Properties

Label 882.3.c.d.685.2
Level $882$
Weight $3$
Character 882.685
Analytic conductor $24.033$
Analytic rank $0$
Dimension $4$
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] = [882,3,Mod(685,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.685");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.c (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: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 685.2
Root \(1.84776i\) of defining polynomial
Character \(\chi\) \(=\) 882.685
Dual form 882.3.c.d.685.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} +2.00000 q^{4} +0.317025i q^{5} -2.82843 q^{8} +O(q^{10})\) \(q-1.41421 q^{2} +2.00000 q^{4} +0.317025i q^{5} -2.82843 q^{8} -0.448342i q^{10} -2.82843 q^{11} +3.11586i q^{13} +4.00000 q^{16} -17.9749i q^{17} +18.7402i q^{19} +0.634051i q^{20} +4.00000 q^{22} -19.5147 q^{23} +24.8995 q^{25} -4.40649i q^{26} -43.5563 q^{29} -36.5838i q^{31} -5.65685 q^{32} +25.4203i q^{34} -13.6985 q^{37} -26.5027i q^{38} -0.896683i q^{40} +53.9564i q^{41} +7.59798 q^{43} -5.65685 q^{44} +27.5980 q^{46} -56.7459i q^{47} -35.2132 q^{50} +6.23172i q^{52} -20.2254 q^{53} -0.896683i q^{55} +61.5980 q^{58} -89.5891i q^{59} +74.4900i q^{61} +51.7373i q^{62} +8.00000 q^{64} -0.987807 q^{65} +4.00000 q^{67} -35.9497i q^{68} -131.799 q^{71} -33.8712i q^{73} +19.3726 q^{74} +37.4804i q^{76} -75.5980 q^{79} +1.26810i q^{80} -76.3059i q^{82} +131.245i q^{83} +5.69848 q^{85} -10.7452 q^{86} +8.00000 q^{88} -111.305i q^{89} -39.0294 q^{92} +80.2509i q^{94} -5.94113 q^{95} -148.106i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 16 q^{16} + 16 q^{22} - 112 q^{23} + 60 q^{25} - 112 q^{29} + 64 q^{37} - 128 q^{43} - 48 q^{46} - 56 q^{50} + 168 q^{53} + 88 q^{58} + 32 q^{64} - 168 q^{65} + 16 q^{67} - 448 q^{71} + 168 q^{74} - 144 q^{79} - 96 q^{85} - 224 q^{86} + 32 q^{88} - 224 q^{92} + 112 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −0.707107
\(3\) 0 0
\(4\) 2.00000 0.500000
\(5\) 0.317025i 0.0634051i 0.999497 + 0.0317025i \(0.0100929\pi\)
−0.999497 + 0.0317025i \(0.989907\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −2.82843 −0.353553
\(9\) 0 0
\(10\) − 0.448342i − 0.0448342i
\(11\) −2.82843 −0.257130 −0.128565 0.991701i \(-0.541037\pi\)
−0.128565 + 0.991701i \(0.541037\pi\)
\(12\) 0 0
\(13\) 3.11586i 0.239682i 0.992793 + 0.119841i \(0.0382384\pi\)
−0.992793 + 0.119841i \(0.961762\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) − 17.9749i − 1.05734i −0.848826 0.528672i \(-0.822690\pi\)
0.848826 0.528672i \(-0.177310\pi\)
\(18\) 0 0
\(19\) 18.7402i 0.986328i 0.869936 + 0.493164i \(0.164160\pi\)
−0.869936 + 0.493164i \(0.835840\pi\)
\(20\) 0.634051i 0.0317025i
\(21\) 0 0
\(22\) 4.00000 0.181818
\(23\) −19.5147 −0.848466 −0.424233 0.905553i \(-0.639456\pi\)
−0.424233 + 0.905553i \(0.639456\pi\)
\(24\) 0 0
\(25\) 24.8995 0.995980
\(26\) − 4.40649i − 0.169480i
\(27\) 0 0
\(28\) 0 0
\(29\) −43.5563 −1.50194 −0.750972 0.660335i \(-0.770414\pi\)
−0.750972 + 0.660335i \(0.770414\pi\)
\(30\) 0 0
\(31\) − 36.5838i − 1.18012i −0.807359 0.590061i \(-0.799104\pi\)
0.807359 0.590061i \(-0.200896\pi\)
\(32\) −5.65685 −0.176777
\(33\) 0 0
\(34\) 25.4203i 0.747655i
\(35\) 0 0
\(36\) 0 0
\(37\) −13.6985 −0.370229 −0.185115 0.982717i \(-0.559266\pi\)
−0.185115 + 0.982717i \(0.559266\pi\)
\(38\) − 26.5027i − 0.697439i
\(39\) 0 0
\(40\) − 0.896683i − 0.0224171i
\(41\) 53.9564i 1.31601i 0.753013 + 0.658005i \(0.228600\pi\)
−0.753013 + 0.658005i \(0.771400\pi\)
\(42\) 0 0
\(43\) 7.59798 0.176697 0.0883486 0.996090i \(-0.471841\pi\)
0.0883486 + 0.996090i \(0.471841\pi\)
\(44\) −5.65685 −0.128565
\(45\) 0 0
\(46\) 27.5980 0.599956
\(47\) − 56.7459i − 1.20736i −0.797227 0.603680i \(-0.793700\pi\)
0.797227 0.603680i \(-0.206300\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −35.2132 −0.704264
\(51\) 0 0
\(52\) 6.23172i 0.119841i
\(53\) −20.2254 −0.381611 −0.190806 0.981628i \(-0.561110\pi\)
−0.190806 + 0.981628i \(0.561110\pi\)
\(54\) 0 0
\(55\) − 0.896683i − 0.0163033i
\(56\) 0 0
\(57\) 0 0
\(58\) 61.5980 1.06203
\(59\) − 89.5891i − 1.51846i −0.650823 0.759230i \(-0.725576\pi\)
0.650823 0.759230i \(-0.274424\pi\)
\(60\) 0 0
\(61\) 74.4900i 1.22115i 0.791959 + 0.610574i \(0.209061\pi\)
−0.791959 + 0.610574i \(0.790939\pi\)
\(62\) 51.7373i 0.834472i
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) −0.987807 −0.0151970
\(66\) 0 0
\(67\) 4.00000 0.0597015 0.0298507 0.999554i \(-0.490497\pi\)
0.0298507 + 0.999554i \(0.490497\pi\)
\(68\) − 35.9497i − 0.528672i
\(69\) 0 0
\(70\) 0 0
