Properties

Label 78.4.a.f.1.1
Level $78$
Weight $4$
Character 78.1
Self dual yes
Analytic conductor $4.602$
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] = [78,4,Mod(1,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("78.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 78.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: \(4.60214898045\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 78.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +4.00000 q^{5} +6.00000 q^{6} +4.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +4.00000 q^{5} +6.00000 q^{6} +4.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +8.00000 q^{10} +2.00000 q^{11} +12.0000 q^{12} -13.0000 q^{13} +8.00000 q^{14} +12.0000 q^{15} +16.0000 q^{16} -6.00000 q^{17} +18.0000 q^{18} -36.0000 q^{19} +16.0000 q^{20} +12.0000 q^{21} +4.00000 q^{22} -20.0000 q^{23} +24.0000 q^{24} -109.000 q^{25} -26.0000 q^{26} +27.0000 q^{27} +16.0000 q^{28} -14.0000 q^{29} +24.0000 q^{30} -152.000 q^{31} +32.0000 q^{32} +6.00000 q^{33} -12.0000 q^{34} +16.0000 q^{35} +36.0000 q^{36} -258.000 q^{37} -72.0000 q^{38} -39.0000 q^{39} +32.0000 q^{40} +84.0000 q^{41} +24.0000 q^{42} -188.000 q^{43} +8.00000 q^{44} +36.0000 q^{45} -40.0000 q^{46} +254.000 q^{47} +48.0000 q^{48} -327.000 q^{49} -218.000 q^{50} -18.0000 q^{51} -52.0000 q^{52} +366.000 q^{53} +54.0000 q^{54} +8.00000 q^{55} +32.0000 q^{56} -108.000 q^{57} -28.0000 q^{58} +550.000 q^{59} +48.0000 q^{60} -14.0000 q^{61} -304.000 q^{62} +36.0000 q^{63} +64.0000 q^{64} -52.0000 q^{65} +12.0000 q^{66} +448.000 q^{67} -24.0000 q^{68} -60.0000 q^{69} +32.0000 q^{70} +926.000 q^{71} +72.0000 q^{72} +254.000 q^{73} -516.000 q^{74} -327.000 q^{75} -144.000 q^{76} +8.00000 q^{77} -78.0000 q^{78} +1328.00 q^{79} +64.0000 q^{80} +81.0000 q^{81} +168.000 q^{82} +186.000 q^{83} +48.0000 q^{84} -24.0000 q^{85} -376.000 q^{86} -42.0000 q^{87} +16.0000 q^{88} -336.000 q^{89} +72.0000 q^{90} -52.0000 q^{91} -80.0000 q^{92} -456.000 q^{93} +508.000 q^{94} -144.000 q^{95} +96.0000 q^{96} +614.000 q^{97} -654.000 q^{98} +18.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 3.00000 0.577350
\(4\) 4.00000 0.500000
\(5\) 4.00000 0.357771 0.178885 0.983870i \(-0.442751\pi\)
0.178885 + 0.983870i \(0.442751\pi\)
\(6\) 6.00000 0.408248
\(7\) 4.00000 0.215980 0.107990 0.994152i \(-0.465559\pi\)
0.107990 + 0.994152i \(0.465559\pi\)
\(8\) 8.00000 0.353553
\(9\) 9.00000 0.333333
\(10\) 8.00000 0.252982
\(11\) 2.00000 0.0548202 0.0274101 0.999624i \(-0.491274\pi\)
0.0274101 + 0.999624i \(0.491274\pi\)
\(12\) 12.0000 0.288675
\(13\) −13.0000 −0.277350
\(14\) 8.00000 0.152721
\(15\) 12.0000 0.206559
\(16\) 16.0000 0.250000
\(17\) −6.00000 −0.0856008 −0.0428004 0.999084i \(-0.513628\pi\)
−0.0428004 + 0.999084i \(0.513628\pi\)
\(18\) 18.0000 0.235702
\(19\) −36.0000 −0.434682 −0.217341 0.976096i \(-0.569738\pi\)
−0.217341 + 0.976096i \(0.569738\pi\)
\(20\) 16.0000 0.178885
\(21\) 12.0000 0.124696
\(22\) 4.00000 0.0387638
\(23\) −20.0000 −0.181317 −0.0906584 0.995882i \(-0.528897\pi\)
−0.0906584 + 0.995882i \(0.528897\pi\)
\(24\) 24.0000 0.204124
\(25\) −109.000 −0.872000
\(26\) −26.0000 −0.196116
\(27\) 27.0000 0.192450
\(28\) 16.0000 0.107990
\(29\) −14.0000 −0.0896460 −0.0448230 0.998995i \(-0.514272\pi\)
−0.0448230 + 0.998995i \(0.514272\pi\)
\(30\) 24.0000 0.146059
\(31\) −152.000 −0.880645 −0.440323 0.897840i \(-0.645136\pi\)
−0.440323 + 0.897840i \(0.645136\pi\)
\(32\) 32.0000 0.176777
\(33\) 6.00000 0.0316505
\(34\) −12.0000 −0.0605289
\(35\) 16.0000 0.0772712
\(36\) 36.0000 0.166667
\(37\) −258.000 −1.14635 −0.573175 0.819433i \(-0.694288\pi\)
−0.573175 + 0.819433i \(0.694288\pi\)
\(38\) −72.0000 −0.307367
\(39\) −39.0000 −0.160128
\(40\) 32.0000 0.126491
\(41\) 84.0000 0.319966 0.159983 0.987120i \(-0.448856\pi\)
0.159983 + 0.987120i \(0.448856\pi\)
\(42\) 24.0000 0.0881733
\(43\) −188.000 −0.666738 −0.333369 0.942796i \(-0.608185\pi\)
−0.333369 + 0.942796i \(0.608185\pi\)
\(44\) 8.00000 0.0274101
\(45\) 36.0000 0.119257
\(46\) −40.0000 −0.128210
\(47\) 254.000 0.788292 0.394146 0.919048i \(-0.371040\pi\)
0.394146 + 0.919048i \(0.371040\pi\)
\(48\) 48.0000 0.144338
\(49\) −327.000 −0.953353
\(50\) −218.000 −0.616597
\(51\) −18.0000 −0.0494217
\(52\) −52.0000 −0.138675
\(53\) 366.000 0.948565 0.474283 0.880373i \(-0.342707\pi\)
0.474283 + 0.880373i \(0.342707\pi\)
\(54\) 54.0000 0.136083
\(55\) 8.00000 0.0196131
\(56\) 32.0000 0.0763604
\(57\) −108.000 −0.250964
\(58\) −28.0000 −0.0633893
\(59\) 550.000 1.21363 0.606813 0.794845i \(-0.292448\pi\)
0.606813 + 0.794845i \(0.292448\pi\)
\(60\) 48.0000 0.103280
\(61\) −14.0000 −0.0293855 −0.0146928 0.999892i \(-0.504677\pi\)
−0.0146928 + 0.999892i \(0.504677\pi\)
\(62\) −304.000 −0.622710
\(63\) 36.0000 0.0719932
\(64\) 64.0000 0.125000
\(65\) −52.0000 −0.0992278
\(66\) 12.0000 0.0223803
\(67\) 448.000 0.816894 0.408447 0.912782i \(-0.366070\pi\)
0.408447 + 0.912782i \(0.366070\pi\)
\(68\) −24.0000 −0.0428004
\(69\) −60.0000 −0.104683
\(70\) 32.0000 0.0546390
\(71\) 926.000 1.54783 0.773915 0.633289i \(-0.218296\pi\)
0.773915 + 0.633289i \(0.218296\pi\)
\(72\) 72.0000 0.117851
