Properties

Label 3822.2.a.r.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: yes
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} -3.00000 q^{20} -1.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +5.00000 q^{29} +3.00000 q^{30} -6.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} +1.00000 q^{38} -1.00000 q^{39} -3.00000 q^{40} +3.00000 q^{43} -1.00000 q^{44} -3.00000 q^{45} -1.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} +3.00000 q^{51} +1.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +3.00000 q^{55} -1.00000 q^{57} +5.00000 q^{58} -2.00000 q^{59} +3.00000 q^{60} +1.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} -3.00000 q^{65} +1.00000 q^{66} +16.0000 q^{67} -3.00000 q^{68} +1.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} +5.00000 q^{73} -1.00000 q^{74} -4.00000 q^{75} +1.00000 q^{76} -1.00000 q^{78} +12.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -6.00000 q^{83} +9.00000 q^{85} +3.00000 q^{86} -5.00000 q^{87} -1.00000 q^{88} -3.00000 q^{90} -1.00000 q^{92} +6.00000 q^{93} +4.00000 q^{94} -3.00000 q^{95} -1.00000 q^{96} +18.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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 −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
\(15\) 3.00000 0.774597
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 3.00000 0.547723
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 1.00000 0.162221
\(39\) −1.00000 −0.160128
\(40\) −3.00000 −0.474342
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 3.00000 0.457496 0.228748 0.973486i \(-0.426537\pi\)
0.228748 + 0.973486i \(0.426537\pi\)
\(44\) −1.00000 −0.150756
\(45\) −3.00000 −0.447214
\(46\) −1.00000 −0.147442
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) 3.00000 0.420084
\(52\) 1.00000 0.138675
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 5.00000 0.656532
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 3.00000 0.387298
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.00000 −0.372104
\(66\) 1.00000 0.123091
\(67\) 16.0000 1.95471 0.977356 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) −3.00000 −0.363803
\(69\) 1.00000 0.120386
\(70\) 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
\(73\) 5.00000 0.585206 0.292603 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) −1.00000 −0.116248
\(75\) −4.00000 −0.461880
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 3.00000 0.323498
\(87\) −5.00000 −0.536056
\(88\) −1.00000 −0.106600
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 6.00000 0.622171
\(94\) 4.00000 0.412568
\(95\) −3.00000 −0.307794
\(96\) −1.00000 −0.102062
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 4.00000 0.400000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 3.00000 0.297044
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 3.00000 0.286039
\(111\) 1.00000 0.0949158
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 3.00000 0.279751
\(116\) 5.00000 0.464238
\(117\) 1.00000 0.0924500
\(118\) −2.00000 −0.184115
\(119\) 0 0
\(120\) 3.00000 0.273861
\(121\) −10.0000 −0.909091
\(122\) 1.00000 0.0905357
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) −3.00000 −0.264135
\(130\) −3.00000 −0.263117
\(131\) 9.00000 0.786334 0.393167 0.919467i \(-0.371379\pi\)
0.393167 + 0.919467i \(0.371379\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 16.0000 1.38219
\(135\) 3.00000 0.258199
\(136\) −3.00000 −0.257248
\(137\) −15.0000 −1.28154 −0.640768 0.767734i \(-0.721384\pi\)
−0.640768 + 0.767734i \(0.721384\pi\)
\(138\) 1.00000 0.0851257
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 6.00000 0.503509
\(143\) −1.00000 −0.0836242
\(144\) 1.00000 0.0833333
\(145\) −15.0000 −1.24568
\(146\) 5.00000 0.413803
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) −4.00000 −0.326599
