Properties

Label 169.4.a.d.1.1
Level $169$
Weight $4$
Character 169.1
Self dual yes
Analytic conductor $9.971$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,4,Mod(1,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 169.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} +2.00000 q^{3} +8.00000 q^{4} +17.0000 q^{5} +8.00000 q^{6} +20.0000 q^{7} -23.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} +2.00000 q^{3} +8.00000 q^{4} +17.0000 q^{5} +8.00000 q^{6} +20.0000 q^{7} -23.0000 q^{9} +68.0000 q^{10} -32.0000 q^{11} +16.0000 q^{12} +80.0000 q^{14} +34.0000 q^{15} -64.0000 q^{16} -13.0000 q^{17} -92.0000 q^{18} +30.0000 q^{19} +136.000 q^{20} +40.0000 q^{21} -128.000 q^{22} +78.0000 q^{23} +164.000 q^{25} -100.000 q^{27} +160.000 q^{28} +197.000 q^{29} +136.000 q^{30} -74.0000 q^{31} -256.000 q^{32} -64.0000 q^{33} -52.0000 q^{34} +340.000 q^{35} -184.000 q^{36} -227.000 q^{37} +120.000 q^{38} -165.000 q^{41} +160.000 q^{42} -156.000 q^{43} -256.000 q^{44} -391.000 q^{45} +312.000 q^{46} -162.000 q^{47} -128.000 q^{48} +57.0000 q^{49} +656.000 q^{50} -26.0000 q^{51} +93.0000 q^{53} -400.000 q^{54} -544.000 q^{55} +60.0000 q^{57} +788.000 q^{58} -864.000 q^{59} +272.000 q^{60} +145.000 q^{61} -296.000 q^{62} -460.000 q^{63} -512.000 q^{64} -256.000 q^{66} +862.000 q^{67} -104.000 q^{68} +156.000 q^{69} +1360.00 q^{70} +654.000 q^{71} +215.000 q^{73} -908.000 q^{74} +328.000 q^{75} +240.000 q^{76} -640.000 q^{77} -76.0000 q^{79} -1088.00 q^{80} +421.000 q^{81} -660.000 q^{82} +628.000 q^{83} +320.000 q^{84} -221.000 q^{85} -624.000 q^{86} +394.000 q^{87} -266.000 q^{89} -1564.00 q^{90} +624.000 q^{92} -148.000 q^{93} -648.000 q^{94} +510.000 q^{95} -512.000 q^{96} +238.000 q^{97} +228.000 q^{98} +736.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 2.00000 0.384900 0.192450 0.981307i \(-0.438357\pi\)
0.192450 + 0.981307i \(0.438357\pi\)
\(4\) 8.00000 1.00000
\(5\) 17.0000 1.52053 0.760263 0.649615i \(-0.225070\pi\)
0.760263 + 0.649615i \(0.225070\pi\)
\(6\) 8.00000 0.544331
\(7\) 20.0000 1.07990 0.539949 0.841698i \(-0.318443\pi\)
0.539949 + 0.841698i \(0.318443\pi\)
\(8\) 0 0
\(9\) −23.0000 −0.851852
\(10\) 68.0000 2.15035
\(11\) −32.0000 −0.877124 −0.438562 0.898701i \(-0.644512\pi\)
−0.438562 + 0.898701i \(0.644512\pi\)
\(12\) 16.0000 0.384900
\(13\) 0 0
\(14\) 80.0000 1.52721
\(15\) 34.0000 0.585251
\(16\) −64.0000 −1.00000
\(17\) −13.0000 −0.185468 −0.0927342 0.995691i \(-0.529561\pi\)
−0.0927342 + 0.995691i \(0.529561\pi\)
\(18\) −92.0000 −1.20470
\(19\) 30.0000 0.362235 0.181118 0.983461i \(-0.442029\pi\)
0.181118 + 0.983461i \(0.442029\pi\)
\(20\) 136.000 1.52053
\(21\) 40.0000 0.415653
\(22\) −128.000 −1.24044
\(23\) 78.0000 0.707136 0.353568 0.935409i \(-0.384968\pi\)
0.353568 + 0.935409i \(0.384968\pi\)
\(24\) 0 0
\(25\) 164.000 1.31200
\(26\) 0 0
\(27\) −100.000 −0.712778
\(28\) 160.000 1.07990
\(29\) 197.000 1.26145 0.630724 0.776007i \(-0.282758\pi\)
0.630724 + 0.776007i \(0.282758\pi\)
\(30\) 136.000 0.827670
\(31\) −74.0000 −0.428735 −0.214368 0.976753i \(-0.568769\pi\)
−0.214368 + 0.976753i \(0.568769\pi\)
\(32\) −256.000 −1.41421
\(33\) −64.0000 −0.337605
\(34\) −52.0000 −0.262292
\(35\) 340.000 1.64201
\(36\) −184.000 −0.851852
\(37\) −227.000 −1.00861 −0.504305 0.863526i \(-0.668251\pi\)
−0.504305 + 0.863526i \(0.668251\pi\)
\(38\) 120.000 0.512278
\(39\) 0 0
\(40\) 0 0
\(41\) −165.000 −0.628504 −0.314252 0.949340i \(-0.601754\pi\)
−0.314252 + 0.949340i \(0.601754\pi\)
\(42\) 160.000 0.587822
\(43\) −156.000 −0.553251 −0.276625 0.960978i \(-0.589216\pi\)
−0.276625 + 0.960978i \(0.589216\pi\)
\(44\) −256.000 −0.877124
\(45\) −391.000 −1.29526
\(46\) 312.000 1.00004
\(47\) −162.000 −0.502769 −0.251384 0.967887i \(-0.580886\pi\)
−0.251384 + 0.967887i \(0.580886\pi\)
\(48\) −128.000 −0.384900
\(49\) 57.0000 0.166181
\(50\) 656.000 1.85545
\(51\) −26.0000 −0.0713868
\(52\) 0 0
\(53\) 93.0000 0.241029 0.120514 0.992712i \(-0.461546\pi\)
0.120514 + 0.992712i \(0.461546\pi\)
\(54\) −400.000 −1.00802
\(55\) −544.000 −1.33369
\(56\) 0 0
\(57\) 60.0000 0.139424
\(58\) 788.000 1.78396
\(59\) −864.000 −1.90650 −0.953248 0.302190i \(-0.902282\pi\)
−0.953248 + 0.302190i \(0.902282\pi\)
\(60\) 272.000 0.585251
\(61\) 145.000 0.304350 0.152175 0.988354i \(-0.451372\pi\)
0.152175 + 0.988354i \(0.451372\pi\)
\(62\) −296.000 −0.606323
\(63\) −460.000 −0.919914
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) −256.000 −0.477446
\(67\) 862.000 1.57179 0.785896 0.618359i \(-0.212202\pi\)
0.785896 + 0.618359i \(0.212202\pi\)
\(68\) −104.000 −0.185468
\(69\) 156.000 0.272177
\(70\) 1360.00 2.32216
\(71\) 654.000 1.09318 0.546588 0.837402i \(-0.315926\pi\)
0.546588 + 0.837402i \(0.315926\pi\)
\(72\) 0 0
\(73\) 215.000 0.344710 0.172355 0.985035i \(-0.444862\pi\)
0.172355 + 0.985035i \(0.444862\pi\)
\(74\) −908.000 −1.42639
\(75\) 328.000 0.504989
