Properties

Label 3234.2.a.g.1.1
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
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] = [3234,2,Mod(1,3234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3234.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.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: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3234.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +3.00000 q^{20} +1.00000 q^{22} +3.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} +3.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} -3.00000 q^{40} +3.00000 q^{41} +2.00000 q^{43} -1.00000 q^{44} +3.00000 q^{45} -3.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} +3.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -3.00000 q^{55} +2.00000 q^{57} +6.00000 q^{58} +12.0000 q^{59} -3.00000 q^{60} -5.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -6.00000 q^{65} -1.00000 q^{66} +5.00000 q^{67} -3.00000 q^{68} -3.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} +16.0000 q^{73} -2.00000 q^{74} -4.00000 q^{75} -2.00000 q^{76} -2.00000 q^{78} +17.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -3.00000 q^{82} -9.00000 q^{83} -9.00000 q^{85} -2.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} -3.00000 q^{90} +3.00000 q^{92} -4.00000 q^{93} -9.00000 q^{94} -6.00000 q^{95} +1.00000 q^{96} -17.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 1.00000 0.204124
\(25\) 4.00000 0.800000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 3.00000 0.547723
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 2.00000 0.324443
\(39\) 2.00000 0.320256
\(40\) −3.00000 −0.474342
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −1.00000 −0.150756
\(45\) 3.00000 0.447214
\(46\) −3.00000 −0.442326
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) 3.00000 0.420084
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 6.00000 0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −3.00000 −0.387298
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) −1.00000 −0.123091
\(67\) 5.00000 0.610847 0.305424 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) −3.00000 −0.363803
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.00000 −0.117851
\(73\) 16.0000 1.87266 0.936329 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) −2.00000 −0.232495
\(75\) −4.00000 −0.461880
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 17.0000 1.91265 0.956325 0.292306i \(-0.0944227\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) −2.00000 −0.215666
\(87\) 6.00000 0.643268
\(88\) 1.00000 0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) 3.00000 0.312772
\(93\) −4.00000 −0.414781
\(94\) −9.00000 −0.928279
\(95\) −6.00000 −0.615587
\(96\) 1.00000 0.102062
\(97\) −17.0000 −1.72609 −0.863044 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 4.00000 0.400000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) −3.00000 −0.297044
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 3.00000 0.286039
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −2.00000 −0.187317
\(115\) 9.00000 0.839254
\(116\) −6.00000 −0.557086
\(117\) −2.00000 −0.184900
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 3.00000 0.273861
\(121\) 1.00000 0.0909091
\(122\) 5.00000 0.452679
\(123\) −3.00000 −0.270501
\(124\) 4.00000 0.359211
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 11.0000 0.976092 0.488046 0.872818i \(-0.337710\pi\)
0.488046 + 0.872818i \(0.337710\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.00000 −0.176090
\(130\) 6.00000 0.526235
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) −5.00000 −0.431934
