Properties

Label 3822.2.a.be.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $1$
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] = [3822,2,Mod(1,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, 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("3822.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.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: \(30.5188236525\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3822.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} +3.00000 q^{29} -1.00000 q^{30} -11.0000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} -4.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} +12.0000 q^{41} -8.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} -6.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} -6.00000 q^{51} -1.00000 q^{52} -5.00000 q^{53} +1.00000 q^{54} +1.00000 q^{55} -4.00000 q^{57} +3.00000 q^{58} -5.00000 q^{59} -1.00000 q^{60} +12.0000 q^{61} -11.0000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -1.00000 q^{66} +16.0000 q^{67} -6.00000 q^{68} -6.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +4.00000 q^{74} -4.00000 q^{75} -4.00000 q^{76} -1.00000 q^{78} +7.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} -17.0000 q^{83} +6.00000 q^{85} -8.00000 q^{86} +3.00000 q^{87} -1.00000 q^{88} -12.0000 q^{89} -1.00000 q^{90} -6.00000 q^{92} -11.0000 q^{93} -8.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} +13.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.00000 −0.800000
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) −1.00000 −0.182574
\(31\) −11.0000 −1.97566 −0.987829 0.155543i \(-0.950287\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −4.00000 −0.648886
\(39\) −1.00000 −0.160128
\(40\) −1.00000 −0.158114
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) −6.00000 −0.884652
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) −6.00000 −0.840168
\(52\) −1.00000 −0.138675
\(53\) −5.00000 −0.686803 −0.343401 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) 1.00000 0.136083
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 3.00000 0.393919
\(59\) −5.00000 −0.650945 −0.325472 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) −1.00000 −0.129099
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) −11.0000 −1.39700
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) −1.00000 −0.123091
\(67\) 16.0000 1.95471 0.977356 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) −6.00000 −0.727607
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 4.00000 0.464991
\(75\) −4.00000 −0.461880
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) 7.00000 0.787562 0.393781 0.919204i \(-0.371167\pi\)
0.393781 + 0.919204i \(0.371167\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) −17.0000 −1.86599 −0.932996 0.359886i \(-0.882816\pi\)
−0.932996 + 0.359886i \(0.882816\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) −8.00000 −0.862662
\(87\) 3.00000 0.321634
\(88\) −1.00000 −0.106600
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) −11.0000 −1.14065
\(94\) −8.00000 −0.825137
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) 13.0000 1.31995 0.659975 0.751288i \(-0.270567\pi\)
0.659975 + 0.751288i \(0.270567\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −4.00000 −0.400000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −6.00000 −0.594089
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −5.00000 −0.485643
\(107\) 7.00000 0.676716 0.338358 0.941018i \(-0.390129\pi\)
0.338358 + 0.941018i \(0.390129\pi\)
\(108\) 1.00000 0.0962250
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 1.00000 0.0953463
