Properties

Label 3822.2.a.bh.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
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] = [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: \(0\)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +3.00000 q^{20} +3.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} -9.00000 q^{29} +3.00000 q^{30} +5.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} -4.00000 q^{38} +1.00000 q^{39} +3.00000 q^{40} -12.0000 q^{41} -4.00000 q^{43} +3.00000 q^{44} +3.00000 q^{45} +6.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +4.00000 q^{50} +6.00000 q^{51} +1.00000 q^{52} -9.00000 q^{53} +1.00000 q^{54} +9.00000 q^{55} -4.00000 q^{57} -9.00000 q^{58} -9.00000 q^{59} +3.00000 q^{60} +8.00000 q^{61} +5.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} +3.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} +6.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} +14.0000 q^{73} -4.00000 q^{74} +4.00000 q^{75} -4.00000 q^{76} +1.00000 q^{78} -1.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} +3.00000 q^{83} +18.0000 q^{85} -4.00000 q^{86} -9.00000 q^{87} +3.00000 q^{88} +3.00000 q^{90} +6.00000 q^{92} +5.00000 q^{93} -12.0000 q^{94} -12.0000 q^{95} +1.00000 q^{96} +5.00000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000 0.204124
\(25\) 4.00000 0.800000
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 3.00000 0.547723
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.00000 0.522233
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −4.00000 −0.648886
\(39\) 1.00000 0.160128
\(40\) 3.00000 0.474342
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.00000 0.452267
\(45\) 3.00000 0.447214
\(46\) 6.00000 0.884652
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) 6.00000 0.840168
\(52\) 1.00000 0.138675
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 1.00000 0.136083
\(55\) 9.00000 1.21356
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) −9.00000 −1.18176
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 3.00000 0.387298
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 5.00000 0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 3.00000 0.369274
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 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
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −4.00000 −0.464991
\(75\) 4.00000 0.461880
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) −12.0000 −1.32518
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 18.0000 1.95237
\(86\) −4.00000 −0.431331
\(87\) −9.00000 −0.964901
\(88\) 3.00000 0.319801
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 5.00000 0.518476
\(94\) −12.0000 −1.23771
\(95\) −12.0000 −1.23117
\(96\) 1.00000 0.102062
\(97\) 5.00000 0.507673 0.253837 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) 4.00000 0.400000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 6.00000 0.594089
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 1.00000 0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 9.00000 0.858116
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) −4.00000 −0.374634
\(115\) 18.0000 1.67851
\(116\) −9.00000 −0.835629
\(117\) 1.00000 0.0924500
\(118\) −9.00000 −0.828517
\(119\) 0 0
\(120\) 3.00000 0.273861
\(121\) −2.00000 −0.181818
\(122\) 8.00000 0.724286
\(123\) −12.0000 −1.08200
\(124\) 5.00000 0.449013
\(125\) −3.00000 −0.268328
\(126\) 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
\(129\) −4.00000 −0.352180
\(130\) 3.00000 0.263117
\(131\) 21.0000 1.83478 0.917389 0.397991i \(-0.130293\pi\)
0.917389 + 0.397991i \(0.130293\pi\)
\(132\) 3.00000 0.261116
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 3.00000 0.258199
\(136\) 6.00000 0.514496
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 6.00000 0.510754
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 6.00000 0.503509
\(143\) 3.00000 0.250873
\(144\) 1.00000 0.0833333
\(145\) −27.0000 −2.24223
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 4.00000 0.326599
