 0
\(281\) 10.6546i 0.635603i 0.948157 + 0.317801i \(0.102945\pi\)
−0.948157 + 0.317801i \(0.897055\pi\)
\(282\) 0 0
\(283\) −9.84325 −0.585120 −0.292560 0.956247i \(-0.594507\pi\)
−0.292560 + 0.956247i \(0.594507\pi\)
\(284\) 0 0
\(285\) 0.379340 0.0224702
\(286\) 0 0
\(287\) −0.740278 −0.0436973
\(288\) 0 0
\(289\) −6.69189 −0.393641
\(290\) 0 0
\(291\) −5.67311 −0.332564
\(292\) 0 0
\(293\) 29.3775i 1.71625i −0.513439 0.858126i \(-0.671629\pi\)
0.513439 0.858126i \(-0.328371\pi\)
\(294\) 0 0
\(295\) 6.57340 0.382718
\(296\) 0 0
\(297\) 4.80659i 0.278907i
\(298\) 0 0
\(299\) 5.51617 22.2579i 0.319008 1.28721i
\(300\) 0 0
\(301\) 0.148148 0.00853909
\(302\) 0 0
\(303\) 17.3244i 0.995260i
\(304\) 0 0
\(305\) 7.49837 0.429355
\(306\) 0 0
\(307\) 15.6395i 0.892591i −0.894886 0.446296i \(-0.852743\pi\)
0.894886 0.446296i \(-0.147257\pi\)
\(308\) 0 0
\(309\) 12.2942i 0.699391i
\(310\) 0 0
\(311\) 16.7393i 0.949201i −0.880201 0.474600i \(-0.842593\pi\)
0.880201 0.474600i \(-0.157407\pi\)
\(312\) 0 0
\(313\) 17.0663i 0.964646i 0.875993 + 0.482323i \(0.160207\pi\)
−0.875993 + 0.482323i \(0.839793\pi\)
\(314\) 0 0
\(315\) 0.143131i 0.00806454i
\(316\) 0 0
\(317\) −29.6407 −1.66479 −0.832394 0.554184i \(-0.813030\pi\)
−0.832394 + 0.554184i \(0.813030\pi\)
\(318\) 0 0
\(319\) 40.3721 2.26041
\(320\) 0 0
\(321\) 1.02774i 0.0573630i
\(322\) 0 0
\(323\) 1.84641i 0.102737i
\(324\) 0 0
\(325\) −4.78151 −0.265230
\(326\) 0 0
\(327\) −16.9971 −0.939939
\(328\) 0 0
\(329\) 1.27886i 0.0705058i
\(330\) 0 0
\(331\) 4.03408i 0.221733i 0.993835 + 0.110866i \(0.0353626\pi\)
−0.993835 + 0.110866i \(0.964637\pi\)
\(332\) 0 0
\(333\) 5.56282i 0.304840i
\(334\) 0 0
\(335\) 4.96888i 0.271479i
\(336\) 0 0
\(337\) 1.54962i 0.0844132i 0.999109 + 0.0422066i \(0.0134388\pi\)
−0.999109 + 0.0422066i \(0.986561\pi\)
\(338\) 0 0
\(339\) −0.637571 −0.0346281
\(340\) 0 0
\(341\) 12.3040i 0.666302i
\(342\) 0 0
\(343\) −2.00091 −0.108039
\(344\) 0 0
\(345\) 1.15365 4.65501i 0.0621103 0.250617i
\(346\) 0 0
\(347\) 9.87984i 0.530377i 0.964197 + 0.265189i \(0.0854343\pi\)
−0.964197 + 0.265189i \(0.914566\pi\)
\(348\) 0 0
\(349\) 6.41716 0.343503 0.171751 0.985140i \(-0.445057\pi\)
0.171751 + 0.985140i \(0.445057\pi\)
\(350\) 0 0
\(351\) 4.78151i 0.255218i
\(352\) 0 0
\(353\) −6.23647 −0.331934 −0.165967 0.986131i \(-0.553074\pi\)
−0.165967 + 0.986131i \(0.553074\pi\)
\(354\) 0 0
\(355\) −6.77385 −0.359519
\(356\) 0 0
\(357\) −0.696682 −0.0368723
\(358\) 0 0
\(359\) 4.19870 0.221599 0.110799 0.993843i \(-0.464659\pi\)
0.110799 + 0.993843i \(0.464659\pi\)
\(360\) 0 0
\(361\) −18.8561 −0.992426
\(362\) 0 0
\(363\) 12.1033i 0.635260i
\(364\) 0 0
\(365\) 8.04529i 0.421110i
\(366\) 0 0
\(367\) −5.21796 −0.272375 −0.136188 0.990683i \(-0.543485\pi\)
−0.136188 + 0.990683i \(0.543485\pi\)
\(368\) 0 0
\(369\) 5.17202 0.269245
\(370\) 0 0
