Properties

Label 882.3.c.d.685.4
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 685.4
Root \(0.765367i\) of defining polynomial
Character \(\chi\) \(=\) 882.685
Dual form 882.3.c.d.685.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +2.00000 q^{4} +4.46088i q^{5} +2.82843 q^{8} +O(q^{10})\) \(q+1.41421 q^{2} +2.00000 q^{4} +4.46088i q^{5} +2.82843 q^{8} +6.30864i q^{10} +2.82843 q^{11} +18.6089i q^{13} +4.00000 q^{16} +12.0376i q^{17} -13.8854i q^{19} +8.92177i q^{20} +4.00000 q^{22} -36.4853 q^{23} +5.10051 q^{25} +26.3170i q^{26} -12.4437 q^{29} +15.1535i q^{31} +5.65685 q^{32} +17.0238i q^{34} +45.6985 q^{37} -19.6369i q^{38} +12.6173i q^{40} +52.6565i q^{41} -71.5980 q^{43} +5.65685 q^{44} -51.5980 q^{46} +84.7343i q^{47} +7.21320 q^{50} +37.2178i q^{52} +104.225 q^{53} +12.6173i q^{55} -17.5980 q^{58} -24.1203i q^{59} +13.5365i q^{61} +21.4303i q^{62} +8.00000 q^{64} -83.0122 q^{65} +4.00000 q^{67} +24.0753i q^{68} -92.2010 q^{71} -85.4678i q^{73} +64.6274 q^{74} -27.7708i q^{76} +3.59798 q^{79} +17.8435i q^{80} +74.4675i q^{82} +80.3410i q^{83} -53.6985 q^{85} -101.255 q^{86} +8.00000 q^{88} +111.925i q^{89} -72.9706 q^{92} +119.832i q^{94} +61.9411 q^{95} +23.0791i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4}+O(q^{10}) \) 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
\(5\) 4.46088i 0.892177i 0.894989 + 0.446088i \(0.147183\pi\)
−0.894989 + 0.446088i \(0.852817\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 2.82843 0.353553
\(9\) 0 0
\(10\) 6.30864i 0.630864i
\(11\) 2.82843 0.257130 0.128565 0.991701i \(-0.458963\pi\)
0.128565 + 0.991701i \(0.458963\pi\)
\(12\) 0 0
\(13\) 18.6089i 1.43145i 0.698380 + 0.715727i \(0.253905\pi\)
−0.698380 + 0.715727i \(0.746095\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 12.0376i 0.708096i 0.935227 + 0.354048i \(0.115195\pi\)
−0.935227 + 0.354048i \(0.884805\pi\)
\(18\) 0 0
\(19\) − 13.8854i − 0.730810i −0.930849 0.365405i \(-0.880931\pi\)
0.930849 0.365405i \(-0.119069\pi\)
\(20\) 8.92177i 0.446088i
\(21\) 0 0
\(22\) 4.00000 0.181818
\(23\) −36.4853 −1.58632 −0.793158 0.609016i \(-0.791565\pi\)
−0.793158 + 0.609016i \(0.791565\pi\)
\(24\) 0 0
\(25\) 5.10051 0.204020
\(26\) 26.3170i 1.01219i
\(27\) 0 0
\(28\) 0 0
\(29\) −12.4437 −0.429091 −0.214546 0.976714i \(-0.568827\pi\)
−0.214546 + 0.976714i \(0.568827\pi\)
\(30\) 0 0
\(31\) 15.1535i 0.488822i 0.969672 + 0.244411i \(0.0785946\pi\)
−0.969672 + 0.244411i \(0.921405\pi\)
\(32\) 5.65685 0.176777
\(33\) 0 0
\(34\) 17.0238i 0.500699i
\(35\) 0 0
\(36\) 0 0
\(37\) 45.6985 1.23509 0.617547 0.786534i \(-0.288126\pi\)
0.617547 + 0.786534i \(0.288126\pi\)
\(38\) − 19.6369i − 0.516761i
\(39\) 0 0
\(40\) 12.6173i 0.315432i
\(41\) 52.6565i 1.28430i 0.766577 + 0.642152i \(0.221958\pi\)
−0.766577 + 0.642152i \(0.778042\pi\)
\(42\) 0 0
\(43\) −71.5980 −1.66507 −0.832535 0.553973i \(-0.813111\pi\)
−0.832535 + 0.553973i \(0.813111\pi\)
\(44\) 5.65685 0.128565
\(45\) 0 0
\(46\) −51.5980 −1.12170
\(47\) 84.7343i 1.80286i 0.432928 + 0.901429i \(0.357480\pi\)
−0.432928 + 0.901429i \(0.642520\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 7.21320 0.144264
\(51\) 0 0
\(52\) 37.2178i 0.715727i
\(53\) 104.225 1.96652 0.983258 0.182216i \(-0.0583271\pi\)
0.983258 + 0.182216i \(0.0583271\pi\)
\(54\) 0 0
\(55\) 12.6173i 0.229405i
\(56\) 0 0
\(57\) 0 0
\(58\) −17.5980 −0.303413
\(59\) − 24.1203i − 0.408819i −0.978885 0.204410i \(-0.934473\pi\)
0.978885 0.204410i \(-0.0655274\pi\)
\(60\) 0 0
\(61\) 13.5365i 0.221910i 0.993825 + 0.110955i \(0.0353909\pi\)
−0.993825 + 0.110955i \(0.964609\pi\)
\(62\) 21.4303i 0.345650i
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) −83.0122 −1.27711
\(66\) 0 0
\(67\) 4.00000 0.0597015 0.0298507 0.999554i \(-0.490497\pi\)
0.0298507 + 0.999554i \(0.490497\pi\)
\(68\) 24.0753i 0.354048i
\(69\) 0 0
\(70\) 0 0
\(71\) −92.2010 −1.29861 −0.649303 0.760530i \(-0.724939\pi\)
−0.649303 + 0.760530i \(0.724939\pi\)
\(72\) 0 0
\(73\) − 85.4678i − 1.17079i −0.810748 0.585396i \(-0.800939\pi\)
0.810748 0.585396i \(-0.199061\pi\)
\(74\) 64.6274 0.873343
\(75\) 0 0
\(76\) − 27.7708i − 0.365405i
\(77\) 0 0
\(78\) 0 0
\(79\) 3.59798 0.0455440 0.0227720 0.999741i \(-0.492751\pi\)
0.0227720 + 0.999741i \(0.492751\pi\)
\(80\) 17.8435i 0.223044i
\(81\) 0 0
\(82\) 74.4675i 0.908140i
\(83\) 80.3410i 0.967964i 0.875078 + 0.483982i \(0.160810\pi\)
−0.875078 + 0.483982i \(0.839190\pi\)
\(84\) 0 0
\(85\) −53.6985 −0.631747