\(135\) −3.00000 −0.258199
\(136\) 3.00000 0.257248
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 3.00000 0.255377
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) −12.0000 −1.00702
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) −18.0000 −1.49482
\(146\) −16.0000 −1.32417
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 4.00000 0.326599
\(151\) −1.00000 −0.0813788 −0.0406894 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 2.00000 0.162221
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 2.00000 0.160128
\(157\) −20.0000 −1.59617 −0.798087 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) −17.0000 −1.35245
\(159\) −6.00000 −0.475831
\(160\) −3.00000 −0.237171
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 3.00000 0.234261
\(165\) 3.00000 0.233550
\(166\) 9.00000 0.698535
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 9.00000 0.690268
\(171\) −2.00000 −0.152944
\(172\) 2.00000 0.152499
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −12.0000 −0.901975
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 3.00000 0.223607
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 5.00000 0.369611
\(184\) −3.00000 −0.221163
\(185\) 6.00000 0.441129
\(186\) 4.00000 0.293294
\(187\) 3.00000 0.219382
\(188\) 9.00000 0.656392
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 17.0000 1.22053
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 1.00000 0.0710669
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) −4.00000 −0.282843
\(201\) −5.00000 −0.352673
\(202\) 12.0000 0.844317
\(203\) 0 0
\(204\) 3.00000 0.210042
\(205\) 9.00000 0.628587
\(206\) 14.0000 0.975426
\(207\) 3.00000 0.208514
\(208\) −2.00000 −0.138675
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 6.00000 0.412082
\(213\) −12.0000 −0.822226
\(214\) −3.00000 −0.205076
\(215\) 6.00000 0.409197
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −11.0000 −0.745014
\(219\) −16.0000 −1.08118
\(220\) −3.00000 −0.202260
\(221\) 6.00000 0.403604
\(222\) 2.00000 0.134231
\(223\) −20.0000 −1.33930 −0.669650 0.742677i \(-0.733556\pi\)
−0.669650 + 0.742677i \(0.733556\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 6.00000 0.399114
\(227\) 21.0000 1.39382 0.696909 0.717159i \(-0.254558\pi\)
0.696909 + 0.717159i \(0.254558\pi\)
\(228\) 2.00000 0.132453
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −9.00000 −0.593442
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 27.0000 1.76883 0.884414 0.466702i \(-0.154558\pi\)
0.884414 + 0.466702i \(0.154558\pi\)
\(234\) 2.00000 0.130744
\(235\) 27.0000 1.76129
\(236\) 12.0000 0.781133
\(237\) −17.0000 −1.10427
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) −3.00000 −0.193649
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −5.00000 −0.320092
\(245\) 0 0
\(246\) 3.00000 0.191273
\(247\) 4.00000 0.254514
\(248\) −4.00000 −0.254000
\(249\) 9.00000 0.570352
\(250\) 3.00000 0.189737
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) −11.0000 −0.690201
\(255\) 9.00000 0.563602
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 2.00000 0.124515
\(259\) 0 0
\(260\) −6.00000 −0.372104
\(261\) −6.00000 −0.371391
\(262\) −12.0000 −0.741362
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 18.0000 1.10573
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 5.00000 0.305424
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 3.00000 0.182574
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) −4.00000 −0.241209
\(276\) −3.00000 −0.180579
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 14.0000 0.839664