\(371\) 1.90955i 0.0991391i
\(372\) 0 0
\(373\) 24.8539i 1.28689i 0.765493 + 0.643444i \(0.222495\pi\)
−0.765493 + 0.643444i \(0.777505\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 40.1614 2.06842
\(378\) 0 0
\(379\) −14.2322 −0.731058 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(380\) 0 0
\(381\) −3.74112 −0.191663
\(382\) 0 0
\(383\) −23.6762 −1.20980 −0.604898 0.796303i \(-0.706786\pi\)
−0.604898 + 0.796303i \(0.706786\pi\)
\(384\) 0 0
\(385\) 0.687974i 0.0350624i
\(386\) 0 0
\(387\) −1.03505 −0.0526144
\(388\) 0 0
\(389\) 6.73328i 0.341391i 0.985324 + 0.170695i \(0.0546014\pi\)
−0.985324 + 0.170695i \(0.945399\pi\)
\(390\) 0 0
\(391\) −22.6579 5.61530i −1.14586 0.283978i
\(392\) 0 0
\(393\) 11.9550 0.603048
\(394\) 0 0
\(395\) 1.53132i 0.0770489i
\(396\) 0 0
\(397\) 9.27378 0.465438 0.232719 0.972544i \(-0.425238\pi\)
0.232719 + 0.972544i \(0.425238\pi\)
\(398\) 0 0
\(399\) 0.0542955i 0.00271817i
\(400\) 0 0
\(401\) 38.6681i 1.93099i 0.260421 + 0.965495i \(0.416139\pi\)
−0.260421 + 0.965495i \(0.583861\pi\)
\(402\) 0 0
\(403\) 12.2398i 0.609709i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 26.7382i 1.32536i
\(408\) 0 0
\(409\) −23.1274 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(410\) 0 0
\(411\) −8.16863 −0.402929
\(412\) 0 0
\(413\) 0.940859i 0.0462967i
\(414\) 0 0
\(415\) 14.3973i 0.706738i
\(416\) 0 0
\(417\) 0.870322 0.0426199
\(418\) 0 0
\(419\) 2.95569 0.144395 0.0721974 0.997390i \(-0.476999\pi\)
0.0721974 + 0.997390i \(0.476999\pi\)
\(420\) 0 0
\(421\) 4.13502i 0.201528i −0.994910 0.100764i \(-0.967871\pi\)
0.994910 0.100764i \(-0.0321288\pi\)
\(422\) 0 0
\(423\) 8.93486i 0.434428i
\(424\) 0 0
\(425\) 4.86743i 0.236105i
\(426\) 0 0
\(427\) 1.07325i 0.0519383i
\(428\) 0 0
\(429\) 22.9827i 1.10962i
\(430\) 0 0
\(431\) 8.46802 0.407890 0.203945 0.978982i \(-0.434624\pi\)
0.203945 + 0.978982i \(0.434624\pi\)
\(432\) 0 0
\(433\) 17.3572i 0.834132i −0.908876 0.417066i \(-0.863058\pi\)
0.908876 0.417066i \(-0.136942\pi\)
\(434\) 0 0
\(435\) 8.39933 0.402717
\(436\) 0 0
\(437\) 0.437625 1.76583i 0.0209344 0.0844712i
\(438\) 0 0
\(439\) 24.8336i 1.18524i −0.805481 0.592622i \(-0.798093\pi\)
0.805481 0.592622i \(-0.201907\pi\)
\(440\) 0 0
\(441\) 6.97951 0.332358
\(442\) 0 0
\(443\) 26.8894i 1.27756i 0.769391 + 0.638778i \(0.220560\pi\)
−0.769391 + 0.638778i \(0.779440\pi\)
\(444\) 0 0
\(445\) −12.6001 −0.597302
\(446\) 0 0
\(447\) 11.4050 0.539440
\(448\) 0 0
\(449\) −33.4193 −1.57716 −0.788578 0.614935i \(-0.789182\pi\)
−0.788578 + 0.614935i \(0.789182\pi\)
\(450\) 0 0
\(451\) −24.8598 −1.17060
\(452\) 0 0
\(453\) −5.66516 −0.266172
\(454\) 0 0
\(455\) 0.684383i 0.0320844i
\(456\) 0 0
\(457\) 7.03968i 0.329302i −0.986352 0.164651i \(-0.947350\pi\)
0.986352 0.164651i \(-0.0526498\pi\)
\(458\) 0 0
\(459\) 4.86743 0.227192
\(460\) 0 0