\(86\) −101.255 −1.17738
\(87\) 0 0
\(88\) 8.00000 0.0909091
\(89\) 111.925i 1.25759i 0.777572 + 0.628794i \(0.216451\pi\)
−0.777572 + 0.628794i \(0.783549\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −72.9706 −0.793158
\(93\) 0 0
\(94\) 119.832i 1.27481i
\(95\) 61.9411 0.652012
\(96\) 0 0
\(97\) 23.0791i 0.237929i 0.992899 + 0.118965i \(0.0379575\pi\)
−0.992899 + 0.118965i \(0.962043\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 10.2010 0.102010
\(101\) − 82.5602i − 0.817427i −0.912663 0.408714i \(-0.865977\pi\)
0.912663 0.408714i \(-0.134023\pi\)
\(102\) 0 0
\(103\) − 162.757i − 1.58016i −0.613003 0.790081i \(-0.710039\pi\)
0.613003 0.790081i \(-0.289961\pi\)
\(104\) 52.6339i 0.506096i
\(105\) 0 0
\(106\) 147.397 1.39054
\(107\) −13.8579 −0.129513 −0.0647564 0.997901i \(-0.520627\pi\)
−0.0647564 + 0.997901i \(0.520627\pi\)
\(108\) 0 0
\(109\) 136.693 1.25407 0.627034 0.778992i \(-0.284269\pi\)
0.627034 + 0.778992i \(0.284269\pi\)
\(110\) 17.8435i 0.162214i
\(111\) 0 0
\(112\) 0 0
\(113\) 132.794 1.17517 0.587584 0.809163i \(-0.300079\pi\)
0.587584 + 0.809163i \(0.300079\pi\)
\(114\) 0 0
\(115\) − 162.757i − 1.41528i
\(116\) −24.8873 −0.214546
\(117\) 0 0
\(118\) − 34.1113i − 0.289079i
\(119\) 0 0
\(120\) 0 0
\(121\) −113.000 −0.933884
\(122\) 19.1435i 0.156914i
\(123\) 0 0
\(124\) 30.3070i 0.244411i
\(125\) 134.275i 1.07420i
\(126\) 0 0
\(127\) 150.392 1.18419 0.592094 0.805869i \(-0.298301\pi\)
0.592094 + 0.805869i \(0.298301\pi\)
\(128\) 11.3137 0.0883883
\(129\) 0 0
\(130\) −117.397 −0.903054
\(131\) 217.709i 1.66190i 0.556345 + 0.830951i \(0.312203\pi\)
−0.556345 + 0.830951i \(0.687797\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 5.65685 0.0422153
\(135\) 0 0
\(136\) 34.0476i 0.250350i
\(137\) 52.0416 0.379866 0.189933 0.981797i \(-0.439173\pi\)
0.189933 + 0.981797i \(0.439173\pi\)
\(138\) 0 0
\(139\) − 143.799i − 1.03452i −0.855827 0.517262i \(-0.826951\pi\)
0.855827 0.517262i \(-0.173049\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −130.392 −0.918253
\(143\) 52.6339i 0.368069i
\(144\) 0 0
\(145\) − 55.5097i − 0.382825i
\(146\) − 120.870i − 0.827875i
\(147\) 0 0
\(148\) 91.3970 0.617547
\(149\) −19.0883 −0.128109 −0.0640547 0.997946i \(-0.520403\pi\)
−0.0640547 + 0.997946i \(0.520403\pi\)
\(150\) 0 0
\(151\) 154.392 1.02246 0.511232 0.859443i \(-0.329189\pi\)
0.511232 + 0.859443i \(0.329189\pi\)
\(152\) − 39.2738i − 0.258380i
\(153\) 0 0
\(154\) 0 0
\(155\) −67.5980 −0.436116
\(156\) 0 0
\(157\) − 117.677i − 0.749535i −0.927119 0.374767i \(-0.877723\pi\)
0.927119 0.374767i \(-0.122277\pi\)
\(158\) 5.08831 0.0322045
\(159\) 0 0
\(160\) 25.2346i 0.157716i
\(161\) 0 0
\(162\) 0 0
\(163\) 185.990 1.14104 0.570521 0.821283i \(-0.306741\pi\)
0.570521 + 0.821283i \(0.306741\pi\)
\(164\) 105.313i 0.642152i
\(165\) 0 0
\(166\) 113.619i 0.684454i
\(167\) − 206.142i − 1.23439i −0.786812 0.617193i \(-0.788270\pi\)
0.786812 0.617193i \(-0.211730\pi\)
\(168\) 0 0
\(169\) −177.291 −1.04906
\(170\) −75.9411 −0.446713
\(171\) 0 0
\(172\) −143.196 −0.832535
\(173\) − 206.075i − 1.19118i −0.803287 0.595592i \(-0.796917\pi\)
0.803287 0.595592i \(-0.203083\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 11.3137 0.0642824
\(177\) 0 0
\(178\) 158.286i 0.889250i
\(179\) −160.083 −0.894320 −0.447160 0.894454i \(-0.647565\pi\)
−0.447160 + 0.894454i \(0.647565\pi\)
\(180\) 0 0
\(181\) − 99.9235i − 0.552064i −0.961149 0.276032i \(-0.910980\pi\)
0.961149 0.276032i \(-0.0890195\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −103.196 −0.560848
\(185\) 203.856i 1.10192i
\(186\) 0 0
\(187\) 34.0476i 0.182073i
\(188\) 169.469i 0.901429i
\(189\) 0 0
\(190\) 87.5980 0.461042
\(191\) 69.5736 0.364260 0.182130 0.983274i \(-0.441701\pi\)
0.182130 + 0.983274i \(0.441701\pi\)
\(192\) 0 0
\(193\) 135.196 0.700497 0.350249 0.936657i \(-0.386097\pi\)
0.350249 + 0.936657i \(0.386097\pi\)
\(194\) 32.6388i 0.168241i
\(195\) 0 0
\(196\) 0 0
\(197\) 21.4903 0.109088 0.0545440 0.998511i \(-0.482629\pi\)
0.0545440 + 0.998511i \(0.482629\pi\)
\(198\) 0 0
\(199\) − 259.827i − 1.30566i −0.757503 0.652831i \(-0.773581\pi\)
0.757503 0.652831i \(-0.226419\pi\)
\(200\) 14.4264 0.0721320
\(201\) 0 0
\(202\) − 116.758i − 0.578008i
\(203\) 0 0
\(204\) 0 0
\(205\) −234.894 −1.14583
\(206\) − 230.173i − 1.11734i
\(207\) 0 0
\(208\) 74.4356i 0.357864i
\(209\) − 39.2738i − 0.187913i
\(210\) 0 0
\(211\) −251.598 −1.19241 −0.596204 0.802833i \(-0.703325\pi\)