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) −4.00000 −0.374634
\(115\) 6.00000 0.559503
\(116\) 3.00000 0.278543
\(117\) −1.00000 −0.0924500
\(118\) −5.00000 −0.460287
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) −10.0000 −0.909091
\(122\) 12.0000 1.08643
\(123\) 12.0000 1.08200
\(124\) −11.0000 −0.987829
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) 1.00000 0.0877058
\(131\) 1.00000 0.0873704 0.0436852 0.999045i \(-0.486090\pi\)
0.0436852 + 0.999045i \(0.486090\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) 16.0000 1.38219
\(135\) −1.00000 −0.0860663
\(136\) −6.00000 −0.514496
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) −6.00000 −0.510754
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 6.00000 0.503509
\(143\) 1.00000 0.0836242
\(144\) 1.00000 0.0833333
\(145\) −3.00000 −0.249136
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −4.00000 −0.326599
\(151\) 13.0000 1.05792 0.528962 0.848645i \(-0.322581\pi\)
0.528962 + 0.848645i \(0.322581\pi\)
\(152\) −4.00000 −0.324443
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 11.0000 0.883541
\(156\) −1.00000 −0.0800641
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 7.00000 0.556890
\(159\) −5.00000 −0.396526
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 12.0000 0.937043
\(165\) 1.00000 0.0778499
\(166\) −17.0000 −1.31946
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 6.00000 0.460179
\(171\) −4.00000 −0.305888
\(172\) −8.00000 −0.609994
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 3.00000 0.227429
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −5.00000 −0.375823
\(178\) −12.0000 −0.899438
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 12.0000 0.887066
\(184\) −6.00000 −0.442326
\(185\) −4.00000 −0.294086
\(186\) −11.0000 −0.806559
\(187\) 6.00000 0.438763
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 1.00000 0.0721688
\(193\) 15.0000 1.07972 0.539862 0.841754i \(-0.318476\pi\)
0.539862 + 0.841754i \(0.318476\pi\)
\(194\) 13.0000 0.933346
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −4.00000 −0.282843
\(201\) 16.0000 1.12855
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) −1.00000 −0.0693375
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −5.00000 −0.343401
\(213\) 6.00000 0.411113
\(214\) 7.00000 0.478510
\(215\) 8.00000 0.545595
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 12.0000 0.812743
\(219\) −10.0000 −0.675737
\(220\) 1.00000 0.0674200
\(221\) 6.00000 0.403604
\(222\) 4.00000 0.268462
\(223\) 7.00000 0.468755 0.234377 0.972146i \(-0.424695\pi\)
0.234377 + 0.972146i \(0.424695\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) −4.00000 −0.266076
\(227\) −15.0000 −0.995585 −0.497792 0.867296i \(-0.665856\pi\)
−0.497792 + 0.867296i \(0.665856\pi\)
\(228\) −4.00000 −0.264906
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 8.00000 0.521862
\(236\) −5.00000 −0.325472
\(237\) 7.00000 0.454699
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −15.0000 −0.966235 −0.483117 0.875556i \(-0.660496\pi\)
−0.483117 + 0.875556i \(0.660496\pi\)
\(242\) −10.0000 −0.642824
\(243\) 1.00000 0.0641500
\(244\) 12.0000 0.768221
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) 4.00000 0.254514
\(248\) −11.0000 −0.698501
\(249\) −17.0000 −1.07733
\(250\) 9.00000 0.569210
\(251\) 17.0000 1.07303 0.536515 0.843891i \(-0.319740\pi\)
0.536515 + 0.843891i \(0.319740\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) −7.00000 −0.439219
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −8.00000 −0.498058
\(259\) 0 0
\(260\) 1.00000 0.0620174
\(261\) 3.00000 0.185695
\(262\) 1.00000 0.0617802
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 5.00000 0.307148