\(151\) −9.00000 −0.732410 −0.366205 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(152\) 1.00000 0.0811107
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 18.0000 1.44579
\(156\) −1.00000 −0.0800641
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) 12.0000 0.954669
\(159\) 6.00000 0.475831
\(160\) −3.00000 −0.237171
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) 0 0
\(165\) −3.00000 −0.233550
\(166\) −6.00000 −0.465690
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 9.00000 0.690268
\(171\) 1.00000 0.0764719
\(172\) 3.00000 0.228748
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) −5.00000 −0.379049
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 2.00000 0.150329
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) −3.00000 −0.223607
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) −1.00000 −0.0737210
\(185\) 3.00000 0.220564
\(186\) 6.00000 0.439941
\(187\) 3.00000 0.219382
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) −3.00000 −0.217643
\(191\) −5.00000 −0.361787 −0.180894 0.983503i \(-0.557899\pi\)
−0.180894 + 0.983503i \(0.557899\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 18.0000 1.29232
\(195\) 3.00000 0.214834
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 3.00000 0.212664 0.106332 0.994331i \(-0.466089\pi\)
0.106332 + 0.994331i \(0.466089\pi\)
\(200\) 4.00000 0.282843
\(201\) −16.0000 −1.12855
\(202\) 14.0000 0.985037
\(203\) 0 0
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) 1.00000 0.0696733
\(207\) −1.00000 −0.0695048
\(208\) 1.00000 0.0693375
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 25.0000 1.72107 0.860535 0.509390i \(-0.170129\pi\)
0.860535 + 0.509390i \(0.170129\pi\)
\(212\) −6.00000 −0.412082
\(213\) −6.00000 −0.411113
\(214\) −8.00000 −0.546869
\(215\) −9.00000 −0.613795
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −7.00000 −0.474100
\(219\) −5.00000 −0.337869
\(220\) 3.00000 0.202260
\(221\) −3.00000 −0.201802
\(222\) 1.00000 0.0671156
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 12.0000 0.798228
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 3.00000 0.197814
\(231\) 0 0
\(232\) 5.00000 0.328266
\(233\) −8.00000 −0.524097 −0.262049 0.965055i \(-0.584398\pi\)
−0.262049 + 0.965055i \(0.584398\pi\)
\(234\) 1.00000 0.0653720
\(235\) −12.0000 −0.782794
\(236\) −2.00000 −0.130189
\(237\) −12.0000 −0.779484
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 3.00000 0.193649
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) −10.0000 −0.642824
\(243\) −1.00000 −0.0641500
\(244\) 1.00000 0.0640184
\(245\) 0 0
\(246\) 0 0
\(247\) 1.00000 0.0636285
\(248\) −6.00000 −0.381000
\(249\) 6.00000 0.380235
\(250\) 3.00000 0.189737
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) 1.00000 0.0628695
\(254\) 16.0000 1.00393
\(255\) −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 \(0.440080\pi\)
\(258\) −3.00000 −0.186772
\(259\) 0 0
\(260\) −3.00000 −0.186052
\(261\) 5.00000 0.309492
\(262\) 9.00000 0.556022
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 1.00000 0.0615457
\(265\) 18.0000 1.10573
\(266\) 0 0
\(267\) 0 0
\(268\) 16.0000 0.977356
\(269\) 16.0000 0.975537 0.487769 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 3.00000 0.182574
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) −15.0000 −0.906183
\(275\) −4.00000 −0.241209
\(276\) 1.00000 0.0601929
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 16.0000 0.959616
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −4.00000 −0.238197
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 6.00000 0.356034
\(285\) 3.00000 0.177705
\(286\) −1.00000 −0.0591312
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) −15.0000 −0.880830
\(291\) −18.0000 −1.05518
\(292\) 5.00000 0.292603
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) −1.00000 −0.0581238
\(297\) 1.00000 0.0580259
\(298\) 4.00000 0.231714
\(299\) −1.00000 −0.0578315