\(71\) −131.799 −1.85632 −0.928162 0.372177i \(-0.878612\pi\)
−0.928162 + 0.372177i \(0.878612\pi\)
\(72\) 0 0
\(73\) − 33.8712i − 0.463989i −0.972717 0.231994i \(-0.925475\pi\)
0.972717 0.231994i \(-0.0745251\pi\)
\(74\) 19.3726 0.261792
\(75\) 0 0
\(76\) 37.4804i 0.493164i
\(77\) 0 0
\(78\) 0 0
\(79\) −75.5980 −0.956936 −0.478468 0.878105i \(-0.658808\pi\)
−0.478468 + 0.878105i \(0.658808\pi\)
\(80\) 1.26810i 0.0158513i
\(81\) 0 0
\(82\) − 76.3059i − 0.930560i
\(83\) 131.245i 1.58127i 0.612289 + 0.790634i \(0.290249\pi\)
−0.612289 + 0.790634i \(0.709751\pi\)
\(84\) 0 0
\(85\) 5.69848 0.0670410
\(86\) −10.7452 −0.124944
\(87\) 0 0
\(88\) 8.00000 0.0909091
\(89\) − 111.305i − 1.25061i −0.780379 0.625306i \(-0.784974\pi\)
0.780379 0.625306i \(-0.215026\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −39.0294 −0.424233
\(93\) 0 0
\(94\) 80.2509i 0.853733i
\(95\) −5.94113 −0.0625382
\(96\) 0 0
\(97\) − 148.106i − 1.52686i −0.645888 0.763432i \(-0.723513\pi\)
0.645888 0.763432i \(-0.276487\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 49.7990 0.497990
\(101\) − 100.019i − 0.990288i −0.868811 0.495144i \(-0.835115\pi\)
0.868811 0.495144i \(-0.164885\pi\)
\(102\) 0 0
\(103\) − 6.18666i − 0.0600647i −0.999549 0.0300323i \(-0.990439\pi\)
0.999549 0.0300323i \(-0.00956103\pi\)
\(104\) − 8.81298i − 0.0847402i
\(105\) 0 0
\(106\) 28.6030 0.269840
\(107\) −42.1421 −0.393852 −0.196926 0.980418i \(-0.563096\pi\)
−0.196926 + 0.980418i \(0.563096\pi\)
\(108\) 0 0
\(109\) −120.693 −1.10728 −0.553640 0.832756i \(-0.686761\pi\)
−0.553640 + 0.832756i \(0.686761\pi\)
\(110\) 1.26810i 0.0115282i
\(111\) 0 0
\(112\) 0 0
\(113\) −104.794 −0.927380 −0.463690 0.885998i \(-0.653475\pi\)
−0.463690 + 0.885998i \(0.653475\pi\)
\(114\) 0 0
\(115\) − 6.18666i − 0.0537970i
\(116\) −87.1127 −0.750972
\(117\) 0 0
\(118\) 126.698i 1.07371i
\(119\) 0 0
\(120\) 0 0
\(121\) −113.000 −0.933884
\(122\) − 105.345i − 0.863482i
\(123\) 0 0
\(124\) − 73.1675i − 0.590061i
\(125\) 15.8194i 0.126555i
\(126\) 0 0
\(127\) −166.392 −1.31017 −0.655086 0.755554i \(-0.727368\pi\)
−0.655086 + 0.755554i \(0.727368\pi\)
\(128\) −11.3137 −0.0883883
\(129\) 0 0
\(130\) 1.39697 0.0107459
\(131\) 65.6864i 0.501423i 0.968062 + 0.250711i \(0.0806645\pi\)
−0.968062 + 0.250711i \(0.919336\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −5.65685 −0.0422153
\(135\) 0 0
\(136\) 50.8406i 0.373828i
\(137\) 3.95837 0.0288932 0.0144466 0.999896i \(-0.495401\pi\)
0.0144466 + 0.999896i \(0.495401\pi\)
\(138\) 0 0
\(139\) − 96.3010i − 0.692813i −0.938084 0.346407i \(-0.887402\pi\)
0.938084 0.346407i \(-0.112598\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 186.392 1.31262
\(143\) − 8.81298i − 0.0616293i
\(144\) 0 0
\(145\) − 13.8085i − 0.0952308i
\(146\) 47.9011i 0.328090i
\(147\) 0 0
\(148\) −27.3970 −0.185115
\(149\) −120.912 −0.811488 −0.405744 0.913987i \(-0.632988\pi\)
−0.405744 + 0.913987i \(0.632988\pi\)
\(150\) 0 0
\(151\) −162.392 −1.07544 −0.537722 0.843122i \(-0.680715\pi\)
−0.537722 + 0.843122i \(0.680715\pi\)
\(152\) − 53.0054i − 0.348719i
\(153\) 0 0
\(154\) 0 0
\(155\) 11.5980 0.0748257
\(156\) 0 0
\(157\) 66.0617i 0.420775i 0.977618 + 0.210387i \(0.0674725\pi\)
−0.977618 + 0.210387i \(0.932527\pi\)
\(158\) 106.912 0.676656
\(159\) 0 0
\(160\) − 1.79337i − 0.0112085i
\(161\) 0 0
\(162\) 0 0
\(163\) −209.990 −1.28828 −0.644141 0.764907i \(-0.722785\pi\)
−0.644141 + 0.764907i \(0.722785\pi\)
\(164\) 107.913i 0.658005i
\(165\) 0 0
\(166\) − 185.609i − 1.11813i
\(167\) − 152.739i − 0.914606i −0.889311 0.457303i \(-0.848815\pi\)
0.889311 0.457303i \(-0.151185\pi\)
\(168\) 0 0
\(169\) 159.291 0.942553
\(170\) −8.05887 −0.0474051
\(171\) 0 0
\(172\) 15.1960 0.0883486
\(173\) 35.5690i 0.205601i 0.994702 + 0.102800i \(0.0327803\pi\)
−0.994702 + 0.102800i \(0.967220\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −11.3137 −0.0642824
\(177\) 0 0
\(178\) 157.408i 0.884317i
\(179\) −63.9167 −0.357077 −0.178538 0.983933i \(-0.557137\pi\)
−0.178538 + 0.983933i \(0.557137\pi\)
\(180\) 0 0
\(181\) − 183.748i − 1.01518i −0.861598 0.507591i \(-0.830536\pi\)
0.861598 0.507591i \(-0.169464\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 55.1960 0.299978
\(185\) − 4.34277i − 0.0234744i
\(186\) 0 0
\(187\) 50.8406i 0.271875i
\(188\) − 113.492i − 0.603680i
\(189\) 0 0
\(190\) 8.40202 0.0442212
\(191\) 154.426 0.808515 0.404258 0.914645i \(-0.367530\pi\)
0.404258 + 0.914645i \(0.367530\pi\)
\(192\) 0 0
\(193\) −23.1960 −0.120186 −0.0600932 0.998193i \(-0.519140\pi\)
−0.0600932 + 0.998193i \(0.519140\pi\)
\(194\) 209.453i 1.07966i