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 9.00000 0.535942
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 12.0000 0.712069
\(285\) 6.00000 0.355409
\(286\) −2.00000 −0.118262
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) 18.0000 1.05700
\(291\) 17.0000 0.996558
\(292\) 16.0000 0.936329
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 36.0000 2.09600
\(296\) −2.00000 −0.116248
\(297\) 1.00000 0.0580259
\(298\) −6.00000 −0.347571
\(299\) −6.00000 −0.346989
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) 1.00000 0.0575435
\(303\) 12.0000 0.689382
\(304\) −2.00000 −0.114708
\(305\) −15.0000 −0.858898
\(306\) 3.00000 0.171499
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) −12.0000 −0.681554
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) −2.00000 −0.113228
\(313\) −2.00000 −0.113047 −0.0565233 0.998401i \(-0.518002\pi\)
−0.0565233 + 0.998401i \(0.518002\pi\)
\(314\) 20.0000 1.12867
\(315\) 0 0
\(316\) 17.0000 0.956325
\(317\) −33.0000 −1.85346 −0.926732 0.375722i \(-0.877395\pi\)
−0.926732 + 0.375722i \(0.877395\pi\)
\(318\) 6.00000 0.336463
\(319\) 6.00000 0.335936
\(320\) 3.00000 0.167705
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 1.00000 0.0555556
\(325\) −8.00000 −0.443760
\(326\) 1.00000 0.0553849
\(327\) −11.0000 −0.608301
\(328\) −3.00000 −0.165647
\(329\) 0 0
\(330\) −3.00000 −0.165145
\(331\) 11.0000 0.604615 0.302307 0.953211i \(-0.402243\pi\)
0.302307 + 0.953211i \(0.402243\pi\)
\(332\) −9.00000 −0.493939
\(333\) 2.00000 0.109599
\(334\) −12.0000 −0.656611
\(335\) 15.0000 0.819538
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 9.00000 0.489535
\(339\) 6.00000 0.325875
\(340\) −9.00000 −0.488094
\(341\) −4.00000 −0.216612
\(342\) 2.00000 0.108148
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) −9.00000 −0.484544
\(346\) 12.0000 0.645124
\(347\) −33.0000 −1.77153 −0.885766 0.464131i \(-0.846367\pi\)
−0.885766 + 0.464131i \(0.846367\pi\)
\(348\) 6.00000 0.321634
\(349\) −11.0000 −0.588817 −0.294408 0.955680i \(-0.595123\pi\)
−0.294408 + 0.955680i \(0.595123\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 1.00000 0.0533002
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 12.0000 0.637793
\(355\) 36.0000 1.91068
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) −3.00000 −0.158114
\(361\) −15.0000 −0.789474
\(362\) −10.0000 −0.525588
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 48.0000 2.51243
\(366\) −5.00000 −0.261354
\(367\) 10.0000 0.521996 0.260998 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) 3.00000 0.156386
\(369\) 3.00000 0.156174
\(370\) −6.00000 −0.311925
\(371\) 0 0
\(372\) −4.00000 −0.207390
\(373\) −1.00000 −0.0517780 −0.0258890 0.999665i \(-0.508242\pi\)
−0.0258890 + 0.999665i \(0.508242\pi\)
\(374\) −3.00000 −0.155126
\(375\) 3.00000 0.154919
\(376\) −9.00000 −0.464140
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 29.0000 1.48963 0.744815 0.667271i \(-0.232538\pi\)
0.744815 + 0.667271i \(0.232538\pi\)
\(380\) −6.00000 −0.307794
\(381\) −11.0000 −0.563547
\(382\) −24.0000 −1.22795
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 2.00000 0.101666
\(388\) −17.0000 −0.863044
\(389\) 21.0000 1.06474 0.532371 0.846511i \(-0.321301\pi\)
0.532371 + 0.846511i \(0.321301\pi\)
\(390\) −6.00000 −0.303822
\(391\) −9.00000 −0.455150
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) −12.0000 −0.604551
\(395\) 51.0000 2.56609
\(396\) −1.00000 −0.0502519
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 2.00000 0.100251
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 5.00000 0.249377
\(403\) −8.00000 −0.398508
\(404\) −12.0000 −0.597022