\(461\) −32.0115 −1.49092 −0.745462 0.666548i \(-0.767771\pi\)
−0.745462 + 0.666548i \(0.767771\pi\)
\(462\) 0 0
\(463\) 16.1154i 0.748945i −0.927238 0.374472i \(-0.877824\pi\)
0.927238 0.374472i \(-0.122176\pi\)
\(464\) 0 0
\(465\) 2.55983i 0.118709i
\(466\) 0 0
\(467\) 40.8400 1.88985 0.944926 0.327285i \(-0.106134\pi\)
0.944926 + 0.327285i \(0.106134\pi\)
\(468\) 0 0
\(469\) 0.711203 0.0328403
\(470\) 0 0
\(471\) −5.94964 −0.274145
\(472\) 0 0
\(473\) 4.97505 0.228753
\(474\) 0 0
\(475\) −0.379340 −0.0174053
\(476\) 0 0
\(477\) 13.3413i 0.610855i
\(478\) 0 0
\(479\) 12.5812 0.574848 0.287424 0.957803i \(-0.407201\pi\)
0.287424 + 0.957803i \(0.407201\pi\)
\(480\) 0 0
\(481\) 26.5986i 1.21279i
\(482\) 0 0
\(483\) −0.666278 0.165123i −0.0303167 0.00751336i
\(484\) 0 0
\(485\) 5.67311 0.257603
\(486\) 0 0
\(487\) 31.0193i 1.40562i 0.711378 + 0.702810i \(0.248072\pi\)
−0.711378 + 0.702810i \(0.751928\pi\)
\(488\) 0 0
\(489\) 12.0905 0.546749
\(490\) 0 0
\(491\) 35.8601i 1.61835i −0.587571 0.809173i \(-0.699916\pi\)
0.587571 0.809173i \(-0.300084\pi\)
\(492\) 0 0
\(493\) 40.8832i 1.84129i
\(494\) 0 0
\(495\) 4.80659i 0.216040i
\(496\) 0 0
\(497\) 0.969550i 0.0434903i
\(498\) 0 0
\(499\) 10.0149i 0.448330i 0.974551 + 0.224165i \(0.0719654\pi\)
−0.974551 + 0.224165i \(0.928035\pi\)
\(500\) 0 0
\(501\) −14.1105 −0.630413
\(502\) 0 0
\(503\) 38.5120 1.71716 0.858582 0.512676i \(-0.171346\pi\)
0.858582 + 0.512676i \(0.171346\pi\)
\(504\) 0 0
\(505\) 17.3244i 0.770925i
\(506\) 0 0
\(507\) 9.86279i 0.438022i
\(508\) 0 0
\(509\) 31.4747 1.39509 0.697546 0.716540i \(-0.254275\pi\)
0.697546 + 0.716540i \(0.254275\pi\)
\(510\) 0 0
\(511\) −1.15153 −0.0509409
\(512\) 0 0
\(513\) 0.379340i 0.0167483i
\(514\) 0 0
\(515\) 12.2942i 0.541746i
\(516\) 0 0
\(517\) 42.9462i 1.88877i
\(518\) 0 0
\(519\) 3.68070i 0.161565i
\(520\) 0 0
\(521\) 21.5620i 0.944647i 0.881425 + 0.472324i \(0.156585\pi\)
−0.881425 + 0.472324i \(0.843415\pi\)
\(522\) 0 0
\(523\) −40.7270 −1.78087 −0.890434 0.455112i \(-0.849599\pi\)
−0.890434 + 0.455112i \(0.849599\pi\)
\(524\) 0 0
\(525\) 0.143131i 0.00624677i
\(526\) 0 0
\(527\) 12.4598 0.542757
\(528\) 0 0
\(529\) −20.3382 10.7405i −0.884269 0.466977i
\(530\) 0 0
\(531\) 6.57340i 0.285261i
\(532\) 0 0
\(533\) −24.7300 −1.07118
\(534\) 0 0
\(535\) 1.02774i 0.0444332i
\(536\) 0 0
\(537\) 9.27393 0.400200
\(538\) 0 0
\(539\) −33.5477 −1.44500
\(540\) 0 0
\(541\) −38.8931 −1.67215 −0.836073 0.548618i \(-0.815154\pi\)
−0.836073 + 0.548618i \(0.815154\pi\)
\(542\) 0 0
\(543\) −5.10162 −0.218931
\(544\) 0 0
\(545\) 16.9971 0.728074
\(546\) 0 0
\(547\) 0.658772i 0.0281671i −0.999901 0.0140835i \(-0.995517\pi\)
0.999901 0.0140835i \(-0.00448308\pi\)
\(548\) 0 0
\(549\) 7.49837i 0.320023i
\(550\) 0 0
\(551\) 3.18620 0.135737
\(552\) 0 0
\(553\) −0.219179 −0.00932046
\(554\) 0 0
\(555\) 5.56282i 0.236128i