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 16.0000 0.977356
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −7.00000 −0.425220 −0.212610 0.977137i \(-0.568196\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −8.00000 −0.483298
\(275\) 4.00000 0.241209
\(276\) −6.00000 −0.361158
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) −11.0000 −0.658553
\(280\) 0 0
\(281\) 4.00000 0.238620 0.119310 0.992857i \(-0.461932\pi\)
0.119310 + 0.992857i \(0.461932\pi\)
\(282\) −8.00000 −0.476393
\(283\) 10.0000 0.594438 0.297219 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) 6.00000 0.356034
\(285\) 4.00000 0.236940
\(286\) 1.00000 0.0591312
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −3.00000 −0.176166
\(291\) 13.0000 0.762073
\(292\) −10.0000 −0.585206
\(293\) −7.00000 −0.408944 −0.204472 0.978872i \(-0.565548\pi\)
−0.204472 + 0.978872i \(0.565548\pi\)
\(294\) 0 0
\(295\) 5.00000 0.291111
\(296\) 4.00000 0.232495
\(297\) −1.00000 −0.0580259
\(298\) −10.0000 −0.579284
\(299\) 6.00000 0.346989
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) 13.0000 0.748066
\(303\) 10.0000 0.574485
\(304\) −4.00000 −0.229416
\(305\) −12.0000 −0.687118
\(306\) −6.00000 −0.342997
\(307\) 6.00000 0.342438 0.171219 0.985233i \(-0.445229\pi\)
0.171219 + 0.985233i \(0.445229\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 11.0000 0.624758
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 13.0000 0.734803 0.367402 0.930062i \(-0.380247\pi\)
0.367402 + 0.930062i \(0.380247\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 7.00000 0.393781
\(317\) −3.00000 −0.168497 −0.0842484 0.996445i \(-0.526849\pi\)
−0.0842484 + 0.996445i \(0.526849\pi\)
\(318\) −5.00000 −0.280386
\(319\) −3.00000 −0.167968
\(320\) −1.00000 −0.0559017
\(321\) 7.00000 0.390702
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) −8.00000 −0.443079
\(327\) 12.0000 0.663602
\(328\) 12.0000 0.662589
\(329\) 0 0
\(330\) 1.00000 0.0550482
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −17.0000 −0.932996
\(333\) 4.00000 0.219199
\(334\) −24.0000 −1.31322
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) 25.0000 1.36184 0.680918 0.732359i \(-0.261581\pi\)
0.680918 + 0.732359i \(0.261581\pi\)
\(338\) 1.00000 0.0543928
\(339\) −4.00000 −0.217250
\(340\) 6.00000 0.325396
\(341\) 11.0000 0.595683
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) 6.00000 0.323029
\(346\) 6.00000 0.322562
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 3.00000 0.160817
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −1.00000 −0.0533002
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) −5.00000 −0.265747
\(355\) −6.00000 −0.318447
\(356\) −12.0000 −0.635999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −22.0000 −1.16112 −0.580558 0.814219i \(-0.697165\pi\)
−0.580558 + 0.814219i \(0.697165\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) −16.0000 −0.840941
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) 12.0000 0.627250
\(367\) −7.00000 −0.365397 −0.182699 0.983169i \(-0.558483\pi\)
−0.182699 + 0.983169i \(0.558483\pi\)
\(368\) −6.00000 −0.312772
\(369\) 12.0000 0.624695
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) −11.0000 −0.570323
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 6.00000 0.310253
\(375\) 9.00000 0.464758
\(376\) −8.00000 −0.412568
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 4.00000 0.205196
\(381\) −7.00000 −0.358621
\(382\) −6.00000 −0.306987
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 15.0000 0.763480
\(387\) −8.00000 −0.406663
\(388\) 13.0000 0.659975
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 1.00000 0.0506370