\(195\) 0 0
\(196\) 0 0
\(197\) 202.510 1.02797 0.513984 0.857800i \(-0.328169\pi\)
0.513984 + 0.857800i \(0.328169\pi\)
\(198\) 0 0
\(199\) − 230.083i − 1.15619i −0.815968 0.578097i \(-0.803796\pi\)
0.815968 0.578097i \(-0.196204\pi\)
\(200\) −70.4264 −0.352132
\(201\) 0 0
\(202\) 141.448i 0.700239i
\(203\) 0 0
\(204\) 0 0
\(205\) −17.1056 −0.0834417
\(206\) 8.74926i 0.0424721i
\(207\) 0 0
\(208\) 12.4634i 0.0599204i
\(209\) − 53.0054i − 0.253614i
\(210\) 0 0
\(211\) −172.402 −0.817071 −0.408536 0.912742i \(-0.633960\pi\)
−0.408536 + 0.912742i \(0.633960\pi\)
\(212\) −40.4508 −0.190806
\(213\) 0 0
\(214\) 59.5980 0.278495
\(215\) 2.40875i 0.0112035i
\(216\) 0 0
\(217\) 0 0
\(218\) 170.686 0.782965
\(219\) 0 0
\(220\) − 1.79337i − 0.00815166i
\(221\) 56.0071 0.253426
\(222\) 0 0
\(223\) − 55.3240i − 0.248090i −0.992277 0.124045i \(-0.960413\pi\)
0.992277 0.124045i \(-0.0395867\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 148.201 0.655757
\(227\) 369.578i 1.62810i 0.580796 + 0.814049i \(0.302741\pi\)
−0.580796 + 0.814049i \(0.697259\pi\)
\(228\) 0 0
\(229\) 294.396i 1.28557i 0.766046 + 0.642786i \(0.222221\pi\)
−0.766046 + 0.642786i \(0.777779\pi\)
\(230\) 8.74926i 0.0380403i
\(231\) 0 0
\(232\) 123.196 0.531017
\(233\) 174.787 0.750158 0.375079 0.926993i \(-0.377616\pi\)
0.375079 + 0.926993i \(0.377616\pi\)
\(234\) 0 0
\(235\) 17.9899 0.0765528
\(236\) − 179.178i − 0.759230i
\(237\) 0 0
\(238\) 0 0
\(239\) 323.279 1.35263 0.676316 0.736611i \(-0.263575\pi\)
0.676316 + 0.736611i \(0.263575\pi\)
\(240\) 0 0
\(241\) 165.501i 0.686726i 0.939203 + 0.343363i \(0.111566\pi\)
−0.939203 + 0.343363i \(0.888434\pi\)
\(242\) 159.806 0.660356
\(243\) 0 0
\(244\) 148.980i 0.610574i
\(245\) 0 0
\(246\) 0 0
\(247\) −58.3919 −0.236405
\(248\) 103.475i 0.417236i
\(249\) 0 0
\(250\) − 22.3720i − 0.0894881i
\(251\) − 99.7339i − 0.397346i −0.980066 0.198673i \(-0.936337\pi\)
0.980066 0.198673i \(-0.0636632\pi\)
\(252\) 0 0
\(253\) 55.1960 0.218166
\(254\) 235.314 0.926432
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 379.976i 1.47851i 0.673427 + 0.739254i \(0.264821\pi\)
−0.673427 + 0.739254i \(0.735179\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.97561 −0.00759851
\(261\) 0 0
\(262\) − 92.8946i − 0.354559i
\(263\) 233.622 0.888298 0.444149 0.895953i \(-0.353506\pi\)
0.444149 + 0.895953i \(0.353506\pi\)
\(264\) 0 0
\(265\) − 6.41196i − 0.0241961i
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.0298507
\(269\) 465.286i 1.72969i 0.502040 + 0.864845i \(0.332583\pi\)
−0.502040 + 0.864845i \(0.667417\pi\)
\(270\) 0 0
\(271\) − 141.671i − 0.522773i −0.965234 0.261386i \(-0.915820\pi\)
0.965234 0.261386i \(-0.0841797\pi\)
\(272\) − 71.8994i − 0.264336i
\(273\) 0 0
\(274\) −5.59798 −0.0204306
\(275\) −70.4264 −0.256096
\(276\) 0 0
\(277\) 314.000 1.13357 0.566787 0.823864i \(-0.308186\pi\)
0.566787 + 0.823864i \(0.308186\pi\)
\(278\) 136.190i 0.489893i
\(279\) 0 0
\(280\) 0 0
\(281\) 323.002 1.14947 0.574737 0.818338i \(-0.305104\pi\)
0.574737 + 0.818338i \(0.305104\pi\)
\(282\) 0 0
\(283\) − 534.500i − 1.88869i −0.328955 0.944346i \(-0.606696\pi\)
0.328955 0.944346i \(-0.393304\pi\)
\(284\) −263.598 −0.928162
\(285\) 0 0
\(286\) 12.4634i 0.0435785i
\(287\) 0 0
\(288\) 0 0
\(289\) −34.0955 −0.117977
\(290\) 19.5281i 0.0673383i
\(291\) 0 0
\(292\) − 67.7424i − 0.231994i
\(293\) − 392.246i − 1.33872i −0.742936 0.669362i \(-0.766567\pi\)
0.742936 0.669362i \(-0.233433\pi\)
\(294\) 0 0
\(295\) 28.4020 0.0962780
\(296\) 38.7452 0.130896
\(297\) 0 0
\(298\) 170.995 0.573809
\(299\) − 60.8051i − 0.203362i
\(300\) 0 0
\(301\) 0 0
\(302\) 229.657 0.760453
\(303\) 0 0
\(304\) 74.9609i 0.246582i
\(305\) −23.6152 −0.0774270
\(306\) 0 0
\(307\) 541.583i 1.76411i 0.471143 + 0.882057i \(0.343842\pi\)
−0.471143 + 0.882057i \(0.656158\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −16.4020 −0.0529097
\(311\) − 57.8866i − 0.186130i −0.995660 0.0930652i \(-0.970333\pi\)
0.995660 0.0930652i \(-0.0296665\pi\)
\(312\) 0 0
\(313\) 461.981i 1.47598i 0.674813 + 0.737989i \(0.264224\pi\)
−0.674813 + 0.737989i \(0.735776\pi\)
\(314\) − 93.4253i − 0.297533i
\(315\) 0 0
\(316\) −151.196 −0.478468
\(317\) 415.196 1.30977 0.654883 0.755730i \(-0.272718\pi\)
0.654883 + 0.755730i \(0.272718\pi\)
\(318\) 0 0
\(319\) 123.196 0.386194
\(320\) 2.53620i 0.00792563i
\(321\) 0 0
\(322\) 0 0
\(323\) 336.853 1.04289
\(324\) 0 0
\(325\) 77.5834i 0.238718i
\(326\) 296.971 0.910953
\(327\) 0 0
\(328\) − 152.612i − 0.465280i
\(329\) 0 0
\(330\) 0 0
\(331\) 76.4020 0.230822 0.115411 0.993318i \(-0.463182\pi\)