\(73\) 254.000 0.407239 0.203620 0.979050i \(-0.434729\pi\)
0.203620 + 0.979050i \(0.434729\pi\)
\(74\) −516.000 −0.810592
\(75\) −327.000 −0.503449
\(76\) −144.000 −0.217341
\(77\) 8.00000 0.0118401
\(78\) −78.0000 −0.113228
\(79\) 1328.00 1.89129 0.945644 0.325205i \(-0.105433\pi\)
0.945644 + 0.325205i \(0.105433\pi\)
\(80\) 64.0000 0.0894427
\(81\) 81.0000 0.111111
\(82\) 168.000 0.226250
\(83\) 186.000 0.245978 0.122989 0.992408i \(-0.460752\pi\)
0.122989 + 0.992408i \(0.460752\pi\)
\(84\) 48.0000 0.0623480
\(85\) −24.0000 −0.0306255
\(86\) −376.000 −0.471455
\(87\) −42.0000 −0.0517572
\(88\) 16.0000 0.0193819
\(89\) −336.000 −0.400179 −0.200089 0.979778i \(-0.564123\pi\)
−0.200089 + 0.979778i \(0.564123\pi\)
\(90\) 72.0000 0.0843274
\(91\) −52.0000 −0.0599020
\(92\) −80.0000 −0.0906584
\(93\) −456.000 −0.508441
\(94\) 508.000 0.557406
\(95\) −144.000 −0.155517
\(96\) 96.0000 0.102062
\(97\) 614.000 0.642704 0.321352 0.946960i \(-0.395863\pi\)
0.321352 + 0.946960i \(0.395863\pi\)
\(98\) −654.000 −0.674122
\(99\) 18.0000 0.0182734
\(100\) −436.000 −0.436000
\(101\) −1606.00 −1.58221 −0.791104 0.611682i \(-0.790493\pi\)
−0.791104 + 0.611682i \(0.790493\pi\)
\(102\) −36.0000 −0.0349464
\(103\) 208.000 0.198979 0.0994896 0.995039i \(-0.468279\pi\)
0.0994896 + 0.995039i \(0.468279\pi\)
\(104\) −104.000 −0.0980581
\(105\) 48.0000 0.0446126
\(106\) 732.000 0.670737
\(107\) −248.000 −0.224066 −0.112033 0.993704i \(-0.535736\pi\)
−0.112033 + 0.993704i \(0.535736\pi\)
\(108\) 108.000 0.0962250
\(109\) −542.000 −0.476277 −0.238138 0.971231i \(-0.576537\pi\)
−0.238138 + 0.971231i \(0.576537\pi\)
\(110\) 16.0000 0.0138685
\(111\) −774.000 −0.661845
\(112\) 64.0000 0.0539949
\(113\) −2042.00 −1.69996 −0.849979 0.526817i \(-0.823385\pi\)
−0.849979 + 0.526817i \(0.823385\pi\)
\(114\) −216.000 −0.177458
\(115\) −80.0000 −0.0648699
\(116\) −56.0000 −0.0448230
\(117\) −117.000 −0.0924500
\(118\) 1100.00 0.858163
\(119\) −24.0000 −0.0184880
\(120\) 96.0000 0.0730297
\(121\) −1327.00 −0.996995
\(122\) −28.0000 −0.0207787
\(123\) 252.000 0.184732
\(124\) −608.000 −0.440323
\(125\) −936.000 −0.669747
\(126\) 72.0000 0.0509069
\(127\) −488.000 −0.340968 −0.170484 0.985360i \(-0.554533\pi\)
−0.170484 + 0.985360i \(0.554533\pi\)
\(128\) 128.000 0.0883883
\(129\) −564.000 −0.384941
\(130\) −104.000 −0.0701646
\(131\) 1744.00 1.16316 0.581580 0.813489i \(-0.302435\pi\)
0.581580 + 0.813489i \(0.302435\pi\)
\(132\) 24.0000 0.0158252
\(133\) −144.000 −0.0938826
\(134\) 896.000 0.577631
\(135\) 108.000 0.0688530
\(136\) −48.0000 −0.0302645
\(137\) −828.000 −0.516356 −0.258178 0.966097i \(-0.583122\pi\)
−0.258178 + 0.966097i \(0.583122\pi\)
\(138\) −120.000 −0.0740223
\(139\) −404.000 −0.246524 −0.123262 0.992374i \(-0.539336\pi\)
−0.123262 + 0.992374i \(0.539336\pi\)
\(140\) 64.0000 0.0386356
\(141\) 762.000 0.455120
\(142\) 1852.00 1.09448
\(143\) −26.0000 −0.0152044
\(144\) 144.000 0.0833333
\(145\) −56.0000 −0.0320727
\(146\) 508.000 0.287962
\(147\) −981.000 −0.550418
\(148\) −1032.00 −0.573175
\(149\) 2928.00 1.60987 0.804937 0.593361i \(-0.202199\pi\)
0.804937 + 0.593361i \(0.202199\pi\)
\(150\) −654.000 −0.355993
\(151\) 1944.00 1.04769 0.523843 0.851815i \(-0.324498\pi\)
0.523843 + 0.851815i \(0.324498\pi\)
\(152\) −288.000 −0.153683
\(153\) −54.0000 −0.0285336
\(154\) 16.0000 0.00837219
\(155\) −608.000 −0.315069
\(156\) −156.000 −0.0800641
\(157\) 3590.00 1.82492 0.912462 0.409161i \(-0.134178\pi\)
0.912462 + 0.409161i \(0.134178\pi\)
\(158\) 2656.00 1.33734
\(159\) 1098.00 0.547654
\(160\) 128.000 0.0632456
\(161\) −80.0000 −0.0391608
\(162\) 162.000 0.0785674
\(163\) −2284.00 −1.09753 −0.548763 0.835978i \(-0.684901\pi\)
−0.548763 + 0.835978i \(0.684901\pi\)
\(164\) 336.000 0.159983
\(165\) 24.0000 0.0113236
\(166\) 372.000 0.173933
\(167\) 3174.00 1.47073 0.735364 0.677673i \(-0.237011\pi\)
0.735364 + 0.677673i \(0.237011\pi\)
\(168\) 96.0000 0.0440867
\(169\) 169.000 0.0769231
\(170\) −48.0000 −0.0216555
\(171\) −324.000 −0.144894
\(172\) −752.000 −0.333369
\(173\) −1358.00 −0.596802 −0.298401 0.954441i \(-0.596453\pi\)
−0.298401 + 0.954441i \(0.596453\pi\)
\(174\) −84.0000 −0.0365978
\(175\) −436.000 −0.188334
\(176\) 32.0000 0.0137051
\(177\) 1650.00 0.700687
\(178\) −672.000 −0.282969
\(179\) 708.000 0.295634 0.147817 0.989015i \(-0.452775\pi\)
0.147817 + 0.989015i \(0.452775\pi\)
\(180\) 144.000 0.0596285
\(181\) −546.000 −0.224220 −0.112110 0.993696i \(-0.535761\pi\)
−0.112110 + 0.993696i \(0.535761\pi\)
\(182\) −104.000 −0.0423571
\(183\) −42.0000 −0.0169657
\(184\) −160.000 −0.0641052
\(185\) −1032.00 −0.410131
\(186\) −912.000 −0.359522
\(187\) −12.0000 −0.00469266
\(188\) 1016.00 0.394146
\(189\) 108.000 0.0415653
\(190\) −288.000 −0.109967
\(191\) −3472.00 −1.31531 −0.657657 0.753317i \(-0.728453\pi\)
−0.657657 + 0.753317i \(0.728453\pi\)
\(192\) 192.000 0.0721688
\(193\) −310.000 −0.115618 −0.0578090 0.998328i \(-0.518411\pi\)
−0.0578090 + 0.998328i \(0.518411\pi\)
\(194\) 1228.00 0.454460
\(195\) −156.000 −0.0572892
\(196\) −1308.00 −0.476676
\(197\) 1020.00 0.368893 0.184447 0.982843i \(-0.440951\pi\)
0.184447 + 0.982843i \(0.440951\pi\)
\(198\) 36.0000 0.0129213