\(76\) 240.000 0.362235
\(77\) −640.000 −0.947205
\(78\) 0 0
\(79\) −76.0000 −0.108236 −0.0541182 0.998535i \(-0.517235\pi\)
−0.0541182 + 0.998535i \(0.517235\pi\)
\(80\) −1088.00 −1.52053
\(81\) 421.000 0.577503
\(82\) −660.000 −0.888839
\(83\) 628.000 0.830505 0.415253 0.909706i \(-0.363693\pi\)
0.415253 + 0.909706i \(0.363693\pi\)
\(84\) 320.000 0.415653
\(85\) −221.000 −0.282010
\(86\) −624.000 −0.782415
\(87\) 394.000 0.485531
\(88\) 0 0
\(89\) −266.000 −0.316808 −0.158404 0.987374i \(-0.550635\pi\)
−0.158404 + 0.987374i \(0.550635\pi\)
\(90\) −1564.00 −1.83178
\(91\) 0 0
\(92\) 624.000 0.707136
\(93\) −148.000 −0.165020
\(94\) −648.000 −0.711022
\(95\) 510.000 0.550788
\(96\) −512.000 −0.544331
\(97\) 238.000 0.249126 0.124563 0.992212i \(-0.460247\pi\)
0.124563 + 0.992212i \(0.460247\pi\)
\(98\) 228.000 0.235015
\(99\) 736.000 0.747180
\(100\) 1312.00 1.31200
\(101\) −819.000 −0.806867 −0.403433 0.915009i \(-0.632183\pi\)
−0.403433 + 0.915009i \(0.632183\pi\)
\(102\) −104.000 −0.100956
\(103\) 1638.00 1.56696 0.783480 0.621417i \(-0.213443\pi\)
0.783480 + 0.621417i \(0.213443\pi\)
\(104\) 0 0
\(105\) 680.000 0.632011
\(106\) 372.000 0.340866
\(107\) 522.000 0.471623 0.235811 0.971799i \(-0.424225\pi\)
0.235811 + 0.971799i \(0.424225\pi\)
\(108\) −800.000 −0.712778
\(109\) −1634.00 −1.43586 −0.717930 0.696115i \(-0.754910\pi\)
−0.717930 + 0.696115i \(0.754910\pi\)
\(110\) −2176.00 −1.88612
\(111\) −454.000 −0.388214
\(112\) −1280.00 −1.07990
\(113\) 327.000 0.272226 0.136113 0.990693i \(-0.456539\pi\)
0.136113 + 0.990693i \(0.456539\pi\)
\(114\) 240.000 0.197176
\(115\) 1326.00 1.07522
\(116\) 1576.00 1.26145
\(117\) 0 0
\(118\) −3456.00 −2.69619
\(119\) −260.000 −0.200287
\(120\) 0 0
\(121\) −307.000 −0.230654
\(122\) 580.000 0.430416
\(123\) −330.000 −0.241911
\(124\) −592.000 −0.428735
\(125\) 663.000 0.474404
\(126\) −1840.00 −1.30095
\(127\) −2158.00 −1.50781 −0.753904 0.656985i \(-0.771831\pi\)
−0.753904 + 0.656985i \(0.771831\pi\)
\(128\) 0 0
\(129\) −312.000 −0.212946
\(130\) 0 0
\(131\) 730.000 0.486873 0.243437 0.969917i \(-0.421725\pi\)
0.243437 + 0.969917i \(0.421725\pi\)
\(132\) −512.000 −0.337605
\(133\) 600.000 0.391177
\(134\) 3448.00 2.22285
\(135\) −1700.00 −1.08380
\(136\) 0 0
\(137\) 1671.00 1.04207 0.521033 0.853536i \(-0.325547\pi\)
0.521033 + 0.853536i \(0.325547\pi\)
\(138\) 624.000 0.384916
\(139\) 912.000 0.556510 0.278255 0.960507i \(-0.410244\pi\)
0.278255 + 0.960507i \(0.410244\pi\)
\(140\) 2720.00 1.64201
\(141\) −324.000 −0.193516
\(142\) 2616.00 1.54598
\(143\) 0 0
\(144\) 1472.00 0.851852
\(145\) 3349.00 1.91806
\(146\) 860.000 0.487494
\(147\) 114.000 0.0639630
\(148\) −1816.00 −1.00861
\(149\) −2115.00 −1.16287 −0.581435 0.813593i \(-0.697508\pi\)
−0.581435 + 0.813593i \(0.697508\pi\)
\(150\) 1312.00 0.714162
\(151\) 514.000 0.277011 0.138506 0.990362i \(-0.455770\pi\)
0.138506 + 0.990362i \(0.455770\pi\)
\(152\) 0 0
\(153\) 299.000 0.157992
\(154\) −2560.00 −1.33955
\(155\) −1258.00 −0.651903
\(156\) 0 0
\(157\) 2901.00 1.47468 0.737341 0.675521i \(-0.236081\pi\)
0.737341 + 0.675521i \(0.236081\pi\)
\(158\) −304.000 −0.153069
\(159\) 186.000 0.0927721
\(160\) −4352.00 −2.15035
\(161\) 1560.00 0.763635
\(162\) 1684.00 0.816713
\(163\) 2360.00 1.13405 0.567023 0.823702i \(-0.308095\pi\)
0.567023 + 0.823702i \(0.308095\pi\)
\(164\) −1320.00 −0.628504
\(165\) −1088.00 −0.513337
\(166\) 2512.00 1.17451
\(167\) 280.000 0.129743 0.0648714 0.997894i \(-0.479336\pi\)
0.0648714 + 0.997894i \(0.479336\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −884.000 −0.398822
\(171\) −690.000 −0.308571
\(172\) −1248.00 −0.553251
\(173\) 1326.00 0.582739 0.291370 0.956611i \(-0.405889\pi\)
0.291370 + 0.956611i \(0.405889\pi\)
\(174\) 1576.00 0.686645
\(175\) 3280.00 1.41683
\(176\) 2048.00 0.877124
\(177\) −1728.00 −0.733810
\(178\) −1064.00 −0.448035
\(179\) 4264.00 1.78048 0.890241 0.455490i \(-0.150536\pi\)
0.890241 + 0.455490i \(0.150536\pi\)
\(180\) −3128.00 −1.29526
\(181\) −403.000 −0.165496 −0.0827479 0.996571i \(-0.526370\pi\)
−0.0827479 + 0.996571i \(0.526370\pi\)
\(182\) 0 0
\(183\) 290.000 0.117144
\(184\) 0 0
\(185\) −3859.00 −1.53362
\(186\) −592.000 −0.233374
\(187\) 416.000 0.162679
\(188\) −1296.00 −0.502769
\(189\) −2000.00 −0.769728
\(190\) 2040.00 0.778932
\(191\) −1246.00 −0.472028 −0.236014 0.971750i \(-0.575841\pi\)
−0.236014 + 0.971750i \(0.575841\pi\)
\(192\) −1024.00 −0.384900
\(193\) 267.000 0.0995807 0.0497904 0.998760i \(-0.484145\pi\)
0.0497904 + 0.998760i \(0.484145\pi\)
\(194\) 952.000 0.352318
\(195\) 0 0
\(196\) 456.000 0.166181
\(197\) 1278.00 0.462202 0.231101 0.972930i \(-0.425767\pi\)
0.231101 + 0.972930i \(0.425767\pi\)
\(198\) 2944.00 1.05667
\(199\) 4238.00 1.50967 0.754834 0.655916i \(-0.227717\pi\)
0.754834 + 0.655916i \(0.227717\pi\)
\(200\) 0 0
\(201\) 1724.00 0.604983
\(202\) −3276.00 −1.14108
\(203\) 3940.00 1.36224