0.115411 + 0.993318i \(0.463182\pi\)
\(332\) 262.491i 0.790634i
\(333\) 0 0
\(334\) 216.006i 0.646724i
\(335\) 1.26810i 0.00378538i
\(336\) 0 0
\(337\) 376.291 1.11659 0.558296 0.829642i \(-0.311455\pi\)
0.558296 + 0.829642i \(0.311455\pi\)
\(338\) −225.272 −0.666485
\(339\) 0 0
\(340\) 11.3970 0.0335205
\(341\) 103.475i 0.303444i
\(342\) 0 0
\(343\) 0 0
\(344\) −21.4903 −0.0624719
\(345\) 0 0
\(346\) − 50.3021i − 0.145382i
\(347\) −78.3289 −0.225732 −0.112866 0.993610i \(-0.536003\pi\)
−0.112866 + 0.993610i \(0.536003\pi\)
\(348\) 0 0
\(349\) − 376.710i − 1.07940i −0.841858 0.539700i \(-0.818538\pi\)
0.841858 0.539700i \(-0.181462\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 16.0000 0.0454545
\(353\) 213.352i 0.604396i 0.953245 + 0.302198i \(0.0977203\pi\)
−0.953245 + 0.302198i \(0.902280\pi\)
\(354\) 0 0
\(355\) − 41.7836i − 0.117700i
\(356\) − 222.609i − 0.625306i
\(357\) 0 0
\(358\) 90.3919 0.252491
\(359\) −28.0143 −0.0780342 −0.0390171 0.999239i \(-0.512423\pi\)
−0.0390171 + 0.999239i \(0.512423\pi\)
\(360\) 0 0
\(361\) 9.80404 0.0271580
\(362\) 259.859i 0.717842i
\(363\) 0 0
\(364\) 0 0
\(365\) 10.7380 0.0294192
\(366\) 0 0
\(367\) 149.742i 0.408015i 0.978969 + 0.204008i \(0.0653967\pi\)
−0.978969 + 0.204008i \(0.934603\pi\)
\(368\) −78.0589 −0.212117
\(369\) 0 0
\(370\) 6.14160i 0.0165989i
\(371\) 0 0
\(372\) 0 0
\(373\) 231.196 0.619828 0.309914 0.950765i \(-0.399700\pi\)
0.309914 + 0.950765i \(0.399700\pi\)
\(374\) − 71.8994i − 0.192244i
\(375\) 0 0
\(376\) 160.502i 0.426866i
\(377\) − 135.716i − 0.359988i
\(378\) 0 0
\(379\) −585.186 −1.54403 −0.772013 0.635607i \(-0.780750\pi\)
−0.772013 + 0.635607i \(0.780750\pi\)
\(380\) −11.8823 −0.0312691
\(381\) 0 0
\(382\) −218.392 −0.571707
\(383\) 533.858i 1.39388i 0.717127 + 0.696942i \(0.245457\pi\)
−0.717127 + 0.696942i \(0.754543\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 32.8040 0.0849846
\(387\) 0 0
\(388\) − 296.212i − 0.763432i
\(389\) −19.7918 −0.0508788 −0.0254394 0.999676i \(-0.508098\pi\)
−0.0254394 + 0.999676i \(0.508098\pi\)
\(390\) 0 0
\(391\) 350.774i 0.897121i
\(392\) 0 0
\(393\) 0 0
\(394\) −286.392 −0.726883
\(395\) − 23.9665i − 0.0606746i
\(396\) 0 0
\(397\) − 367.340i − 0.925291i −0.886543 0.462645i \(-0.846900\pi\)
0.886543 0.462645i \(-0.153100\pi\)
\(398\) 325.386i 0.817552i
\(399\) 0 0
\(400\) 99.5980 0.248995
\(401\) −692.943 −1.72804 −0.864019 0.503459i \(-0.832060\pi\)
−0.864019 + 0.503459i \(0.832060\pi\)
\(402\) 0 0
\(403\) 113.990 0.282853
\(404\) − 200.038i − 0.495144i
\(405\) 0 0
\(406\) 0 0
\(407\) 38.7452 0.0951970
\(408\) 0 0
\(409\) 290.899i 0.711245i 0.934630 + 0.355623i \(0.115731\pi\)
−0.934630 + 0.355623i \(0.884269\pi\)
\(410\) 24.1909 0.0590022
\(411\) 0 0
\(412\) − 12.3733i − 0.0300323i
\(413\) 0 0
\(414\) 0 0
\(415\) −41.6081 −0.100260
\(416\) − 17.6260i − 0.0423701i
\(417\) 0 0
\(418\) 74.9609i 0.179332i
\(419\) − 624.333i − 1.49005i −0.667034 0.745027i \(-0.732437\pi\)
0.667034 0.745027i \(-0.267563\pi\)
\(420\) 0 0
\(421\) 753.980 1.79093 0.895463 0.445136i \(-0.146845\pi\)
0.895463 + 0.445136i \(0.146845\pi\)
\(422\) 243.813 0.577757
\(423\) 0 0
\(424\) 57.2061 0.134920
\(425\) − 447.565i − 1.05309i
\(426\) 0 0
\(427\) 0 0
\(428\) −84.2843 −0.196926
\(429\) 0 0
\(430\) − 3.40649i − 0.00792207i
\(431\) −1.15137 −0.00267139 −0.00133570 0.999999i \(-0.500425\pi\)
−0.00133570 + 0.999999i \(0.500425\pi\)
\(432\) 0 0
\(433\) − 193.834i − 0.447655i −0.974629 0.223827i \(-0.928145\pi\)
0.974629 0.223827i \(-0.0718552\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −241.387 −0.553640
\(437\) − 365.710i − 0.836865i
\(438\) 0 0
\(439\) − 444.295i − 1.01206i −0.862515 0.506031i \(-0.831112\pi\)
0.862515 0.506031i \(-0.168888\pi\)
\(440\) 2.53620i 0.00576410i
\(441\) 0 0
\(442\) −79.2061 −0.179199
\(443\) 403.897 0.911730 0.455865 0.890049i \(-0.349330\pi\)
0.455865 + 0.890049i \(0.349330\pi\)
\(444\) 0 0
\(445\) 35.2864 0.0792952
\(446\) 78.2399i 0.175426i
\(447\) 0 0
\(448\) 0 0
\(449\) −242.520 −0.540133 −0.270067 0.962842i \(-0.587046\pi\)
−0.270067 + 0.962842i \(0.587046\pi\)
\(450\) 0 0
\(451\) − 152.612i − 0.338385i
\(452\) −209.588 −0.463690
\(453\) 0 0
\(454\) − 522.663i − 1.15124i
\(455\) 0 0
\(456\) 0 0
\(457\) 744.764 1.62968 0.814840 0.579686i \(-0.196825\pi\)
0.814840 + 0.579686i \(0.196825\pi\)
\(458\) − 416.339i − 0.909036i
\(459\) 0 0
\(460\) − 12.3733i − 0.0268985i
\(461\) 764.329i 1.65798i 0.559263 + 0.828990i \(0.311084\pi\)
−0.559263 + 0.828990i \(0.688916\pi\)
\(462\) 0 0