\(556\) 0 0
\(557\) 4.89839i 0.207551i −0.994601 0.103776i \(-0.966908\pi\)
0.994601 0.103776i \(-0.0330924\pi\)
\(558\) 0 0
\(559\) 4.94908 0.209324
\(560\) 0 0
\(561\) −23.3958 −0.987770
\(562\) 0 0
\(563\) 8.22969 0.346840 0.173420 0.984848i \(-0.444518\pi\)
0.173420 + 0.984848i \(0.444518\pi\)
\(564\) 0 0
\(565\) 0.637571 0.0268228
\(566\) 0 0
\(567\) 0.143131 0.00601095
\(568\) 0 0
\(569\) 4.84217i 0.202994i 0.994836 + 0.101497i \(0.0323632\pi\)
−0.994836 + 0.101497i \(0.967637\pi\)
\(570\) 0 0
\(571\) 18.6944 0.782335 0.391167 0.920320i \(-0.372071\pi\)
0.391167 + 0.920320i \(0.372071\pi\)
\(572\) 0 0
\(573\) 20.1633i 0.842336i
\(574\) 0 0
\(575\) −1.15365 + 4.65501i −0.0481104 + 0.194127i
\(576\) 0 0
\(577\) 37.9160 1.57847 0.789233 0.614094i \(-0.210479\pi\)
0.789233 + 0.614094i \(0.210479\pi\)
\(578\) 0 0
\(579\) 1.06748i 0.0443631i
\(580\) 0 0
\(581\) 2.06071 0.0854927
\(582\) 0 0
\(583\) 64.1260i 2.65583i
\(584\) 0 0
\(585\) 4.78151i 0.197691i
\(586\) 0 0
\(587\) 4.52386i 0.186720i −0.995632 0.0933599i \(-0.970239\pi\)
0.995632 0.0933599i \(-0.0297607\pi\)
\(588\) 0 0
\(589\) 0.971046i 0.0400112i
\(590\) 0 0
\(591\) 4.31277i 0.177404i
\(592\) 0 0
\(593\) 21.4620 0.881338 0.440669 0.897670i \(-0.354741\pi\)
0.440669 + 0.897670i \(0.354741\pi\)
\(594\) 0 0
\(595\) 0.696682 0.0285612
\(596\) 0 0
\(597\) 9.35032i 0.382683i
\(598\) 0 0
\(599\) 43.6864i 1.78498i −0.451069 0.892489i \(-0.648957\pi\)
0.451069 0.892489i \(-0.351043\pi\)
\(600\) 0 0
\(601\) −43.6463 −1.78037 −0.890185 0.455599i \(-0.849425\pi\)
−0.890185 + 0.455599i \(0.849425\pi\)
\(602\) 0 0
\(603\) −4.96888 −0.202349
\(604\) 0 0
\(605\) 12.1033i 0.492070i
\(606\) 0 0
\(607\) 13.8775i 0.563269i 0.959522 + 0.281635i \(0.0908766\pi\)
−0.959522 + 0.281635i \(0.909123\pi\)
\(608\) 0 0
\(609\) 1.20221i 0.0487159i
\(610\) 0 0
\(611\) 42.7221i 1.72835i
\(612\) 0 0
\(613\) 18.7574i 0.757606i −0.925477 0.378803i \(-0.876336\pi\)
0.925477 0.378803i \(-0.123664\pi\)
\(614\) 0 0
\(615\) −5.17202 −0.208556
\(616\) 0 0
\(617\) 28.7105i 1.15584i 0.816092 + 0.577922i \(0.196136\pi\)
−0.816092 + 0.577922i \(0.803864\pi\)
\(618\) 0 0
\(619\) 0.458137 0.0184141 0.00920703 0.999958i \(-0.497069\pi\)
0.00920703 + 0.999958i \(0.497069\pi\)
\(620\) 0 0
\(621\) 4.65501 + 1.15365i 0.186799 + 0.0462943i
\(622\) 0 0
\(623\) 1.80347i 0.0722545i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.82333i 0.0728169i
\(628\) 0 0
\(629\) 27.0766 1.07962
\(630\) 0 0
\(631\) 37.7755 1.50382 0.751910 0.659265i \(-0.229133\pi\)
0.751910 + 0.659265i \(0.229133\pi\)
\(632\) 0 0
\(633\) −19.8715 −0.789820
\(634\) 0 0
\(635\) 3.74112 0.148462
\(636\) 0 0
\(637\) −33.3726 −1.32227
\(638\) 0 0
\(639\) 6.77385i 0.267969i
\(640\) 0 0
\(641\) 29.5885i 1.16868i 0.811510 + 0.584338i \(0.198646\pi\)
−0.811510 + 0.584338i \(0.801354\pi\)
\(642\) 0 0