\(204\) −208.000 −0.0713868
\(205\) −2805.00 −0.955657
\(206\) 6552.00 2.21602
\(207\) −1794.00 −0.602375
\(208\) 0 0
\(209\) −960.000 −0.317725
\(210\) 2720.00 0.893799
\(211\) 3070.00 1.00165 0.500823 0.865549i \(-0.333031\pi\)
0.500823 + 0.865549i \(0.333031\pi\)
\(212\) 744.000 0.241029
\(213\) 1308.00 0.420764
\(214\) 2088.00 0.666975
\(215\) −2652.00 −0.841232
\(216\) 0 0
\(217\) −1480.00 −0.462991
\(218\) −6536.00 −2.03061
\(219\) 430.000 0.132679
\(220\) −4352.00 −1.33369
\(221\) 0 0
\(222\) −1816.00 −0.549018
\(223\) −5378.00 −1.61497 −0.807483 0.589891i \(-0.799171\pi\)
−0.807483 + 0.589891i \(0.799171\pi\)
\(224\) −5120.00 −1.52721
\(225\) −3772.00 −1.11763
\(226\) 1308.00 0.384986
\(227\) −3974.00 −1.16195 −0.580977 0.813920i \(-0.697329\pi\)
−0.580977 + 0.813920i \(0.697329\pi\)
\(228\) 480.000 0.139424
\(229\) −6298.00 −1.81740 −0.908698 0.417455i \(-0.862922\pi\)
−0.908698 + 0.417455i \(0.862922\pi\)
\(230\) 5304.00 1.52059
\(231\) −1280.00 −0.364579
\(232\) 0 0
\(233\) 4030.00 1.13311 0.566554 0.824025i \(-0.308276\pi\)
0.566554 + 0.824025i \(0.308276\pi\)
\(234\) 0 0
\(235\) −2754.00 −0.764473
\(236\) −6912.00 −1.90650
\(237\) −152.000 −0.0416602
\(238\) −1040.00 −0.283249
\(239\) −984.000 −0.266317 −0.133158 0.991095i \(-0.542512\pi\)
−0.133158 + 0.991095i \(0.542512\pi\)
\(240\) −2176.00 −0.585251
\(241\) 943.000 0.252050 0.126025 0.992027i \(-0.459778\pi\)
0.126025 + 0.992027i \(0.459778\pi\)
\(242\) −1228.00 −0.326194
\(243\) 3542.00 0.935059
\(244\) 1160.00 0.304350
\(245\) 969.000 0.252682
\(246\) −1320.00 −0.342114
\(247\) 0 0
\(248\) 0 0
\(249\) 1256.00 0.319662
\(250\) 2652.00 0.670909
\(251\) −2730.00 −0.686518 −0.343259 0.939241i \(-0.611531\pi\)
−0.343259 + 0.939241i \(0.611531\pi\)
\(252\) −3680.00 −0.919914
\(253\) −2496.00 −0.620246
\(254\) −8632.00 −2.13236
\(255\) −442.000 −0.108546
\(256\) 4096.00 1.00000
\(257\) −1885.00 −0.457522 −0.228761 0.973483i \(-0.573467\pi\)
−0.228761 + 0.973483i \(0.573467\pi\)
\(258\) −1248.00 −0.301151
\(259\) −4540.00 −1.08920
\(260\) 0 0
\(261\) −4531.00 −1.07457
\(262\) 2920.00 0.688543
\(263\) 4032.00 0.945338 0.472669 0.881240i \(-0.343291\pi\)
0.472669 + 0.881240i \(0.343291\pi\)
\(264\) 0 0
\(265\) 1581.00 0.366491
\(266\) 2400.00 0.553208
\(267\) −532.000 −0.121940
\(268\) 6896.00 1.57179
\(269\) 4006.00 0.907993 0.453997 0.891003i \(-0.349998\pi\)
0.453997 + 0.891003i \(0.349998\pi\)
\(270\) −6800.00 −1.53272
\(271\) −4296.00 −0.962965 −0.481482 0.876456i \(-0.659901\pi\)
−0.481482 + 0.876456i \(0.659901\pi\)
\(272\) 832.000 0.185468
\(273\) 0 0
\(274\) 6684.00 1.47371
\(275\) −5248.00 −1.15079
\(276\) 1248.00 0.272177
\(277\) −5551.00 −1.20407 −0.602035 0.798470i \(-0.705643\pi\)
−0.602035 + 0.798470i \(0.705643\pi\)
\(278\) 3648.00 0.787023
\(279\) 1702.00 0.365219
\(280\) 0 0
\(281\) −5557.00 −1.17973 −0.589863 0.807504i \(-0.700818\pi\)
−0.589863 + 0.807504i \(0.700818\pi\)
\(282\) −1296.00 −0.273673
\(283\) 3120.00 0.655352 0.327676 0.944790i \(-0.393734\pi\)
0.327676 + 0.944790i \(0.393734\pi\)
\(284\) 5232.00 1.09318
\(285\) 1020.00 0.211999
\(286\) 0 0
\(287\) −3300.00 −0.678721
\(288\) 5888.00 1.20470
\(289\) −4744.00 −0.965601
\(290\) 13396.0 2.71255
\(291\) 476.000 0.0958887
\(292\) 1720.00 0.344710
\(293\) 8301.00 1.65512 0.827559 0.561379i \(-0.189729\pi\)
0.827559 + 0.561379i \(0.189729\pi\)
\(294\) 456.000 0.0904573
\(295\) −14688.0 −2.89888
\(296\) 0 0
\(297\) 3200.00 0.625195
\(298\) −8460.00 −1.64455
\(299\) 0 0
\(300\) 2624.00 0.504989
\(301\) −3120.00 −0.597455
\(302\) 2056.00 0.391753
\(303\) −1638.00 −0.310563
\(304\) −1920.00 −0.362235
\(305\) 2465.00 0.462772
\(306\) 1196.00 0.223434
\(307\) 8678.00 1.61329 0.806644 0.591037i \(-0.201281\pi\)
0.806644 + 0.591037i \(0.201281\pi\)
\(308\) −5120.00 −0.947205
\(309\) 3276.00 0.603123
\(310\) −5032.00 −0.921930
\(311\) 8658.00 1.57862 0.789309 0.613996i \(-0.210439\pi\)
0.789309 + 0.613996i \(0.210439\pi\)
\(312\) 0 0
\(313\) −5250.00 −0.948075 −0.474038 0.880505i \(-0.657204\pi\)
−0.474038 + 0.880505i \(0.657204\pi\)
\(314\) 11604.0 2.08551
\(315\) −7820.00 −1.39875
\(316\) −608.000 −0.108236
\(317\) 6413.00 1.13625 0.568123 0.822944i \(-0.307670\pi\)
0.568123 + 0.822944i \(0.307670\pi\)
\(318\) 744.000 0.131200
\(319\) −6304.00 −1.10645
\(320\) −8704.00 −1.52053
\(321\) 1044.00 0.181528
\(322\) 6240.00 1.07994
\(323\) −390.000 −0.0671832
\(324\) 3368.00 0.577503
\(325\) 0 0
\(326\) 9440.00 1.60378
\(327\) −3268.00 −0.552663
\(328\) 0 0
\(329\) −3240.00 −0.542939
\(330\) −4352.00 −0.725969
\(331\) 3488.00 0.579208 0.289604 0.957147i \(-0.406476\pi\)
0.289604 + 0.957147i \(0.406476\pi\)
\(332\) 5024.00 0.830505
\(333\) 5221.00 0.859186
\(334\) 1120.00 0.183484
\(335\) 14654.0 2.38995
\(336\) −2560.00 −0.415653
\(337\) −1833.00 −0.296290 −0.148145 0.988966i \(-0.547330\pi\)
−0.148145 + 0.988966i \(0.547330\pi\)
\(338\) 0 0