\(463\) −658.774 −1.42284 −0.711419 0.702768i \(-0.751947\pi\)
−0.711419 + 0.702768i \(0.751947\pi\)
\(464\) −174.225 −0.375486
\(465\) 0 0
\(466\) −247.186 −0.530442
\(467\) − 65.6226i − 0.140520i −0.997529 0.0702598i \(-0.977617\pi\)
0.997529 0.0702598i \(-0.0223828\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −25.4416 −0.0541310
\(471\) 0 0
\(472\) 253.396i 0.536857i
\(473\) −21.4903 −0.0454341
\(474\) 0 0
\(475\) 466.622i 0.982362i
\(476\) 0 0
\(477\) 0 0
\(478\) −457.186 −0.956456
\(479\) 248.478i 0.518743i 0.965778 + 0.259371i \(0.0835153\pi\)
−0.965778 + 0.259371i \(0.916485\pi\)
\(480\) 0 0
\(481\) − 42.6826i − 0.0887371i
\(482\) − 234.054i − 0.485589i
\(483\) 0 0
\(484\) −226.000 −0.466942
\(485\) 46.9533 0.0968110
\(486\) 0 0
\(487\) −764.402 −1.56961 −0.784807 0.619740i \(-0.787238\pi\)
−0.784807 + 0.619740i \(0.787238\pi\)
\(488\) − 210.690i − 0.431741i
\(489\) 0 0
\(490\) 0 0
\(491\) −782.621 −1.59393 −0.796967 0.604023i \(-0.793564\pi\)
−0.796967 + 0.604023i \(0.793564\pi\)
\(492\) 0 0
\(493\) 782.919i 1.58807i
\(494\) 82.5786 0.167163
\(495\) 0 0
\(496\) − 146.335i − 0.295030i
\(497\) 0 0
\(498\) 0 0
\(499\) −131.980 −0.264489 −0.132244 0.991217i \(-0.542218\pi\)
−0.132244 + 0.991217i \(0.542218\pi\)
\(500\) 31.6388i 0.0632776i
\(501\) 0 0
\(502\) 141.045i 0.280966i
\(503\) 756.910i 1.50479i 0.658712 + 0.752395i \(0.271102\pi\)
−0.658712 + 0.752395i \(0.728898\pi\)
\(504\) 0 0
\(505\) 31.7086 0.0627893
\(506\) −78.0589 −0.154267
\(507\) 0 0
\(508\) −332.784 −0.655086
\(509\) − 233.418i − 0.458582i −0.973358 0.229291i \(-0.926359\pi\)
0.973358 0.229291i \(-0.0736407\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −22.6274 −0.0441942
\(513\) 0 0
\(514\) − 537.368i − 1.04546i
\(515\) 1.96133 0.00380840
\(516\) 0 0
\(517\) 160.502i 0.310448i
\(518\) 0 0
\(519\) 0 0
\(520\) 2.79394 0.00537296
\(521\) − 596.691i − 1.14528i −0.819807 0.572640i \(-0.805919\pi\)
0.819807 0.572640i \(-0.194081\pi\)
\(522\) 0 0
\(523\) − 815.152i − 1.55861i −0.626646 0.779304i \(-0.715573\pi\)
0.626646 0.779304i \(-0.284427\pi\)
\(524\) 131.373i 0.250711i
\(525\) 0 0
\(526\) −330.392 −0.628122
\(527\) −657.588 −1.24779
\(528\) 0 0
\(529\) −148.176 −0.280105
\(530\) 9.06789i 0.0171092i
\(531\) 0 0
\(532\) 0 0
\(533\) −168.121 −0.315423
\(534\) 0 0
\(535\) − 13.3601i − 0.0249722i
\(536\) −11.3137 −0.0211077
\(537\) 0 0
\(538\) − 658.014i − 1.22307i
\(539\) 0 0
\(540\) 0 0
\(541\) 233.568 0.431733 0.215867 0.976423i \(-0.430742\pi\)
0.215867 + 0.976423i \(0.430742\pi\)
\(542\) 200.354i 0.369656i
\(543\) 0 0
\(544\) 101.681i 0.186914i
\(545\) − 38.2629i − 0.0702071i
\(546\) 0 0
\(547\) −27.1758 −0.0496815 −0.0248407 0.999691i \(-0.507908\pi\)
−0.0248407 + 0.999691i \(0.507908\pi\)
\(548\) 7.91674 0.0144466
\(549\) 0 0
\(550\) 99.5980 0.181087
\(551\) − 816.256i − 1.48141i
\(552\) 0 0
\(553\) 0 0
\(554\) −444.063 −0.801558
\(555\) 0 0
\(556\) − 192.602i − 0.346407i
\(557\) 290.745 0.521984 0.260992 0.965341i \(-0.415950\pi\)
0.260992 + 0.965341i \(0.415950\pi\)
\(558\) 0 0
\(559\) 23.6742i 0.0423511i
\(560\) 0 0
\(561\) 0 0
\(562\) −456.794 −0.812801
\(563\) − 976.476i − 1.73442i −0.497946 0.867208i \(-0.665912\pi\)
0.497946 0.867208i \(-0.334088\pi\)
\(564\) 0 0
\(565\) − 33.2223i − 0.0588006i
\(566\) 755.897i 1.33551i
\(567\) 0 0
\(568\) 372.784 0.656310
\(569\) 635.252 1.11644 0.558218 0.829694i \(-0.311485\pi\)
0.558218 + 0.829694i \(0.311485\pi\)
\(570\) 0 0
\(571\) 453.186 0.793671 0.396835 0.917890i \(-0.370108\pi\)
0.396835 + 0.917890i \(0.370108\pi\)
\(572\) − 17.6260i − 0.0308146i
\(573\) 0 0
\(574\) 0 0
\(575\) −485.907 −0.845055
\(576\) 0 0
\(577\) 1093.00i 1.89429i 0.320807 + 0.947144i \(0.396046\pi\)
−0.320807 + 0.947144i \(0.603954\pi\)
\(578\) 48.2183 0.0834226
\(579\) 0 0
\(580\) − 27.6169i − 0.0476154i
\(581\) 0 0
\(582\) 0 0
\(583\) 57.2061 0.0981236
\(584\) 95.8022i 0.164045i
\(585\) 0 0
\(586\) 554.720i 0.946621i
\(587\) 883.401i 1.50494i 0.658625 + 0.752471i \(0.271138\pi\)
−0.658625 + 0.752471i \(0.728862\pi\)
\(588\) 0 0
\(589\) 685.588 1.16399
\(590\) −40.1665 −0.0680789
\(591\) 0 0
\(592\) −54.7939 −0.0925573
\(593\) − 100.462i − 0.169413i −0.996406 0.0847065i \(-0.973005\pi\)
0.996406 0.0847065i \(-0.0269953\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −241.823 −0.405744
\(597\) 0 0
\(598\) 85.9915i 0.143798i
\(599\) −1034.05 −1.72630 −0.863149 0.504949i \(-0.831511\pi\)
−0.863149 + 0.504949i \(0.831511\pi\)
\(600\) 0 0
\(601\) − 401.682i − 0.668357i −0.942510 0.334178i \(-0.891541\pi\)
0.942510 0.334178i \(-0.108459\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −324.784 −0.537722