\(643\) 47.0949 1.85724 0.928620 0.371032i \(-0.120996\pi\)
0.928620 + 0.371032i \(0.120996\pi\)
\(644\) 0 0
\(645\) 1.03505 0.0407549
\(646\) 0 0
\(647\) 26.6440i 1.04748i −0.851877 0.523741i \(-0.824536\pi\)
0.851877 0.523741i \(-0.175464\pi\)
\(648\) 0 0
\(649\) 31.5956i 1.24024i
\(650\) 0 0
\(651\) 0.366392 0.0143600
\(652\) 0 0
\(653\) 32.1606 1.25854 0.629272 0.777185i \(-0.283353\pi\)
0.629272 + 0.777185i \(0.283353\pi\)
\(654\) 0 0
\(655\) −11.9550 −0.467119
\(656\) 0 0
\(657\) 8.04529 0.313877
\(658\) 0 0
\(659\) −1.56508 −0.0609670 −0.0304835 0.999535i \(-0.509705\pi\)
−0.0304835 + 0.999535i \(0.509705\pi\)
\(660\) 0 0
\(661\) 11.9297i 0.464010i −0.972715 0.232005i \(-0.925471\pi\)
0.972715 0.232005i \(-0.0745285\pi\)
\(662\) 0 0
\(663\) −23.2737 −0.903874
\(664\) 0 0
\(665\) 0.0542955i 0.00210549i
\(666\) 0 0
\(667\) 9.68987 39.0989i 0.375193 1.51392i
\(668\) 0 0
\(669\) 13.4715 0.520837
\(670\) 0 0
\(671\) 36.0416i 1.39137i
\(672\) 0 0
\(673\) 27.9834 1.07868 0.539340 0.842088i \(-0.318674\pi\)
0.539340 + 0.842088i \(0.318674\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 49.5307i 1.90362i 0.306691 + 0.951809i \(0.400778\pi\)
−0.306691 + 0.951809i \(0.599222\pi\)
\(678\) 0 0
\(679\) 0.812001i 0.0311617i
\(680\) 0 0
\(681\) 3.56731i 0.136700i
\(682\) 0 0
\(683\) 18.2648i 0.698884i 0.936958 + 0.349442i \(0.113629\pi\)
−0.936958 + 0.349442i \(0.886371\pi\)
\(684\) 0 0
\(685\) 8.16863 0.312107
\(686\) 0 0
\(687\) −9.73343 −0.371354
\(688\) 0 0
\(689\) 63.7913i 2.43026i
\(690\) 0 0
\(691\) 24.6191i 0.936554i −0.883582 0.468277i \(-0.844875\pi\)
0.883582 0.468277i \(-0.155125\pi\)
\(692\) 0 0
\(693\) −0.687974 −0.0261340
\(694\) 0 0
\(695\) −0.870322 −0.0330132
\(696\) 0 0
\(697\) 25.1745i 0.953550i
\(698\) 0 0
\(699\) 19.7008i 0.745154i
\(700\) 0 0
\(701\) 35.5895i 1.34420i −0.740461 0.672099i \(-0.765393\pi\)
0.740461 0.672099i \(-0.234607\pi\)
\(702\) 0 0
\(703\) 2.11020i 0.0795877i
\(704\) 0 0
\(705\) 8.93486i 0.336506i
\(706\) 0 0
\(707\) −2.47966 −0.0932574
\(708\) 0 0
\(709\) 8.00880i 0.300777i −0.988627 0.150388i \(-0.951948\pi\)
0.988627 0.150388i \(-0.0480524\pi\)
\(710\) 0 0
\(711\) 1.53132 0.0574289
\(712\) 0 0
\(713\) 11.9160 + 2.95314i 0.446258 + 0.110596i
\(714\) 0 0
\(715\) 22.9827i 0.859506i
\(716\) 0 0
\(717\) 9.71151 0.362683
\(718\) 0 0
\(719\) 40.1847i 1.49864i 0.662210 + 0.749318i \(0.269619\pi\)
−0.662210 + 0.749318i \(0.730381\pi\)
\(720\) 0 0
\(721\) 1.75968 0.0655340
\(722\) 0 0
\(723\) −11.3349 −0.421549
\(724\) 0 0
\(725\) −8.39933 −0.311943
\(726\) 0 0
\(727\) −46.1258 −1.71071 −0.855355 0.518042i \(-0.826661\pi\)
−0.855355 + 0.518042i \(0.826661\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 5.03802i 0.186338i
\(732\) 0 0
\(733\) 11.4344i 0.422340i 0.977449 + 0.211170i \(0.0677274\pi\)
−0.977449 + 0.211170i \(0.932273\pi\)
\(734\) 0 0