\(199\) −3256.00 −1.15986 −0.579929 0.814667i \(-0.696920\pi\)
−0.579929 + 0.814667i \(0.696920\pi\)
\(200\) −872.000 −0.308299
\(201\) 1344.00 0.471634
\(202\) −3212.00 −1.11879
\(203\) −56.0000 −0.0193617
\(204\) −72.0000 −0.0247108
\(205\) 336.000 0.114474
\(206\) 416.000 0.140699
\(207\) −180.000 −0.0604390
\(208\) −208.000 −0.0693375
\(209\) −72.0000 −0.0238294
\(210\) 96.0000 0.0315459
\(211\) −4564.00 −1.48909 −0.744547 0.667570i \(-0.767334\pi\)
−0.744547 + 0.667570i \(0.767334\pi\)
\(212\) 1464.00 0.474283
\(213\) 2778.00 0.893640
\(214\) −496.000 −0.158439
\(215\) −752.000 −0.238539
\(216\) 216.000 0.0680414
\(217\) −608.000 −0.190202
\(218\) −1084.00 −0.336779
\(219\) 762.000 0.235120
\(220\) 32.0000 0.00980654
\(221\) 78.0000 0.0237414
\(222\) −1548.00 −0.467995
\(223\) −72.0000 −0.0216210 −0.0108105 0.999942i \(-0.503441\pi\)
−0.0108105 + 0.999942i \(0.503441\pi\)
\(224\) 128.000 0.0381802
\(225\) −981.000 −0.290667
\(226\) −4084.00 −1.20205
\(227\) 2694.00 0.787696 0.393848 0.919176i \(-0.371144\pi\)
0.393848 + 0.919176i \(0.371144\pi\)
\(228\) −432.000 −0.125482
\(229\) 5922.00 1.70889 0.854447 0.519538i \(-0.173896\pi\)
0.854447 + 0.519538i \(0.173896\pi\)
\(230\) −160.000 −0.0458699
\(231\) 24.0000 0.00683586
\(232\) −112.000 −0.0316947
\(233\) −5122.00 −1.44014 −0.720072 0.693900i \(-0.755891\pi\)
−0.720072 + 0.693900i \(0.755891\pi\)
\(234\) −234.000 −0.0653720
\(235\) 1016.00 0.282028
\(236\) 2200.00 0.606813
\(237\) 3984.00 1.09194
\(238\) −48.0000 −0.0130730
\(239\) 5022.00 1.35919 0.679595 0.733588i \(-0.262156\pi\)
0.679595 + 0.733588i \(0.262156\pi\)
\(240\) 192.000 0.0516398
\(241\) −1218.00 −0.325553 −0.162777 0.986663i \(-0.552045\pi\)
−0.162777 + 0.986663i \(0.552045\pi\)
\(242\) −2654.00 −0.704982
\(243\) 243.000 0.0641500
\(244\) −56.0000 −0.0146928
\(245\) −1308.00 −0.341082
\(246\) 504.000 0.130625
\(247\) 468.000 0.120559
\(248\) −1216.00 −0.311355
\(249\) 558.000 0.142015
\(250\) −1872.00 −0.473583
\(251\) −2112.00 −0.531109 −0.265554 0.964096i \(-0.585555\pi\)
−0.265554 + 0.964096i \(0.585555\pi\)
\(252\) 144.000 0.0359966
\(253\) −40.0000 −0.00993984
\(254\) −976.000 −0.241101
\(255\) −72.0000 −0.0176816
\(256\) 256.000 0.0625000
\(257\) 2814.00 0.683006 0.341503 0.939881i \(-0.389064\pi\)
0.341503 + 0.939881i \(0.389064\pi\)
\(258\) −1128.00 −0.272195
\(259\) −1032.00 −0.247588
\(260\) −208.000 −0.0496139
\(261\) −126.000 −0.0298820
\(262\) 3488.00 0.822478
\(263\) −4044.00 −0.948151 −0.474076 0.880484i \(-0.657218\pi\)
−0.474076 + 0.880484i \(0.657218\pi\)
\(264\) 48.0000 0.0111901
\(265\) 1464.00 0.339369
\(266\) −288.000 −0.0663850
\(267\) −1008.00 −0.231043
\(268\) 1792.00 0.408447
\(269\) −1470.00 −0.333188 −0.166594 0.986026i \(-0.553277\pi\)
−0.166594 + 0.986026i \(0.553277\pi\)
\(270\) 216.000 0.0486864
\(271\) −1844.00 −0.413340 −0.206670 0.978411i \(-0.566263\pi\)
−0.206670 + 0.978411i \(0.566263\pi\)
\(272\) −96.0000 −0.0214002
\(273\) −156.000 −0.0345844
\(274\) −1656.00 −0.365119
\(275\) −218.000 −0.0478033
\(276\) −240.000 −0.0523417
\(277\) 5766.00 1.25071 0.625353 0.780342i \(-0.284955\pi\)
0.625353 + 0.780342i \(0.284955\pi\)
\(278\) −808.000 −0.174319
\(279\) −1368.00 −0.293548
\(280\) 128.000 0.0273195
\(281\) −7468.00 −1.58542 −0.792711 0.609598i \(-0.791331\pi\)
−0.792711 + 0.609598i \(0.791331\pi\)
\(282\) 1524.00 0.321819
\(283\) 1228.00 0.257940 0.128970 0.991648i \(-0.458833\pi\)
0.128970 + 0.991648i \(0.458833\pi\)
\(284\) 3704.00 0.773915
\(285\) −432.000 −0.0897876
\(286\) −52.0000 −0.0107511
\(287\) 336.000 0.0691061
\(288\) 288.000 0.0589256
\(289\) −4877.00 −0.992673
\(290\) −112.000 −0.0226788
\(291\) 1842.00 0.371065
\(292\) 1016.00 0.203620
\(293\) 6608.00 1.31755 0.658777 0.752338i \(-0.271074\pi\)
0.658777 + 0.752338i \(0.271074\pi\)
\(294\) −1962.00 −0.389205
\(295\) 2200.00 0.434200
\(296\) −2064.00 −0.405296
\(297\) 54.0000 0.0105502
\(298\) 5856.00 1.13835
\(299\) 260.000 0.0502883
\(300\) −1308.00 −0.251725
\(301\) −752.000 −0.144002
\(302\) 3888.00 0.740825
\(303\) −4818.00 −0.913488
\(304\) −576.000 −0.108671
\(305\) −56.0000 −0.0105133
\(306\) −108.000 −0.0201763
\(307\) 7664.00 1.42478 0.712390 0.701784i \(-0.247613\pi\)
0.712390 + 0.701784i \(0.247613\pi\)
\(308\) 32.0000 0.00592003
\(309\) 624.000 0.114881
\(310\) −1216.00 −0.222788
\(311\) −2340.00 −0.426653 −0.213327 0.976981i \(-0.568430\pi\)
−0.213327 + 0.976981i \(0.568430\pi\)
\(312\) −312.000 −0.0566139
\(313\) 6710.00 1.21173 0.605865 0.795567i \(-0.292827\pi\)
0.605865 + 0.795567i \(0.292827\pi\)
\(314\) 7180.00 1.29042
\(315\) 144.000 0.0257571
\(316\) 5312.00 0.945644
\(317\) 4164.00 0.737771 0.368886 0.929475i \(-0.379739\pi\)
0.368886 + 0.929475i \(0.379739\pi\)
\(318\) 2196.00 0.387250
\(319\) −28.0000 −0.00491442
\(320\) 256.000 0.0447214
\(321\) −744.000 −0.129365
\(322\) −160.000 −0.0276908
\(323\) 216.000 0.0372092
\(324\) 324.000 0.0555556
\(325\) 1417.00 0.241849
\(326\) −4568.00 −0.776068
\(327\) −1626.00 −0.274979
\(328\) 672.000 0.113125
\(329\) 1016.00 0.170255
\(330\) 48.0000 0.00800701
\(331\) −10072.0 −1.67253 −0.836265 0.548326i \(-0.815265\pi\)
−0.836265 + 0.548326i \(0.815265\pi\)
\(332\) 744.000 0.122989