−0.596204 + 0.802833i \(0.703325\pi\)
\(212\) 208.451 0.983258
\(213\) 0 0
\(214\) −19.5980 −0.0915793
\(215\) − 319.390i − 1.48554i
\(216\) 0 0
\(217\) 0 0
\(218\) 193.314 0.886760
\(219\) 0 0
\(220\) 25.2346i 0.114703i
\(221\) −224.007 −1.01361
\(222\) 0 0
\(223\) 29.0389i 0.130219i 0.997878 + 0.0651096i \(0.0207397\pi\)
−0.997878 + 0.0651096i \(0.979260\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 187.799 0.830969
\(227\) − 275.543i − 1.21385i −0.794761 0.606923i \(-0.792404\pi\)
0.794761 0.606923i \(-0.207596\pi\)
\(228\) 0 0
\(229\) 41.8457i 0.182733i 0.995817 + 0.0913663i \(0.0291234\pi\)
−0.995817 + 0.0913663i \(0.970877\pi\)
\(230\) − 230.173i − 1.00075i
\(231\) 0 0
\(232\) −35.1960 −0.151707
\(233\) 217.213 0.932246 0.466123 0.884720i \(-0.345651\pi\)
0.466123 + 0.884720i \(0.345651\pi\)
\(234\) 0 0
\(235\) −377.990 −1.60847
\(236\) − 48.2406i − 0.204410i
\(237\) 0 0
\(238\) 0 0
\(239\) 68.7208 0.287535 0.143767 0.989612i \(-0.454078\pi\)
0.143767 + 0.989612i \(0.454078\pi\)
\(240\) 0 0
\(241\) − 18.0386i − 0.0748489i −0.999299 0.0374244i \(-0.988085\pi\)
0.999299 0.0374244i \(-0.0119154\pi\)
\(242\) −159.806 −0.660356
\(243\) 0 0
\(244\) 27.0730i 0.110955i
\(245\) 0 0
\(246\) 0 0
\(247\) 258.392 1.04612
\(248\) 42.8605i 0.172825i
\(249\) 0 0
\(250\) 189.893i 0.759573i
\(251\) − 166.869i − 0.664815i −0.943136 0.332408i \(-0.892139\pi\)
0.943136 0.332408i \(-0.107861\pi\)
\(252\) 0 0
\(253\) −103.196 −0.407889
\(254\) 212.686 0.837348
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) − 305.873i − 1.19017i −0.803664 0.595083i \(-0.797119\pi\)
0.803664 0.595083i \(-0.202881\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −166.024 −0.638555
\(261\) 0 0
\(262\) 307.887i 1.17514i
\(263\) −9.62237 −0.0365869 −0.0182935 0.999833i \(-0.505823\pi\)
−0.0182935 + 0.999833i \(0.505823\pi\)
\(264\) 0 0
\(265\) 464.938i 1.75448i
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.0298507
\(269\) 99.6422i 0.370417i 0.982699 + 0.185209i \(0.0592961\pi\)
−0.982699 + 0.185209i \(0.940704\pi\)
\(270\) 0 0
\(271\) − 504.628i − 1.86210i −0.364898 0.931048i \(-0.618896\pi\)
0.364898 0.931048i \(-0.381104\pi\)
\(272\) 48.1505i 0.177024i
\(273\) 0 0
\(274\) 73.5980 0.268606
\(275\) 14.4264 0.0524597
\(276\) 0 0
\(277\) 314.000 1.13357 0.566787 0.823864i \(-0.308186\pi\)
0.566787 + 0.823864i \(0.308186\pi\)
\(278\) − 203.362i − 0.731519i
\(279\) 0 0
\(280\) 0 0
\(281\) −155.002 −0.551609 −0.275804 0.961214i \(-0.588944\pi\)
−0.275804 + 0.961214i \(0.588944\pi\)
\(282\) 0 0
\(283\) 276.503i 0.977044i 0.872552 + 0.488522i \(0.162464\pi\)
−0.872552 + 0.488522i \(0.837536\pi\)
\(284\) −184.402 −0.649303
\(285\) 0 0
\(286\) 74.4356i 0.260264i
\(287\) 0 0
\(288\) 0 0
\(289\) 144.095 0.498600
\(290\) − 78.5026i − 0.270698i
\(291\) 0 0
\(292\) − 170.936i − 0.585396i
\(293\) − 485.027i − 1.65538i −0.561185 0.827690i \(-0.689654\pi\)
0.561185 0.827690i \(-0.310346\pi\)
\(294\) 0 0
\(295\) 107.598 0.364739
\(296\) 129.255 0.436672
\(297\) 0 0
\(298\) −26.9949 −0.0905871
\(299\) − 678.951i − 2.27074i
\(300\) 0 0
\(301\) 0 0
\(302\) 218.343 0.722991
\(303\) 0 0
\(304\) − 55.5416i − 0.182702i
\(305\) −60.3848 −0.197983
\(306\) 0 0
\(307\) − 126.364i − 0.411609i −0.978593 0.205805i \(-0.934019\pi\)
0.978593 0.205805i \(-0.0659811\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −95.5980 −0.308381
\(311\) 421.968i 1.35681i 0.734688 + 0.678405i \(0.237329\pi\)
−0.734688 + 0.678405i \(0.762671\pi\)
\(312\) 0 0
\(313\) 431.650i 1.37907i 0.724251 + 0.689536i \(0.242186\pi\)
−0.724251 + 0.689536i \(0.757814\pi\)
\(314\) − 166.420i − 0.530001i
\(315\) 0 0
\(316\) 7.19596 0.0227720
\(317\) 256.804 0.810107 0.405054 0.914293i \(-0.367253\pi\)
0.405054 + 0.914293i \(0.367253\pi\)
\(318\) 0 0
\(319\) −35.1960 −0.110332
\(320\) 35.6871i 0.111522i
\(321\) 0 0
\(322\) 0 0
\(323\) 167.147 0.517484
\(324\) 0 0
\(325\) 94.9148i 0.292046i
\(326\) 263.029 0.806839
\(327\) 0 0
\(328\) 148.935i 0.454070i
\(329\) 0 0
\(330\) 0 0
\(331\) 155.598 0.470085 0.235042 0.971985i \(-0.424477\pi\)
0.235042 + 0.971985i \(0.424477\pi\)
\(332\) 160.682i 0.483982i
\(333\) 0 0
\(334\) − 291.529i − 0.872843i
\(335\) 17.8435i 0.0532643i
\(336\) 0 0
\(337\) 39.7086 0.117830 0.0589148 0.998263i \(-0.481236\pi\)
0.0589148 + 0.998263i \(0.481236\pi\)
\(338\) −250.728 −0.741799
\(339\) 0 0
\(340\) −107.397 −0.315873
\(341\) 42.8605i 0.125691i
\(342\) 0 0
\(343\) 0 0
\(344\) −202.510 −0.588691
\(345\) 0 0