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) −4.00000 −0.324443
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 15.0000 1.20483
\(156\) 1.00000 0.0800641
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −1.00000 −0.0795557
\(159\) −9.00000 −0.713746
\(160\) 3.00000 0.237171
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) −12.0000 −0.937043
\(165\) 9.00000 0.700649
\(166\) 3.00000 0.232845
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 18.0000 1.38054
\(171\) −4.00000 −0.305888
\(172\) −4.00000 −0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −9.00000 −0.682288
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) −9.00000 −0.676481
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 3.00000 0.223607
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 6.00000 0.442326
\(185\) −12.0000 −0.882258
\(186\) 5.00000 0.366618
\(187\) 18.0000 1.31629
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) −12.0000 −0.870572
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 1.00000 0.0721688
\(193\) −1.00000 −0.0719816 −0.0359908 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) 5.00000 0.358979
\(195\) 3.00000 0.214834
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 3.00000 0.213201
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 4.00000 0.282843
\(201\) −4.00000 −0.282138
\(202\) 18.0000 1.26648
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) −36.0000 −2.51435
\(206\) 8.00000 0.557386
\(207\) 6.00000 0.417029
\(208\) 1.00000 0.0693375
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −9.00000 −0.618123
\(213\) 6.00000 0.411113
\(214\) 3.00000 0.205076
\(215\) −12.0000 −0.818393
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −16.0000 −1.08366
\(219\) 14.0000 0.946032
\(220\) 9.00000 0.606780
\(221\) 6.00000 0.403604
\(222\) −4.00000 −0.268462
\(223\) −1.00000 −0.0669650 −0.0334825 0.999439i \(-0.510660\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) −12.0000 −0.798228
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) −4.00000 −0.264906
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 18.0000 1.18688
\(231\) 0 0
\(232\) −9.00000 −0.590879
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 1.00000 0.0653720
\(235\) −36.0000 −2.34838
\(236\) −9.00000 −0.585850
\(237\) −1.00000 −0.0649570
\(238\) 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
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) −2.00000 −0.128565
\(243\) 1.00000 0.0641500
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) −12.0000 −0.765092
\(247\) −4.00000 −0.254514
\(248\) 5.00000 0.317500
\(249\) 3.00000 0.190117
\(250\) −3.00000 −0.189737
\(251\) −3.00000 −0.189358 −0.0946792 0.995508i \(-0.530183\pi\)
−0.0946792 + 0.995508i \(0.530183\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) −7.00000 −0.439219
\(255\) 18.0000 1.12720
\(256\) 1.00000 0.0625000
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 3.00000 0.186052
\(261\) −9.00000 −0.557086
\(262\) 21.0000 1.29738
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 3.00000 0.184637
\(265\) −27.0000 −1.65860
\(266\) 0 0
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 3.00000 0.182913 0.0914566 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 3.00000 0.182574
\(271\) −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
\(274\) −12.0000 −0.724947
\(275\) 12.0000 0.723627
\(276\) 6.00000 0.361158
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −4.00000 −0.239904
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −12.0000 −0.714590
\(283\) 2.00000 0.118888 0.0594438 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) 6.00000 0.356034
\(285\) −12.0000 −0.710819
\(286\) 3.00000 0.177394
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −27.0000 −1.58549
\(291\) 5.00000 0.293105
\(292\) 14.0000 0.819288
\(293\) 21.0000 1.22683 0.613417 0.789760i \(-0.289795\pi\)
0.613417 + 0.789760i \(0.289795\pi\)
\(294\) 0 0
\(295\) −27.0000 −1.57200
\(296\) −4.00000 −0.232495
\(297\) 3.00000 0.174078
\(298\) 6.00000 0.347571