\(333\) −2322.00 −0.382117
\(334\) 6348.00 1.03996
\(335\) 1792.00 0.292261
\(336\) 192.000 0.0311740
\(337\) 2990.00 0.483311 0.241655 0.970362i \(-0.422310\pi\)
0.241655 + 0.970362i \(0.422310\pi\)
\(338\) 338.000 0.0543928
\(339\) −6126.00 −0.981471
\(340\) −96.0000 −0.0153127
\(341\) −304.000 −0.0482772
\(342\) −648.000 −0.102456
\(343\) −2680.00 −0.421885
\(344\) −1504.00 −0.235727
\(345\) −240.000 −0.0374527
\(346\) −2716.00 −0.422003
\(347\) 6564.00 1.01549 0.507743 0.861508i \(-0.330480\pi\)
0.507743 + 0.861508i \(0.330480\pi\)
\(348\) −168.000 −0.0258786
\(349\) −674.000 −0.103376 −0.0516882 0.998663i \(-0.516460\pi\)
−0.0516882 + 0.998663i \(0.516460\pi\)
\(350\) −872.000 −0.133172
\(351\) −351.000 −0.0533761
\(352\) 64.0000 0.00969094
\(353\) −10732.0 −1.61815 −0.809075 0.587706i \(-0.800031\pi\)
−0.809075 + 0.587706i \(0.800031\pi\)
\(354\) 3300.00 0.495461
\(355\) 3704.00 0.553769
\(356\) −1344.00 −0.200089
\(357\) −72.0000 −0.0106741
\(358\) 1416.00 0.209044
\(359\) −4842.00 −0.711841 −0.355921 0.934516i \(-0.615833\pi\)
−0.355921 + 0.934516i \(0.615833\pi\)
\(360\) 288.000 0.0421637
\(361\) −5563.00 −0.811051
\(362\) −1092.00 −0.158548
\(363\) −3981.00 −0.575615
\(364\) −208.000 −0.0299510
\(365\) 1016.00 0.145698
\(366\) −84.0000 −0.0119966
\(367\) −6280.00 −0.893224 −0.446612 0.894728i \(-0.647370\pi\)
−0.446612 + 0.894728i \(0.647370\pi\)
\(368\) −320.000 −0.0453292
\(369\) 756.000 0.106655
\(370\) −2064.00 −0.290006
\(371\) 1464.00 0.204871
\(372\) −1824.00 −0.254220
\(373\) 6434.00 0.893136 0.446568 0.894750i \(-0.352646\pi\)
0.446568 + 0.894750i \(0.352646\pi\)
\(374\) −24.0000 −0.00331821
\(375\) −2808.00 −0.386679
\(376\) 2032.00 0.278703
\(377\) 182.000 0.0248633
\(378\) 216.000 0.0293911
\(379\) −9068.00 −1.22900 −0.614501 0.788916i \(-0.710643\pi\)
−0.614501 + 0.788916i \(0.710643\pi\)
\(380\) −576.000 −0.0777584
\(381\) −1464.00 −0.196858
\(382\) −6944.00 −0.930068
\(383\) 3162.00 0.421855 0.210928 0.977502i \(-0.432352\pi\)
0.210928 + 0.977502i \(0.432352\pi\)
\(384\) 384.000 0.0510310
\(385\) 32.0000 0.00423603
\(386\) −620.000 −0.0817543
\(387\) −1692.00 −0.222246
\(388\) 2456.00 0.321352
\(389\) −3666.00 −0.477824 −0.238912 0.971041i \(-0.576791\pi\)
−0.238912 + 0.971041i \(0.576791\pi\)
\(390\) −312.000 −0.0405096
\(391\) 120.000 0.0155209
\(392\) −2616.00 −0.337061
\(393\) 5232.00 0.671551
\(394\) 2040.00 0.260847
\(395\) 5312.00 0.676647
\(396\) 72.0000 0.00913671
\(397\) 11054.0 1.39744 0.698721 0.715394i \(-0.253753\pi\)
0.698721 + 0.715394i \(0.253753\pi\)
\(398\) −6512.00 −0.820143
\(399\) −432.000 −0.0542031
\(400\) −1744.00 −0.218000
\(401\) −5328.00 −0.663510 −0.331755 0.943366i \(-0.607641\pi\)
−0.331755 + 0.943366i \(0.607641\pi\)
\(402\) 2688.00 0.333496
\(403\) 1976.00 0.244247
\(404\) −6424.00 −0.791104
\(405\) 324.000 0.0397523
\(406\) −112.000 −0.0136908
\(407\) −516.000 −0.0628432
\(408\) −144.000 −0.0174732
\(409\) −12074.0 −1.45971 −0.729854 0.683603i \(-0.760412\pi\)
−0.729854 + 0.683603i \(0.760412\pi\)
\(410\) 672.000 0.0809456
\(411\) −2484.00 −0.298118
\(412\) 832.000 0.0994896
\(413\) 2200.00 0.262118
\(414\) −360.000 −0.0427368
\(415\) 744.000 0.0880037
\(416\) −416.000 −0.0490290
\(417\) −1212.00 −0.142331
\(418\) −144.000 −0.0168499
\(419\) 13584.0 1.58382 0.791911 0.610636i \(-0.209086\pi\)
0.791911 + 0.610636i \(0.209086\pi\)
\(420\) 192.000 0.0223063
\(421\) −7406.00 −0.857355 −0.428677 0.903458i \(-0.641020\pi\)
−0.428677 + 0.903458i \(0.641020\pi\)
\(422\) −9128.00 −1.05295
\(423\) 2286.00 0.262764
\(424\) 2928.00 0.335369
\(425\) 654.000 0.0746439
\(426\) 5556.00 0.631899
\(427\) −56.0000 −0.00634667
\(428\) −992.000 −0.112033
\(429\) −78.0000 −0.00877826
\(430\) −1504.00 −0.168673
\(431\) −10134.0 −1.13257 −0.566285 0.824210i \(-0.691620\pi\)
−0.566285 + 0.824210i \(0.691620\pi\)
\(432\) 432.000 0.0481125
\(433\) 9406.00 1.04393 0.521967 0.852966i \(-0.325198\pi\)
0.521967 + 0.852966i \(0.325198\pi\)
\(434\) −1216.00 −0.134493
\(435\) −168.000 −0.0185172
\(436\) −2168.00 −0.238138
\(437\) 720.000 0.0788153
\(438\) 1524.00 0.166255
\(439\) 4088.00 0.444441 0.222220 0.974996i \(-0.428670\pi\)
0.222220 + 0.974996i \(0.428670\pi\)
\(440\) 64.0000 0.00693427
\(441\) −2943.00 −0.317784
\(442\) 156.000 0.0167877
\(443\) −5328.00 −0.571424 −0.285712 0.958315i \(-0.592230\pi\)
−0.285712 + 0.958315i \(0.592230\pi\)
\(444\) −3096.00 −0.330923
\(445\) −1344.00 −0.143172
\(446\) −144.000 −0.0152883
\(447\) 8784.00 0.929461
\(448\) 256.000 0.0269975
\(449\) 13160.0 1.38320 0.691602 0.722279i \(-0.256905\pi\)
0.691602 + 0.722279i \(0.256905\pi\)
\(450\) −1962.00 −0.205532
\(451\) 168.000 0.0175406
\(452\) −8168.00 −0.849979
\(453\) 5832.00 0.604881
\(454\) 5388.00 0.556985
\(455\) −208.000 −0.0214312
\(456\) −864.000 −0.0887292
\(457\) −9146.00 −0.936175 −0.468087 0.883682i \(-0.655057\pi\)
−0.468087 + 0.883682i \(0.655057\pi\)
\(458\) 11844.0 1.20837
\(459\) −162.000 −0.0164739
\(460\) −320.000 −0.0324349
\(461\) 5580.00 0.563745 0.281873 0.959452i \(-0.409044\pi\)
0.281873 + 0.959452i \(0.409044\pi\)
\(462\) 48.0000 0.00483368
\(463\) 14788.0 1.48436 0.742178 0.670203i \(-0.233793\pi\)