\(346\) − 291.434i − 0.842294i
\(347\) −649.671 −1.87225 −0.936126 0.351666i \(-0.885615\pi\)
−0.936126 + 0.351666i \(0.885615\pi\)
\(348\) 0 0
\(349\) 616.789i 1.76730i 0.468143 + 0.883652i \(0.344923\pi\)
−0.468143 + 0.883652i \(0.655077\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 16.0000 0.0454545
\(353\) − 530.755i − 1.50356i −0.659416 0.751778i \(-0.729197\pi\)
0.659416 0.751778i \(-0.270803\pi\)
\(354\) 0 0
\(355\) − 411.298i − 1.15859i
\(356\) 223.851i 0.628794i
\(357\) 0 0
\(358\) −226.392 −0.632380
\(359\) 532.014 1.48193 0.740967 0.671541i \(-0.234367\pi\)
0.740967 + 0.671541i \(0.234367\pi\)
\(360\) 0 0
\(361\) 168.196 0.465917
\(362\) − 141.313i − 0.390368i
\(363\) 0 0
\(364\) 0 0
\(365\) 381.262 1.04455
\(366\) 0 0
\(367\) 391.072i 1.06559i 0.846244 + 0.532796i \(0.178859\pi\)
−0.846244 + 0.532796i \(0.821141\pi\)
\(368\) −145.941 −0.396579
\(369\) 0 0
\(370\) 288.295i 0.779177i
\(371\) 0 0
\(372\) 0 0
\(373\) 72.8040 0.195185 0.0975925 0.995226i \(-0.468886\pi\)
0.0975925 + 0.995226i \(0.468886\pi\)
\(374\) 48.1505i 0.128745i
\(375\) 0 0
\(376\) 239.665i 0.637406i
\(377\) − 231.563i − 0.614225i
\(378\) 0 0
\(379\) −30.8141 −0.0813038 −0.0406519 0.999173i \(-0.512943\pi\)
−0.0406519 + 0.999173i \(0.512943\pi\)
\(380\) 123.882 0.326006
\(381\) 0 0
\(382\) 98.3919 0.257570
\(383\) 446.269i 1.16519i 0.812762 + 0.582596i \(0.197963\pi\)
−0.812762 + 0.582596i \(0.802037\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 191.196 0.495326
\(387\) 0 0
\(388\) 46.1582i 0.118965i
\(389\) −260.208 −0.668916 −0.334458 0.942411i \(-0.608553\pi\)
−0.334458 + 0.942411i \(0.608553\pi\)
\(390\) 0 0
\(391\) − 439.196i − 1.12326i
\(392\) 0 0
\(393\) 0 0
\(394\) 30.3919 0.0771369
\(395\) 16.0502i 0.0406334i
\(396\) 0 0
\(397\) 609.847i 1.53614i 0.640367 + 0.768069i \(0.278782\pi\)
−0.640367 + 0.768069i \(0.721218\pi\)
\(398\) − 367.451i − 0.923243i
\(399\) 0 0
\(400\) 20.4020 0.0510051
\(401\) −147.057 −0.366725 −0.183363 0.983045i \(-0.558698\pi\)
−0.183363 + 0.983045i \(0.558698\pi\)
\(402\) 0 0
\(403\) −281.990 −0.699727
\(404\) − 165.120i − 0.408714i
\(405\) 0 0
\(406\) 0 0
\(407\) 129.255 0.317579
\(408\) 0 0
\(409\) − 158.763i − 0.388173i −0.980984 0.194087i \(-0.937826\pi\)
0.980984 0.194087i \(-0.0621743\pi\)
\(410\) −332.191 −0.810222
\(411\) 0 0
\(412\) − 325.513i − 0.790081i
\(413\) 0 0
\(414\) 0 0
\(415\) −358.392 −0.863595
\(416\) 105.268i 0.253048i
\(417\) 0 0
\(418\) − 55.5416i − 0.132875i
\(419\) 577.000i 1.37709i 0.725195 + 0.688544i \(0.241750\pi\)
−0.725195 + 0.688544i \(0.758250\pi\)
\(420\) 0 0
\(421\) −37.9798 −0.0902133 −0.0451066 0.998982i \(-0.514363\pi\)
−0.0451066 + 0.998982i \(0.514363\pi\)
\(422\) −355.813 −0.843159
\(423\) 0 0
\(424\) 294.794 0.695269
\(425\) 61.3980i 0.144466i
\(426\) 0 0
\(427\) 0 0
\(428\) −27.7157 −0.0647564
\(429\) 0 0
\(430\) − 451.686i − 1.05043i
\(431\) 785.151 1.82170 0.910848 0.412741i \(-0.135429\pi\)
0.910848 + 0.412741i \(0.135429\pi\)
\(432\) 0 0
\(433\) − 582.519i − 1.34531i −0.739957 0.672655i \(-0.765154\pi\)
0.739957 0.672655i \(-0.234846\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 273.387 0.627034
\(437\) 506.612i 1.15930i
\(438\) 0 0
\(439\) 6.46796i 0.0147334i 0.999973 + 0.00736669i \(0.00234491\pi\)
−0.999973 + 0.00736669i \(0.997655\pi\)
\(440\) 35.6871i 0.0811070i
\(441\) 0 0
\(442\) −316.794 −0.716728
\(443\) −291.897 −0.658909 −0.329454 0.944172i \(-0.606865\pi\)
−0.329454 + 0.944172i \(0.606865\pi\)
\(444\) 0 0
\(445\) −499.286 −1.12199
\(446\) 41.0672i 0.0920789i
\(447\) 0 0
\(448\) 0 0
\(449\) −457.480 −1.01889 −0.509443 0.860504i \(-0.670149\pi\)
−0.509443 + 0.860504i \(0.670149\pi\)
\(450\) 0 0
\(451\) 148.935i 0.330233i
\(452\) 265.588 0.587584
\(453\) 0 0
\(454\) − 389.677i − 0.858319i
\(455\) 0 0
\(456\) 0 0
\(457\) −680.764 −1.48964 −0.744818 0.667268i \(-0.767464\pi\)
−0.744818 + 0.667268i \(0.767464\pi\)
\(458\) 59.1788i 0.129211i
\(459\) 0 0
\(460\) − 325.513i − 0.707638i
\(461\) 156.401i 0.339265i 0.985507 + 0.169633i \(0.0542581\pi\)
−0.985507 + 0.169633i \(0.945742\pi\)
\(462\) 0 0
\(463\) 370.774 0.800807 0.400404 0.916339i \(-0.368870\pi\)
0.400404 + 0.916339i \(0.368870\pi\)
\(464\) −49.7746 −0.107273
\(465\) 0 0
\(466\) 307.186 0.659197
\(467\) − 40.1705i − 0.0860182i −0.999075 0.0430091i \(-0.986306\pi\)
0.999075 0.0430091i \(-0.0136944\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −534.558 −1.13736
\(471\) 0 0
\(472\) − 68.2226i − 0.144539i