\(339\) 654.000 0.104780
\(340\) −1768.00 −0.282010
\(341\) 2368.00 0.376054
\(342\) −2760.00 −0.436385
\(343\) −5720.00 −0.900440
\(344\) 0 0
\(345\) 2652.00 0.413852
\(346\) 5304.00 0.824118
\(347\) 7230.00 1.11852 0.559260 0.828992i \(-0.311085\pi\)
0.559260 + 0.828992i \(0.311085\pi\)
\(348\) 3152.00 0.485531
\(349\) −5258.00 −0.806459 −0.403230 0.915099i \(-0.632112\pi\)
−0.403230 + 0.915099i \(0.632112\pi\)
\(350\) 13120.0 2.00370
\(351\) 0 0
\(352\) 8192.00 1.24044
\(353\) 3163.00 0.476911 0.238455 0.971153i \(-0.423359\pi\)
0.238455 + 0.971153i \(0.423359\pi\)
\(354\) −6912.00 −1.03776
\(355\) 11118.0 1.66220
\(356\) −2128.00 −0.316808
\(357\) −520.000 −0.0770905
\(358\) 17056.0 2.51798
\(359\) −10068.0 −1.48014 −0.740068 0.672532i \(-0.765207\pi\)
−0.740068 + 0.672532i \(0.765207\pi\)
\(360\) 0 0
\(361\) −5959.00 −0.868786
\(362\) −1612.00 −0.234047
\(363\) −614.000 −0.0887786
\(364\) 0 0
\(365\) 3655.00 0.524141
\(366\) 1160.00 0.165667
\(367\) 7438.00 1.05793 0.528965 0.848644i \(-0.322580\pi\)
0.528965 + 0.848644i \(0.322580\pi\)
\(368\) −4992.00 −0.707136
\(369\) 3795.00 0.535392
\(370\) −15436.0 −2.16886
\(371\) 1860.00 0.260287
\(372\) −1184.00 −0.165020
\(373\) −9683.00 −1.34415 −0.672073 0.740485i \(-0.734596\pi\)
−0.672073 + 0.740485i \(0.734596\pi\)
\(374\) 1664.00 0.230063
\(375\) 1326.00 0.182598
\(376\) 0 0
\(377\) 0 0
\(378\) −8000.00 −1.08856
\(379\) −1062.00 −0.143935 −0.0719674 0.997407i \(-0.522928\pi\)
−0.0719674 + 0.997407i \(0.522928\pi\)
\(380\) 4080.00 0.550788
\(381\) −4316.00 −0.580355
\(382\) −4984.00 −0.667549
\(383\) −3532.00 −0.471219 −0.235609 0.971848i \(-0.575709\pi\)
−0.235609 + 0.971848i \(0.575709\pi\)
\(384\) 0 0
\(385\) −10880.0 −1.44025
\(386\) 1068.00 0.140828
\(387\) 3588.00 0.471288
\(388\) 1904.00 0.249126
\(389\) −11063.0 −1.44194 −0.720972 0.692964i \(-0.756304\pi\)
−0.720972 + 0.692964i \(0.756304\pi\)
\(390\) 0 0
\(391\) −1014.00 −0.131151
\(392\) 0 0
\(393\) 1460.00 0.187398
\(394\) 5112.00 0.653652
\(395\) −1292.00 −0.164576
\(396\) 5888.00 0.747180
\(397\) −5986.00 −0.756747 −0.378374 0.925653i \(-0.623517\pi\)
−0.378374 + 0.925653i \(0.623517\pi\)
\(398\) 16952.0 2.13499
\(399\) 1200.00 0.150564
\(400\) −10496.0 −1.31200
\(401\) 5935.00 0.739102 0.369551 0.929211i \(-0.379512\pi\)
0.369551 + 0.929211i \(0.379512\pi\)
\(402\) 6896.00 0.855575
\(403\) 0 0
\(404\) −6552.00 −0.806867
\(405\) 7157.00 0.878109
\(406\) 15760.0 1.92649
\(407\) 7264.00 0.884676
\(408\) 0 0
\(409\) −15089.0 −1.82421 −0.912106 0.409954i \(-0.865545\pi\)
−0.912106 + 0.409954i \(0.865545\pi\)
\(410\) −11220.0 −1.35150
\(411\) 3342.00 0.401092
\(412\) 13104.0 1.56696
\(413\) −17280.0 −2.05882
\(414\) −7176.00 −0.851887
\(415\) 10676.0 1.26281
\(416\) 0 0
\(417\) 1824.00 0.214201
\(418\) −3840.00 −0.449331
\(419\) −10814.0 −1.26086 −0.630428 0.776248i \(-0.717120\pi\)
−0.630428 + 0.776248i \(0.717120\pi\)
\(420\) 5440.00 0.632011
\(421\) −6535.00 −0.756524 −0.378262 0.925699i \(-0.623478\pi\)
−0.378262 + 0.925699i \(0.623478\pi\)
\(422\) 12280.0 1.41654
\(423\) 3726.00 0.428284
\(424\) 0 0
\(425\) −2132.00 −0.243335
\(426\) 5232.00 0.595050
\(427\) 2900.00 0.328667
\(428\) 4176.00 0.471623
\(429\) 0 0
\(430\) −10608.0 −1.18968
\(431\) 1980.00 0.221284 0.110642 0.993860i \(-0.464709\pi\)
0.110642 + 0.993860i \(0.464709\pi\)
\(432\) 6400.00 0.712778
\(433\) −6929.00 −0.769022 −0.384511 0.923120i \(-0.625630\pi\)
−0.384511 + 0.923120i \(0.625630\pi\)
\(434\) −5920.00 −0.654767
\(435\) 6698.00 0.738263
\(436\) −13072.0 −1.43586
\(437\) 2340.00 0.256150
\(438\) 1720.00 0.187636
\(439\) −4576.00 −0.497496 −0.248748 0.968568i \(-0.580019\pi\)
−0.248748 + 0.968568i \(0.580019\pi\)
\(440\) 0 0
\(441\) −1311.00 −0.141561
\(442\) 0 0
\(443\) −8812.00 −0.945081 −0.472540 0.881309i \(-0.656663\pi\)
−0.472540 + 0.881309i \(0.656663\pi\)
\(444\) −3632.00 −0.388214
\(445\) −4522.00 −0.481715
\(446\) −21512.0 −2.28391
\(447\) −4230.00 −0.447589
\(448\) −10240.0 −1.07990
\(449\) 1918.00 0.201595 0.100797 0.994907i \(-0.467861\pi\)
0.100797 + 0.994907i \(0.467861\pi\)
\(450\) −15088.0 −1.58057
\(451\) 5280.00 0.551276
\(452\) 2616.00 0.272226
\(453\) 1028.00 0.106622
\(454\) −15896.0 −1.64325
\(455\) 0 0
\(456\) 0 0
\(457\) −11761.0 −1.20384 −0.601922 0.798555i \(-0.705598\pi\)
−0.601922 + 0.798555i \(0.705598\pi\)
\(458\) −25192.0 −2.57019
\(459\) 1300.00 0.132198
\(460\) 10608.0 1.07522
\(461\) 901.000 0.0910277 0.0455138 0.998964i \(-0.485507\pi\)
0.0455138 + 0.998964i \(0.485507\pi\)
\(462\) −5120.00 −0.515593
\(463\) 1372.00 0.137715 0.0688577 0.997626i \(-0.478065\pi\)
0.0688577 + 0.997626i \(0.478065\pi\)
\(464\) −12608.0 −1.26145
\(465\) −2516.00 −0.250918
\(466\) 16120.0 1.60246
\(467\) −6396.00 −0.633772 −0.316886 0.948464i \(-0.602637\pi\)
−0.316886 + 0.948464i \(0.602637\pi\)
\(468\) 0 0
\(469\) 17240.0 1.69738
\(470\) −11016.0 −1.08113
\(471\) 5802.00 0.567605