0.742178 + 0.670203i \(0.233793\pi\)
\(464\) −224.000 −0.0224115
\(465\) −1824.00 −0.181905
\(466\) −10244.0 −1.01834
\(467\) 12376.0 1.22632 0.613162 0.789957i \(-0.289897\pi\)
0.613162 + 0.789957i \(0.289897\pi\)
\(468\) −468.000 −0.0462250
\(469\) 1792.00 0.176433
\(470\) 2032.00 0.199424
\(471\) 10770.0 1.05362
\(472\) 4400.00 0.429081
\(473\) −376.000 −0.0365507
\(474\) 7968.00 0.772115
\(475\) 3924.00 0.379043
\(476\) −96.0000 −0.00924402
\(477\) 3294.00 0.316188
\(478\) 10044.0 0.961092
\(479\) 834.000 0.0795541 0.0397771 0.999209i \(-0.487335\pi\)
0.0397771 + 0.999209i \(0.487335\pi\)
\(480\) 384.000 0.0365148
\(481\) 3354.00 0.317940
\(482\) −2436.00 −0.230201
\(483\) −240.000 −0.0226095
\(484\) −5308.00 −0.498497
\(485\) 2456.00 0.229941
\(486\) 486.000 0.0453609
\(487\) −13192.0 −1.22749 −0.613744 0.789505i \(-0.710337\pi\)
−0.613744 + 0.789505i \(0.710337\pi\)
\(488\) −112.000 −0.0103893
\(489\) −6852.00 −0.633657
\(490\) −2616.00 −0.241181
\(491\) 16568.0 1.52282 0.761409 0.648272i \(-0.224508\pi\)
0.761409 + 0.648272i \(0.224508\pi\)
\(492\) 1008.00 0.0923662
\(493\) 84.0000 0.00767377
\(494\) 936.000 0.0852482
\(495\) 72.0000 0.00653770
\(496\) −2432.00 −0.220161
\(497\) 3704.00 0.334300
\(498\) 1116.00 0.100420
\(499\) −10136.0 −0.909318 −0.454659 0.890666i \(-0.650239\pi\)
−0.454659 + 0.890666i \(0.650239\pi\)
\(500\) −3744.00 −0.334874
\(501\) 9522.00 0.849125
\(502\) −4224.00 −0.375550
\(503\) 10412.0 0.922959 0.461479 0.887151i \(-0.347319\pi\)
0.461479 + 0.887151i \(0.347319\pi\)
\(504\) 288.000 0.0254535
\(505\) −6424.00 −0.566068
\(506\) −80.0000 −0.00702853
\(507\) 507.000 0.0444116
\(508\) −1952.00 −0.170484
\(509\) −4180.00 −0.363999 −0.181999 0.983299i \(-0.558257\pi\)
−0.181999 + 0.983299i \(0.558257\pi\)
\(510\) −144.000 −0.0125028
\(511\) 1016.00 0.0879554
\(512\) 512.000 0.0441942
\(513\) −972.000 −0.0836547
\(514\) 5628.00 0.482958
\(515\) 832.000 0.0711889
\(516\) −2256.00 −0.192471
\(517\) 508.000 0.0432143
\(518\) −2064.00 −0.175071
\(519\) −4074.00 −0.344564
\(520\) −416.000 −0.0350823
\(521\) −14610.0 −1.22855 −0.614276 0.789091i \(-0.710552\pi\)
−0.614276 + 0.789091i \(0.710552\pi\)
\(522\) −252.000 −0.0211298
\(523\) −2172.00 −0.181596 −0.0907982 0.995869i \(-0.528942\pi\)
−0.0907982 + 0.995869i \(0.528942\pi\)
\(524\) 6976.00 0.581580
\(525\) −1308.00 −0.108735
\(526\) −8088.00 −0.670444
\(527\) 912.000 0.0753840
\(528\) 96.0000 0.00791262
\(529\) −11767.0 −0.967124
\(530\) 2928.00 0.239970
\(531\) 4950.00 0.404542
\(532\) −576.000 −0.0469413
\(533\) −1092.00 −0.0887425
\(534\) −2016.00 −0.163372
\(535\) −992.000 −0.0801643
\(536\) 3584.00 0.288816
\(537\) 2124.00 0.170684
\(538\) −2940.00 −0.235599
\(539\) −654.000 −0.0522630
\(540\) 432.000 0.0344265
\(541\) −11758.0 −0.934410 −0.467205 0.884149i \(-0.654739\pi\)
−0.467205 + 0.884149i \(0.654739\pi\)
\(542\) −3688.00 −0.292275
\(543\) −1638.00 −0.129454
\(544\) −192.000 −0.0151322
\(545\) −2168.00 −0.170398
\(546\) −312.000 −0.0244549
\(547\) 340.000 0.0265765 0.0132883 0.999912i \(-0.495770\pi\)
0.0132883 + 0.999912i \(0.495770\pi\)
\(548\) −3312.00 −0.258178
\(549\) −126.000 −0.00979517
\(550\) −436.000 −0.0338020
\(551\) 504.000 0.0389676
\(552\) −480.000 −0.0370112
\(553\) 5312.00 0.408480
\(554\) 11532.0 0.884382
\(555\) −3096.00 −0.236789
\(556\) −1616.00 −0.123262
\(557\) −3768.00 −0.286634 −0.143317 0.989677i \(-0.545777\pi\)
−0.143317 + 0.989677i \(0.545777\pi\)
\(558\) −2736.00 −0.207570
\(559\) 2444.00 0.184920
\(560\) 256.000 0.0193178
\(561\) −36.0000 −0.00270931
\(562\) −14936.0 −1.12106
\(563\) −10172.0 −0.761454 −0.380727 0.924687i \(-0.624326\pi\)
−0.380727 + 0.924687i \(0.624326\pi\)
\(564\) 3048.00 0.227560
\(565\) −8168.00 −0.608195
\(566\) 2456.00 0.182391
\(567\) 324.000 0.0239977
\(568\) 7408.00 0.547241
\(569\) −5506.00 −0.405665 −0.202833 0.979213i \(-0.565015\pi\)
−0.202833 + 0.979213i \(0.565015\pi\)
\(570\) −864.000 −0.0634894
\(571\) 2340.00 0.171499 0.0857495 0.996317i \(-0.472672\pi\)
0.0857495 + 0.996317i \(0.472672\pi\)
\(572\) −104.000 −0.00760220
\(573\) −10416.0 −0.759397
\(574\) 672.000 0.0488654
\(575\) 2180.00 0.158108
\(576\) 576.000 0.0416667
\(577\) −20094.0 −1.44978 −0.724891 0.688864i \(-0.758110\pi\)
−0.724891 + 0.688864i \(0.758110\pi\)
\(578\) −9754.00 −0.701925
\(579\) −930.000 −0.0667521
\(580\) −224.000 −0.0160364
\(581\) 744.000 0.0531262
\(582\) 3684.00 0.262383
\(583\) 732.000 0.0520006
\(584\) 2032.00 0.143981
\(585\) −468.000 −0.0330759
\(586\) 13216.0 0.931652
\(587\) −7118.00 −0.500496 −0.250248 0.968182i \(-0.580512\pi\)
−0.250248 + 0.968182i \(0.580512\pi\)
\(588\) −3924.00 −0.275209
\(589\) 5472.00 0.382801
\(590\) 4400.00 0.307026
\(591\) 3060.00 0.212981
\(592\) −4128.00 −0.286587
\(593\) −10328.0 −0.715211 −0.357606 0.933873i \(-0.616407\pi\)
−0.357606 + 0.933873i \(0.616407\pi\)
\(594\) 108.000 0.00746009
\(595\) −96.0000 −0.00661448
\(596\) 11712.0 0.804937
\(597\) −9768.00 −0.669644
\(598\) 520.000 0.0355592
\(599\) −19732.0 −1.34596 −0.672978 0.739662i \(-0.734985\pi\)
−0.672978 + 0.739662i \(0.734985\pi\)
\(600\) −2616.00 −0.177996
\(601\) −12026.0 −0.816224 −0.408112 0.912932i \(-0.633813\pi\)
