Properties

Label 169.4.a.a.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 \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 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 \(-0.774930\pi\)
−0.760263 + 0.649615i \(0.774930\pi\)
\(6\) −8.00000 −0.544331
\(7\) −20.0000 −1.07990 −0.539949 0.841698i \(-0.681557\pi\)
−0.539949 + 0.841698i \(0.681557\pi\)
\(8\) 0 0
\(9\) −23.0000 −0.851852
\(10\) 68.0000 2.15035
\(11\) 32.0000 0.877124 0.438562 0.898701i \(-0.355488\pi\)
0.438562 + 0.898701i \(0.355488\pi\)
\(12\) 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 \(-0.557971\pi\)
−0.181118 + 0.983461i \(0.557971\pi\)
\(20\) −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 \(-0.431231\pi\)
0.214368 + 0.976753i \(0.431231\pi\)
\(32\) 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 \(-0.331749\pi\)
0.504305 + 0.863526i \(0.331749\pi\)
\(38\) 120.000 0.512278
\(39\) 0 0
\(40\) 0 0
\(41\) 165.000 0.628504 0.314252 0.949340i \(-0.398246\pi\)
0.314252 + 0.949340i \(0.398246\pi\)
\(42\) 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 \(-0.419114\pi\)
0.251384 + 0.967887i \(0.419114\pi\)
\(48\) −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 \(-0.0977178\pi\)
0.953248 + 0.302190i \(0.0977178\pi\)
\(60\) −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 \(-0.787798\pi\)
−0.785896 + 0.618359i \(0.787798\pi\)
\(68\) −104.000 −0.185468
\(69\) 156.000 0.272177
\(70\) −1360.00 −2.32216
\(71\) −654.000 −1.09318 −0.546588 0.837402i \(-0.684074\pi\)
−0.546588 + 0.837402i \(0.684074\pi\)
\(72\) 0 0
\(73\) −215.000 −0.344710 −0.172355 0.985035i \(-0.555138\pi\)
−0.172355 + 0.985035i \(0.555138\pi\)
\(74\) −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 \(-0.636307\pi\)
−0.415253 + 0.909706i \(0.636307\pi\)
\(84\) −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 \(-0.449365\pi\)
0.158404 + 0.987374i \(0.449365\pi\)
\(90\) −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 \(-0.539753\pi\)
−0.124563 + 0.992212i \(0.539753\pi\)
\(98\) −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 \(-0.245090\pi\)
0.717930 + 0.696115i \(0.245090\pi\)
\(110\) 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 \(-0.674453\pi\)
−0.521033 + 0.853536i \(0.674453\pi\)
\(138\) −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 \(-0.302492\pi\)
0.581435 + 0.813593i \(0.302492\pi\)
\(150\) −1312.00 −0.714162
\(151\) −514.000 −0.277011 −0.138506 0.990362i \(-0.544230\pi\)
−0.138506 + 0.990362i \(0.544230\pi\)
\(152\) 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 \(-0.691905\pi\)
−0.567023 + 0.823702i \(0.691905\pi\)
\(164\) 1320.00 0.628504
\(165\) −1088.00 −0.513337
\(166\) 2512.00 1.17451
\(167\) −280.000 −0.129743 −0.0648714 0.997894i \(-0.520664\pi\)
−0.0648714 + 0.997894i \(0.520664\pi\)
\(168\) 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 \(-0.515855\pi\)
−0.0497904 + 0.998760i \(0.515855\pi\)
\(194\) 952.000 0.352318
\(195\) 0 0
\(196\) 456.000 0.166181
\(197\) −1278.00 −0.462202 −0.231101 0.972930i \(-0.574233\pi\)
−0.231101 + 0.972930i \(0.574233\pi\)
\(198\) 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 \(-0.200829\pi\)
0.807483 + 0.589891i \(0.200829\pi\)
\(224\) −5120.00 −1.52721
\(225\) −3772.00 −1.11763
\(226\) −1308.00 −0.384986
\(227\) 3974.00 1.16195 0.580977 0.813920i \(-0.302671\pi\)
0.580977 + 0.813920i \(0.302671\pi\)
\(228\) −480.000 −0.139424
\(229\) 6298.00 1.81740 0.908698 0.417455i \(-0.137078\pi\)
0.908698 + 0.417455i \(0.137078\pi\)
\(230\) 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 \(-0.457488\pi\)
0.133158 + 0.991095i \(0.457488\pi\)
\(240\) 2176.00 0.585251
\(241\) −943.000 −0.252050 −0.126025 0.992027i \(-0.540222\pi\)
−0.126025 + 0.992027i \(0.540222\pi\)
\(242\) 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 \(-0.340099\pi\)
0.481482 + 0.876456i \(0.340099\pi\)
\(272\) 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 \(-0.299182\pi\)
0.589863 + 0.807504i \(0.299182\pi\)
\(282\) −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 \(-0.810271\pi\)