\(472\) 0 0
\(473\) 4992.00 0.485269
\(474\) −608.000 −0.0589164
\(475\) 4920.00 0.475253
\(476\) −2080.00 −0.200287
\(477\) −2139.00 −0.205321
\(478\) −3936.00 −0.376629
\(479\) 3270.00 0.311921 0.155960 0.987763i \(-0.450153\pi\)
0.155960 + 0.987763i \(0.450153\pi\)
\(480\) −8704.00 −0.827670
\(481\) 0 0
\(482\) 3772.00 0.356452
\(483\) 3120.00 0.293923
\(484\) −2456.00 −0.230654
\(485\) 4046.00 0.378803
\(486\) 14168.0 1.32237
\(487\) 19920.0 1.85351 0.926757 0.375661i \(-0.122584\pi\)
0.926757 + 0.375661i \(0.122584\pi\)
\(488\) 0 0
\(489\) 4720.00 0.436494
\(490\) 3876.00 0.357347
\(491\) 6552.00 0.602215 0.301108 0.953590i \(-0.402644\pi\)
0.301108 + 0.953590i \(0.402644\pi\)
\(492\) −2640.00 −0.241911
\(493\) −2561.00 −0.233959
\(494\) 0 0
\(495\) 12512.0 1.13611
\(496\) 4736.00 0.428735
\(497\) 13080.0 1.18052
\(498\) 5024.00 0.452070
\(499\) 1746.00 0.156637 0.0783183 0.996928i \(-0.475045\pi\)
0.0783183 + 0.996928i \(0.475045\pi\)
\(500\) 5304.00 0.474404
\(501\) 560.000 0.0499380
\(502\) −10920.0 −0.970883
\(503\) 14692.0 1.30235 0.651177 0.758926i \(-0.274276\pi\)
0.651177 + 0.758926i \(0.274276\pi\)
\(504\) 0 0
\(505\) −13923.0 −1.22686
\(506\) −9984.00 −0.877160
\(507\) 0 0
\(508\) −17264.0 −1.50781
\(509\) 8077.00 0.703353 0.351677 0.936122i \(-0.385612\pi\)
0.351677 + 0.936122i \(0.385612\pi\)
\(510\) −1768.00 −0.153507
\(511\) 4300.00 0.372252
\(512\) 16384.0 1.41421
\(513\) −3000.00 −0.258193
\(514\) −7540.00 −0.647033
\(515\) 27846.0 2.38260
\(516\) −2496.00 −0.212946
\(517\) 5184.00 0.440990
\(518\) −18160.0 −1.54036
\(519\) 2652.00 0.224296
\(520\) 0 0
\(521\) 11247.0 0.945758 0.472879 0.881127i \(-0.343215\pi\)
0.472879 + 0.881127i \(0.343215\pi\)
\(522\) −18124.0 −1.51967
\(523\) 2732.00 0.228417 0.114208 0.993457i \(-0.463567\pi\)
0.114208 + 0.993457i \(0.463567\pi\)
\(524\) 5840.00 0.486873
\(525\) 6560.00 0.545337
\(526\) 16128.0 1.33691
\(527\) 962.000 0.0795168
\(528\) 4096.00 0.337605
\(529\) −6083.00 −0.499959
\(530\) 6324.00 0.518296
\(531\) 19872.0 1.62405
\(532\) 4800.00 0.391177
\(533\) 0 0
\(534\) −2128.00 −0.172449
\(535\) 8874.00 0.717115
\(536\) 0 0
\(537\) 8528.00 0.685308
\(538\) 16024.0 1.28410
\(539\) −1824.00 −0.145761
\(540\) −13600.0 −1.08380
\(541\) −18375.0 −1.46026 −0.730132 0.683306i \(-0.760542\pi\)
−0.730132 + 0.683306i \(0.760542\pi\)
\(542\) −17184.0 −1.36184
\(543\) −806.000 −0.0636994
\(544\) 3328.00 0.262292
\(545\) −27778.0 −2.18326
\(546\) 0 0
\(547\) −10346.0 −0.808708 −0.404354 0.914603i \(-0.632504\pi\)
−0.404354 + 0.914603i \(0.632504\pi\)
\(548\) 13368.0 1.04207
\(549\) −3335.00 −0.259261
\(550\) −20992.0 −1.62746
\(551\) 5910.00 0.456941
\(552\) 0 0
\(553\) −1520.00 −0.116884
\(554\) −22204.0 −1.70281
\(555\) −7718.00 −0.590290
\(556\) 7296.00 0.556510
\(557\) 345.000 0.0262444 0.0131222 0.999914i \(-0.495823\pi\)
0.0131222 + 0.999914i \(0.495823\pi\)
\(558\) 6808.00 0.516498
\(559\) 0 0
\(560\) −21760.0 −1.64201
\(561\) 832.000 0.0626151
\(562\) −22228.0 −1.66838
\(563\) −8580.00 −0.642280 −0.321140 0.947032i \(-0.604066\pi\)
−0.321140 + 0.947032i \(0.604066\pi\)
\(564\) −2592.00 −0.193516
\(565\) 5559.00 0.413927
\(566\) 12480.0 0.926808
\(567\) 8420.00 0.623645
\(568\) 0 0
\(569\) −19682.0 −1.45011 −0.725055 0.688691i \(-0.758186\pi\)
−0.725055 + 0.688691i \(0.758186\pi\)
\(570\) 4080.00 0.299811
\(571\) 26624.0 1.95128 0.975639 0.219382i \(-0.0704042\pi\)
0.975639 + 0.219382i \(0.0704042\pi\)
\(572\) 0 0
\(573\) −2492.00 −0.181684
\(574\) −13200.0 −0.959856
\(575\) 12792.0 0.927762
\(576\) 11776.0 0.851852
\(577\) −14101.0 −1.01739 −0.508694 0.860948i \(-0.669871\pi\)
−0.508694 + 0.860948i \(0.669871\pi\)
\(578\) −18976.0 −1.36557
\(579\) 534.000 0.0383286
\(580\) 26792.0 1.91806
\(581\) 12560.0 0.896862
\(582\) 1904.00 0.135607
\(583\) −2976.00 −0.211412
\(584\) 0 0
\(585\) 0 0
\(586\) 33204.0 2.34069
\(587\) 1408.00 0.0990023 0.0495012 0.998774i \(-0.484237\pi\)
0.0495012 + 0.998774i \(0.484237\pi\)
\(588\) 912.000 0.0639630
\(589\) −2220.00 −0.155303
\(590\) −58752.0 −4.09963
\(591\) 2556.00 0.177902
\(592\) 14528.0 1.00861
\(593\) −1241.00 −0.0859389 −0.0429694 0.999076i \(-0.513682\pi\)
−0.0429694 + 0.999076i \(0.513682\pi\)
\(594\) 12800.0 0.884159
\(595\) −4420.00 −0.304542
\(596\) −16920.0 −1.16287
\(597\) 8476.00 0.581071
\(598\) 0 0
\(599\) 11078.0 0.755651 0.377825 0.925877i \(-0.376672\pi\)
0.377825 + 0.925877i \(0.376672\pi\)
\(600\) 0 0
\(601\) −13817.0 −0.937782 −0.468891 0.883256i \(-0.655346\pi\)
−0.468891 + 0.883256i \(0.655346\pi\)
\(602\) −12480.0 −0.844928
\(603\) −19826.0 −1.33893
\(604\) 4112.00 0.277011
\(605\) −5219.00 −0.350715
\(606\) −6552.00 −0.439203
\(607\) 8270.00 0.552997 0.276498 0.961014i \(-0.410826\pi\)
0.276498 + 0.961014i \(0.410826\pi\)
\(608\) −7680.00 −0.512278
\(609\) 7880.00 0.524325
\(610\) 9860.00 0.654459
\(611\) 0 0
\(612\) 2392.00 0.157992