\(391\) 36.0000 1.82060
\(392\) 0 0
\(393\) 1.00000 0.0504433
\(394\) −18.0000 −0.906827
\(395\) −7.00000 −0.352208
\(396\) −1.00000 −0.0502519
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 16.0000 0.798007
\(403\) 11.0000 0.547949
\(404\) 10.0000 0.497519
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) −6.00000 −0.297044
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) −12.0000 −0.592638
\(411\) −8.00000 −0.394611
\(412\) 0 0
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) 17.0000 0.834497
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 4.00000 0.195646
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) −36.0000 −1.75453 −0.877266 0.480004i \(-0.840635\pi\)
−0.877266 + 0.480004i \(0.840635\pi\)
\(422\) 12.0000 0.584151
\(423\) −8.00000 −0.388973
\(424\) −5.00000 −0.242821
\(425\) 24.0000 1.16417
\(426\) 6.00000 0.290701
\(427\) 0 0
\(428\) 7.00000 0.338358
\(429\) 1.00000 0.0482805
\(430\) 8.00000 0.385794
\(431\) −38.0000 −1.83040 −0.915198 0.403005i \(-0.867966\pi\)
−0.915198 + 0.403005i \(0.867966\pi\)
\(432\) 1.00000 0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −3.00000 −0.143839
\(436\) 12.0000 0.574696
\(437\) 24.0000 1.14808
\(438\) −10.0000 −0.477818
\(439\) −17.0000 −0.811366 −0.405683 0.914014i \(-0.632966\pi\)
−0.405683 + 0.914014i \(0.632966\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 6.00000 0.285391
\(443\) −39.0000 −1.85295 −0.926473 0.376361i \(-0.877175\pi\)
−0.926473 + 0.376361i \(0.877175\pi\)
\(444\) 4.00000 0.189832
\(445\) 12.0000 0.568855
\(446\) 7.00000 0.331460
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) −4.00000 −0.188562
\(451\) −12.0000 −0.565058
\(452\) −4.00000 −0.188144
\(453\) 13.0000 0.610793
\(454\) −15.0000 −0.703985
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) 6.00000 0.280362
\(459\) −6.00000 −0.280056
\(460\) 6.00000 0.279751
\(461\) 34.0000 1.58354 0.791769 0.610821i \(-0.209160\pi\)
0.791769 + 0.610821i \(0.209160\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 3.00000 0.139272
\(465\) 11.0000 0.510113
\(466\) 6.00000 0.277945
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 8.00000 0.369012
\(471\) −18.0000 −0.829396
\(472\) −5.00000 −0.230144
\(473\) 8.00000 0.367840
\(474\) 7.00000 0.321521
\(475\) 16.0000 0.734130
\(476\) 0 0
\(477\) −5.00000 −0.228934
\(478\) −6.00000 −0.274434
\(479\) 14.0000 0.639676 0.319838 0.947472i \(-0.396371\pi\)
0.319838 + 0.947472i \(0.396371\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −4.00000 −0.182384
\(482\) −15.0000 −0.683231
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) −13.0000 −0.590300
\(486\) 1.00000 0.0453609
\(487\) 23.0000 1.04223 0.521115 0.853487i \(-0.325516\pi\)
0.521115 + 0.853487i \(0.325516\pi\)
\(488\) 12.0000 0.543214
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −35.0000 −1.57953 −0.789764 0.613411i \(-0.789797\pi\)
−0.789764 + 0.613411i \(0.789797\pi\)
\(492\) 12.0000 0.541002
\(493\) −18.0000 −0.810679
\(494\) 4.00000 0.179969
\(495\) 1.00000 0.0449467
\(496\) −11.0000 −0.493915
\(497\) 0 0
\(498\) −17.0000 −0.761788
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 9.00000 0.402492
\(501\) −24.0000 −1.07224
\(502\) 17.0000 0.758747
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) 6.00000 0.266733
\(507\) 1.00000 0.0444116
\(508\) −7.00000 −0.310575
\(509\) 13.0000 0.576215 0.288107 0.957598i \(-0.406974\pi\)
0.288107 + 0.957598i \(0.406974\pi\)
\(510\) 6.00000 0.265684
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 1.00000 0.0438529
\(521\) 28.0000 1.22670 0.613351 0.789810i \(-0.289821\pi\)
0.613351 + 0.789810i \(0.289821\pi\)
\(522\) 3.00000 0.131306