\(605\) − 35.8239i − 0.0592130i
\(606\) 0 0
\(607\) − 876.217i − 1.44352i −0.692143 0.721760i \(-0.743333\pi\)
0.692143 0.721760i \(-0.256667\pi\)
\(608\) − 106.011i − 0.174360i
\(609\) 0 0
\(610\) 33.3970 0.0547491
\(611\) 176.812 0.289382
\(612\) 0 0
\(613\) −85.4773 −0.139441 −0.0697205 0.997567i \(-0.522211\pi\)
−0.0697205 + 0.997567i \(0.522211\pi\)
\(614\) − 765.914i − 1.24742i
\(615\) 0 0
\(616\) 0 0
\(617\) −522.115 −0.846215 −0.423108 0.906079i \(-0.639061\pi\)
−0.423108 + 0.906079i \(0.639061\pi\)
\(618\) 0 0
\(619\) − 653.218i − 1.05528i −0.849468 0.527640i \(-0.823077\pi\)
0.849468 0.527640i \(-0.176923\pi\)
\(620\) 23.1960 0.0374128
\(621\) 0 0
\(622\) 81.8640i 0.131614i
\(623\) 0 0
\(624\) 0 0
\(625\) 617.472 0.987956
\(626\) − 653.340i − 1.04367i
\(627\) 0 0
\(628\) 132.123i 0.210387i
\(629\) 246.228i 0.391460i
\(630\) 0 0
\(631\) −728.764 −1.15493 −0.577467 0.816414i \(-0.695959\pi\)
−0.577467 + 0.816414i \(0.695959\pi\)
\(632\) 213.823 0.338328
\(633\) 0 0
\(634\) −587.176 −0.926145
\(635\) − 52.7505i − 0.0830716i
\(636\) 0 0
\(637\) 0 0
\(638\) −174.225 −0.273081
\(639\) 0 0
\(640\) − 3.58673i − 0.00560427i
\(641\) 190.606 0.297357 0.148679 0.988886i \(-0.452498\pi\)
0.148679 + 0.988886i \(0.452498\pi\)
\(642\) 0 0
\(643\) 910.647i 1.41625i 0.706089 + 0.708124i \(0.250458\pi\)
−0.706089 + 0.708124i \(0.749542\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −476.382 −0.737433
\(647\) − 355.757i − 0.549856i −0.961465 0.274928i \(-0.911346\pi\)
0.961465 0.274928i \(-0.0886539\pi\)
\(648\) 0 0
\(649\) 253.396i 0.390441i
\(650\) − 109.719i − 0.168799i
\(651\) 0 0
\(652\) −419.980 −0.644141
\(653\) −1130.75 −1.73163 −0.865813 0.500367i \(-0.833198\pi\)
−0.865813 + 0.500367i \(0.833198\pi\)
\(654\) 0 0
\(655\) −20.8242 −0.0317927
\(656\) 215.826i 0.329003i
\(657\) 0 0
\(658\) 0 0
\(659\) −520.388 −0.789663 −0.394831 0.918754i \(-0.629197\pi\)
−0.394831 + 0.918754i \(0.629197\pi\)
\(660\) 0 0
\(661\) 351.243i 0.531381i 0.964058 + 0.265691i \(0.0855999\pi\)
−0.964058 + 0.265691i \(0.914400\pi\)
\(662\) −108.049 −0.163216
\(663\) 0 0
\(664\) − 371.218i − 0.559063i
\(665\) 0 0
\(666\) 0 0
\(667\) 849.990 1.27435
\(668\) − 305.479i − 0.457303i
\(669\) 0 0
\(670\) − 1.79337i − 0.00267667i
\(671\) − 210.690i − 0.313993i
\(672\) 0 0
\(673\) 397.075 0.590008 0.295004 0.955496i \(-0.404679\pi\)
0.295004 + 0.955496i \(0.404679\pi\)
\(674\) −532.156 −0.789550
\(675\) 0 0
\(676\) 318.583 0.471276
\(677\) − 361.210i − 0.533545i −0.963760 0.266772i \(-0.914043\pi\)
0.963760 0.266772i \(-0.0859572\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −16.1177 −0.0237026
\(681\) 0 0
\(682\) − 146.335i − 0.214568i
\(683\) 938.455 1.37402 0.687010 0.726648i \(-0.258923\pi\)
0.687010 + 0.726648i \(0.258923\pi\)
\(684\) 0 0
\(685\) 1.25490i 0.00183198i
\(686\) 0 0
\(687\) 0 0
\(688\) 30.3919 0.0441743
\(689\) − 63.0195i − 0.0914652i
\(690\) 0 0
\(691\) 381.888i 0.552660i 0.961063 + 0.276330i \(0.0891182\pi\)
−0.961063 + 0.276330i \(0.910882\pi\)
\(692\) 71.1379i 0.102800i
\(693\) 0 0
\(694\) 110.774 0.159616
\(695\) 30.5299 0.0439279
\(696\) 0 0
\(697\) 969.859 1.39148
\(698\) 532.749i 0.763251i
\(699\) 0 0
\(700\) 0 0
\(701\) −713.325 −1.01758 −0.508791 0.860890i \(-0.669907\pi\)
−0.508791 + 0.860890i \(0.669907\pi\)
\(702\) 0 0
\(703\) − 256.713i − 0.365167i
\(704\) −22.6274 −0.0321412
\(705\) 0 0
\(706\) − 301.725i − 0.427372i
\(707\) 0 0
\(708\) 0 0
\(709\) −388.894 −0.548511 −0.274256 0.961657i \(-0.588431\pi\)
−0.274256 + 0.961657i \(0.588431\pi\)
\(710\) 59.0910i 0.0832267i
\(711\) 0 0
\(712\) 314.817i 0.442158i
\(713\) 713.922i 1.00129i
\(714\) 0 0
\(715\) 2.79394 0.00390761
\(716\) −127.833 −0.178538
\(717\) 0 0
\(718\) 39.6182 0.0551785
\(719\) − 460.435i − 0.640383i −0.947353 0.320192i \(-0.896253\pi\)
0.947353 0.320192i \(-0.103747\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −13.8650 −0.0192036
\(723\) 0 0
\(724\) − 367.496i − 0.507591i
\(725\) −1084.53 −1.49590
\(726\) 0 0
\(727\) 1005.79i 1.38347i 0.722150 + 0.691737i \(0.243154\pi\)
−0.722150 + 0.691737i \(0.756846\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −15.1859 −0.0208025
\(731\) − 136.573i − 0.186830i
\(732\) 0 0
\(733\) 779.623i 1.06361i 0.846868 + 0.531803i \(0.178485\pi\)
−0.846868 + 0.531803i \(0.821515\pi\)
\(734\) − 211.767i − 0.288510i
\(735\) 0 0
\(736\) 110.392 0.149989
\(737\) −11.3137 −0.0153510
\(738\) 0 0
\(739\) 72.3818 0.0979456 0.0489728 0.998800i \(-0.484405\pi\)
0.0489728 + 0.998800i \(0.484405\pi\)
\(740\) − 8.68553i − 0.0117372i