\(735\) −6.97951 −0.257443
\(736\) 0 0
\(737\) 23.8834 0.879756
\(738\) 0 0
\(739\) 10.7005i 0.393625i −0.980441 0.196813i \(-0.936941\pi\)
0.980441 0.196813i \(-0.0630591\pi\)
\(740\) 0 0
\(741\) 1.81382i 0.0666322i
\(742\) 0 0
\(743\) 44.7313 1.64103 0.820515 0.571624i \(-0.193687\pi\)
0.820515 + 0.571624i \(0.193687\pi\)
\(744\) 0 0
\(745\) −11.4050 −0.417848
\(746\) 0 0
\(747\) −14.3973 −0.526771
\(748\) 0 0
\(749\) 0.147102 0.00537500
\(750\) 0 0
\(751\) −15.5462 −0.567289 −0.283644 0.958930i \(-0.591544\pi\)
−0.283644 + 0.958930i \(0.591544\pi\)
\(752\) 0 0
\(753\) 12.4895i 0.455142i
\(754\) 0 0
\(755\) 5.66516 0.206176
\(756\) 0 0
\(757\) 15.3689i 0.558594i −0.960205 0.279297i \(-0.909899\pi\)
0.960205 0.279297i \(-0.0901014\pi\)
\(758\) 0 0
\(759\) −22.3747 5.54511i −0.812151 0.201275i
\(760\) 0 0
\(761\) −26.4309 −0.958121 −0.479060 0.877782i \(-0.659023\pi\)
−0.479060 + 0.877782i \(0.659023\pi\)
\(762\) 0 0
\(763\) 2.43281i 0.0880737i
\(764\) 0 0
\(765\) −4.86743 −0.175982
\(766\) 0 0
\(767\) 31.4307i 1.13490i
\(768\) 0 0
\(769\) 51.7442i 1.86594i 0.359948 + 0.932972i \(0.382794\pi\)
−0.359948 + 0.932972i \(0.617206\pi\)
\(770\) 0 0
\(771\) 4.82467i 0.173756i
\(772\) 0 0
\(773\) 11.3165i 0.407026i 0.979072 + 0.203513i \(0.0652360\pi\)
−0.979072 + 0.203513i \(0.934764\pi\)
\(774\) 0 0
\(775\) 2.55983i 0.0919517i
\(776\) 0 0
\(777\) 0.796214 0.0285640
\(778\) 0 0
\(779\) −1.96195 −0.0702943
\(780\) 0 0
\(781\) 32.5591i 1.16506i
\(782\) 0 0
\(783\) 8.39933i 0.300168i
\(784\) 0 0
\(785\) 5.94964 0.212352
\(786\) 0 0
\(787\) −20.8240 −0.742296 −0.371148 0.928574i \(-0.621036\pi\)
−0.371148 + 0.928574i \(0.621036\pi\)
\(788\) 0 0
\(789\) 6.75865i 0.240614i
\(790\) 0 0
\(791\) 0.0912564i 0.00324470i
\(792\) 0 0
\(793\) 35.8535i 1.27319i
\(794\) 0 0
\(795\) 13.3413i 0.473166i
\(796\) 0 0
\(797\) 34.5644i 1.22433i 0.790729 + 0.612166i \(0.209702\pi\)
−0.790729 + 0.612166i \(0.790298\pi\)
\(798\) 0 0
\(799\) −43.4898 −1.53856
\(800\) 0 0
\(801\) 12.6001i 0.445203i
\(802\) 0 0
\(803\) −38.6704 −1.36465
\(804\) 0 0
\(805\) 0.666278 + 0.165123i 0.0234832 + 0.00581983i
\(806\) 0 0
\(807\) 9.49855i 0.334365i
\(808\) 0 0
\(809\) −14.4517 −0.508095 −0.254047 0.967192i \(-0.581762\pi\)
−0.254047 + 0.967192i \(0.581762\pi\)
\(810\) 0 0
\(811\) 24.2239i 0.850616i 0.905049 + 0.425308i \(0.139834\pi\)
−0.905049 + 0.425308i \(0.860166\pi\)
\(812\) 0 0
\(813\) −17.6288 −0.618268
\(814\) 0 0
\(815\) −12.0905 −0.423510
\(816\) 0 0
\(817\) 0.392635 0.0137365
\(818\) 0 0
\(819\) −0.684383 −0.0239143
\(820\) 0 0
\(821\) 19.2778 0.672799 0.336400 0.941719i \(-0.390791\pi\)
0.336400 + 0.941719i \(0.390791\pi\)
\(822\) 0 0
\(823\) 32.9410i 1.14825i −0.818767 0.574126i \(-0.805342\pi\)
0.818767 0.574126i \(-0.194658\pi\)
\(824\) 0 0
\(825\) 4.80659i 0.167344i
\(826\) 0 0
\(827\) −26.6343 −0.926164 −0.463082 0.886316i \(-0.653256\pi\)