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 1.00000 0.0436852
\(525\) 0 0
\(526\) −18.0000 −0.784837
\(527\) 66.0000 2.87501
\(528\) −1.00000 −0.0435194
\(529\) 13.0000 0.565217
\(530\) 5.00000 0.217186
\(531\) −5.00000 −0.216982
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) −12.0000 −0.519291
\(535\) −7.00000 −0.302636
\(536\) 16.0000 0.691095
\(537\) 12.0000 0.517838
\(538\) −9.00000 −0.388018
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) −7.00000 −0.300676
\(543\) −16.0000 −0.686626
\(544\) −6.00000 −0.257248
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) −8.00000 −0.341743
\(549\) 12.0000 0.512148
\(550\) 4.00000 0.170561
\(551\) −12.0000 −0.511217
\(552\) −6.00000 −0.255377
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) −4.00000 −0.169791
\(556\) 0 0
\(557\) −17.0000 −0.720313 −0.360157 0.932892i \(-0.617277\pi\)
−0.360157 + 0.932892i \(0.617277\pi\)
\(558\) −11.0000 −0.465667
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 4.00000 0.168730
\(563\) 21.0000 0.885044 0.442522 0.896758i \(-0.354084\pi\)
0.442522 + 0.896758i \(0.354084\pi\)
\(564\) −8.00000 −0.336861
\(565\) 4.00000 0.168281
\(566\) 10.0000 0.420331
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) −4.00000 −0.167689 −0.0838444 0.996479i \(-0.526720\pi\)
−0.0838444 + 0.996479i \(0.526720\pi\)
\(570\) 4.00000 0.167542
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 1.00000 0.0418121
\(573\) −6.00000 −0.250654
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 1.00000 0.0416667
\(577\) 1.00000 0.0416305 0.0208153 0.999783i \(-0.493374\pi\)
0.0208153 + 0.999783i \(0.493374\pi\)
\(578\) 19.0000 0.790296
\(579\) 15.0000 0.623379
\(580\) −3.00000 −0.124568
\(581\) 0 0
\(582\) 13.0000 0.538867
\(583\) 5.00000 0.207079
\(584\) −10.0000 −0.413803
\(585\) 1.00000 0.0413449
\(586\) −7.00000 −0.289167
\(587\) 5.00000 0.206372 0.103186 0.994662i \(-0.467096\pi\)
0.103186 + 0.994662i \(0.467096\pi\)
\(588\) 0 0
\(589\) 44.0000 1.81299
\(590\) 5.00000 0.205847
\(591\) −18.0000 −0.740421
\(592\) 4.00000 0.164399
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 8.00000 0.327418
\(598\) 6.00000 0.245358
\(599\) −10.0000 −0.408589 −0.204294 0.978909i \(-0.565490\pi\)
−0.204294 + 0.978909i \(0.565490\pi\)
\(600\) −4.00000 −0.163299
\(601\) 11.0000 0.448699 0.224350 0.974509i \(-0.427974\pi\)
0.224350 + 0.974509i \(0.427974\pi\)
\(602\) 0 0
\(603\) 16.0000 0.651570
\(604\) 13.0000 0.528962
\(605\) 10.0000 0.406558
\(606\) 10.0000 0.406222
\(607\) 29.0000 1.17707 0.588537 0.808470i \(-0.299704\pi\)
0.588537 + 0.808470i \(0.299704\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) −12.0000 −0.485866
\(611\) 8.00000 0.323645
\(612\) −6.00000 −0.242536
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 6.00000 0.242140
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) −36.0000 −1.44931 −0.724653 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 0 0
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 11.0000 0.441771
\(621\) −6.00000 −0.240772
\(622\) 2.00000 0.0801927
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 11.0000 0.440000
\(626\) 13.0000 0.519584
\(627\) 4.00000 0.159745
\(628\) −18.0000 −0.718278
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −29.0000 −1.15447 −0.577236 0.816577i \(-0.695869\pi\)
−0.577236 + 0.816577i \(0.695869\pi\)
\(632\) 7.00000 0.278445
\(633\) 12.0000 0.476957
\(634\) −3.00000 −0.119145
\(635\) 7.00000 0.277787
\(636\) −5.00000 −0.198263
\(637\) 0 0
\(638\) −3.00000 −0.118771
\(639\) 6.00000 0.237356
\(640\) −1.00000 −0.0395285
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 7.00000 0.276268