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) −3.00000 −0.148522
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) −9.00000 −0.444478
\(411\) −18.0000 −0.887875
\(412\) −14.0000 −0.689730
\(413\) 0 0
\(414\) −3.00000 −0.147442
\(415\) −27.0000 −1.32538
\(416\) 2.00000 0.0980581
\(417\) 14.0000 0.685583
\(418\) −2.00000 −0.0978232
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 22.0000 1.07094
\(423\) 9.00000 0.437595
\(424\) −6.00000 −0.291386
\(425\) −12.0000 −0.582086
\(426\) 12.0000 0.581402
\(427\) 0 0
\(428\) 3.00000 0.145010
\(429\) −2.00000 −0.0965609
\(430\) −6.00000 −0.289346
\(431\) −6.00000 −0.289010 −0.144505 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −35.0000 −1.68199 −0.840996 0.541041i \(-0.818030\pi\)
−0.840996 + 0.541041i \(0.818030\pi\)
\(434\) 0 0
\(435\) 18.0000 0.863034
\(436\) 11.0000 0.526804
\(437\) −6.00000 −0.287019
\(438\) 16.0000 0.764510
\(439\) 19.0000 0.906821 0.453410 0.891302i \(-0.350207\pi\)
0.453410 + 0.891302i \(0.350207\pi\)
\(440\) 3.00000 0.143019
\(441\) 0 0
\(442\) −6.00000 −0.285391
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 18.0000 0.853282
\(446\) 20.0000 0.947027
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) −4.00000 −0.188562
\(451\) −3.00000 −0.141264
\(452\) −6.00000 −0.282216
\(453\) 1.00000 0.0469841
\(454\) −21.0000 −0.985579
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) −10.0000 −0.467269
\(459\) 3.00000 0.140028
\(460\) 9.00000 0.419627
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −6.00000 −0.278543
\(465\) −12.0000 −0.556487
\(466\) −27.0000 −1.25075
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) −27.0000 −1.24542
\(471\) 20.0000 0.921551
\(472\) −12.0000 −0.552345
\(473\) −2.00000 −0.0919601
\(474\) 17.0000 0.780836
\(475\) −8.00000 −0.367065
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) −6.00000 −0.274434
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 3.00000 0.136931
\(481\) −4.00000 −0.182384
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −51.0000 −2.31579
\(486\) 1.00000 0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 5.00000 0.226339
\(489\) 1.00000 0.0452216
\(490\) 0 0
\(491\) −3.00000 −0.135388 −0.0676941 0.997706i \(-0.521564\pi\)
−0.0676941 + 0.997706i \(0.521564\pi\)
\(492\) −3.00000 −0.135250
\(493\) 18.0000 0.810679
\(494\) −4.00000 −0.179969
\(495\) −3.00000 −0.134840
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) −9.00000 −0.403300
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −3.00000 −0.134164
\(501\) −12.0000 −0.536120
\(502\) −24.0000 −1.07117
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) −36.0000 −1.60198
\(506\) 3.00000 0.133366
\(507\) 9.00000 0.399704
\(508\) 11.0000 0.488046
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) −9.00000 −0.398527
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) 6.00000 0.264649
\(515\) −42.0000 −1.85074
\(516\) −2.00000 −0.0880451
\(517\) −9.00000 −0.395820
\(518\) 0 0
\(519\) 12.0000 0.526742
\(520\) 6.00000 0.263117
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 6.00000 0.262613
\(523\) −26.0000 −1.13690 −0.568450 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −12.0000 −0.522728
\(528\) 1.00000 0.0435194
\(529\) −14.0000 −0.608696
\(530\) −18.0000 −0.781870
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 6.00000 0.259645
\(535\) 9.00000 0.389104
\(536\) −5.00000 −0.215967
\(537\) −12.0000 −0.517838
\(538\) 9.00000 0.388018
\(539\) 0 0
\(540\) −3.00000 −0.129099
\(541\) 23.0000 0.988847 0.494424 0.869221i \(-0.335379\pi\)
0.494424 + 0.869221i \(0.335379\pi\)
\(542\) −16.0000 −0.687259