\(473\) −202.510 −0.428139
\(474\) 0 0
\(475\) − 70.8225i − 0.149100i
\(476\) 0 0
\(477\) 0 0
\(478\) 97.1859 0.203318
\(479\) − 213.136i − 0.444960i −0.974937 0.222480i \(-0.928585\pi\)
0.974937 0.222480i \(-0.0714152\pi\)
\(480\) 0 0
\(481\) 850.399i 1.76798i
\(482\) − 25.5104i − 0.0529262i
\(483\) 0 0
\(484\) −226.000 −0.466942
\(485\) −102.953 −0.212275
\(486\) 0 0
\(487\) −843.598 −1.73223 −0.866117 0.499841i \(-0.833392\pi\)
−0.866117 + 0.499841i \(0.833392\pi\)
\(488\) 38.2870i 0.0784570i
\(489\) 0 0
\(490\) 0 0
\(491\) 614.621 1.25177 0.625887 0.779913i \(-0.284737\pi\)
0.625887 + 0.779913i \(0.284737\pi\)
\(492\) 0 0
\(493\) − 149.792i − 0.303838i
\(494\) 365.421 0.739719
\(495\) 0 0
\(496\) 60.6140i 0.122206i
\(497\) 0 0
\(498\) 0 0
\(499\) 659.980 1.32260 0.661302 0.750119i \(-0.270004\pi\)
0.661302 + 0.750119i \(0.270004\pi\)
\(500\) 268.550i 0.537100i
\(501\) 0 0
\(502\) − 235.988i − 0.470095i
\(503\) − 301.276i − 0.598959i −0.954103 0.299480i \(-0.903187\pi\)
0.954103 0.299480i \(-0.0968130\pi\)
\(504\) 0 0
\(505\) 368.291 0.729290
\(506\) −145.941 −0.288421
\(507\) 0 0
\(508\) 300.784 0.592094
\(509\) 866.644i 1.70264i 0.524646 + 0.851320i \(0.324198\pi\)
−0.524646 + 0.851320i \(0.675802\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274 0.0441942
\(513\) 0 0
\(514\) − 432.569i − 0.841574i
\(515\) 726.039 1.40978
\(516\) 0 0
\(517\) 239.665i 0.463568i
\(518\) 0 0
\(519\) 0 0
\(520\) −234.794 −0.451527
\(521\) − 623.830i − 1.19737i −0.800984 0.598685i \(-0.795690\pi\)
0.800984 0.598685i \(-0.204310\pi\)
\(522\) 0 0
\(523\) − 770.604i − 1.47343i −0.676203 0.736715i \(-0.736376\pi\)
0.676203 0.736715i \(-0.263624\pi\)
\(524\) 435.418i 0.830951i
\(525\) 0 0
\(526\) −13.6081 −0.0258709
\(527\) −182.412 −0.346133
\(528\) 0 0
\(529\) 802.176 1.51640
\(530\) 657.521i 1.24061i
\(531\) 0 0
\(532\) 0 0
\(533\) −979.879 −1.83842
\(534\) 0 0
\(535\) − 61.8183i − 0.115548i
\(536\) 11.3137 0.0211077
\(537\) 0 0
\(538\) 140.915i 0.261924i
\(539\) 0 0
\(540\) 0 0
\(541\) −1033.57 −1.91048 −0.955238 0.295838i \(-0.904401\pi\)
−0.955238 + 0.295838i \(0.904401\pi\)
\(542\) − 713.651i − 1.31670i
\(543\) 0 0
\(544\) 68.0951i 0.125175i
\(545\) 609.774i 1.11885i
\(546\) 0 0
\(547\) 923.176 1.68771 0.843854 0.536574i \(-0.180282\pi\)
0.843854 + 0.536574i \(0.180282\pi\)
\(548\) 104.083 0.189933
\(549\) 0 0
\(550\) 20.4020 0.0370946
\(551\) 172.785i 0.313584i
\(552\) 0 0
\(553\) 0 0
\(554\) 444.063 0.801558
\(555\) 0 0
\(556\) − 287.598i − 0.517262i
\(557\) 381.255 0.684479 0.342240 0.939613i \(-0.388815\pi\)
0.342240 + 0.939613i \(0.388815\pi\)
\(558\) 0 0
\(559\) − 1332.36i − 2.38347i
\(560\) 0 0
\(561\) 0 0
\(562\) −219.206 −0.390046
\(563\) 214.659i 0.381277i 0.981660 + 0.190638i \(0.0610557\pi\)
−0.981660 + 0.190638i \(0.938944\pi\)
\(564\) 0 0
\(565\) 592.378i 1.04846i
\(566\) 391.035i 0.690874i
\(567\) 0 0
\(568\) −260.784 −0.459126
\(569\) −131.252 −0.230671 −0.115336 0.993327i \(-0.536794\pi\)
−0.115336 + 0.993327i \(0.536794\pi\)
\(570\) 0 0
\(571\) −101.186 −0.177208 −0.0886041 0.996067i \(-0.528241\pi\)
−0.0886041 + 0.996067i \(0.528241\pi\)
\(572\) 105.268i 0.184035i
\(573\) 0 0
\(574\) 0 0
\(575\) −186.093 −0.323641
\(576\) 0 0
\(577\) 920.331i 1.59503i 0.603301 + 0.797514i \(0.293852\pi\)
−0.603301 + 0.797514i \(0.706148\pi\)
\(578\) 203.782 0.352564
\(579\) 0 0
\(580\) − 111.019i − 0.191413i
\(581\) 0 0
\(582\) 0 0
\(583\) 294.794 0.505650
\(584\) − 241.739i − 0.413937i
\(585\) 0 0
\(586\) − 685.931i − 1.17053i
\(587\) 595.384i 1.01428i 0.861863 + 0.507141i \(0.169298\pi\)
−0.861863 + 0.507141i \(0.830702\pi\)
\(588\) 0 0
\(589\) 210.412 0.357236
\(590\) 152.167 0.257909
\(591\) 0 0
\(592\) 182.794 0.308774
\(593\) 441.134i 0.743903i 0.928252 + 0.371951i \(0.121311\pi\)
−0.928252 + 0.371951i \(0.878689\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −38.1766 −0.0640547
\(597\) 0 0
\(598\) − 960.182i − 1.60566i
\(599\) 250.053 0.417451 0.208725 0.977974i \(-0.433069\pi\)
0.208725 + 0.977974i \(0.433069\pi\)
\(600\) 0 0
\(601\) 593.457i 0.987449i 0.869618 + 0.493725i \(0.164365\pi\)
−0.869618 + 0.493725i \(0.835635\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 308.784 0.511232
\(605\) − 504.080i − 0.833190i
\(606\) 0 0
\(607\) 338.449i 0.557577i 0.960353 + 0.278788i \(0.0899328\pi\)
−0.960353 + 0.278788i \(0.910067\pi\)
\(608\) − 78.5476i − 0.129190i
\(609\) 0 0
\(610\) −85.3970 −0.139995
\(611\) −1576.81 −2.58071
\(612\) 0 0