\(613\) 22273.0 1.46753 0.733767 0.679402i \(-0.237761\pi\)
0.733767 + 0.679402i \(0.237761\pi\)
\(614\) 34712.0 2.28153
\(615\) −5610.00 −0.367833
\(616\) 0 0
\(617\) −18989.0 −1.23901 −0.619504 0.784993i \(-0.712666\pi\)
−0.619504 + 0.784993i \(0.712666\pi\)
\(618\) 13104.0 0.852945
\(619\) 72.0000 0.00467516 0.00233758 0.999997i \(-0.499256\pi\)
0.00233758 + 0.999997i \(0.499256\pi\)
\(620\) −10064.0 −0.651903
\(621\) −7800.00 −0.504031
\(622\) 34632.0 2.23250
\(623\) −5320.00 −0.342121
\(624\) 0 0
\(625\) −9229.00 −0.590656
\(626\) −21000.0 −1.34078
\(627\) −1920.00 −0.122293
\(628\) 23208.0 1.47468
\(629\) 2951.00 0.187065
\(630\) −31280.0 −1.97813
\(631\) −23380.0 −1.47503 −0.737514 0.675331i \(-0.764001\pi\)
−0.737514 + 0.675331i \(0.764001\pi\)
\(632\) 0 0
\(633\) 6140.00 0.385534
\(634\) 25652.0 1.60689
\(635\) −36686.0 −2.29266
\(636\) 1488.00 0.0927721
\(637\) 0 0
\(638\) −25216.0 −1.56475
\(639\) −15042.0 −0.931224
\(640\) 0 0
\(641\) 6383.00 0.393313 0.196656 0.980472i \(-0.436992\pi\)
0.196656 + 0.980472i \(0.436992\pi\)
\(642\) 4176.00 0.256719
\(643\) −17104.0 −1.04901 −0.524507 0.851406i \(-0.675750\pi\)
−0.524507 + 0.851406i \(0.675750\pi\)
\(644\) 12480.0 0.763635
\(645\) −5304.00 −0.323790
\(646\) −1560.00 −0.0950114
\(647\) 6994.00 0.424981 0.212490 0.977163i \(-0.431843\pi\)
0.212490 + 0.977163i \(0.431843\pi\)
\(648\) 0 0
\(649\) 27648.0 1.67223
\(650\) 0 0
\(651\) −2960.00 −0.178205
\(652\) 18880.0 1.13405
\(653\) −5250.00 −0.314622 −0.157311 0.987549i \(-0.550283\pi\)
−0.157311 + 0.987549i \(0.550283\pi\)
\(654\) −13072.0 −0.781584
\(655\) 12410.0 0.740304
\(656\) 10560.0 0.628504
\(657\) −4945.00 −0.293642
\(658\) −12960.0 −0.767832
\(659\) −4340.00 −0.256544 −0.128272 0.991739i \(-0.540943\pi\)
−0.128272 + 0.991739i \(0.540943\pi\)
\(660\) −8704.00 −0.513337
\(661\) −4179.00 −0.245907 −0.122953 0.992412i \(-0.539237\pi\)
−0.122953 + 0.992412i \(0.539237\pi\)
\(662\) 13952.0 0.819124
\(663\) 0 0
\(664\) 0 0
\(665\) 10200.0 0.594796
\(666\) 20884.0 1.21507
\(667\) 15366.0 0.892015
\(668\) 2240.00 0.129743
\(669\) −10756.0 −0.621601
\(670\) 58616.0 3.37990
\(671\) −4640.00 −0.266953
\(672\) −10240.0 −0.587822
\(673\) 22867.0 1.30974 0.654872 0.755740i \(-0.272722\pi\)
0.654872 + 0.755740i \(0.272722\pi\)
\(674\) −7332.00 −0.419018
\(675\) −16400.0 −0.935165
\(676\) 0 0
\(677\) 5410.00 0.307124 0.153562 0.988139i \(-0.450925\pi\)
0.153562 + 0.988139i \(0.450925\pi\)
\(678\) 2616.00 0.148181
\(679\) 4760.00 0.269031
\(680\) 0 0
\(681\) −7948.00 −0.447236
\(682\) 9472.00 0.531821
\(683\) −13578.0 −0.760685 −0.380342 0.924846i \(-0.624194\pi\)
−0.380342 + 0.924846i \(0.624194\pi\)
\(684\) −5520.00 −0.308571
\(685\) 28407.0 1.58449
\(686\) −22880.0 −1.27341
\(687\) −12596.0 −0.699516
\(688\) 9984.00 0.553251
\(689\) 0 0
\(690\) 10608.0 0.585275
\(691\) 12744.0 0.701599 0.350799 0.936451i \(-0.385910\pi\)
0.350799 + 0.936451i \(0.385910\pi\)
\(692\) 10608.0 0.582739
\(693\) 14720.0 0.806878
\(694\) 28920.0 1.58183
\(695\) 15504.0 0.846187
\(696\) 0 0
\(697\) 2145.00 0.116568
\(698\) −21032.0 −1.14051
\(699\) 8060.00 0.436133
\(700\) 26240.0 1.41683
\(701\) 16406.0 0.883946 0.441973 0.897028i \(-0.354279\pi\)
0.441973 + 0.897028i \(0.354279\pi\)
\(702\) 0 0
\(703\) −6810.00 −0.365354
\(704\) 16384.0 0.877124
\(705\) −5508.00 −0.294246
\(706\) 12652.0 0.674454
\(707\) −16380.0 −0.871334
\(708\) −13824.0 −0.733810
\(709\) 709.000 0.0375558 0.0187779 0.999824i \(-0.494022\pi\)
0.0187779 + 0.999824i \(0.494022\pi\)
\(710\) 44472.0 2.35071
\(711\) 1748.00 0.0922013
\(712\) 0 0
\(713\) −5772.00 −0.303174
\(714\) −2080.00 −0.109022
\(715\) 0 0
\(716\) 34112.0 1.78048
\(717\) −1968.00 −0.102505
\(718\) −40272.0 −2.09323
\(719\) −7644.00 −0.396486 −0.198243 0.980153i \(-0.563523\pi\)
−0.198243 + 0.980153i \(0.563523\pi\)
\(720\) 25024.0 1.29526
\(721\) 32760.0 1.69216
\(722\) −23836.0 −1.22865
\(723\) 1886.00 0.0970140
\(724\) −3224.00 −0.165496
\(725\) 32308.0 1.65502
\(726\) −2456.00 −0.125552
\(727\) −15808.0 −0.806446 −0.403223 0.915102i \(-0.632110\pi\)
−0.403223 + 0.915102i \(0.632110\pi\)
\(728\) 0 0
\(729\) −4283.00 −0.217599
\(730\) 14620.0 0.741247
\(731\) 2028.00 0.102611
\(732\) 2320.00 0.117144
\(733\) −2583.00 −0.130157 −0.0650786 0.997880i \(-0.520730\pi\)
−0.0650786 + 0.997880i \(0.520730\pi\)
\(734\) 29752.0 1.49614
\(735\) 1938.00 0.0972574
\(736\) −19968.0 −1.00004
\(737\) −27584.0 −1.37866
\(738\) 15180.0 0.757159
\(739\) 4076.00 0.202893 0.101447 0.994841i \(-0.467653\pi\)
0.101447 + 0.994841i \(0.467653\pi\)
\(740\) −30872.0 −1.53362
\(741\) 0 0
\(742\) 7440.00 0.368101
\(743\) −34056.0 −1.68155 −0.840776 0.541383i \(-0.817901\pi\)
−0.840776 + 0.541383i \(0.817901\pi\)
\(744\) 0 0
\(745\) −35955.0 −1.76817
\(746\) −38732.0 −1.90091
\(747\) −14444.0 −0.707468
\(748\) 3328.00 0.162679
\(749\) 10440.0 0.509305
\(750\) 5304.00 0.258233
\(751\) −364.000 −0.0176865 −0.00884324 0.999961i \(-0.502815\pi\)