\(543\) −10.0000 −0.429141
\(544\) 3.00000 0.128624
\(545\) 33.0000 1.41356
\(546\) 0 0
\(547\) −46.0000 −1.96682 −0.983409 0.181402i \(-0.941936\pi\)
−0.983409 + 0.181402i \(0.941936\pi\)
\(548\) 18.0000 0.768922
\(549\) −5.00000 −0.213395
\(550\) 4.00000 0.170561
\(551\) 12.0000 0.511217
\(552\) 3.00000 0.127688
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) −6.00000 −0.254686
\(556\) −14.0000 −0.593732
\(557\) −36.0000 −1.52537 −0.762684 0.646771i \(-0.776119\pi\)
−0.762684 + 0.646771i \(0.776119\pi\)
\(558\) −4.00000 −0.169334
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) −3.00000 −0.126547
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −9.00000 −0.378968
\(565\) −18.0000 −0.757266
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −6.00000 −0.251312
\(571\) 38.0000 1.59025 0.795125 0.606445i \(-0.207405\pi\)
0.795125 + 0.606445i \(0.207405\pi\)
\(572\) 2.00000 0.0836242
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 1.00000 0.0416667
\(577\) −17.0000 −0.707719 −0.353860 0.935299i \(-0.615131\pi\)
−0.353860 + 0.935299i \(0.615131\pi\)
\(578\) 8.00000 0.332756
\(579\) −2.00000 −0.0831172
\(580\) −18.0000 −0.747409
\(581\) 0 0
\(582\) −17.0000 −0.704673
\(583\) −6.00000 −0.248495
\(584\) −16.0000 −0.662085
\(585\) −6.00000 −0.248069
\(586\) −24.0000 −0.991431
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) −36.0000 −1.48210
\(591\) −12.0000 −0.493614
\(592\) 2.00000 0.0821995
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 2.00000 0.0818546
\(598\) 6.00000 0.245358
\(599\) 27.0000 1.10319 0.551595 0.834112i \(-0.314019\pi\)
0.551595 + 0.834112i \(0.314019\pi\)
\(600\) 4.00000 0.163299
\(601\) 28.0000 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 0 0
\(603\) 5.00000 0.203616
\(604\) −1.00000 −0.0406894
\(605\) 3.00000 0.121967
\(606\) −12.0000 −0.487467
\(607\) 1.00000 0.0405887 0.0202944 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 15.0000 0.607332
\(611\) −18.0000 −0.728202
\(612\) −3.00000 −0.121268
\(613\) −31.0000 −1.25208 −0.626039 0.779792i \(-0.715325\pi\)
−0.626039 + 0.779792i \(0.715325\pi\)
\(614\) −22.0000 −0.887848
\(615\) −9.00000 −0.362915
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) −14.0000 −0.563163
\(619\) −11.0000 −0.442127 −0.221064 0.975259i \(-0.570953\pi\)
−0.221064 + 0.975259i \(0.570953\pi\)
\(620\) 12.0000 0.481932
\(621\) −3.00000 −0.120386
\(622\) 21.0000 0.842023
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −29.0000 −1.16000
\(626\) 2.00000 0.0799361
\(627\) −2.00000 −0.0798723
\(628\) −20.0000 −0.798087
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −17.0000 −0.676224
\(633\) 22.0000 0.874421
\(634\) 33.0000 1.31060
\(635\) 33.0000 1.30957
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) −6.00000 −0.237542
\(639\) 12.0000 0.474713
\(640\) −3.00000 −0.118585
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 3.00000 0.118401
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 0 0
\(645\) −6.00000 −0.236250
\(646\) −6.00000 −0.236067
\(647\) −39.0000 −1.53325 −0.766624 0.642096i \(-0.778065\pi\)
−0.766624 + 0.642096i \(0.778065\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −12.0000 −0.471041
\(650\) 8.00000 0.313786
\(651\) 0 0
\(652\) −1.00000 −0.0391630
\(653\) −21.0000 −0.821794 −0.410897 0.911682i \(-0.634784\pi\)
−0.410897 + 0.911682i \(0.634784\pi\)
\(654\) 11.0000 0.430134
\(655\) 36.0000 1.40664
\(656\) 3.00000 0.117130
\(657\) 16.0000 0.624219
\(658\) 0 0
\(659\) −33.0000 −1.28550 −0.642749 0.766077i \(-0.722206\pi\)
−0.642749 + 0.766077i \(0.722206\pi\)
\(660\) 3.00000 0.116775