\(299\) 6.00000 0.346989
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) −19.0000 −1.09333
\(303\) 18.0000 1.03407
\(304\) −4.00000 −0.229416
\(305\) 24.0000 1.37424
\(306\) 6.00000 0.342997
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 15.0000 0.851943
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 1.00000 0.0566139
\(313\) −19.0000 −1.07394 −0.536972 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) −15.0000 −0.842484 −0.421242 0.906948i \(-0.638406\pi\)
−0.421242 + 0.906948i \(0.638406\pi\)
\(318\) −9.00000 −0.504695
\(319\) −27.0000 −1.51171
\(320\) 3.00000 0.167705
\(321\) 3.00000 0.167444
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) 8.00000 0.443079
\(327\) −16.0000 −0.884802
\(328\) −12.0000 −0.662589
\(329\) 0 0
\(330\) 9.00000 0.495434
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) 3.00000 0.164646
\(333\) −4.00000 −0.219199
\(334\) 12.0000 0.656611
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −31.0000 −1.68868 −0.844339 0.535810i \(-0.820006\pi\)
−0.844339 + 0.535810i \(0.820006\pi\)
\(338\) 1.00000 0.0543928
\(339\) −12.0000 −0.651751
\(340\) 18.0000 0.976187
\(341\) 15.0000 0.812296
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 18.0000 0.969087
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −9.00000 −0.482451
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 3.00000 0.159901
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) −9.00000 −0.478345
\(355\) 18.0000 0.955341
\(356\) 0 0
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 3.00000 0.158114
\(361\) −3.00000 −0.157895
\(362\) −16.0000 −0.840941
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 42.0000 2.19838
\(366\) 8.00000 0.418167
\(367\) −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
\(370\) −12.0000 −0.623850
\(371\) 0 0
\(372\) 5.00000 0.259238
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 18.0000 0.930758
\(375\) −3.00000 −0.154919
\(376\) −12.0000 −0.618853
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −12.0000 −0.615587
\(381\) −7.00000 −0.358621
\(382\) 6.00000 0.306987
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −1.00000 −0.0508987
\(387\) −4.00000 −0.203331
\(388\) 5.00000 0.253837
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 3.00000 0.151911
\(391\) 36.0000 1.82060
\(392\) 0 0
\(393\) 21.0000 1.05931
\(394\) −18.0000 −0.906827
\(395\) −3.00000 −0.150946
\(396\) 3.00000 0.150756
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −4.00000 −0.199502
\(403\) 5.00000 0.249068
\(404\) 18.0000 0.895533
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 6.00000 0.297044
\(409\) −7.00000 −0.346128 −0.173064 0.984911i \(-0.555367\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) −36.0000 −1.77791
\(411\) −12.0000 −0.591916
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 9.00000 0.441793
\(416\) 1.00000 0.0490290
\(417\) −4.00000 −0.195881
\(418\) −12.0000 −0.586939
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 8.00000 0.389434
\(423\) −12.0000 −0.583460
\(424\) −9.00000 −0.437079
\(425\) 24.0000 1.16417
\(426\) 6.00000 0.290701
\(427\) 0 0
\(428\) 3.00000 0.145010
\(429\) 3.00000 0.144841
\(430\) −12.0000 −0.578691
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) −27.0000 −1.29455
\(436\) −16.0000 −0.766261
\(437\) −24.0000 −1.14808
\(438\) 14.0000 0.668946
\(439\) −1.00000 −0.0477274 −0.0238637 0.999715i \(-0.507597\pi\)
−0.0238637 + 0.999715i \(0.507597\pi\)
\(440\) 9.00000 0.429058
\(441\) 0 0
\(442\) 6.00000 0.285391
\(443\) −3.00000 −0.142534 −0.0712672 0.997457i \(-0.522704\pi\)
−0.0712672 + 0.997457i \(0.522704\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −1.00000 −0.0473514
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 4.00000 0.188562
\(451\) −36.0000 −1.69517
\(452\) −12.0000 −0.564433
\(453\) −19.0000 −0.892698
\(454\) −3.00000 −0.140797
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −19.0000 −0.888783 −0.444391 0.895833i \(-0.646580\pi\)
−0.444391 + 0.895833i \(0.646580\pi\)
\(458\) 26.0000 1.21490
\(459\) 6.00000 0.280056