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 24.0000 0.944267
\(647\) −4.00000 −0.157256 −0.0786281 0.996904i \(-0.525054\pi\)
−0.0786281 + 0.996904i \(0.525054\pi\)
\(648\) 1.00000 0.0392837
\(649\) 5.00000 0.196267
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) −11.0000 −0.430463 −0.215232 0.976563i \(-0.569051\pi\)
−0.215232 + 0.976563i \(0.569051\pi\)
\(654\) 12.0000 0.469237
\(655\) −1.00000 −0.0390732
\(656\) 12.0000 0.468521
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 1.00000 0.0389249
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −8.00000 −0.310929
\(663\) 6.00000 0.233021
\(664\) −17.0000 −0.659728
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) −18.0000 −0.696963
\(668\) −24.0000 −0.928588
\(669\) 7.00000 0.270636
\(670\) −16.0000 −0.618134
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −39.0000 −1.50334 −0.751670 0.659540i \(-0.770751\pi\)
−0.751670 + 0.659540i \(0.770751\pi\)
\(674\) 25.0000 0.962964
\(675\) −4.00000 −0.153960
\(676\) 1.00000 0.0384615
\(677\) −35.0000 −1.34516 −0.672580 0.740025i \(-0.734814\pi\)
−0.672580 + 0.740025i \(0.734814\pi\)
\(678\) −4.00000 −0.153619
\(679\) 0 0
\(680\) 6.00000 0.230089
\(681\) −15.0000 −0.574801
\(682\) 11.0000 0.421212
\(683\) −27.0000 −1.03313 −0.516563 0.856249i \(-0.672789\pi\)
−0.516563 + 0.856249i \(0.672789\pi\)
\(684\) −4.00000 −0.152944
\(685\) 8.00000 0.305664
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) −8.00000 −0.304997
\(689\) 5.00000 0.190485
\(690\) 6.00000 0.228416
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 0 0
\(696\) 3.00000 0.113715
\(697\) −72.0000 −2.72719
\(698\) 14.0000 0.529908
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −13.0000 −0.491003 −0.245502 0.969396i \(-0.578953\pi\)
−0.245502 + 0.969396i \(0.578953\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −16.0000 −0.603451
\(704\) −1.00000 −0.0376889
\(705\) 8.00000 0.301297
\(706\) 30.0000 1.12906
\(707\) 0 0
\(708\) −5.00000 −0.187912
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −6.00000 −0.225176
\(711\) 7.00000 0.262521
\(712\) −12.0000 −0.449719
\(713\) 66.0000 2.47172
\(714\) 0 0
\(715\) −1.00000 −0.0373979
\(716\) 12.0000 0.448461
\(717\) −6.00000 −0.224074
\(718\) −22.0000 −0.821033
\(719\) 46.0000 1.71551 0.857755 0.514058i \(-0.171858\pi\)
0.857755 + 0.514058i \(0.171858\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −15.0000 −0.557856
\(724\) −16.0000 −0.594635
\(725\) −12.0000 −0.445669
\(726\) −10.0000 −0.371135
\(727\) 19.0000 0.704671 0.352335 0.935874i \(-0.385388\pi\)
0.352335 + 0.935874i \(0.385388\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) 48.0000 1.77534
\(732\) 12.0000 0.443533
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) −7.00000 −0.258375
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −16.0000 −0.589368
\(738\) 12.0000 0.441726
\(739\) −26.0000 −0.956425 −0.478213 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) −4.00000 −0.147043
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) −32.0000 −1.17397 −0.586983 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) −11.0000 −0.403280
\(745\) 10.0000 0.366372
\(746\) 26.0000 0.951928
\(747\) −17.0000 −0.621997
\(748\) 6.00000 0.219382
\(749\) 0 0
\(750\) 9.00000 0.328634
\(751\) 25.0000 0.912263 0.456131 0.889912i \(-0.349235\pi\)
0.456131 + 0.889912i \(0.349235\pi\)
\(752\) −8.00000 −0.291730
\(753\) 17.0000 0.619514
\(754\) −3.00000 −0.109254
\(755\) −13.0000 −0.473118
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 16.0000 0.581146
\(759\) 6.00000 0.217786
\(760\) 4.00000 0.145095
\(761\) −28.0000 −1.01500 −0.507500 0.861652i \(-0.669430\pi\)
−0.507500 + 0.861652i \(0.669430\pi\)