−0.408112 + 0.912932i \(0.633813\pi\)
\(602\) −1504.00 −0.101825
\(603\) 4032.00 0.272298
\(604\) 7776.00 0.523843
\(605\) −5308.00 −0.356696
\(606\) −9636.00 −0.645934
\(607\) 17016.0 1.13782 0.568911 0.822399i \(-0.307365\pi\)
0.568911 + 0.822399i \(0.307365\pi\)
\(608\) −1152.00 −0.0768417
\(609\) −168.000 −0.0111785
\(610\) −112.000 −0.00743401
\(611\) −3302.00 −0.218633
\(612\) −216.000 −0.0142668
\(613\) 11654.0 0.767864 0.383932 0.923361i \(-0.374570\pi\)
0.383932 + 0.923361i \(0.374570\pi\)
\(614\) 15328.0 1.00747
\(615\) 1008.00 0.0660918
\(616\) 64.0000 0.00418609
\(617\) 11612.0 0.757669 0.378834 0.925465i \(-0.376325\pi\)
0.378834 + 0.925465i \(0.376325\pi\)
\(618\) 1248.00 0.0812329
\(619\) 4024.00 0.261290 0.130645 0.991429i \(-0.458295\pi\)
0.130645 + 0.991429i \(0.458295\pi\)
\(620\) −2432.00 −0.157535
\(621\) −540.000 −0.0348945
\(622\) −4680.00 −0.301690
\(623\) −1344.00 −0.0864305
\(624\) −624.000 −0.0400320
\(625\) 9881.00 0.632384
\(626\) 13420.0 0.856823
\(627\) −216.000 −0.0137579
\(628\) 14360.0 0.912462
\(629\) 1548.00 0.0981285
\(630\) 288.000 0.0182130
\(631\) −1088.00 −0.0686412 −0.0343206 0.999411i \(-0.510927\pi\)
−0.0343206 + 0.999411i \(0.510927\pi\)
\(632\) 10624.0 0.668671
\(633\) −13692.0 −0.859729
\(634\) 8328.00 0.521683
\(635\) −1952.00 −0.121989
\(636\) 4392.00 0.273827
\(637\) 4251.00 0.264412
\(638\) −56.0000 −0.00347502
\(639\) 8334.00 0.515944
\(640\) 512.000 0.0316228
\(641\) −7078.00 −0.436138 −0.218069 0.975933i \(-0.569976\pi\)
−0.218069 + 0.975933i \(0.569976\pi\)
\(642\) −1488.00 −0.0914746
\(643\) 8336.00 0.511259 0.255630 0.966775i \(-0.417717\pi\)
0.255630 + 0.966775i \(0.417717\pi\)
\(644\) −320.000 −0.0195804
\(645\) −2256.00 −0.137721
\(646\) 432.000 0.0263109
\(647\) 32.0000 0.00194444 0.000972218 1.00000i \(-0.499691\pi\)
0.000972218 1.00000i \(0.499691\pi\)
\(648\) 648.000 0.0392837
\(649\) 1100.00 0.0665312
\(650\) 2834.00 0.171013
\(651\) −1824.00 −0.109813
\(652\) −9136.00 −0.548763
\(653\) −15822.0 −0.948182 −0.474091 0.880476i \(-0.657223\pi\)
−0.474091 + 0.880476i \(0.657223\pi\)
\(654\) −3252.00 −0.194439
\(655\) 6976.00 0.416145
\(656\) 1344.00 0.0799914
\(657\) 2286.00 0.135746
\(658\) 2032.00 0.120388
\(659\) 21540.0 1.27326 0.636631 0.771169i \(-0.280328\pi\)
0.636631 + 0.771169i \(0.280328\pi\)
\(660\) 96.0000 0.00566181
\(661\) 8270.00 0.486635 0.243317 0.969947i \(-0.421764\pi\)
0.243317 + 0.969947i \(0.421764\pi\)
\(662\) −20144.0 −1.18266
\(663\) 234.000 0.0137071
\(664\) 1488.00 0.0869663
\(665\) −576.000 −0.0335885
\(666\) −4644.00 −0.270197
\(667\) 280.000 0.0162543
\(668\) 12696.0 0.735364
\(669\) −216.000 −0.0124829
\(670\) 3584.00 0.206660
\(671\) −28.0000 −0.00161092
\(672\) 384.000 0.0220433
\(673\) 8482.00 0.485820 0.242910 0.970049i \(-0.421898\pi\)
0.242910 + 0.970049i \(0.421898\pi\)
\(674\) 5980.00 0.341752
\(675\) −2943.00 −0.167816
\(676\) 676.000 0.0384615
\(677\) 2550.00 0.144763 0.0723814 0.997377i \(-0.476940\pi\)
0.0723814 + 0.997377i \(0.476940\pi\)
\(678\) −12252.0 −0.694005
\(679\) 2456.00 0.138811
\(680\) −192.000 −0.0108277
\(681\) 8082.00 0.454777
\(682\) −608.000 −0.0341371
\(683\) −31534.0 −1.76664 −0.883320 0.468771i \(-0.844697\pi\)
−0.883320 + 0.468771i \(0.844697\pi\)
\(684\) −1296.00 −0.0724471
\(685\) −3312.00 −0.184737
\(686\) −5360.00 −0.298317
\(687\) 17766.0 0.986631
\(688\) −3008.00 −0.166684
\(689\) −4758.00 −0.263085
\(690\) −480.000 −0.0264830
\(691\) 33832.0 1.86256 0.931281 0.364302i \(-0.118693\pi\)
0.931281 + 0.364302i \(0.118693\pi\)
\(692\) −5432.00 −0.298401
\(693\) 72.0000 0.00394669
\(694\) 13128.0 0.718058
\(695\) −1616.00 −0.0881991
\(696\) −336.000 −0.0182989
\(697\) −504.000 −0.0273893
\(698\) −1348.00 −0.0730982
\(699\) −15366.0 −0.831467
\(700\) −1744.00 −0.0941671
\(701\) 19422.0 1.04645 0.523223 0.852196i \(-0.324729\pi\)
0.523223 + 0.852196i \(0.324729\pi\)
\(702\) −702.000 −0.0377426
\(703\) 9288.00 0.498298
\(704\) 128.000 0.00685253
\(705\) 3048.00 0.162829
\(706\) −21464.0 −1.14420
\(707\) −6424.00 −0.341725
\(708\) 6600.00 0.350343
\(709\) −1894.00 −0.100325 −0.0501627 0.998741i \(-0.515974\pi\)
−0.0501627 + 0.998741i \(0.515974\pi\)
\(710\) 7408.00 0.391574
\(711\) 11952.0 0.630429
\(712\) −2688.00 −0.141485
\(713\) 3040.00 0.159676
\(714\) −144.000 −0.00754771
\(715\) −104.000 −0.00543969
\(716\) 2832.00 0.147817
\(717\) 15066.0 0.784728
\(718\) −9684.00 −0.503348
\(719\) −20156.0 −1.04547 −0.522734 0.852496i \(-0.675088\pi\)
−0.522734 + 0.852496i \(0.675088\pi\)
\(720\) 576.000 0.0298142
\(721\) 832.000 0.0429754
\(722\) −11126.0 −0.573500
\(723\) −3654.00 −0.187958
\(724\) −2184.00 −0.112110
\(725\) 1526.00 0.0781713
\(726\) −7962.00 −0.407021
\(727\) 11128.0 0.567696 0.283848 0.958869i \(-0.408389\pi\)
0.283848 + 0.958869i \(0.408389\pi\)
\(728\) −416.000 −0.0211786
\(729\) 729.000 0.0370370
\(730\) 2032.00 0.103024
\(731\) 1128.00 0.0570733
\(732\) −168.000 −0.00848287
\(733\) 16202.0 0.816418 0.408209 0.912888i \(-0.366153\pi\)
0.408209 + 0.912888i \(0.366153\pi\)
\(734\) −12560.0 −0.631605
\(735\) −3924.00 −0.196924
\(736\) −640.000 −0.0320526
\(737\) 896.000 0.0447823
\(738\) 1512.00 0.0754167