\(460\) 18.0000 0.839254
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −9.00000 −0.417815
\(465\) 15.0000 0.695608
\(466\) −6.00000 −0.277945
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) −36.0000 −1.66056
\(471\) 2.00000 0.0921551
\(472\) −9.00000 −0.414259
\(473\) −12.0000 −0.551761
\(474\) −1.00000 −0.0459315
\(475\) −16.0000 −0.734130
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) 6.00000 0.274434
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 3.00000 0.136931
\(481\) −4.00000 −0.182384
\(482\) 17.0000 0.774329
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 15.0000 0.681115
\(486\) 1.00000 0.0453609
\(487\) 23.0000 1.04223 0.521115 0.853487i \(-0.325516\pi\)
0.521115 + 0.853487i \(0.325516\pi\)
\(488\) 8.00000 0.362143
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) −12.0000 −0.541002
\(493\) −54.0000 −2.43204
\(494\) −4.00000 −0.179969
\(495\) 9.00000 0.404520
\(496\) 5.00000 0.224507
\(497\) 0 0
\(498\) 3.00000 0.134433
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) −3.00000 −0.134164
\(501\) 12.0000 0.536120
\(502\) −3.00000 −0.133897
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) 54.0000 2.40297
\(506\) 18.0000 0.800198
\(507\) 1.00000 0.0444116
\(508\) −7.00000 −0.310575
\(509\) 9.00000 0.398918 0.199459 0.979906i \(-0.436082\pi\)
0.199459 + 0.979906i \(0.436082\pi\)
\(510\) 18.0000 0.797053
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −30.0000 −1.32324
\(515\) 24.0000 1.05757
\(516\) −4.00000 −0.176090
\(517\) −36.0000 −1.58328
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 3.00000 0.131559
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −9.00000 −0.393919
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 21.0000 0.917389
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) 30.0000 1.30682
\(528\) 3.00000 0.130558
\(529\) 13.0000 0.565217
\(530\) −27.0000 −1.17281
\(531\) −9.00000 −0.390567
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 9.00000 0.389104
\(536\) −4.00000 −0.172774
\(537\) 12.0000 0.517838
\(538\) 3.00000 0.129339
\(539\) 0 0
\(540\) 3.00000 0.129099
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) −7.00000 −0.300676
\(543\) −16.0000 −0.686626
\(544\) 6.00000 0.257248
\(545\) −48.0000 −2.05609
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) −12.0000 −0.512615
\(549\) 8.00000 0.341432
\(550\) 12.0000 0.511682
\(551\) 36.0000 1.53365
\(552\) 6.00000 0.255377
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) −12.0000 −0.509372
\(556\) −4.00000 −0.169638
\(557\) −21.0000 −0.889799 −0.444899 0.895581i \(-0.646761\pi\)
−0.444899 + 0.895581i \(0.646761\pi\)
\(558\) 5.00000 0.211667
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) 0 0
\(563\) 33.0000 1.39078 0.695392 0.718631i \(-0.255231\pi\)
0.695392 + 0.718631i \(0.255231\pi\)
\(564\) −12.0000 −0.505291
\(565\) −36.0000 −1.51453
\(566\) 2.00000 0.0840663
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 12.0000 0.503066 0.251533 0.967849i \(-0.419065\pi\)
0.251533 + 0.967849i \(0.419065\pi\)
\(570\) −12.0000 −0.502625
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 3.00000 0.125436
\(573\) 6.00000 0.250654
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 1.00000 0.0416667
\(577\) 17.0000 0.707719 0.353860 0.935299i \(-0.384869\pi\)
0.353860 + 0.935299i \(0.384869\pi\)
\(578\) 19.0000 0.790296
\(579\) −1.00000 −0.0415586
\(580\) −27.0000 −1.12111
\(581\) 0 0
\(582\) 5.00000 0.207257
\(583\) −27.0000 −1.11823
\(584\) 14.0000 0.579324
\(585\) 3.00000 0.124035
\(586\) 21.0000 0.867502
\(587\) 33.0000 1.36206 0.681028 0.732257i \(-0.261533\pi\)
0.681028 + 0.732257i \(0.261533\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) −27.0000 −1.11157
\(591\) −18.0000 −0.740421
\(592\) −4.00000 −0.164399
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 8.00000 0.327418
\(598\) 6.00000 0.245358
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 4.00000 0.163299
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −19.0000 −0.773099
\(605\) −6.00000 −0.243935