\(613\) 805.477 1.31399 0.656996 0.753894i \(-0.271827\pi\)
0.656996 + 0.753894i \(0.271827\pi\)
\(614\) − 178.706i − 0.291052i
\(615\) 0 0
\(616\) 0 0
\(617\) 18.1148 0.0293595 0.0146797 0.999892i \(-0.495327\pi\)
0.0146797 + 0.999892i \(0.495327\pi\)
\(618\) 0 0
\(619\) 448.137i 0.723969i 0.932184 + 0.361984i \(0.117901\pi\)
−0.932184 + 0.361984i \(0.882099\pi\)
\(620\) −135.196 −0.218058
\(621\) 0 0
\(622\) 596.753i 0.959410i
\(623\) 0 0
\(624\) 0 0
\(625\) −471.472 −0.754356
\(626\) 610.445i 0.975152i
\(627\) 0 0
\(628\) − 235.354i − 0.374767i
\(629\) 550.101i 0.874565i
\(630\) 0 0
\(631\) 696.764 1.10422 0.552111 0.833771i \(-0.313823\pi\)
0.552111 + 0.833771i \(0.313823\pi\)
\(632\) 10.1766 0.0161023
\(633\) 0 0
\(634\) 363.176 0.572832
\(635\) 670.881i 1.05651i
\(636\) 0 0
\(637\) 0 0
\(638\) −49.7746 −0.0780166
\(639\) 0 0
\(640\) 50.4692i 0.0788581i
\(641\) 985.394 1.53728 0.768638 0.639684i \(-0.220935\pi\)
0.768638 + 0.639684i \(0.220935\pi\)
\(642\) 0 0
\(643\) 675.943i 1.05123i 0.850722 + 0.525616i \(0.176165\pi\)
−0.850722 + 0.525616i \(0.823835\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 236.382 0.365916
\(647\) 116.745i 0.180440i 0.995922 + 0.0902199i \(0.0287570\pi\)
−0.995922 + 0.0902199i \(0.971243\pi\)
\(648\) 0 0
\(649\) − 68.2226i − 0.105120i
\(650\) 134.230i 0.206507i
\(651\) 0 0
\(652\) 371.980 0.570521
\(653\) −941.248 −1.44142 −0.720710 0.693236i \(-0.756184\pi\)
−0.720710 + 0.693236i \(0.756184\pi\)
\(654\) 0 0
\(655\) −971.176 −1.48271
\(656\) 210.626i 0.321076i
\(657\) 0 0
\(658\) 0 0
\(659\) −1159.61 −1.75965 −0.879827 0.475294i \(-0.842342\pi\)
−0.879827 + 0.475294i \(0.842342\pi\)
\(660\) 0 0
\(661\) 740.805i 1.12073i 0.828244 + 0.560367i \(0.189340\pi\)
−0.828244 + 0.560367i \(0.810660\pi\)
\(662\) 220.049 0.332400
\(663\) 0 0
\(664\) 227.239i 0.342227i
\(665\) 0 0
\(666\) 0 0
\(667\) 454.010 0.680675
\(668\) − 412.285i − 0.617193i
\(669\) 0 0
\(670\) 25.2346i 0.0376635i
\(671\) 38.2870i 0.0570596i
\(672\) 0 0
\(673\) −573.075 −0.851523 −0.425762 0.904835i \(-0.639994\pi\)
−0.425762 + 0.904835i \(0.639994\pi\)
\(674\) 56.1564 0.0833181
\(675\) 0 0
\(676\) −354.583 −0.524531
\(677\) − 136.629i − 0.201816i −0.994896 0.100908i \(-0.967825\pi\)
0.994896 0.100908i \(-0.0321747\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −151.882 −0.223356
\(681\) 0 0
\(682\) 60.6140i 0.0888768i
\(683\) −266.455 −0.390124 −0.195062 0.980791i \(-0.562491\pi\)
−0.195062 + 0.980791i \(0.562491\pi\)
\(684\) 0 0
\(685\) 232.152i 0.338908i
\(686\) 0 0
\(687\) 0 0
\(688\) −286.392 −0.416267
\(689\) 1939.52i 2.81498i
\(690\) 0 0
\(691\) − 127.568i − 0.184614i −0.995731 0.0923071i \(-0.970576\pi\)
0.995731 0.0923071i \(-0.0294242\pi\)
\(692\) − 412.150i − 0.595592i
\(693\) 0 0
\(694\) −918.774 −1.32388
\(695\) 641.470 0.922979
\(696\) 0 0
\(697\) −633.859 −0.909410
\(698\) 872.272i 1.24967i
\(699\) 0 0
\(700\) 0 0
\(701\) 545.325 0.777924 0.388962 0.921254i \(-0.372834\pi\)
0.388962 + 0.921254i \(0.372834\pi\)
\(702\) 0 0
\(703\) − 634.541i − 0.902619i
\(704\) 22.6274 0.0321412
\(705\) 0 0
\(706\) − 750.601i − 1.06317i
\(707\) 0 0
\(708\) 0 0
\(709\) −171.106 −0.241334 −0.120667 0.992693i \(-0.538503\pi\)
−0.120667 + 0.992693i \(0.538503\pi\)
\(710\) − 581.663i − 0.819244i
\(711\) 0 0
\(712\) 316.573i 0.444625i
\(713\) − 552.879i − 0.775427i
\(714\) 0 0
\(715\) −234.794 −0.328383
\(716\) −320.167 −0.447160
\(717\) 0 0
\(718\) 752.382 1.04789
\(719\) 233.579i 0.324867i 0.986720 + 0.162433i \(0.0519342\pi\)
−0.986720 + 0.162433i \(0.948066\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 237.865 0.329453
\(723\) 0 0
\(724\) − 199.847i − 0.276032i
\(725\) −63.4689 −0.0875433
\(726\) 0 0
\(727\) − 912.568i − 1.25525i −0.778515 0.627626i \(-0.784027\pi\)
0.778515 0.627626i \(-0.215973\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 539.186 0.738611
\(731\) − 861.870i − 1.17903i
\(732\) 0 0
\(733\) 333.754i 0.455327i 0.973740 + 0.227663i \(0.0731085\pi\)
−0.973740 + 0.227663i \(0.926891\pi\)
\(734\) 553.060i 0.753487i
\(735\) 0 0
\(736\) −206.392 −0.280424
\(737\) 11.3137 0.0153510
\(738\) 0 0
\(739\) −640.382 −0.866552 −0.433276 0.901261i \(-0.642642\pi\)
−0.433276 + 0.901261i \(0.642642\pi\)
\(740\) 407.711i 0.550961i
\(741\) 0 0
\(742\) 0 0
\(743\) 750.642 1.01028 0.505142 0.863036i \(-0.331440\pi\)
0.505142 + 0.863036i \(0.331440\pi\)
\(744\) 0 0
\(745\) − 85.1508i − 0.114296i
\(746\) 102.960 0.138017
\(747\) 0 0