−0.00884324 + 0.999961i \(0.502815\pi\)
\(752\) 10368.0 0.502769
\(753\) −5460.00 −0.264241
\(754\) 0 0
\(755\) 8738.00 0.421203
\(756\) −16000.0 −0.769728
\(757\) −6914.00 −0.331960 −0.165980 0.986129i \(-0.553079\pi\)
−0.165980 + 0.986129i \(0.553079\pi\)
\(758\) −4248.00 −0.203554
\(759\) −4992.00 −0.238733
\(760\) 0 0
\(761\) 13982.0 0.666028 0.333014 0.942922i \(-0.391934\pi\)
0.333014 + 0.942922i \(0.391934\pi\)
\(762\) −17264.0 −0.820746
\(763\) −32680.0 −1.55058
\(764\) −9968.00 −0.472028
\(765\) 5083.00 0.240230
\(766\) −14128.0 −0.666404
\(767\) 0 0
\(768\) 8192.00 0.384900
\(769\) −18066.0 −0.847174 −0.423587 0.905855i \(-0.639229\pi\)
−0.423587 + 0.905855i \(0.639229\pi\)
\(770\) −43520.0 −2.03682
\(771\) −3770.00 −0.176100
\(772\) 2136.00 0.0995807
\(773\) 14434.0 0.671610 0.335805 0.941931i \(-0.390992\pi\)
0.335805 + 0.941931i \(0.390992\pi\)
\(774\) 14352.0 0.666501
\(775\) −12136.0 −0.562501
\(776\) 0 0
\(777\) −9080.00 −0.419232
\(778\) −44252.0 −2.03922
\(779\) −4950.00 −0.227666
\(780\) 0 0
\(781\) −20928.0 −0.958851
\(782\) −4056.00 −0.185476
\(783\) −19700.0 −0.899132
\(784\) −3648.00 −0.166181
\(785\) 49317.0 2.24229
\(786\) 5840.00 0.265020
\(787\) −15398.0 −0.697433 −0.348716 0.937228i \(-0.613382\pi\)
−0.348716 + 0.937228i \(0.613382\pi\)
\(788\) 10224.0 0.462202
\(789\) 8064.00 0.363861
\(790\) −5168.00 −0.232746
\(791\) 6540.00 0.293977
\(792\) 0 0
\(793\) 0 0
\(794\) −23944.0 −1.07020
\(795\) 3162.00 0.141062
\(796\) 33904.0 1.50967
\(797\) −36842.0 −1.63740 −0.818702 0.574219i \(-0.805306\pi\)
−0.818702 + 0.574219i \(0.805306\pi\)
\(798\) 4800.00 0.212930
\(799\) 2106.00 0.0932477
\(800\) −41984.0 −1.85545
\(801\) 6118.00 0.269874
\(802\) 23740.0 1.04525
\(803\) −6880.00 −0.302354
\(804\) 13792.0 0.604983
\(805\) 26520.0 1.16113
\(806\) 0 0
\(807\) 8012.00 0.349487
\(808\) 0 0
\(809\) 41511.0 1.80402 0.902008 0.431719i \(-0.142093\pi\)
0.902008 + 0.431719i \(0.142093\pi\)
\(810\) 28628.0 1.24183
\(811\) 23066.0 0.998714 0.499357 0.866396i \(-0.333570\pi\)
0.499357 + 0.866396i \(0.333570\pi\)
\(812\) 31520.0 1.36224
\(813\) −8592.00 −0.370645
\(814\) 29056.0 1.25112
\(815\) 40120.0 1.72435
\(816\) 1664.00 0.0713868
\(817\) −4680.00 −0.200407
\(818\) −60356.0 −2.57983
\(819\) 0 0
\(820\) −22440.0 −0.955657
\(821\) 28838.0 1.22589 0.612943 0.790127i \(-0.289985\pi\)
0.612943 + 0.790127i \(0.289985\pi\)
\(822\) 13368.0 0.567229
\(823\) 27456.0 1.16289 0.581443 0.813587i \(-0.302488\pi\)
0.581443 + 0.813587i \(0.302488\pi\)
\(824\) 0 0
\(825\) −10496.0 −0.442938
\(826\) −69120.0 −2.91161
\(827\) −33572.0 −1.41162 −0.705812 0.708399i \(-0.749418\pi\)
−0.705812 + 0.708399i \(0.749418\pi\)
\(828\) −14352.0 −0.602375
\(829\) −45799.0 −1.91878 −0.959388 0.282090i \(-0.908972\pi\)
−0.959388 + 0.282090i \(0.908972\pi\)
\(830\) 42704.0 1.78588
\(831\) −11102.0 −0.463447
\(832\) 0 0
\(833\) −741.000 −0.0308213
\(834\) 7296.00 0.302925
\(835\) 4760.00 0.197277
\(836\) −7680.00 −0.317725
\(837\) 7400.00 0.305593
\(838\) −43256.0 −1.78312
\(839\) 32286.0 1.32853 0.664265 0.747497i \(-0.268745\pi\)
0.664265 + 0.747497i \(0.268745\pi\)
\(840\) 0 0
\(841\) 14420.0 0.591250
\(842\) −26140.0 −1.06989
\(843\) −11114.0 −0.454077
\(844\) 24560.0 1.00165
\(845\) 0 0
\(846\) 14904.0 0.605686
\(847\) −6140.00 −0.249083
\(848\) −5952.00 −0.241029
\(849\) 6240.00 0.252245
\(850\) −8528.00 −0.344127
\(851\) −17706.0 −0.713224
\(852\) 10464.0 0.420764
\(853\) 20937.0 0.840409 0.420205 0.907429i \(-0.361958\pi\)
0.420205 + 0.907429i \(0.361958\pi\)
\(854\) 11600.0 0.464805
\(855\) −11730.0 −0.469190
\(856\) 0 0
\(857\) −7189.00 −0.286548 −0.143274 0.989683i \(-0.545763\pi\)
−0.143274 + 0.989683i \(0.545763\pi\)
\(858\) 0 0
\(859\) −32498.0 −1.29082 −0.645412 0.763835i \(-0.723314\pi\)
−0.645412 + 0.763835i \(0.723314\pi\)
\(860\) −21216.0 −0.841232
\(861\) −6600.00 −0.261240
\(862\) 7920.00 0.312942
\(863\) 8428.00 0.332436 0.166218 0.986089i \(-0.446844\pi\)
0.166218 + 0.986089i \(0.446844\pi\)
\(864\) 25600.0 1.00802
\(865\) 22542.0 0.886071
\(866\) −27716.0 −1.08756
\(867\) −9488.00 −0.371660
\(868\) −11840.0 −0.462991
\(869\) 2432.00 0.0949367
\(870\) 26792.0 1.04406
\(871\) 0 0
\(872\) 0 0
\(873\) −5474.00 −0.212219
\(874\) 9360.00 0.362250
\(875\) 13260.0 0.512308
\(876\) 3440.00 0.132679
\(877\) −6847.00 −0.263634 −0.131817 0.991274i \(-0.542081\pi\)
−0.131817 + 0.991274i \(0.542081\pi\)
\(878\) −18304.0 −0.703565
\(879\) 16602.0 0.637055
\(880\) 34816.0 1.33369
\(881\) 29731.0 1.13696 0.568481 0.822697i \(-0.307531\pi\)
0.568481 + 0.822697i \(0.307531\pi\)
\(882\) −5244.00 −0.200198
\(883\) −23738.0 −0.904697 −0.452348 0.891841i \(-0.649414\pi\)
−0.452348 + 0.891841i \(0.649414\pi\)
\(884\) 0 0
\(885\) −29376.0 −1.11578
\(886\) −35248.0 −1.33655
\(887\) 27588.0 1.04432 0.522161 0.852847i \(-0.325126\pi\)
0.522161 + 0.852847i \(0.325126\pi\)