\(606\) 18.0000 0.731200
\(607\) 5.00000 0.202944 0.101472 0.994838i \(-0.467645\pi\)
0.101472 + 0.994838i \(0.467645\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 24.0000 0.971732
\(611\) −12.0000 −0.485468
\(612\) 6.00000 0.242536
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 2.00000 0.0807134
\(615\) −36.0000 −1.45166
\(616\) 0 0
\(617\) 48.0000 1.93241 0.966204 0.257780i \(-0.0829910\pi\)
0.966204 + 0.257780i \(0.0829910\pi\)
\(618\) 8.00000 0.321807
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 15.0000 0.602414
\(621\) 6.00000 0.240772
\(622\) −18.0000 −0.721734
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) −29.0000 −1.16000
\(626\) −19.0000 −0.759393
\(627\) −12.0000 −0.479234
\(628\) 2.00000 0.0798087
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −13.0000 −0.517522 −0.258761 0.965941i \(-0.583314\pi\)
−0.258761 + 0.965941i \(0.583314\pi\)
\(632\) −1.00000 −0.0397779
\(633\) 8.00000 0.317971
\(634\) −15.0000 −0.595726
\(635\) −21.0000 −0.833360
\(636\) −9.00000 −0.356873
\(637\) 0 0
\(638\) −27.0000 −1.06894
\(639\) 6.00000 0.237356
\(640\) 3.00000 0.118585
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 3.00000 0.118401
\(643\) 8.00000 0.315489 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) 0 0
\(645\) −12.0000 −0.472500
\(646\) −24.0000 −0.944267
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 1.00000 0.0392837
\(649\) −27.0000 −1.05984
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 9.00000 0.352197 0.176099 0.984373i \(-0.443652\pi\)
0.176099 + 0.984373i \(0.443652\pi\)
\(654\) −16.0000 −0.625650
\(655\) 63.0000 2.46161
\(656\) −12.0000 −0.468521
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 9.00000 0.350325
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 32.0000 1.24372
\(663\) 6.00000 0.233021
\(664\) 3.00000 0.116423
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) −54.0000 −2.09089
\(668\) 12.0000 0.464294
\(669\) −1.00000 −0.0386622
\(670\) −12.0000 −0.463600
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −31.0000 −1.19496 −0.597481 0.801883i \(-0.703832\pi\)
−0.597481 + 0.801883i \(0.703832\pi\)
\(674\) −31.0000 −1.19408
\(675\) 4.00000 0.153960
\(676\) 1.00000 0.0384615
\(677\) 9.00000 0.345898 0.172949 0.984931i \(-0.444670\pi\)
0.172949 + 0.984931i \(0.444670\pi\)
\(678\) −12.0000 −0.460857
\(679\) 0 0
\(680\) 18.0000 0.690268
\(681\) −3.00000 −0.114960
\(682\) 15.0000 0.574380
\(683\) −39.0000 −1.49229 −0.746147 0.665782i \(-0.768098\pi\)
−0.746147 + 0.665782i \(0.768098\pi\)
\(684\) −4.00000 −0.152944
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) 26.0000 0.991962
\(688\) −4.00000 −0.152499
\(689\) −9.00000 −0.342873
\(690\) 18.0000 0.685248
\(691\) −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
\(694\) 12.0000 0.455514
\(695\) −12.0000 −0.455186
\(696\) −9.00000 −0.341144
\(697\) −72.0000 −2.72719
\(698\) −34.0000 −1.28692
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 15.0000 0.566542 0.283271 0.959040i \(-0.408580\pi\)
0.283271 + 0.959040i \(0.408580\pi\)
\(702\) 1.00000 0.0377426
\(703\) 16.0000 0.603451
\(704\) 3.00000 0.113067
\(705\) −36.0000 −1.35584
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) −9.00000 −0.338241
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 18.0000 0.675528
\(711\) −1.00000 −0.0375029
\(712\) 0 0
\(713\) 30.0000 1.12351
\(714\) 0 0
\(715\) 9.00000 0.336581
\(716\) 12.0000 0.448461
\(717\) 6.00000 0.224074
\(718\) −6.00000 −0.223918
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 3.00000 0.111803
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 17.0000 0.632237
\(724\) −16.0000 −0.594635
\(725\) −36.0000 −1.33701
\(726\) −2.00000 −0.0742270
\(727\) −37.0000 −1.37225 −0.686127 0.727482i \(-0.740691\pi\)
−0.686127 + 0.727482i \(0.740691\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 42.0000 1.55449
\(731\) −24.0000 −0.887672
\(732\) 8.00000 0.295689
\(733\) 50.0000 1.84679 0.923396 0.383849i \(-0.125402\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) −7.00000 −0.258375