\(741\) 0 0
\(742\) 0 0
\(743\) 145.358 0.195637 0.0978185 0.995204i \(-0.468814\pi\)
0.0978185 + 0.995204i \(0.468814\pi\)
\(744\) 0 0
\(745\) − 38.3321i − 0.0514524i
\(746\) −326.960 −0.438285
\(747\) 0 0
\(748\) 101.681i 0.135937i
\(749\) 0 0
\(750\) 0 0
\(751\) 4.02020 0.00535313 0.00267657 0.999996i \(-0.499148\pi\)
0.00267657 + 0.999996i \(0.499148\pi\)
\(752\) − 226.984i − 0.301840i
\(753\) 0 0
\(754\) 191.931i 0.254550i
\(755\) − 51.4824i − 0.0681885i
\(756\) 0 0
\(757\) 705.678 0.932204 0.466102 0.884731i \(-0.345658\pi\)
0.466102 + 0.884731i \(0.345658\pi\)
\(758\) 827.578 1.09179
\(759\) 0 0
\(760\) 16.8040 0.0221106
\(761\) 883.814i 1.16138i 0.814123 + 0.580692i \(0.197218\pi\)
−0.814123 + 0.580692i \(0.802782\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 308.853 0.404258
\(765\) 0 0
\(766\) − 754.989i − 0.985625i
\(767\) 279.147 0.363947
\(768\) 0 0
\(769\) − 52.5686i − 0.0683597i −0.999416 0.0341799i \(-0.989118\pi\)
0.999416 0.0341799i \(-0.0108819\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −46.3919 −0.0600932
\(773\) − 613.176i − 0.793242i −0.917982 0.396621i \(-0.870183\pi\)
0.917982 0.396621i \(-0.129817\pi\)
\(774\) 0 0
\(775\) − 910.917i − 1.17538i
\(776\) 418.907i 0.539828i
\(777\) 0 0
\(778\) 27.9899 0.0359767
\(779\) −1011.16 −1.29802
\(780\) 0 0
\(781\) 372.784 0.477316
\(782\) − 496.070i − 0.634360i
\(783\) 0 0
\(784\) 0 0
\(785\) −20.9432 −0.0266793
\(786\) 0 0
\(787\) − 1058.60i − 1.34511i −0.740048 0.672554i \(-0.765197\pi\)
0.740048 0.672554i \(-0.234803\pi\)
\(788\) 405.019 0.513984
\(789\) 0 0
\(790\) 33.8937i 0.0429034i
\(791\) 0 0
\(792\) 0 0
\(793\) −232.101 −0.292687
\(794\) 519.498i 0.654279i
\(795\) 0 0
\(796\) − 460.165i − 0.578097i
\(797\) − 1128.99i − 1.41655i −0.705934 0.708277i \(-0.749473\pi\)
0.705934 0.708277i \(-0.250527\pi\)
\(798\) 0 0
\(799\) −1020.00 −1.27660
\(800\) −140.853 −0.176066
\(801\) 0 0
\(802\) 979.970 1.22191
\(803\) 95.8022i 0.119305i
\(804\) 0 0
\(805\) 0 0
\(806\) −161.206 −0.200008
\(807\) 0 0
\(808\) 282.897i 0.350120i
\(809\) −604.118 −0.746746 −0.373373 0.927681i \(-0.621799\pi\)
−0.373373 + 0.927681i \(0.621799\pi\)
\(810\) 0 0
\(811\) 194.215i 0.239476i 0.992806 + 0.119738i \(0.0382055\pi\)
−0.992806 + 0.119738i \(0.961795\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −54.7939 −0.0673144
\(815\) − 66.5721i − 0.0816836i
\(816\) 0 0
\(817\) 142.388i 0.174281i
\(818\) − 411.394i − 0.502926i
\(819\) 0 0
\(820\) −34.2111 −0.0417209
\(821\) 776.098 0.945308 0.472654 0.881248i \(-0.343296\pi\)
0.472654 + 0.881248i \(0.343296\pi\)
\(822\) 0 0
\(823\) −148.804 −0.180807 −0.0904034 0.995905i \(-0.528816\pi\)
−0.0904034 + 0.995905i \(0.528816\pi\)
\(824\) 17.4985i 0.0212361i
\(825\) 0 0
\(826\) 0 0
\(827\) 1018.50 1.23156 0.615782 0.787916i \(-0.288840\pi\)
0.615782 + 0.787916i \(0.288840\pi\)
\(828\) 0 0
\(829\) 310.313i 0.374322i 0.982329 + 0.187161i \(0.0599286\pi\)
−0.982329 + 0.187161i \(0.940071\pi\)
\(830\) 58.8427 0.0708948
\(831\) 0 0
\(832\) 24.9269i 0.0299602i
\(833\) 0 0
\(834\) 0 0
\(835\) 48.4222 0.0579907
\(836\) − 106.011i − 0.126807i
\(837\) 0 0
\(838\) 882.940i 1.05363i
\(839\) − 331.790i − 0.395459i −0.980257 0.197729i \(-0.936643\pi\)
0.980257 0.197729i \(-0.0633568\pi\)
\(840\) 0 0
\(841\) 1056.16 1.25583
\(842\) −1066.29 −1.26638
\(843\) 0 0
\(844\) −344.804 −0.408536
\(845\) 50.4994i 0.0597626i
\(846\) 0 0
\(847\) 0 0
\(848\) −80.9016 −0.0954028
\(849\) 0 0
\(850\) 632.952i 0.744650i
\(851\) 267.322 0.314127
\(852\) 0 0
\(853\) − 358.597i − 0.420395i −0.977659 0.210197i \(-0.932589\pi\)
0.977659 0.210197i \(-0.0674106\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 119.196 0.139248
\(857\) 832.835i 0.971803i 0.874014 + 0.485901i \(0.161509\pi\)
−0.874014 + 0.485901i \(0.838491\pi\)
\(858\) 0 0
\(859\) 1550.24i 1.80470i 0.431002 + 0.902351i \(0.358160\pi\)
−0.431002 + 0.902351i \(0.641840\pi\)
\(860\) 4.81750i 0.00560175i
\(861\) 0 0
\(862\) 1.62828 0.00188896
\(863\) 917.249 1.06286 0.531430 0.847102i \(-0.321655\pi\)
0.531430 + 0.847102i \(0.321655\pi\)
\(864\) 0 0
\(865\) −11.2763 −0.0130361
\(866\) 274.123i 0.316540i
\(867\) 0 0
\(868\) 0 0
\(869\) 213.823 0.246057
\(870\) 0 0
\(871\) 12.4634i 0.0143093i
\(872\) 341.373 0.391482
\(873\) 0 0
\(874\) 517.192i 0.591753i
\(875\) 0 0
\(876\) 0 0
\(877\) 1490.26 1.69927 0.849636 0.527370i \(-0.176822\pi\)
0.849636 + 0.527370i \(0.176822\pi\)
\(878\) 628.328i 0.715636i
\(879\) 0 0
\(880\) − 3.58673i − 0.00407583i
\(881\) 156.004i 0.177076i 0.996073 + 0.0885378i \(0.0282194\pi\)