−0.827559 + 0.561379i \(0.810271\pi\)
\(294\) −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 \(-0.798719\pi\)
−0.806644 + 0.591037i \(0.798719\pi\)
\(308\) −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 \(-0.692330\pi\)
−0.568123 + 0.822944i \(0.692330\pi\)
\(318\) −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 \(-0.593524\pi\)
−0.289604 + 0.957147i \(0.593524\pi\)
\(332\) −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 \(-0.367888\pi\)
0.403230 + 0.915099i \(0.367888\pi\)
\(350\) 13120.0 2.00370
\(351\) 0 0
\(352\) 8192.00 1.24044
\(353\) −3163.00 −0.476911 −0.238455 0.971153i \(-0.576641\pi\)
−0.238455 + 0.971153i \(0.576641\pi\)
\(354\) −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 \(-0.234793\pi\)
0.740068 + 0.672532i \(0.234793\pi\)
\(360\) 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 \(-0.477072\pi\)
0.0719674 + 0.997407i \(0.477072\pi\)
\(380\) 4080.00 0.550788
\(381\) −4316.00 −0.580355
\(382\) 4984.00 0.667549
\(383\) 3532.00 0.471219 0.235609 0.971848i \(-0.424291\pi\)
0.235609 + 0.971848i \(0.424291\pi\)
\(384\) 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 \(-0.376483\pi\)
0.378374 + 0.925653i \(0.376483\pi\)
\(398\) −16952.0 −2.13499
\(399\) 1200.00 0.150564
\(400\) −10496.0 −1.31200
\(401\) −5935.00 −0.739102 −0.369551 0.929211i \(-0.620488\pi\)
−0.369551 + 0.929211i \(0.620488\pi\)
\(402\) 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 \(-0.134455\pi\)
0.912106 + 0.409954i \(0.134455\pi\)
\(410\) 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 \(-0.376522\pi\)
0.378262 + 0.925699i \(0.376522\pi\)
\(422\) −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 \(-0.535291\pi\)
−0.110642 + 0.993860i \(0.535291\pi\)
\(432\) 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 \(-0.532139\pi\)
−0.100797 + 0.994907i \(0.532139\pi\)
\(450\) 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 \(-0.294402\pi\)
0.601922 + 0.798555i \(0.294402\pi\)
\(458\) −25192.0 −2.57019
\(459\) 1300.00 0.132198
\(460\) −10608.0 −1.07522
\(461\) −901.000 −0.0910277 −0.0455138 0.998964i \(-0.514493\pi\)
−0.0455138 + 0.998964i \(0.514493\pi\)
\(462\) 5120.00 0.515593
\(463\) −1372.00 −0.137715 −0.0688577 0.997626i \(-0.521935\pi\)
−0.0688577 + 0.997626i \(0.521935\pi\)
\(464\) −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 \(-0.549847\pi\)
−0.155960 + 0.987763i \(0.549847\pi\)
\(480\) −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 \(-0.877416\pi\)
−0.926757 + 0.375661i \(0.877416\pi\)
\(488\) 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 \(-0.524955\pi\)
−0.0783183 + 0.996928i \(0.524955\pi\)
\(500\) −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 \(-0.614388\pi\)
−0.351677 + 0.936122i \(0.614388\pi\)
\(510\) −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 \(-0.239458\pi\)
0.730132 + 0.683306i \(0.239458\pi\)
\(542\) −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 \(-0.504177\pi\)
−0.0131222 + 0.999914i \(0.504177\pi\)
\(558\) 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 \(-0.330129\pi\)
0.508694 + 0.860948i \(0.330129\pi\)
\(578\) 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 \(-0.515763\pi\)
−0.0495012 + 0.998774i \(0.515763\pi\)
\(588\) 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 \(-0.486318\pi\)
0.0429694 + 0.999076i \(0.486318\pi\)
\(594\) 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 \(-0.762239\pi\)
−0.733767 + 0.679402i \(0.762239\pi\)
\(614\) 34712.0 2.28153
\(615\) −5610.00 −0.367833
\(616\) 0 0
\(617\) 18989.0 1.23901 0.619504 0.784993i \(-0.287334\pi\)
0.619504 + 0.784993i \(0.287334\pi\)
\(618\) −13104.0 −0.852945
\(619\) −72.0000 −0.00467516 −0.00233758 0.999997i \(-0.500744\pi\)
−0.00233758 + 0.999997i \(0.500744\pi\)
\(620\) −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 \(-0.235999\pi\)
0.737514 + 0.675331i \(0.235999\pi\)
\(632\) 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 \(-0.324250\pi\)
0.524507 + 0.851406i \(0.324250\pi\)