\(762\) −7.00000 −0.253583
\(763\) 0 0
\(764\) −6.00000 −0.217072
\(765\) 6.00000 0.216930
\(766\) −14.0000 −0.505841
\(767\) 5.00000 0.180540
\(768\) 1.00000 0.0360844
\(769\) 31.0000 1.11789 0.558944 0.829205i \(-0.311207\pi\)
0.558944 + 0.829205i \(0.311207\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 15.0000 0.539862
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) −8.00000 −0.287554
\(775\) 44.0000 1.58053
\(776\) 13.0000 0.466673
\(777\) 0 0
\(778\) 22.0000 0.788738
\(779\) −48.0000 −1.71978
\(780\) 1.00000 0.0358057
\(781\) −6.00000 −0.214697
\(782\) 36.0000 1.28736
\(783\) 3.00000 0.107211
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 1.00000 0.0356688
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) −18.0000 −0.641223
\(789\) −18.0000 −0.640817
\(790\) −7.00000 −0.249049
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) −12.0000 −0.426132
\(794\) −18.0000 −0.638796
\(795\) 5.00000 0.177332
\(796\) 8.00000 0.283552
\(797\) −3.00000 −0.106265 −0.0531327 0.998587i \(-0.516921\pi\)
−0.0531327 + 0.998587i \(0.516921\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) −4.00000 −0.141421
\(801\) −12.0000 −0.423999
\(802\) −18.0000 −0.635602
\(803\) 10.0000 0.352892
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 11.0000 0.387458
\(807\) −9.00000 −0.316815
\(808\) 10.0000 0.351799
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 0 0
\(813\) −7.00000 −0.245501
\(814\) −4.00000 −0.140200
\(815\) 8.00000 0.280228
\(816\) −6.00000 −0.210042
\(817\) 32.0000 1.11954
\(818\) 25.0000 0.874105
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) −11.0000 −0.383903 −0.191951 0.981404i \(-0.561482\pi\)
−0.191951 + 0.981404i \(0.561482\pi\)
\(822\) −8.00000 −0.279032
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −25.0000 −0.869335 −0.434668 0.900591i \(-0.643134\pi\)
−0.434668 + 0.900591i \(0.643134\pi\)
\(828\) −6.00000 −0.208514
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 17.0000 0.590079
\(831\) −10.0000 −0.346896
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) 0 0
\(835\) 24.0000 0.830554
\(836\) 4.00000 0.138343
\(837\) −11.0000 −0.380216
\(838\) 16.0000 0.552711
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −36.0000 −1.24064
\(843\) 4.00000 0.137767
\(844\) 12.0000 0.413057
\(845\) −1.00000 −0.0344010
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) −5.00000 −0.171701
\(849\) 10.0000 0.343199
\(850\) 24.0000 0.823193
\(851\) −24.0000 −0.822709
\(852\) 6.00000 0.205557
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 7.00000 0.239255
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 1.00000 0.0341394
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) −38.0000 −1.29429
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.00000 −0.204006
\(866\) −14.0000 −0.475739
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) −7.00000 −0.237459
\(870\) −3.00000 −0.101710
\(871\) −16.0000 −0.542139
\(872\) 12.0000 0.406371
\(873\) 13.0000 0.439983
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) −17.0000 −0.573722
\(879\) −7.00000 −0.236104
\(880\) 1.00000 0.0337100
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) 30.0000 1.00958 0.504790 0.863242i \(-0.331570\pi\)
0.504790 + 0.863242i \(0.331570\pi\)
\(884\) 6.00000 0.201802
\(885\) 5.00000 0.168073
\(886\) −39.0000 −1.31023
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 4.00000 0.134231
\(889\) 0 0
\(890\) 12.0000 0.402241
\(891\) −1.00000 −0.0335013
\(892\) 7.00000 0.234377
\(893\) 32.0000 1.07084
\(894\) −10.0000 −0.334450
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) 20.0000 0.667409
\(899\) −33.0000 −1.10061
\(900\) −4.00000 −0.133333