−0.463082 + 0.886316i \(0.653256\pi\)
\(828\) 0 0
\(829\) −5.05373 −0.175523 −0.0877616 0.996142i \(-0.527971\pi\)
−0.0877616 + 0.996142i \(0.527971\pi\)
\(830\) 0 0
\(831\) 8.47253i 0.293909i
\(832\) 0 0
\(833\) 33.9723i 1.17707i
\(834\) 0 0
\(835\) 14.1105 0.488316
\(836\) 0 0
\(837\) −2.55983 −0.0884806
\(838\) 0 0
\(839\) −11.1301 −0.384254 −0.192127 0.981370i \(-0.561539\pi\)
−0.192127 + 0.981370i \(0.561539\pi\)
\(840\) 0 0
\(841\) 41.5487 1.43272
\(842\) 0 0
\(843\) 10.6546 0.366965
\(844\) 0 0
\(845\) 9.86279i 0.339290i
\(846\) 0 0
\(847\) 1.73237 0.0595248
\(848\) 0 0
\(849\) 9.84325i 0.337819i
\(850\) 0 0
\(851\) 25.8950 + 6.41753i 0.887668 + 0.219990i
\(852\) 0 0
\(853\) 17.8102 0.609809 0.304904 0.952383i \(-0.401375\pi\)
0.304904 + 0.952383i \(0.401375\pi\)
\(854\) 0 0
\(855\) 0.379340i 0.0129732i
\(856\) 0 0
\(857\) −42.1930 −1.44129 −0.720643 0.693307i \(-0.756153\pi\)
−0.720643 + 0.693307i \(0.756153\pi\)
\(858\) 0 0
\(859\) 48.2853i 1.64747i 0.566972 + 0.823737i \(0.308115\pi\)
−0.566972 + 0.823737i \(0.691885\pi\)
\(860\) 0 0
\(861\) 0.740278i 0.0252286i
\(862\) 0 0
\(863\) 56.5320i 1.92437i 0.272392 + 0.962186i \(0.412185\pi\)
−0.272392 + 0.962186i \(0.587815\pi\)
\(864\) 0 0
\(865\) 3.68070i 0.125148i
\(866\) 0 0
\(867\) 6.69189i 0.227269i
\(868\) 0 0
\(869\) −7.36041 −0.249685
\(870\) 0 0
\(871\) 23.7587 0.805034
\(872\) 0 0
\(873\) 5.67311i 0.192006i
\(874\) 0 0
\(875\) 0.143131i 0.00483872i
\(876\) 0 0
\(877\) 29.6117 0.999915 0.499958 0.866050i \(-0.333349\pi\)
0.499958 + 0.866050i \(0.333349\pi\)
\(878\) 0 0
\(879\) −29.3775 −0.990879
\(880\) 0 0
\(881\) 10.6754i 0.359662i −0.983698 0.179831i \(-0.942445\pi\)
0.983698 0.179831i \(-0.0575551\pi\)
\(882\) 0 0
\(883\) 13.9342i 0.468922i 0.972126 + 0.234461i \(0.0753325\pi\)
−0.972126 + 0.234461i \(0.924667\pi\)
\(884\) 0 0
\(885\) 6.57340i 0.220962i
\(886\) 0 0
\(887\) 47.6768i 1.60083i −0.599446 0.800415i \(-0.704613\pi\)
0.599446 0.800415i \(-0.295387\pi\)
\(888\) 0 0
\(889\) 0.535471i 0.0179591i
\(890\) 0 0
\(891\) 4.80659 0.161027
\(892\) 0 0
\(893\) 3.38935i 0.113420i
\(894\) 0 0
\(895\) −9.27393 −0.309993
\(896\) 0 0
\(897\) −22.2579 5.51617i −0.743171 0.184180i
\(898\) 0 0
\(899\) 21.5008i 0.717093i
\(900\) 0 0
\(901\) 64.9377 2.16339
\(902\) 0 0
\(903\) 0.148148i 0.00493004i
\(904\) 0 0
\(905\) 5.10162 0.169583
\(906\) 0 0
\(907\) 12.0207 0.399141 0.199571 0.979883i \(-0.436045\pi\)
0.199571 + 0.979883i \(0.436045\pi\)
\(908\) 0 0
\(909\) 17.3244 0.574614
\(910\) 0 0
\(911\) 9.73661 0.322588 0.161294 0.986906i \(-0.448433\pi\)
0.161294 + 0.986906i \(0.448433\pi\)
\(912\) 0 0
\(913\) 69.2021 2.29026
\(914\) 0 0
\(915\) 7.49837i 0.247888i
\(916\) 0 0
\(917\) 1.71113i 0.0565065i
\(918\) 0 0
\(919\) −38.9231 −1.28396 −0.641978 0.766723i \(-0.721886\pi\)
−0.641978 + 0.766723i \(0.721886\pi\)