\(748\) 68.0951i 0.0910363i
\(749\) 0 0
\(750\) 0 0
\(751\) 795.980 1.05989 0.529947 0.848031i \(-0.322212\pi\)
0.529947 + 0.848031i \(0.322212\pi\)
\(752\) 338.937i 0.450714i
\(753\) 0 0
\(754\) − 327.479i − 0.434323i
\(755\) 688.725i 0.912218i
\(756\) 0 0
\(757\) −145.678 −0.192442 −0.0962208 0.995360i \(-0.530675\pi\)
−0.0962208 + 0.995360i \(0.530675\pi\)
\(758\) −43.5778 −0.0574905
\(759\) 0 0
\(760\) 175.196 0.230521
\(761\) 866.153i 1.13818i 0.822276 + 0.569089i \(0.192704\pi\)
−0.822276 + 0.569089i \(0.807296\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 139.147 0.182130
\(765\) 0 0
\(766\) 631.119i 0.823916i
\(767\) 448.853 0.585206
\(768\) 0 0
\(769\) 1051.96i 1.36796i 0.729502 + 0.683978i \(0.239752\pi\)
−0.729502 + 0.683978i \(0.760248\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 270.392 0.350249
\(773\) − 855.796i − 1.10711i −0.832813 0.553555i \(-0.813271\pi\)
0.832813 0.553555i \(-0.186729\pi\)
\(774\) 0 0
\(775\) 77.2905i 0.0997296i
\(776\) 65.2776i 0.0841206i
\(777\) 0 0
\(778\) −367.990 −0.472995
\(779\) 731.156 0.938582
\(780\) 0 0
\(781\) −260.784 −0.333910
\(782\) − 621.117i − 0.794268i
\(783\) 0 0
\(784\) 0 0
\(785\) 524.943 0.668717
\(786\) 0 0
\(787\) 1405.91i 1.78642i 0.449643 + 0.893208i \(0.351551\pi\)
−0.449643 + 0.893208i \(0.648449\pi\)
\(788\) 42.9807 0.0545440
\(789\) 0 0
\(790\) 22.6984i 0.0287321i
\(791\) 0 0
\(792\) 0 0
\(793\) −251.899 −0.317654
\(794\) 862.454i 1.08621i
\(795\) 0 0
\(796\) − 519.654i − 0.652831i
\(797\) − 253.331i − 0.317856i −0.987290 0.158928i \(-0.949196\pi\)
0.987290 0.158928i \(-0.0508037\pi\)
\(798\) 0 0
\(799\) −1020.00 −1.27660
\(800\) 28.8528 0.0360660
\(801\) 0 0
\(802\) −207.970 −0.259314
\(803\) − 241.739i − 0.301045i
\(804\) 0 0
\(805\) 0 0
\(806\) −398.794 −0.494782
\(807\) 0 0
\(808\) − 233.515i − 0.289004i
\(809\) −739.882 −0.914564 −0.457282 0.889322i \(-0.651177\pi\)
−0.457282 + 0.889322i \(0.651177\pi\)
\(810\) 0 0
\(811\) 764.518i 0.942686i 0.881950 + 0.471343i \(0.156231\pi\)
−0.881950 + 0.471343i \(0.843769\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 182.794 0.224563
\(815\) 829.680i 1.01801i
\(816\) 0 0
\(817\) 994.166i 1.21685i
\(818\) − 224.524i − 0.274480i
\(819\) 0 0
\(820\) −469.789 −0.572913
\(821\) 119.902 0.146044 0.0730222 0.997330i \(-0.476736\pi\)
0.0730222 + 0.997330i \(0.476736\pi\)
\(822\) 0 0
\(823\) −307.196 −0.373264 −0.186632 0.982430i \(-0.559757\pi\)
−0.186632 + 0.982430i \(0.559757\pi\)
\(824\) − 460.345i − 0.558671i
\(825\) 0 0
\(826\) 0 0
\(827\) −514.504 −0.622133 −0.311066 0.950388i \(-0.600686\pi\)
−0.311066 + 0.950388i \(0.600686\pi\)
\(828\) 0 0
\(829\) − 806.651i − 0.973041i −0.873669 0.486520i \(-0.838266\pi\)
0.873669 0.486520i \(-0.161734\pi\)
\(830\) −506.843 −0.610654
\(831\) 0 0
\(832\) 148.871i 0.178932i
\(833\) 0 0
\(834\) 0 0
\(835\) 919.578 1.10129
\(836\) − 78.5476i − 0.0939565i
\(837\) 0 0
\(838\) 816.001i 0.973748i
\(839\) 100.694i 0.120017i 0.998198 + 0.0600085i \(0.0191128\pi\)
−0.998198 + 0.0600085i \(0.980887\pi\)
\(840\) 0 0
\(841\) −686.156 −0.815881
\(842\) −53.7115 −0.0637904
\(843\) 0 0
\(844\) −503.196 −0.596204
\(845\) − 790.877i − 0.935949i
\(846\) 0 0
\(847\) 0 0
\(848\) 416.902 0.491629
\(849\) 0 0
\(850\) 86.8299i 0.102153i
\(851\) −1667.32 −1.95925
\(852\) 0 0
\(853\) − 137.712i − 0.161444i −0.996737 0.0807219i \(-0.974277\pi\)
0.996737 0.0807219i \(-0.0257226\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −39.1960 −0.0457897
\(857\) − 557.744i − 0.650809i −0.945575 0.325405i \(-0.894499\pi\)
0.945575 0.325405i \(-0.105501\pi\)
\(858\) 0 0
\(859\) − 1578.94i − 1.83811i −0.394126 0.919056i \(-0.628953\pi\)
0.394126 0.919056i \(-0.371047\pi\)
\(860\) − 638.781i − 0.742768i
\(861\) 0 0
\(862\) 1110.37 1.28813
\(863\) −525.249 −0.608631 −0.304316 0.952571i \(-0.598428\pi\)
−0.304316 + 0.952571i \(0.598428\pi\)
\(864\) 0 0
\(865\) 919.276 1.06275
\(866\) − 823.806i − 0.951277i
\(867\) 0 0
\(868\) 0 0
\(869\) 10.1766 0.0117107
\(870\) 0 0
\(871\) 74.4356i 0.0854600i
\(872\) 386.627 0.443380
\(873\) 0 0
\(874\) 716.458i 0.819746i
\(875\) 0 0
\(876\) 0 0
\(877\) −34.2611 −0.0390663 −0.0195331 0.999809i \(-0.506218\pi\)
−0.0195331 + 0.999809i \(0.506218\pi\)
\(878\) 9.14707i 0.0104181i
\(879\) 0 0
\(880\) 50.4692i 0.0573513i
\(881\) − 366.173i − 0.415634i −0.978168 0.207817i \(-0.933364\pi\)
0.978168 0.207817i \(-0.0666358\pi\)
\(882\) 0 0
\(883\) 493.186 0.558534 0.279267 0.960213i \(-0.409908\pi\)