\(888\) 0 0
\(889\) −43160.0 −1.62828
\(890\) −18088.0 −0.681248
\(891\) −13472.0 −0.506542
\(892\) −43024.0 −1.61497
\(893\) −4860.00 −0.182121
\(894\) −16920.0 −0.632986
\(895\) 72488.0 2.70727
\(896\) 0 0
\(897\) 0 0
\(898\) 7672.00 0.285098
\(899\) −14578.0 −0.540827
\(900\) −30176.0 −1.11763
\(901\) −1209.00 −0.0447033
\(902\) 21120.0 0.779622
\(903\) −6240.00 −0.229960
\(904\) 0 0
\(905\) −6851.00 −0.251641
\(906\) 4112.00 0.150786
\(907\) −37128.0 −1.35922 −0.679611 0.733572i \(-0.737852\pi\)
−0.679611 + 0.733572i \(0.737852\pi\)
\(908\) −31792.0 −1.16195
\(909\) 18837.0 0.687331
\(910\) 0 0
\(911\) 20516.0 0.746131 0.373066 0.927805i \(-0.378307\pi\)
0.373066 + 0.927805i \(0.378307\pi\)
\(912\) −3840.00 −0.139424
\(913\) −20096.0 −0.728456
\(914\) −47044.0 −1.70249
\(915\) 4930.00 0.178121
\(916\) −50384.0 −1.81740
\(917\) 14600.0 0.525774
\(918\) 5200.00 0.186956
\(919\) −21006.0 −0.753998 −0.376999 0.926214i \(-0.623044\pi\)
−0.376999 + 0.926214i \(0.623044\pi\)
\(920\) 0 0
\(921\) 17356.0 0.620955
\(922\) 3604.00 0.128733
\(923\) 0 0
\(924\) −10240.0 −0.364579
\(925\) −37228.0 −1.32330
\(926\) 5488.00 0.194759
\(927\) −37674.0 −1.33482
\(928\) −50432.0 −1.78396
\(929\) 20427.0 0.721408 0.360704 0.932680i \(-0.382536\pi\)
0.360704 + 0.932680i \(0.382536\pi\)
\(930\) −10064.0 −0.354851
\(931\) 1710.00 0.0601965
\(932\) 32240.0 1.13311
\(933\) 17316.0 0.607610
\(934\) −25584.0 −0.896289
\(935\) 7072.00 0.247357
\(936\) 0 0
\(937\) 33191.0 1.15721 0.578603 0.815609i \(-0.303598\pi\)
0.578603 + 0.815609i \(0.303598\pi\)
\(938\) 68960.0 2.40045
\(939\) −10500.0 −0.364914
\(940\) −22032.0 −0.764473
\(941\) −36422.0 −1.26177 −0.630884 0.775877i \(-0.717308\pi\)
−0.630884 + 0.775877i \(0.717308\pi\)
\(942\) 23208.0 0.802715
\(943\) −12870.0 −0.444438
\(944\) 55296.0 1.90650
\(945\) −34000.0 −1.17039
\(946\) 19968.0 0.686275
\(947\) −39630.0 −1.35988 −0.679938 0.733270i \(-0.737993\pi\)
−0.679938 + 0.733270i \(0.737993\pi\)
\(948\) −1216.00 −0.0416602
\(949\) 0 0
\(950\) 19680.0 0.672109
\(951\) 12826.0 0.437341
\(952\) 0 0
\(953\) 57642.0 1.95929 0.979647 0.200727i \(-0.0643305\pi\)
0.979647 + 0.200727i \(0.0643305\pi\)
\(954\) −8556.00 −0.290368
\(955\) −21182.0 −0.717731
\(956\) −7872.00 −0.266317
\(957\) −12608.0 −0.425871
\(958\) 13080.0 0.441123
\(959\) 33420.0 1.12533
\(960\) −17408.0 −0.585251
\(961\) −24315.0 −0.816186
\(962\) 0 0
\(963\) −12006.0 −0.401753
\(964\) 7544.00 0.252050
\(965\) 4539.00 0.151415
\(966\) 12480.0 0.415670
\(967\) 2162.00 0.0718979 0.0359489 0.999354i \(-0.488555\pi\)
0.0359489 + 0.999354i \(0.488555\pi\)
\(968\) 0 0
\(969\) −780.000 −0.0258588
\(970\) 16184.0 0.535708
\(971\) −19758.0 −0.653001 −0.326501 0.945197i \(-0.605870\pi\)
−0.326501 + 0.945197i \(0.605870\pi\)
\(972\) 28336.0 0.935059
\(973\) 18240.0 0.600974
\(974\) 79680.0 2.62126
\(975\) 0 0
\(976\) −9280.00 −0.304350
\(977\) −12489.0 −0.408965 −0.204482 0.978870i \(-0.565551\pi\)
−0.204482 + 0.978870i \(0.565551\pi\)
\(978\) 18880.0 0.617296
\(979\) 8512.00 0.277880
\(980\) 7752.00 0.252682
\(981\) 37582.0 1.22314
\(982\) 26208.0 0.851661
\(983\) −28658.0 −0.929856 −0.464928 0.885349i \(-0.653920\pi\)
−0.464928 + 0.885349i \(0.653920\pi\)
\(984\) 0 0
\(985\) 21726.0 0.702790
\(986\) −10244.0 −0.330868
\(987\) −6480.00 −0.208977
\(988\) 0 0
\(989\) −12168.0 −0.391223
\(990\) 50048.0 1.60670
\(991\) −42794.0 −1.37174 −0.685871 0.727723i \(-0.740579\pi\)
−0.685871 + 0.727723i \(0.740579\pi\)
\(992\) 18944.0 0.606323
\(993\) 6976.00 0.222937
\(994\) 52320.0 1.66951
\(995\) 72046.0 2.29549
\(996\) 10048.0 0.319662
\(997\) −52583.0 −1.67033 −0.835166 0.549998i \(-0.814628\pi\)
−0.835166 + 0.549998i \(0.814628\pi\)
\(998\) 6984.00 0.221518
\(999\) 22700.0 0.718915
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 169.4.a.d.1.1 1
3.2 odd 2 1521.4.a.b.1.1 1
13.2 odd 12 169.4.e.c.147.2 4
13.3 even 3 13.4.c.a.9.1 yes 2
13.4 even 6 169.4.c.d.146.1 2
13.5 odd 4 169.4.b.c.168.1 2
13.6 odd 12 169.4.e.c.23.1 4
13.7 odd 12 169.4.e.c.23.2 4
13.8 odd 4 169.4.b.c.168.2 2
13.9 even 3 13.4.c.a.3.1 2
13.10 even 6 169.4.c.d.22.1 2
13.11 odd 12 169.4.e.c.147.1 4
13.12 even 2 169.4.a.a.1.1 1
39.29 odd 6 117.4.g.c.100.1 2
39.35 odd 6 117.4.g.c.55.1 2
39.38 odd 2 1521.4.a.k.1.1 1
52.3 odd 6 208.4.i.b.113.1 2
52.35 odd 6 208.4.i.b.81.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.c.a.3.1 2 13.9 even 3
13.4.c.a.9.1 yes 2 13.3 even 3
117.4.g.c.55.1 2 39.35 odd 6
117.4.g.c.100.1 2 39.29 odd 6
169.4.a.a.1.1 1 13.12 even 2
169.4.a.d.1.1 1 1.1 even 1 trivial
169.4.b.c.168.1 2 13.5 odd 4
169.4.b.c.168.2 2 13.8 odd 4
169.4.c.d.22.1 2 13.10 even 6
169.4.c.d.146.1 2 13.4 even 6
169.4.e.c.23.1 4 13.6 odd 12
169.4.e.c.23.2 4 13.7 odd 12
169.4.e.c.147.1 4 13.11 odd 12
169.4.e.c.147.2 4 13.2 odd 12
208.4.i.b.81.1 2 52.35 odd 6
208.4.i.b.113.1 2 52.3 odd 6
1521.4.a.b.1.1 1 3.2 odd 2
1521.4.a.k.1.1 1 39.38 odd 2