\(920\) 0 0
\(921\) −15.6395 −0.515338
\(922\) 0 0
\(923\) 32.3892i 1.06610i
\(924\) 0 0
\(925\) 5.56282i 0.182904i
\(926\) 0 0
\(927\) −12.2942 −0.403794
\(928\) 0 0
\(929\) 43.8793 1.43963 0.719817 0.694164i \(-0.244226\pi\)
0.719817 + 0.694164i \(0.244226\pi\)
\(930\) 0 0
\(931\) −2.64761 −0.0867719
\(932\) 0 0
\(933\) −16.7393 −0.548021
\(934\) 0 0
\(935\) 23.3958 0.765123
\(936\) 0 0
\(937\) 39.3915i 1.28686i 0.765504 + 0.643432i \(0.222490\pi\)
−0.765504 + 0.643432i \(0.777510\pi\)
\(938\) 0 0
\(939\) 17.0663 0.556939
\(940\) 0 0
\(941\) 45.6529i 1.48824i 0.668045 + 0.744120i \(0.267131\pi\)
−0.668045 + 0.744120i \(0.732869\pi\)
\(942\) 0 0
\(943\) −5.96669 + 24.0758i −0.194302 + 0.784016i
\(944\) 0 0
\(945\) −0.143131 −0.00465606
\(946\) 0 0
\(947\) 0.780609i 0.0253664i 0.999920 + 0.0126832i \(0.00403729\pi\)
−0.999920 + 0.0126832i \(0.995963\pi\)
\(948\) 0 0
\(949\) −38.4686 −1.24874
\(950\) 0 0
\(951\) 29.6407i 0.961166i
\(952\) 0 0
\(953\) 0.373697i 0.0121052i −0.999982 0.00605261i \(-0.998073\pi\)
0.999982 0.00605261i \(-0.00192662\pi\)
\(954\) 0 0
\(955\) 20.1633i 0.652471i
\(956\) 0 0
\(957\) 40.3721i 1.30505i
\(958\) 0 0
\(959\) 1.16919i 0.0377550i
\(960\) 0 0
\(961\) 24.4473 0.788622
\(962\) 0 0
\(963\) −1.02774 −0.0331186
\(964\) 0 0
\(965\) 1.06748i 0.0343635i
\(966\) 0 0
\(967\) 11.2527i 0.361862i 0.983496 + 0.180931i \(0.0579110\pi\)
−0.983496 + 0.180931i \(0.942089\pi\)
\(968\) 0 0
\(969\) −1.84641 −0.0593153
\(970\) 0 0
\(971\) −61.0936 −1.96059 −0.980294 0.197544i \(-0.936703\pi\)
−0.980294 + 0.197544i \(0.936703\pi\)
\(972\) 0 0
\(973\) 0.124570i 0.00399354i
\(974\) 0 0
\(975\) 4.78151i 0.153131i
\(976\) 0 0
\(977\) 10.4612i 0.334685i −0.985899 0.167342i \(-0.946481\pi\)
0.985899 0.167342i \(-0.0535185\pi\)
\(978\) 0 0
\(979\) 60.5636i 1.93562i
\(980\) 0 0
\(981\) 16.9971i 0.542674i
\(982\) 0 0
\(983\) 57.0974 1.82112 0.910562 0.413373i \(-0.135649\pi\)
0.910562 + 0.413373i \(0.135649\pi\)
\(984\) 0 0
\(985\) 4.31277i 0.137416i
\(986\) 0 0
\(987\) −1.27886 −0.0407065
\(988\) 0 0
\(989\) 1.19408 4.81815i 0.0379695 0.153208i
\(990\) 0 0
\(991\) 8.99297i 0.285671i −0.989746 0.142836i \(-0.954378\pi\)
0.989746 0.142836i \(-0.0456220\pi\)
\(992\) 0 0
\(993\) 4.03408 0.128018
\(994\) 0 0
\(995\) 9.35032i 0.296425i
\(996\) 0 0
\(997\) −39.7082 −1.25757 −0.628786 0.777578i \(-0.716448\pi\)
−0.628786 + 0.777578i \(0.716448\pi\)
\(998\) 0 0
\(999\) −5.56282 −0.176000
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 5520.2.be.d.1471.17 yes 32
4.3 odd 2 5520.2.be.c.1471.18 yes 32
23.22 odd 2 5520.2.be.c.1471.17 32
92.91 even 2 inner 5520.2.be.d.1471.18 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5520.2.be.c.1471.17 32 23.22 odd 2
5520.2.be.c.1471.18 yes 32 4.3 odd 2
5520.2.be.d.1471.17 yes 32 1.1 even 1 trivial
5520.2.be.d.1471.18 yes 32 92.91 even 2 inner