\(661\) −32.0000 −1.24466 −0.622328 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) −11.0000 −0.427527
\(663\) −6.00000 −0.233021
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −18.0000 −0.696963
\(668\) 12.0000 0.464294
\(669\) 20.0000 0.773245
\(670\) −15.0000 −0.579501
\(671\) 5.00000 0.193023
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 22.0000 0.847408
\(675\) −4.00000 −0.153960
\(676\) −9.00000 −0.346154
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 9.00000 0.345134
\(681\) −21.0000 −0.804722
\(682\) 4.00000 0.153168
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 54.0000 2.06323
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 2.00000 0.0762493
\(689\) −12.0000 −0.457164
\(690\) 9.00000 0.342624
\(691\) 19.0000 0.722794 0.361397 0.932412i \(-0.382300\pi\)
0.361397 + 0.932412i \(0.382300\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 33.0000 1.25266
\(695\) −42.0000 −1.59315
\(696\) −6.00000 −0.227429
\(697\) −9.00000 −0.340899
\(698\) 11.0000 0.416356
\(699\) −27.0000 −1.02123
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −4.00000 −0.150863
\(704\) −1.00000 −0.0376889
\(705\) −27.0000 −1.01688
\(706\) −12.0000 −0.451626
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) −36.0000 −1.35106
\(711\) 17.0000 0.637550
\(712\) −6.00000 −0.224860
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) 12.0000 0.448461
\(717\) −6.00000 −0.224074
\(718\) −30.0000 −1.11959
\(719\) 21.0000 0.783168 0.391584 0.920142i \(-0.371927\pi\)
0.391584 + 0.920142i \(0.371927\pi\)
\(720\) 3.00000 0.111803
\(721\) 0 0
\(722\) 15.0000 0.558242
\(723\) −10.0000 −0.371904
\(724\) 10.0000 0.371647
\(725\) −24.0000 −0.891338
\(726\) 1.00000 0.0371135
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −48.0000 −1.77656
\(731\) −6.00000 −0.221918
\(732\) 5.00000 0.184805
\(733\) 49.0000 1.80986 0.904928 0.425564i \(-0.139924\pi\)
0.904928 + 0.425564i \(0.139924\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) −5.00000 −0.184177
\(738\) −3.00000 −0.110432
\(739\) 38.0000 1.39785 0.698926 0.715194i \(-0.253662\pi\)
0.698926 + 0.715194i \(0.253662\pi\)
\(740\) 6.00000 0.220564
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) −30.0000 −1.10059 −0.550297 0.834969i \(-0.685485\pi\)
−0.550297 + 0.834969i \(0.685485\pi\)
\(744\) 4.00000 0.146647
\(745\) 18.0000 0.659469
\(746\) 1.00000 0.0366126
\(747\) −9.00000 −0.329293
\(748\) 3.00000 0.109691
\(749\) 0 0
\(750\) −3.00000 −0.109545
\(751\) 2.00000 0.0729810 0.0364905 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) 9.00000 0.328196
\(753\) −24.0000 −0.874609
\(754\) −12.0000 −0.437014
\(755\) −3.00000 −0.109181
\(756\) 0 0
\(757\) 14.0000 0.508839 0.254419 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(758\) −29.0000 −1.05333
\(759\) 3.00000 0.108893
\(760\) 6.00000 0.217643
\(761\) −15.0000 −0.543750 −0.271875 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) 11.0000 0.398488
\(763\) 0 0
\(764\) 24.0000 0.868290
\(765\) −9.00000 −0.325396
\(766\) 24.0000 0.867155
\(767\) −24.0000 −0.866590
\(768\) −1.00000 −0.0360844
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 2.00000 0.0719816
\(773\) 27.0000 0.971123 0.485561 0.874203i \(-0.338615\pi\)
0.485561 + 0.874203i \(0.338615\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 16.0000 0.574737
\(776\) 17.0000 0.610264
\(777\) 0 0
\(778\) −21.0000 −0.752886
\(779\) −6.00000 −0.214972
\(780\) 6.00000 0.214834
\(781\) −12.0000 −0.429394
\(782\) 9.00000 0.321839
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) −60.0000 −2.14149
\(786\) 12.0000 0.428026
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 12.0000 0.427482
