Properties

Label 3234.2.a.h.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 \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) −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 \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −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 \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −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 \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 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 \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 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 \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −3.00000 −0.387298
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 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 \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −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 \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 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 \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 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 \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 4.00000 0.400000
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) −3.00000 −0.297044
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −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 \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −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 \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 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 \(-0.205846\pi\)
0.798087 + 0.602542i \(0.205846\pi\)
\(158\) −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 \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 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 \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 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 \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 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 \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) −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 \(-0.266444\pi\)
0.669650 + 0.742677i \(0.266444\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 6.00000 0.399114
\(227\) −21.0000 −1.39382 −0.696909 0.717159i \(-0.745442\pi\)
−0.696909 + 0.717159i \(0.745442\pi\)
\(228\) 2.00000 0.132453
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 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 \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −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 \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 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 \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −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 \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) 3.00000 0.182574
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 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 \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 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 \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 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 \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) −12.0000 −0.681554
\(311\) 21.0000 1.19080 0.595400 0.803429i \(-0.296993\pi\)
0.595400 + 0.803429i \(0.296993\pi\)
\(312\) −2.00000 −0.113228
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −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 \(-0.404877\pi\)
0.294408 + 0.955680i \(0.404877\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 1.00000 0.0533002
\(353\) −12.0000 −0.638696 −0.319348 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 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 \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 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 \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −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 \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −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 \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −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 \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 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 \(-0.181970\pi\)
0.840996 + 0.541041i \(0.181970\pi\)
\(434\) 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 \(-0.649793\pi\)
−0.453410 + 0.891302i \(0.649793\pi\)
\(440\) −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 \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 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 \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 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 \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 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 \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 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 \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 6.00000 0.262613
\(523\) 26.0000 1.13690 0.568450 0.822718i \(-0.307543\pi\)
0.568450 + 0.822718i \(0.307543\pi\)
\(524\) −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 \(-0.384869\pi\)
0.353860 + 0.935299i \(0.384869\pi\)
\(578\) 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 \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 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 \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 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 \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 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 \(-0.506460\pi\)
−0.0202944 + 0.999794i \(0.506460\pi\)
\(608\) −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 \(-0.429047\pi\)
0.221064 + 0.975259i \(0.429047\pi\)
\(620\) 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 \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) −6.00000 −0.236250
\(646\) −6.00000 −0.236067
\(647\) 39.0000 1.53325 0.766624 0.642096i \(-0.221935\pi\)
0.766624 + 0.642096i \(0.221935\pi\)
\(648\) −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 \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) −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 \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 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 \(-0.617700\pi\)
−0.361397 + 0.932412i \(0.617700\pi\)
\(692\) 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 \(-0.628073\pi\)
−0.391584 + 0.920142i \(0.628073\pi\)
\(720\) −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 \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 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 \(-0.860076\pi\)
−0.904928 + 0.425564i \(0.860076\pi\)
\(734\) 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 \(-0.412356\pi\)
0.271875 + 0.962333i \(0.412356\pi\)
\(762\) −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 \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 2.00000 0.0719816
\(773\) −27.0000 −0.971123 −0.485561 0.874203i \(-0.661385\pi\)
−0.485561 + 0.874203i \(0.661385\pi\)
\(774\) −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 \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 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 \(-0.658709\pi\)
−0.478195 + 0.878254i \(0.658709\pi\)
\(798\) 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 \(-0.349106\pi\)
0.456492 + 0.889728i \(0.349106\pi\)
\(812\) 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 \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 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 \(-0.307081\pi\)
0.569643 + 0.821892i \(0.307081\pi\)
\(840\) 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 \(-0.406001\pi\)
0.291034 + 0.956713i \(0.406001\pi\)
\(854\) 0 0
\(855\) −6.00000 −0.205196
\(856\) −3.00000 −0.102538
\(857\) −21.0000 −0.717346 −0.358673 0.933463i \(-0.616771\pi\)
−0.358673 + 0.933463i \(0.616771\pi\)
\(858\) 2.00000 0.0682789
\(859\) 35.0000 1.19418 0.597092 0.802173i \(-0.296323\pi\)
0.597092 + 0.802173i \(0.296323\pi\)
\(860\) −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 \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 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 \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) −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 \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) −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 \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) 2.00000 0.0652675
\(940\) 27.0000 0.880643
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) −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 \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) 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 \(-0.754777\pi\)
−0.717639 + 0.696416i \(0.754777\pi\)
\(984\) 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 \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 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.h.1.1 1
3.2 odd 2 9702.2.a.cf.1.1 1
7.2 even 3 462.2.i.c.67.1 2
7.4 even 3 462.2.i.c.331.1 yes 2
7.6 odd 2 3234.2.a.g.1.1 1
21.2 odd 6 1386.2.k.c.991.1 2
21.11 odd 6 1386.2.k.c.793.1 2
21.20 even 2 9702.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.c.67.1 2 7.2 even 3
462.2.i.c.331.1 yes 2 7.4 even 3
1386.2.k.c.793.1 2 21.11 odd 6
1386.2.k.c.991.1 2 21.2 odd 6
3234.2.a.g.1.1 1 7.6 odd 2
3234.2.a.h.1.1 1 1.1 even 1 trivial
9702.2.a.bd.1.1 1 21.20 even 2
9702.2.a.cf.1.1 1 3.2 odd 2