\(644\) −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 \(-0.460763\pi\)
0.122953 + 0.992412i \(0.460763\pi\)
\(662\) 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 \(-0.375806\pi\)
0.380342 + 0.924846i \(0.375806\pi\)
\(684\) 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 \(-0.614090\pi\)
−0.350799 + 0.936451i \(0.614090\pi\)
\(692\) 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 \(-0.505978\pi\)
−0.0187779 + 0.999824i \(0.505978\pi\)
\(710\) −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 \(-0.479270\pi\)
0.0650786 + 0.997880i \(0.479270\pi\)
\(734\) −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 \(-0.532347\pi\)
−0.101447 + 0.994841i \(0.532347\pi\)
\(740\) −30872.0 −1.53362
\(741\) 0 0
\(742\) 7440.00 0.368101
\(743\) 34056.0 1.68155 0.840776 0.541383i \(-0.182099\pi\)
0.840776 + 0.541383i \(0.182099\pi\)
\(744\) 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 \(-0.608066\pi\)
−0.333014 + 0.942922i \(0.608066\pi\)
\(762\) 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 \(-0.360771\pi\)
0.423587 + 0.905855i \(0.360771\pi\)
\(770\) −43520.0 −2.03682
\(771\) −3770.00 −0.176100
\(772\) −2136.00 −0.0995807
\(773\) −14434.0 −0.671610 −0.335805 0.941931i \(-0.609008\pi\)
−0.335805 + 0.941931i \(0.609008\pi\)
\(774\) −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 \(-0.386618\pi\)
0.348716 + 0.937228i \(0.386618\pi\)
\(788\) −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 \(-0.666430\pi\)
−0.499357 + 0.866396i \(0.666430\pi\)
\(812\) −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 \(-0.710015\pi\)
−0.612943 + 0.790127i \(0.710015\pi\)
\(822\) 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 \(-0.250582\pi\)
0.705812 + 0.708399i \(0.250582\pi\)
\(828\) −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 \(-0.731255\pi\)
−0.664265 + 0.747497i \(0.731255\pi\)
\(840\) 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 \(-0.638042\pi\)
−0.420205 + 0.907429i \(0.638042\pi\)
\(854\) 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 \(-0.553156\pi\)
−0.166218 + 0.986089i \(0.553156\pi\)
\(864\) −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 \(-0.457919\pi\)
0.131817 + 0.991274i \(0.457919\pi\)
\(878\) 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 \(-0.617464\pi\)
−0.360704 + 0.932680i \(0.617464\pi\)
\(930\) 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 \(-0.282692\pi\)
0.630884 + 0.775877i \(0.282692\pi\)
\(942\) −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 \(-0.262007\pi\)
0.679938 + 0.733270i \(0.262007\pi\)
\(948\) −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 \(-0.511445\pi\)
−0.0359489 + 0.999354i \(0.511445\pi\)
\(968\) 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 \(-0.434449\pi\)
0.204482 + 0.978870i \(0.434449\pi\)
\(978\) 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 \(-0.346080\pi\)
0.464928 + 0.885349i \(0.346080\pi\)
\(984\) 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.a.1.1 1
3.2 odd 2 1521.4.a.k.1.1 1
13.2 odd 12 169.4.e.c.147.1 4
13.3 even 3 169.4.c.d.22.1 2
13.4 even 6 13.4.c.a.3.1 2
13.5 odd 4 169.4.b.c.168.2 2
13.6 odd 12 169.4.e.c.23.2 4
13.7 odd 12 169.4.e.c.23.1 4
13.8 odd 4 169.4.b.c.168.1 2
13.9 even 3 169.4.c.d.146.1 2
13.10 even 6 13.4.c.a.9.1 yes 2
13.11 odd 12 169.4.e.c.147.2 4
13.12 even 2 169.4.a.d.1.1 1
39.17 odd 6 117.4.g.c.55.1 2
39.23 odd 6 117.4.g.c.100.1 2
39.38 odd 2 1521.4.a.b.1.1 1
52.23 odd 6 208.4.i.b.113.1 2
52.43 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 13.4 even 6
13.4.c.a.9.1 yes 2 13.10 even 6
117.4.g.c.55.1 2 39.17 odd 6
117.4.g.c.100.1 2 39.23 odd 6
169.4.a.a.1.1 1 1.1 even 1 trivial
169.4.a.d.1.1 1 13.12 even 2
169.4.b.c.168.1 2 13.8 odd 4
169.4.b.c.168.2 2 13.5 odd 4
169.4.c.d.22.1 2 13.3 even 3
169.4.c.d.146.1 2 13.9 even 3
169.4.e.c.23.1 4 13.7 odd 12
169.4.e.c.23.2 4 13.6 odd 12
169.4.e.c.147.1 4 13.2 odd 12
169.4.e.c.147.2 4 13.11 odd 12
208.4.i.b.81.1 2 52.43 odd 6
208.4.i.b.113.1 2 52.23 odd 6
1521.4.a.b.1.1 1 39.38 odd 2
1521.4.a.k.1.1 1 3.2 odd 2