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) −12.0000 −0.442026
\(738\) −12.0000 −0.441726
\(739\) 38.0000 1.39785 0.698926 0.715194i \(-0.253662\pi\)
0.698926 + 0.715194i \(0.253662\pi\)
\(740\) −12.0000 −0.441129
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 5.00000 0.183309
\(745\) 18.0000 0.659469
\(746\) 14.0000 0.512576
\(747\) 3.00000 0.109764
\(748\) 18.0000 0.658145
\(749\) 0 0
\(750\) −3.00000 −0.109545
\(751\) −31.0000 −1.13121 −0.565603 0.824678i \(-0.691357\pi\)
−0.565603 + 0.824678i \(0.691357\pi\)
\(752\) −12.0000 −0.437595
\(753\) −3.00000 −0.109326
\(754\) −9.00000 −0.327761
\(755\) −57.0000 −2.07444
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −16.0000 −0.581146
\(759\) 18.0000 0.653359
\(760\) −12.0000 −0.435286
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) −7.00000 −0.253583
\(763\) 0 0
\(764\) 6.00000 0.217072
\(765\) 18.0000 0.650791
\(766\) −6.00000 −0.216789
\(767\) −9.00000 −0.324971
\(768\) 1.00000 0.0360844
\(769\) −1.00000 −0.0360609 −0.0180305 0.999837i \(-0.505740\pi\)
−0.0180305 + 0.999837i \(0.505740\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) −1.00000 −0.0359908
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) −4.00000 −0.143777
\(775\) 20.0000 0.718421
\(776\) 5.00000 0.179490
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 48.0000 1.71978
\(780\) 3.00000 0.107417
\(781\) 18.0000 0.644091
\(782\) 36.0000 1.28736
\(783\) −9.00000 −0.321634
\(784\) 0 0
\(785\) 6.00000 0.214149
\(786\) 21.0000 0.749045
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −18.0000 −0.641223
\(789\) 6.00000 0.213606
\(790\) −3.00000 −0.106735
\(791\) 0 0
\(792\) 3.00000 0.106600
\(793\) 8.00000 0.284088
\(794\) 2.00000 0.0709773
\(795\) −27.0000 −0.957591
\(796\) 8.00000 0.283552
\(797\) 9.00000 0.318796 0.159398 0.987214i \(-0.449045\pi\)
0.159398 + 0.987214i \(0.449045\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 42.0000 1.48215
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 5.00000 0.176117
\(807\) 3.00000 0.105605
\(808\) 18.0000 0.633238
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 3.00000 0.105409
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) −7.00000 −0.245501
\(814\) −12.0000 −0.420600
\(815\) 24.0000 0.840683
\(816\) 6.00000 0.210042
\(817\) 16.0000 0.559769
\(818\) −7.00000 −0.244749
\(819\) 0 0
\(820\) −36.0000 −1.25717
\(821\) −39.0000 −1.36111 −0.680555 0.732697i \(-0.738261\pi\)
−0.680555 + 0.732697i \(0.738261\pi\)
\(822\) −12.0000 −0.418548
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 8.00000 0.278693
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) 6.00000 0.208514
\(829\) −46.0000 −1.59765 −0.798823 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 9.00000 0.312395
\(831\) 2.00000 0.0693792
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) −4.00000 −0.138509
\(835\) 36.0000 1.24583
\(836\) −12.0000 −0.415029
\(837\) 5.00000 0.172825
\(838\) 24.0000 0.829066
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 8.00000 0.275698
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 3.00000 0.103203
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) −9.00000 −0.309061
\(849\) 2.00000 0.0686398
\(850\) 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
\(855\) −12.0000 −0.410391
\(856\) 3.00000 0.102538
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 3.00000 0.102418
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −12.0000 −0.409197
\(861\) 0 0
\(862\) 30.0000 1.02180
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 1.00000 0.0340207
\(865\) 18.0000 0.612018
\(866\) 2.00000 0.0679628
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) −3.00000 −0.101768
\(870\) −27.0000 −0.915386
\(871\) −4.00000 −0.135535
\(872\) −16.0000 −0.541828
\(873\) 5.00000 0.169224
\(874\) −24.0000 −0.811812
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) −58.0000 −1.95852 −0.979260 0.202606i \(-0.935059\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) −1.00000 −0.0337484
\(879\) 21.0000 0.708312
\(880\) 9.00000 0.303390