\(789\) 12.0000 0.427211
\(790\) −51.0000 −1.81450
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 10.0000 0.355110
\(794\) 2.00000 0.0709773
\(795\) −18.0000 −0.638394
\(796\) −2.00000 −0.0708881
\(797\) 27.0000 0.956389 0.478195 0.878254i \(-0.341291\pi\)
0.478195 + 0.878254i \(0.341291\pi\)
\(798\) 0 0
\(799\) −27.0000 −0.955191
\(800\) −4.00000 −0.141421
\(801\) 6.00000 0.212000
\(802\) −30.0000 −1.05934
\(803\) −16.0000 −0.564628
\(804\) −5.00000 −0.176336
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 9.00000 0.316815
\(808\) 12.0000 0.422159
\(809\) −9.00000 −0.316423 −0.158212 0.987405i \(-0.550573\pi\)
−0.158212 + 0.987405i \(0.550573\pi\)
\(810\) −3.00000 −0.105409
\(811\) −26.0000 −0.912983 −0.456492 0.889728i \(-0.650894\pi\)
−0.456492 + 0.889728i \(0.650894\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 2.00000 0.0701000
\(815\) −3.00000 −0.105085
\(816\) 3.00000 0.105021
\(817\) −4.00000 −0.139942
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) 9.00000 0.314294
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 18.0000 0.627822
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) 14.0000 0.487713
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 33.0000 1.14752 0.573761 0.819023i \(-0.305484\pi\)
0.573761 + 0.819023i \(0.305484\pi\)
\(828\) 3.00000 0.104257
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 27.0000 0.937184
\(831\) 10.0000 0.346896
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) −14.0000 −0.484780
\(835\) 36.0000 1.24583
\(836\) 2.00000 0.0691714
\(837\) −4.00000 −0.138260
\(838\) −24.0000 −0.829066
\(839\) −33.0000 −1.13929 −0.569643 0.821892i \(-0.692919\pi\)
−0.569643 + 0.821892i \(0.692919\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −20.0000 −0.689246
\(843\) −3.00000 −0.103325
\(844\) −22.0000 −0.757271
\(845\) −27.0000 −0.928828
\(846\) −9.00000 −0.309426
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 14.0000 0.480479
\(850\) 12.0000 0.411597
\(851\) 6.00000 0.205677
\(852\) −12.0000 −0.411113
\(853\) −17.0000 −0.582069 −0.291034 0.956713i \(-0.593999\pi\)
−0.291034 + 0.956713i \(0.593999\pi\)
\(854\) 0 0
\(855\) −6.00000 −0.205196
\(856\) −3.00000 −0.102538
\(857\) 21.0000 0.717346 0.358673 0.933463i \(-0.383229\pi\)
0.358673 + 0.933463i \(0.383229\pi\)
\(858\) 2.00000 0.0682789
\(859\) −35.0000 −1.19418 −0.597092 0.802173i \(-0.703677\pi\)
−0.597092 + 0.802173i \(0.703677\pi\)
\(860\) 6.00000 0.204598
\(861\) 0 0
\(862\) 6.00000 0.204361
\(863\) 39.0000 1.32758 0.663788 0.747921i \(-0.268948\pi\)
0.663788 + 0.747921i \(0.268948\pi\)
\(864\) 1.00000 0.0340207
\(865\) −36.0000 −1.22404
\(866\) 35.0000 1.18935
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −17.0000 −0.576686
\(870\) −18.0000 −0.610257
\(871\) −10.0000 −0.338837
\(872\) −11.0000 −0.372507
\(873\) −17.0000 −0.575363
\(874\) 6.00000 0.202953
\(875\) 0 0
\(876\) −16.0000 −0.540590
\(877\) −37.0000 −1.24940 −0.624701 0.780864i \(-0.714779\pi\)
−0.624701 + 0.780864i \(0.714779\pi\)
\(878\) −19.0000 −0.641219
\(879\) −24.0000 −0.809500
\(880\) −3.00000 −0.101130
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) −13.0000 −0.437485 −0.218742 0.975783i \(-0.570195\pi\)
−0.218742 + 0.975783i \(0.570195\pi\)
\(884\) 6.00000 0.201802
\(885\) −36.0000 −1.21013
\(886\) −18.0000 −0.604722
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) −18.0000 −0.603361
\(891\) −1.00000 −0.0335013
\(892\) −20.0000 −0.669650
\(893\) −18.0000 −0.602347
\(894\) 6.00000 0.200670
\(895\) 36.0000 1.20335
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) −30.0000 −1.00111
\(899\) −24.0000 −0.800445