\(901\) 30.0000 0.999445
\(902\) −12.0000 −0.399556
\(903\) 0 0
\(904\) −4.00000 −0.133038
\(905\) 16.0000 0.531858
\(906\) 13.0000 0.431896
\(907\) −36.0000 −1.19536 −0.597680 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) −15.0000 −0.497792
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −10.0000 −0.331315 −0.165657 0.986183i \(-0.552975\pi\)
−0.165657 + 0.986183i \(0.552975\pi\)
\(912\) −4.00000 −0.132453
\(913\) 17.0000 0.562618
\(914\) −11.0000 −0.363848
\(915\) −12.0000 −0.396708
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) −6.00000 −0.198030
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 6.00000 0.197814
\(921\) 6.00000 0.197707
\(922\) 34.0000 1.11973
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) −16.0000 −0.526077
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) 3.00000 0.0984798
\(929\) −20.0000 −0.656179 −0.328089 0.944647i \(-0.606405\pi\)
−0.328089 + 0.944647i \(0.606405\pi\)
\(930\) 11.0000 0.360704
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 2.00000 0.0654771
\(934\) 12.0000 0.392652
\(935\) −6.00000 −0.196221
\(936\) −1.00000 −0.0326860
\(937\) −13.0000 −0.424691 −0.212346 0.977195i \(-0.568110\pi\)
−0.212346 + 0.977195i \(0.568110\pi\)
\(938\) 0 0
\(939\) 13.0000 0.424239
\(940\) 8.00000 0.260931
\(941\) 15.0000 0.488986 0.244493 0.969651i \(-0.421378\pi\)
0.244493 + 0.969651i \(0.421378\pi\)
\(942\) −18.0000 −0.586472
\(943\) −72.0000 −2.34464
\(944\) −5.00000 −0.162736
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 7.00000 0.227349
\(949\) 10.0000 0.324614
\(950\) 16.0000 0.519109
\(951\) −3.00000 −0.0972817
\(952\) 0 0
\(953\) −44.0000 −1.42530 −0.712650 0.701520i \(-0.752505\pi\)
−0.712650 + 0.701520i \(0.752505\pi\)
\(954\) −5.00000 −0.161881
\(955\) 6.00000 0.194155
\(956\) −6.00000 −0.194054
\(957\) −3.00000 −0.0969762
\(958\) 14.0000 0.452319
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) 90.0000 2.90323
\(962\) −4.00000 −0.128965
\(963\) 7.00000 0.225572
\(964\) −15.0000 −0.483117
\(965\) −15.0000 −0.482867
\(966\) 0 0
\(967\) −11.0000 −0.353736 −0.176868 0.984235i \(-0.556597\pi\)
−0.176868 + 0.984235i \(0.556597\pi\)
\(968\) −10.0000 −0.321412
\(969\) 24.0000 0.770991
\(970\) −13.0000 −0.417405
\(971\) −29.0000 −0.930654 −0.465327 0.885139i \(-0.654063\pi\)
−0.465327 + 0.885139i \(0.654063\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 23.0000 0.736968
\(975\) 4.00000 0.128103
\(976\) 12.0000 0.384111
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) −8.00000 −0.255812
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) −35.0000 −1.11689
\(983\) −26.0000 −0.829271 −0.414636 0.909988i \(-0.636091\pi\)
−0.414636 + 0.909988i \(0.636091\pi\)
\(984\) 12.0000 0.382546
\(985\) 18.0000 0.573528
\(986\) −18.0000 −0.573237
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 48.0000 1.52631
\(990\) 1.00000 0.0317821
\(991\) −5.00000 −0.158830 −0.0794151 0.996842i \(-0.525305\pi\)
−0.0794151 + 0.996842i \(0.525305\pi\)
\(992\) −11.0000 −0.349250
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) −17.0000 −0.538666
\(997\) −28.0000 −0.886769 −0.443384 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) −28.0000 −0.886325
\(999\) 4.00000 0.126554
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 3822.2.a.be.1.1 1
7.2 even 3 546.2.i.b.235.1 yes 2
7.4 even 3 546.2.i.b.79.1 2
7.6 odd 2 3822.2.a.v.1.1 1
21.2 odd 6 1638.2.j.h.235.1 2
21.11 odd 6 1638.2.j.h.1171.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.b.79.1 2 7.4 even 3
546.2.i.b.235.1 yes 2 7.2 even 3
1638.2.j.h.235.1 2 21.2 odd 6
1638.2.j.h.1171.1 2 21.11 odd 6
3822.2.a.v.1.1 1 7.6 odd 2
3822.2.a.be.1.1 1 1.1 even 1 trivial