0.279267 + 0.960213i \(0.409908\pi\)
\(884\) −448.014 −0.506803
\(885\) 0 0
\(886\) −412.804 −0.465919
\(887\) 204.619i 0.230687i 0.993326 + 0.115344i \(0.0367969\pi\)
−0.993326 + 0.115344i \(0.963203\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −706.098 −0.793368
\(891\) 0 0
\(892\) 58.0778i 0.0651096i
\(893\) 1176.57 1.31755
\(894\) 0 0
\(895\) − 714.113i − 0.797892i
\(896\) 0 0
\(897\) 0 0
\(898\) −646.975 −0.720462
\(899\) − 188.565i − 0.209749i
\(900\) 0 0
\(901\) 1254.63i 1.39248i
\(902\) 210.626i 0.233510i
\(903\) 0 0
\(904\) 375.598 0.415484
\(905\) 445.747 0.492538
\(906\) 0 0
\(907\) −136.764 −0.150787 −0.0753934 0.997154i \(-0.524021\pi\)
−0.0753934 + 0.997154i \(0.524021\pi\)
\(908\) − 551.086i − 0.606923i
\(909\) 0 0
\(910\) 0 0
\(911\) −676.563 −0.742659 −0.371330 0.928501i \(-0.621098\pi\)
−0.371330 + 0.928501i \(0.621098\pi\)
\(912\) 0 0
\(913\) 227.239i 0.248892i
\(914\) −962.745 −1.05333
\(915\) 0 0
\(916\) 83.6915i 0.0913663i
\(917\) 0 0
\(918\) 0 0
\(919\) −1022.41 −1.11253 −0.556263 0.831006i \(-0.687765\pi\)
−0.556263 + 0.831006i \(0.687765\pi\)
\(920\) − 460.345i − 0.500375i
\(921\) 0 0
\(922\) 221.185i 0.239897i
\(923\) − 1715.76i − 1.85889i
\(924\) 0 0
\(925\) 233.085 0.251984
\(926\) 524.353 0.566256
\(927\) 0 0
\(928\) −70.3919 −0.0758534
\(929\) − 1797.60i − 1.93499i −0.252894 0.967494i \(-0.581382\pi\)
0.252894 0.967494i \(-0.418618\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 434.426 0.466123
\(933\) 0 0
\(934\) − 56.8097i − 0.0608240i
\(935\) −151.882 −0.162441
\(936\) 0 0
\(937\) 494.548i 0.527800i 0.964550 + 0.263900i \(0.0850088\pi\)
−0.964550 + 0.263900i \(0.914991\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −755.980 −0.804234
\(941\) 39.6996i 0.0421888i 0.999777 + 0.0210944i \(0.00671505\pi\)
−0.999777 + 0.0210944i \(0.993285\pi\)
\(942\) 0 0
\(943\) − 1921.19i − 2.03731i
\(944\) − 96.4813i − 0.102205i
\(945\) 0 0
\(946\) −286.392 −0.302740
\(947\) 1517.96 1.60291 0.801455 0.598055i \(-0.204060\pi\)
0.801455 + 0.598055i \(0.204060\pi\)
\(948\) 0 0
\(949\) 1590.46 1.67593
\(950\) − 100.158i − 0.105430i
\(951\) 0 0
\(952\) 0 0
\(953\) 1512.57 1.58717 0.793583 0.608462i \(-0.208213\pi\)
0.793583 + 0.608462i \(0.208213\pi\)
\(954\) 0 0
\(955\) 310.360i 0.324984i
\(956\) 137.442 0.143767
\(957\) 0 0
\(958\) − 301.419i − 0.314634i
\(959\) 0 0
\(960\) 0 0
\(961\) 731.372 0.761053
\(962\) 1202.65i 1.25015i
\(963\) 0 0
\(964\) − 36.0772i − 0.0374244i
\(965\) 603.094i 0.624967i
\(966\) 0 0
\(967\) 308.382 0.318906 0.159453 0.987206i \(-0.449027\pi\)
0.159453 + 0.987206i \(0.449027\pi\)
\(968\) −319.612 −0.330178
\(969\) 0 0
\(970\) −145.598 −0.150101
\(971\) − 683.164i − 0.703568i −0.936081 0.351784i \(-0.885575\pi\)
0.936081 0.351784i \(-0.114425\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −1193.03 −1.22487
\(975\) 0 0
\(976\) 54.1460i 0.0554775i
\(977\) −1339.53 −1.37106 −0.685532 0.728042i \(-0.740430\pi\)
−0.685532 + 0.728042i \(0.740430\pi\)
\(978\) 0 0
\(979\) 316.573i 0.323363i
\(980\) 0 0
\(981\) 0 0
\(982\) 869.206 0.885139
\(983\) − 952.611i − 0.969085i −0.874768 0.484543i \(-0.838986\pi\)
0.874768 0.484543i \(-0.161014\pi\)
\(984\) 0 0
\(985\) 95.8659i 0.0973258i
\(986\) − 211.838i − 0.214846i
\(987\) 0 0
\(988\) 516.784 0.523061
\(989\) 2612.27 2.64133
\(990\) 0 0
\(991\) 1307.92 1.31980 0.659899 0.751355i \(-0.270599\pi\)
0.659899 + 0.751355i \(0.270599\pi\)
\(992\) 85.7211i 0.0864124i
\(993\) 0 0
\(994\) 0 0
\(995\) 1159.06 1.16488
\(996\) 0 0
\(997\) − 248.288i − 0.249035i −0.992217 0.124518i \(-0.960262\pi\)
0.992217 0.124518i \(-0.0397383\pi\)
\(998\) 933.352 0.935223
\(999\) 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.685.4 yes 4
3.2 odd 2 882.3.c.e.685.1 yes 4
7.2 even 3 882.3.n.h.325.2 8
7.3 odd 6 882.3.n.h.19.2 8
7.4 even 3 882.3.n.h.19.1 8
7.5 odd 6 882.3.n.h.325.1 8
7.6 odd 2 inner 882.3.c.d.685.3 4
21.2 odd 6 882.3.n.g.325.3 8
21.5 even 6 882.3.n.g.325.4 8
21.11 odd 6 882.3.n.g.19.4 8
21.17 even 6 882.3.n.g.19.3 8
21.20 even 2 882.3.c.e.685.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.3.c.d.685.3 4 7.6 odd 2 inner
882.3.c.d.685.4 yes 4 1.1 even 1 trivial
882.3.c.e.685.1 yes 4 3.2 odd 2
882.3.c.e.685.2 yes 4 21.20 even 2
882.3.n.g.19.3 8 21.17 even 6
882.3.n.g.19.4 8 21.11 odd 6
882.3.n.g.325.3 8 21.2 odd 6
882.3.n.g.325.4 8 21.5 even 6
882.3.n.h.19.1 8 7.4 even 3
882.3.n.h.19.2 8 7.3 odd 6
882.3.n.h.325.1 8 7.5 odd 6
882.3.n.h.325.2 8 7.2 even 3