−0.996073 + 0.0885378i \(0.971781\pi\)
\(882\) 0 0
\(883\) −61.1859 −0.0692932 −0.0346466 0.999400i \(-0.511031\pi\)
−0.0346466 + 0.999400i \(0.511031\pi\)
\(884\) 112.014 0.126713
\(885\) 0 0
\(886\) −571.196 −0.644691
\(887\) 880.738i 0.992940i 0.868054 + 0.496470i \(0.165371\pi\)
−0.868054 + 0.496470i \(0.834629\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −49.9025 −0.0560702
\(891\) 0 0
\(892\) − 110.648i − 0.124045i
\(893\) 1063.43 1.19085
\(894\) 0 0
\(895\) − 20.2632i − 0.0226405i
\(896\) 0 0
\(897\) 0 0
\(898\) 342.975 0.381932
\(899\) 1593.46i 1.77248i
\(900\) 0 0
\(901\) 363.549i 0.403495i
\(902\) 215.826i 0.239275i
\(903\) 0 0
\(904\) 296.402 0.327878
\(905\) 58.2527 0.0643677
\(906\) 0 0
\(907\) 1288.76 1.42091 0.710454 0.703744i \(-0.248490\pi\)
0.710454 + 0.703744i \(0.248490\pi\)
\(908\) 739.156i 0.814049i
\(909\) 0 0
\(910\) 0 0
\(911\) 788.563 0.865601 0.432801 0.901490i \(-0.357525\pi\)
0.432801 + 0.901490i \(0.357525\pi\)
\(912\) 0 0
\(913\) − 371.218i − 0.406591i
\(914\) −1053.25 −1.15236
\(915\) 0 0
\(916\) 588.792i 0.642786i
\(917\) 0 0
\(918\) 0 0
\(919\) −1497.59 −1.62958 −0.814792 0.579753i \(-0.803149\pi\)
−0.814792 + 0.579753i \(0.803149\pi\)
\(920\) 17.4985i 0.0190201i
\(921\) 0 0
\(922\) − 1080.92i − 1.17237i
\(923\) − 410.667i − 0.444927i
\(924\) 0 0
\(925\) −341.085 −0.368741
\(926\) 931.647 1.00610
\(927\) 0 0
\(928\) 246.392 0.265509
\(929\) − 1000.22i − 1.07667i −0.842732 0.538334i \(-0.819054\pi\)
0.842732 0.538334i \(-0.180946\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 349.574 0.375079
\(933\) 0 0
\(934\) 92.8044i 0.0993623i
\(935\) −16.1177 −0.0172382
\(936\) 0 0
\(937\) − 776.352i − 0.828550i −0.910152 0.414275i \(-0.864035\pi\)
0.910152 0.414275i \(-0.135965\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 35.9798 0.0382764
\(941\) − 3.45542i − 0.00367207i −0.999998 0.00183603i \(-0.999416\pi\)
0.999998 0.00183603i \(-0.000584428\pi\)
\(942\) 0 0
\(943\) − 1052.94i − 1.11659i
\(944\) − 358.356i − 0.379615i
\(945\) 0 0
\(946\) 30.3919 0.0321268
\(947\) 890.045 0.939857 0.469929 0.882704i \(-0.344280\pi\)
0.469929 + 0.882704i \(0.344280\pi\)
\(948\) 0 0
\(949\) 105.538 0.111210
\(950\) − 659.903i − 0.694635i
\(951\) 0 0
\(952\) 0 0
\(953\) 1399.43 1.46845 0.734224 0.678907i \(-0.237546\pi\)
0.734224 + 0.678907i \(0.237546\pi\)
\(954\) 0 0
\(955\) 48.9571i 0.0512640i
\(956\) 646.558 0.676316
\(957\) 0 0
\(958\) − 351.401i − 0.366806i
\(959\) 0 0
\(960\) 0 0
\(961\) −377.372 −0.392686
\(962\) 60.3623i 0.0627466i
\(963\) 0 0
\(964\) 331.002i 0.343363i
\(965\) − 7.35371i − 0.00762042i
\(966\) 0 0
\(967\) −404.382 −0.418182 −0.209091 0.977896i \(-0.567050\pi\)
−0.209091 + 0.977896i \(0.567050\pi\)
\(968\) 319.612 0.330178
\(969\) 0 0
\(970\) −66.4020 −0.0684557
\(971\) 629.341i 0.648137i 0.946034 + 0.324069i \(0.105051\pi\)
−0.946034 + 0.324069i \(0.894949\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1081.03 1.10988
\(975\) 0 0
\(976\) 297.960i 0.305287i
\(977\) 835.530 0.855200 0.427600 0.903968i \(-0.359359\pi\)
0.427600 + 0.903968i \(0.359359\pi\)
\(978\) 0 0
\(979\) 314.817i 0.321570i
\(980\) 0 0
\(981\) 0 0
\(982\) 1106.79 1.12708
\(983\) 585.085i 0.595204i 0.954690 + 0.297602i \(0.0961868\pi\)
−0.954690 + 0.297602i \(0.903813\pi\)
\(984\) 0 0
\(985\) 64.2007i 0.0651784i
\(986\) − 1107.21i − 1.12294i
\(987\) 0 0
\(988\) −116.784 −0.118202
\(989\) −148.272 −0.149922
\(990\) 0 0
\(991\) −1859.92 −1.87681 −0.938405 0.345537i \(-0.887697\pi\)
−0.938405 + 0.345537i \(0.887697\pi\)
\(992\) 206.949i 0.208618i
\(993\) 0 0
\(994\) 0 0
\(995\) 72.9420 0.0733085
\(996\) 0 0
\(997\) 137.481i 0.137895i 0.997620 + 0.0689473i \(0.0219640\pi\)
−0.997620 + 0.0689473i \(0.978036\pi\)
\(998\) 186.648 0.187022
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 882.3.c.d.685.2 yes 4
3.2 odd 2 882.3.c.e.685.3 yes 4
7.2 even 3 882.3.n.h.325.4 8
7.3 odd 6 882.3.n.h.19.4 8
7.4 even 3 882.3.n.h.19.3 8
7.5 odd 6 882.3.n.h.325.3 8
7.6 odd 2 inner 882.3.c.d.685.1 4
21.2 odd 6 882.3.n.g.325.1 8
21.5 even 6 882.3.n.g.325.2 8
21.11 odd 6 882.3.n.g.19.2 8
21.17 even 6 882.3.n.g.19.1 8
21.20 even 2 882.3.c.e.685.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.3.c.d.685.1 4 7.6 odd 2 inner
882.3.c.d.685.2 yes 4 1.1 even 1 trivial
882.3.c.e.685.3 yes 4 3.2 odd 2
882.3.c.e.685.4 yes 4 21.20 even 2
882.3.n.g.19.1 8 21.17 even 6
882.3.n.g.19.2 8 21.11 odd 6
882.3.n.g.325.1 8 21.2 odd 6
882.3.n.g.325.2 8 21.5 even 6
882.3.n.h.19.3 8 7.4 even 3
882.3.n.h.19.4 8 7.3 odd 6
882.3.n.h.325.3 8 7.5 odd 6
882.3.n.h.325.4 8 7.2 even 3