\(739\) −5328.00 −0.265215 −0.132607 0.991169i \(-0.542335\pi\)
−0.132607 + 0.991169i \(0.542335\pi\)
\(740\) −4128.00 −0.205065
\(741\) 1404.00 0.0696049
\(742\) 2928.00 0.144866
\(743\) −20482.0 −1.01132 −0.505661 0.862732i \(-0.668751\pi\)
−0.505661 + 0.862732i \(0.668751\pi\)
\(744\) −3648.00 −0.179761
\(745\) 11712.0 0.575966
\(746\) 12868.0 0.631543
\(747\) 1674.00 0.0819926
\(748\) −48.0000 −0.00234633
\(749\) −992.000 −0.0483937
\(750\) −5616.00 −0.273423
\(751\) 8040.00 0.390657 0.195329 0.980738i \(-0.437423\pi\)
0.195329 + 0.980738i \(0.437423\pi\)
\(752\) 4064.00 0.197073
\(753\) −6336.00 −0.306636
\(754\) 364.000 0.0175810
\(755\) 7776.00 0.374831
\(756\) 432.000 0.0207827
\(757\) −15822.0 −0.759657 −0.379829 0.925057i \(-0.624017\pi\)
−0.379829 + 0.925057i \(0.624017\pi\)
\(758\) −18136.0 −0.869036
\(759\) −120.000 −0.00573877
\(760\) −1152.00 −0.0549835
\(761\) −1452.00 −0.0691655 −0.0345828 0.999402i \(-0.511010\pi\)
−0.0345828 + 0.999402i \(0.511010\pi\)
\(762\) −2928.00 −0.139200
\(763\) −2168.00 −0.102866
\(764\) −13888.0 −0.657657
\(765\) −216.000 −0.0102085
\(766\) 6324.00 0.298297
\(767\) −7150.00 −0.336599
\(768\) 768.000 0.0360844
\(769\) 32298.0 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 64.0000 0.00299532
\(771\) 8442.00 0.394334
\(772\) −1240.00 −0.0578090
\(773\) 18736.0 0.871781 0.435891 0.900000i \(-0.356433\pi\)
0.435891 + 0.900000i \(0.356433\pi\)
\(774\) −3384.00 −0.157152
\(775\) 16568.0 0.767923
\(776\) 4912.00 0.227230
\(777\) −3096.00 −0.142945
\(778\) −7332.00 −0.337873
\(779\) −3024.00 −0.139083
\(780\) −624.000 −0.0286446
\(781\) 1852.00 0.0848525
\(782\) 240.000 0.0109749
\(783\) −378.000 −0.0172524
\(784\) −5232.00 −0.238338
\(785\) 14360.0 0.652905
\(786\) 10464.0 0.474858
\(787\) −40816.0 −1.84871 −0.924354 0.381536i \(-0.875395\pi\)
−0.924354 + 0.381536i \(0.875395\pi\)
\(788\) 4080.00 0.184447
\(789\) −12132.0 −0.547415
\(790\) 10624.0 0.478462
\(791\) −8168.00 −0.367156
\(792\) 144.000 0.00646063
\(793\) 182.000 0.00815008
\(794\) 22108.0 0.988141
\(795\) 4392.00 0.195935
\(796\) −13024.0 −0.579929
\(797\) 4518.00 0.200798 0.100399 0.994947i \(-0.467988\pi\)
0.100399 + 0.994947i \(0.467988\pi\)
\(798\) −864.000 −0.0383274
\(799\) −1524.00 −0.0674784
\(800\) −3488.00 −0.154149
\(801\) −3024.00 −0.133393
\(802\) −10656.0 −0.469173
\(803\) 508.000 0.0223249
\(804\) 5376.00 0.235817
\(805\) −320.000 −0.0140106
\(806\) 3952.00 0.172709
\(807\) −4410.00 −0.192366
\(808\) −12848.0 −0.559395
\(809\) −5058.00 −0.219814 −0.109907 0.993942i \(-0.535055\pi\)
−0.109907 + 0.993942i \(0.535055\pi\)
\(810\) 648.000 0.0281091
\(811\) −22564.0 −0.976978 −0.488489 0.872570i \(-0.662452\pi\)
−0.488489 + 0.872570i \(0.662452\pi\)
\(812\) −224.000 −0.00968086
\(813\) −5532.00 −0.238642
\(814\) −1032.00 −0.0444368
\(815\) −9136.00 −0.392663
\(816\) −288.000 −0.0123554
\(817\) 6768.00 0.289819
\(818\) −24148.0 −1.03217
\(819\) −468.000 −0.0199673
\(820\) 1344.00 0.0572372
\(821\) 32584.0 1.38513 0.692564 0.721357i \(-0.256481\pi\)
0.692564 + 0.721357i \(0.256481\pi\)
\(822\) −4968.00 −0.210802
\(823\) −9288.00 −0.393389 −0.196695 0.980465i \(-0.563021\pi\)
−0.196695 + 0.980465i \(0.563021\pi\)
\(824\) 1664.00 0.0703497
\(825\) −654.000 −0.0275992
\(826\) 4400.00 0.185346
\(827\) 20586.0 0.865593 0.432796 0.901492i \(-0.357527\pi\)
0.432796 + 0.901492i \(0.357527\pi\)
\(828\) −720.000 −0.0302195
\(829\) −46118.0 −1.93214 −0.966070 0.258280i \(-0.916844\pi\)
−0.966070 + 0.258280i \(0.916844\pi\)
\(830\) 1488.00 0.0622280
\(831\) 17298.0 0.722095
\(832\) −832.000 −0.0346688
\(833\) 1962.00 0.0816078
\(834\) −2424.00 −0.100643
\(835\) 12696.0 0.526183
\(836\) −288.000 −0.0119147
\(837\) −4104.00 −0.169480
\(838\) 27168.0 1.11993
\(839\) −39230.0 −1.61427 −0.807133 0.590369i \(-0.798982\pi\)
−0.807133 + 0.590369i \(0.798982\pi\)
\(840\) 384.000 0.0157729
\(841\) −24193.0 −0.991964
\(842\) −14812.0 −0.606241
\(843\) −22404.0 −0.915344
\(844\) −18256.0 −0.744547
\(845\) 676.000 0.0275208
\(846\) 4572.00 0.185802
\(847\) −5308.00 −0.215331
\(848\) 5856.00 0.237141
\(849\) 3684.00 0.148922
\(850\) 1308.00 0.0527812
\(851\) 5160.00 0.207853
\(852\) 11112.0 0.446820
\(853\) −18674.0 −0.749573 −0.374786 0.927111i \(-0.622284\pi\)
−0.374786 + 0.927111i \(0.622284\pi\)
\(854\) −112.000 −0.00448778
\(855\) −1296.00 −0.0518389
\(856\) −1984.00 −0.0792193
\(857\) 41678.0 1.66125 0.830626 0.556830i \(-0.187983\pi\)
0.830626 + 0.556830i \(0.187983\pi\)
\(858\) −156.000 −0.00620717
\(859\) −14740.0 −0.585474 −0.292737 0.956193i \(-0.594566\pi\)
−0.292737 + 0.956193i \(0.594566\pi\)
\(860\) −3008.00 −0.119270
\(861\) 1008.00 0.0398984
\(862\) −20268.0 −0.800848
\(863\) −24982.0 −0.985396 −0.492698 0.870200i \(-0.663989\pi\)
−0.492698 + 0.870200i \(0.663989\pi\)
\(864\) 864.000 0.0340207
\(865\) −5432.00 −0.213519
\(866\) 18812.0 0.738173
\(867\) −14631.0 −0.573120
\(868\) −2432.00 −0.0951008
\(869\) 2656.00 0.103681
\(870\) −336.000 −0.0130936
\(871\) −5824.00 −0.226566
\(872\) −4336.00 −0.168389
\(873\) 5526.00 0.214235
\(874\) 1440.00 0.0557308
\(875\) −3744.00 −0.144652
\(876\) 3048.00 0.117560
\(877\) 1134.00 0.0436630 0.0218315 0.999762i \(-0.493050\pi\)