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) 26.0000 0.874970 0.437485 0.899226i \(-0.355869\pi\)
0.437485 + 0.899226i \(0.355869\pi\)
\(884\) 6.00000 0.201802
\(885\) −27.0000 −0.907595
\(886\) −3.00000 −0.100787
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) −4.00000 −0.134231
\(889\) 0 0
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) −1.00000 −0.0334825
\(893\) 48.0000 1.60626
\(894\) 6.00000 0.200670
\(895\) 36.0000 1.20335
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) −12.0000 −0.400445
\(899\) −45.0000 −1.50083
\(900\) 4.00000 0.133333
\(901\) −54.0000 −1.79900
\(902\) −36.0000 −1.19867
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) −48.0000 −1.59557
\(906\) −19.0000 −0.631233
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −3.00000 −0.0995585
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 54.0000 1.78910 0.894550 0.446968i \(-0.147496\pi\)
0.894550 + 0.446968i \(0.147496\pi\)
\(912\) −4.00000 −0.132453
\(913\) 9.00000 0.297857
\(914\) −19.0000 −0.628464
\(915\) 24.0000 0.793416
\(916\) 26.0000 0.859064
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 18.0000 0.593442
\(921\) 2.00000 0.0659022
\(922\) 18.0000 0.592798
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) −16.0000 −0.526077
\(926\) −16.0000 −0.525793
\(927\) 8.00000 0.262754
\(928\) −9.00000 −0.295439
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 15.0000 0.491869
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) −18.0000 −0.589294
\(934\) −36.0000 −1.17796
\(935\) 54.0000 1.76599
\(936\) 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
\(939\) −19.0000 −0.620042
\(940\) −36.0000 −1.17419
\(941\) −45.0000 −1.46696 −0.733479 0.679712i \(-0.762105\pi\)
−0.733479 + 0.679712i \(0.762105\pi\)
\(942\) 2.00000 0.0651635
\(943\) −72.0000 −2.34464
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 60.0000 1.94974 0.974869 0.222779i \(-0.0715128\pi\)
0.974869 + 0.222779i \(0.0715128\pi\)
\(948\) −1.00000 −0.0324785
\(949\) 14.0000 0.454459
\(950\) −16.0000 −0.519109
\(951\) −15.0000 −0.486408
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) −9.00000 −0.291386
\(955\) 18.0000 0.582466
\(956\) 6.00000 0.194054
\(957\) −27.0000 −0.872786
\(958\) 6.00000 0.193851
\(959\) 0 0
\(960\) 3.00000 0.0968246
\(961\) −6.00000 −0.193548
\(962\) −4.00000 −0.128965
\(963\) 3.00000 0.0966736
\(964\) 17.0000 0.547533
\(965\) −3.00000 −0.0965734
\(966\) 0 0
\(967\) 29.0000 0.932577 0.466289 0.884633i \(-0.345591\pi\)
0.466289 + 0.884633i \(0.345591\pi\)
\(968\) −2.00000 −0.0642824
\(969\) −24.0000 −0.770991
\(970\) 15.0000 0.481621
\(971\) −33.0000 −1.05902 −0.529510 0.848304i \(-0.677624\pi\)
−0.529510 + 0.848304i \(0.677624\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 23.0000 0.736968
\(975\) 4.00000 0.128103
\(976\) 8.00000 0.256074
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 8.00000 0.255812
\(979\) 0 0
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) −15.0000 −0.478669
\(983\) 18.0000 0.574111 0.287055 0.957914i \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) −12.0000 −0.382546
\(985\) −54.0000 −1.72058
\(986\) −54.0000 −1.71971
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) −24.0000 −0.763156
\(990\) 9.00000 0.286039
\(991\) 35.0000 1.11181 0.555906 0.831245i \(-0.312372\pi\)
0.555906 + 0.831245i \(0.312372\pi\)
\(992\) 5.00000 0.158750
\(993\) 32.0000 1.01549
\(994\) 0 0
\(995\) 24.0000 0.760851
\(996\) 3.00000 0.0950586
\(997\) −16.0000 −0.506725 −0.253363 0.967371i \(-0.581537\pi\)
−0.253363 + 0.967371i \(0.581537\pi\)
\(998\) −16.0000 −0.506471
\(999\) −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.bh.1.1 1
7.2 even 3 546.2.i.a.235.1 yes 2
7.4 even 3 546.2.i.a.79.1 2
7.6 odd 2 3822.2.a.s.1.1 1
21.2 odd 6 1638.2.j.j.235.1 2
21.11 odd 6 1638.2.j.j.1171.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.a.79.1 2 7.4 even 3
546.2.i.a.235.1 yes 2 7.2 even 3
1638.2.j.j.235.1 2 21.2 odd 6
1638.2.j.j.1171.1 2 21.11 odd 6
3822.2.a.s.1.1 1 7.6 odd 2
3822.2.a.bh.1.1 1 1.1 even 1 trivial