\(900\) 4.00000 0.133333
\(901\) −18.0000 −0.599667
\(902\) 3.00000 0.0998891
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 30.0000 0.997234
\(906\) −1.00000 −0.0332228
\(907\) 47.0000 1.56061 0.780305 0.625400i \(-0.215064\pi\)
0.780305 + 0.625400i \(0.215064\pi\)
\(908\) 21.0000 0.696909
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −27.0000 −0.894550 −0.447275 0.894397i \(-0.647605\pi\)
−0.447275 + 0.894397i \(0.647605\pi\)
\(912\) 2.00000 0.0662266
\(913\) 9.00000 0.297857
\(914\) −38.0000 −1.25693
\(915\) 15.0000 0.495885
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) −3.00000 −0.0990148
\(919\) −7.00000 −0.230909 −0.115454 0.993313i \(-0.536832\pi\)
−0.115454 + 0.993313i \(0.536832\pi\)
\(920\) −9.00000 −0.296721
\(921\) −22.0000 −0.724925
\(922\) 6.00000 0.197599
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 16.0000 0.525793
\(927\) −14.0000 −0.459820
\(928\) 6.00000 0.196960
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 12.0000 0.393496
\(931\) 0 0
\(932\) 27.0000 0.884414
\(933\) 21.0000 0.687509
\(934\) 0 0
\(935\) 9.00000 0.294331
\(936\) 2.00000 0.0653720
\(937\) 34.0000 1.11073 0.555366 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 0 0
\(939\) 2.00000 0.0652675
\(940\) 27.0000 0.880643
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −20.0000 −0.651635
\(943\) 9.00000 0.293080
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) −17.0000 −0.552134
\(949\) −32.0000 −1.03876
\(950\) 8.00000 0.259554
\(951\) 33.0000 1.07010
\(952\) 0 0
\(953\) 27.0000 0.874616 0.437308 0.899312i \(-0.355932\pi\)
0.437308 + 0.899312i \(0.355932\pi\)
\(954\) −6.00000 −0.194257
\(955\) 72.0000 2.32987
\(956\) 6.00000 0.194054
\(957\) −6.00000 −0.193952
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) −3.00000 −0.0968246
\(961\) −15.0000 −0.483871
\(962\) 4.00000 0.128965
\(963\) 3.00000 0.0966736
\(964\) 10.0000 0.322078
\(965\) 6.00000 0.193147
\(966\) 0 0
\(967\) −43.0000 −1.38279 −0.691393 0.722478i \(-0.743003\pi\)
−0.691393 + 0.722478i \(0.743003\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −6.00000 −0.192748
\(970\) 51.0000 1.63751
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 8.00000 0.256205
\(976\) −5.00000 −0.160046
\(977\) −36.0000 −1.15174 −0.575871 0.817541i \(-0.695337\pi\)
−0.575871 + 0.817541i \(0.695337\pi\)
\(978\) −1.00000 −0.0319765
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 11.0000 0.351203
\(982\) 3.00000 0.0957338
\(983\) 45.0000 1.43528 0.717639 0.696416i \(-0.245223\pi\)
0.717639 + 0.696416i \(0.245223\pi\)
\(984\) 3.00000 0.0956365
\(985\) 36.0000 1.14706
\(986\) −18.0000 −0.573237
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 6.00000 0.190789
\(990\) 3.00000 0.0953463
\(991\) −22.0000 −0.698853 −0.349427 0.936964i \(-0.613624\pi\)
−0.349427 + 0.936964i \(0.613624\pi\)
\(992\) −4.00000 −0.127000
\(993\) −11.0000 −0.349074
\(994\) 0 0
\(995\) −6.00000 −0.190213
\(996\) 9.00000 0.285176
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 28.0000 0.886325
\(999\) −2.00000 −0.0632772
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 3234.2.a.g.1.1 1
3.2 odd 2 9702.2.a.bd.1.1 1
7.3 odd 6 462.2.i.c.331.1 yes 2
7.5 odd 6 462.2.i.c.67.1 2
7.6 odd 2 3234.2.a.h.1.1 1
21.5 even 6 1386.2.k.c.991.1 2
21.17 even 6 1386.2.k.c.793.1 2
21.20 even 2 9702.2.a.cf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.c.67.1 2 7.5 odd 6
462.2.i.c.331.1 yes 2 7.3 odd 6
1386.2.k.c.793.1 2 21.17 even 6
1386.2.k.c.991.1 2 21.5 even 6
3234.2.a.g.1.1 1 1.1 even 1 trivial
3234.2.a.h.1.1 1 7.6 odd 2
9702.2.a.bd.1.1 1 3.2 odd 2
9702.2.a.cf.1.1 1 21.20 even 2