0.0218315 + 0.999762i \(0.493050\pi\)
\(878\) 8176.00 0.314267
\(879\) 19824.0 0.760690
\(880\) 128.000 0.00490327
\(881\) 34950.0 1.33654 0.668272 0.743917i \(-0.267034\pi\)
0.668272 + 0.743917i \(0.267034\pi\)
\(882\) −5886.00 −0.224707
\(883\) −3068.00 −0.116927 −0.0584634 0.998290i \(-0.518620\pi\)
−0.0584634 + 0.998290i \(0.518620\pi\)
\(884\) 312.000 0.0118707
\(885\) 6600.00 0.250685
\(886\) −10656.0 −0.404058
\(887\) −14080.0 −0.532988 −0.266494 0.963837i \(-0.585865\pi\)
−0.266494 + 0.963837i \(0.585865\pi\)
\(888\) −6192.00 −0.233998
\(889\) −1952.00 −0.0736423
\(890\) −2688.00 −0.101238
\(891\) 162.000 0.00609114
\(892\) −288.000 −0.0108105
\(893\) −9144.00 −0.342657
\(894\) 17568.0 0.657228
\(895\) 2832.00 0.105769
\(896\) 512.000 0.0190901
\(897\) 780.000 0.0290339
\(898\) 26320.0 0.978073
\(899\) 2128.00 0.0789464
\(900\) −3924.00 −0.145333
\(901\) −2196.00 −0.0811980
\(902\) 336.000 0.0124031
\(903\) −2256.00 −0.0831395
\(904\) −16336.0 −0.601026
\(905\) −2184.00 −0.0802195
\(906\) 11664.0 0.427716
\(907\) −24876.0 −0.910688 −0.455344 0.890316i \(-0.650484\pi\)
−0.455344 + 0.890316i \(0.650484\pi\)
\(908\) 10776.0 0.393848
\(909\) −14454.0 −0.527403
\(910\) −416.000 −0.0151541
\(911\) 51456.0 1.87136 0.935682 0.352843i \(-0.114785\pi\)
0.935682 + 0.352843i \(0.114785\pi\)
\(912\) −1728.00 −0.0627410
\(913\) 372.000 0.0134846
\(914\) −18292.0 −0.661975
\(915\) −168.000 −0.00606985
\(916\) 23688.0 0.854447
\(917\) 6976.00 0.251219
\(918\) −324.000 −0.0116488
\(919\) −31032.0 −1.11388 −0.556938 0.830554i \(-0.688024\pi\)
−0.556938 + 0.830554i \(0.688024\pi\)
\(920\) −640.000 −0.0229350
\(921\) 22992.0 0.822597
\(922\) 11160.0 0.398628
\(923\) −12038.0 −0.429291
\(924\) 96.0000 0.00341793
\(925\) 28122.0 0.999617
\(926\) 29576.0 1.04960
\(927\) 1872.00 0.0663264
\(928\) −448.000 −0.0158473
\(929\) 50820.0 1.79478 0.897390 0.441239i \(-0.145461\pi\)
0.897390 + 0.441239i \(0.145461\pi\)
\(930\) −3648.00 −0.128626
\(931\) 11772.0 0.414406
\(932\) −20488.0 −0.720072
\(933\) −7020.00 −0.246328
\(934\) 24752.0 0.867142
\(935\) −48.0000 −0.00167890
\(936\) −936.000 −0.0326860
\(937\) 5982.00 0.208563 0.104281 0.994548i \(-0.466746\pi\)
0.104281 + 0.994548i \(0.466746\pi\)
\(938\) 3584.00 0.124757
\(939\) 20130.0 0.699593
\(940\) 4064.00 0.141014
\(941\) −20224.0 −0.700620 −0.350310 0.936634i \(-0.613924\pi\)
−0.350310 + 0.936634i \(0.613924\pi\)
\(942\) 21540.0 0.745022
\(943\) −1680.00 −0.0580152
\(944\) 8800.00 0.303406
\(945\) 432.000 0.0148709
\(946\) −752.000 −0.0258453
\(947\) 8478.00 0.290917 0.145458 0.989364i \(-0.453534\pi\)
0.145458 + 0.989364i \(0.453534\pi\)
\(948\) 15936.0 0.545968
\(949\) −3302.00 −0.112948
\(950\) 7848.00 0.268024
\(951\) 12492.0 0.425953
\(952\) −192.000 −0.00653651
\(953\) 40918.0 1.39083 0.695417 0.718607i \(-0.255220\pi\)
0.695417 + 0.718607i \(0.255220\pi\)
\(954\) 6588.00 0.223579
\(955\) −13888.0 −0.470581
\(956\) 20088.0 0.679595
\(957\) −84.0000 −0.00283734
\(958\) 1668.00 0.0562533
\(959\) −3312.00 −0.111522
\(960\) 768.000 0.0258199
\(961\) −6687.00 −0.224464
\(962\) 6708.00 0.224818
\(963\) −2232.00 −0.0746887
\(964\) −4872.00 −0.162777
\(965\) −1240.00 −0.0413648
\(966\) −480.000 −0.0159873
\(967\) −4624.00 −0.153772 −0.0768862 0.997040i \(-0.524498\pi\)
−0.0768862 + 0.997040i \(0.524498\pi\)
\(968\) −10616.0 −0.352491
\(969\) 648.000 0.0214827
\(970\) 4912.00 0.162593
\(971\) 15300.0 0.505665 0.252832 0.967510i \(-0.418638\pi\)
0.252832 + 0.967510i \(0.418638\pi\)
\(972\) 972.000 0.0320750
\(973\) −1616.00 −0.0532442
\(974\) −26384.0 −0.867965
\(975\) 4251.00 0.139632
\(976\) −224.000 −0.00734638
\(977\) 19584.0 0.641298 0.320649 0.947198i \(-0.396099\pi\)
0.320649 + 0.947198i \(0.396099\pi\)
\(978\) −13704.0 −0.448063
\(979\) −672.000 −0.0219379
\(980\) −5232.00 −0.170541
\(981\) −4878.00 −0.158759
\(982\) 33136.0 1.07679
\(983\) −17582.0 −0.570477 −0.285238 0.958457i \(-0.592073\pi\)
−0.285238 + 0.958457i \(0.592073\pi\)
\(984\) 2016.00 0.0653127
\(985\) 4080.00 0.131979
\(986\) 168.000 0.00542618
\(987\) 3048.00 0.0982968
\(988\) 1872.00 0.0602796
\(989\) 3760.00 0.120891
\(990\) 144.000 0.00462285
\(991\) 47904.0 1.53554 0.767770 0.640725i \(-0.221366\pi\)
0.767770 + 0.640725i \(0.221366\pi\)
\(992\) −4864.00 −0.155678
\(993\) −30216.0 −0.965635
\(994\) 7408.00 0.236386
\(995\) −13024.0 −0.414963
\(996\) 2232.00 0.0710077
\(997\) −44578.0 −1.41605 −0.708024 0.706189i \(-0.750413\pi\)
−0.708024 + 0.706189i \(0.750413\pi\)
\(998\) −20272.0 −0.642985
\(999\) −6966.00 −0.220615
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 78.4.a.f.1.1 1
3.2 odd 2 234.4.a.c.1.1 1
4.3 odd 2 624.4.a.c.1.1 1
5.4 even 2 1950.4.a.a.1.1 1
8.3 odd 2 2496.4.a.l.1.1 1
8.5 even 2 2496.4.a.c.1.1 1
12.11 even 2 1872.4.a.f.1.1 1
13.5 odd 4 1014.4.b.g.337.1 2
13.8 odd 4 1014.4.b.g.337.2 2
13.12 even 2 1014.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.f.1.1 1 1.1 even 1 trivial
234.4.a.c.1.1 1 3.2 odd 2
624.4.a.c.1.1 1 4.3 odd 2
1014.4.a.e.1.1 1 13.12 even 2
1014.4.b.g.337.1 2 13.5 odd 4
1014.4.b.g.337.2 2 13.8 odd 4
1872.4.a.f.1.1 1 12.11 even 2
1950.4.a.a.1.1 1 5.4 even 2
2496.4.a.c.1.1 1 8.5 even 2
2496.4.a.l.1.1 1 8.3 odd 2