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) −9.00000 −0.517892
\(303\) −14.0000 −0.804279
\(304\) 1.00000 0.0573539
\(305\) −3.00000 −0.171780
\(306\) −3.00000 −0.171499
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) 18.0000 1.02233
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 13.0000 0.733632
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 6.00000 0.336463
\(319\) −5.00000 −0.279946
\(320\) −3.00000 −0.167705
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) 6.00000 0.332309
\(327\) 7.00000 0.387101
\(328\) 0 0
\(329\) 0 0
\(330\) −3.00000 −0.165145
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) −6.00000 −0.329293
\(333\) −1.00000 −0.0547997
\(334\) −3.00000 −0.164153
\(335\) −48.0000 −2.62252
\(336\) 0 0
\(337\) 31.0000 1.68868 0.844339 0.535810i \(-0.179994\pi\)
0.844339 + 0.535810i \(0.179994\pi\)
\(338\) 1.00000 0.0543928
\(339\) −12.0000 −0.651751
\(340\) 9.00000 0.488094
\(341\) 6.00000 0.324918
\(342\) 1.00000 0.0540738
\(343\) 0 0
\(344\) 3.00000 0.161749
\(345\) −3.00000 −0.161515
\(346\) 4.00000 0.215041
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) −5.00000 −0.268028
\(349\) −4.00000 −0.214115 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −1.00000 −0.0533002
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 2.00000 0.106299
\(355\) −18.0000 −0.955341
\(356\) 0 0
\(357\) 0 0
\(358\) −18.0000 −0.951330
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) −3.00000 −0.158114
\(361\) −18.0000 −0.947368
\(362\) 14.0000 0.735824
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) −15.0000 −0.785136
\(366\) −1.00000 −0.0522708
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 3.00000 0.155963
\(371\) 0 0
\(372\) 6.00000 0.311086
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 3.00000 0.155126
\(375\) −3.00000 −0.154919
\(376\) 4.00000 0.206284
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) 22.0000 1.13006 0.565032 0.825069i \(-0.308864\pi\)
0.565032 + 0.825069i \(0.308864\pi\)
\(380\) −3.00000 −0.153897
\(381\) −16.0000 −0.819705
\(382\) −5.00000 −0.255822
\(383\) 33.0000 1.68622 0.843111 0.537740i \(-0.180722\pi\)
0.843111 + 0.537740i \(0.180722\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 3.00000 0.152499
\(388\) 18.0000 0.913812
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 3.00000 0.151911
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) −9.00000 −0.453990
\(394\) 18.0000 0.906827
\(395\) −36.0000 −1.81136
\(396\) −1.00000 −0.0502519
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 3.00000 0.150376
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) −16.0000 −0.798007
\(403\) −6.00000 −0.298881
\(404\) 14.0000 0.696526
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) 1.00000 0.0495682
\(408\) 3.00000 0.148522
\(409\) 15.0000 0.741702 0.370851 0.928692i \(-0.379066\pi\)
0.370851 + 0.928692i \(0.379066\pi\)
\(410\) 0 0
\(411\) 15.0000 0.739895
\(412\) 1.00000 0.0492665
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 18.0000 0.883585
\(416\) 1.00000 0.0490290
\(417\) −16.0000 −0.783523
\(418\) −1.00000 −0.0489116
\(419\) −21.0000 −1.02592 −0.512959 0.858413i \(-0.671451\pi\)
−0.512959 + 0.858413i \(0.671451\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) 25.0000 1.21698
\(423\) 4.00000 0.194487
\(424\) −6.00000 −0.291386
\(425\) −12.0000 −0.582086
\(426\) −6.00000 −0.290701
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 1.00000 0.0482805
\(430\) −9.00000 −0.434019
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 0 0
\(435\) 15.0000 0.719195
\(436\) −7.00000 −0.335239
\(437\) −1.00000 −0.0478365
\(438\) −5.00000 −0.238909
\(439\) −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
\(442\) −3.00000 −0.142695
\(443\) 42.0000 1.99548 0.997740 0.0671913i \(-0.0214038\pi\)
0.997740 + 0.0671913i \(0.0214038\pi\)
\(444\) 1.00000 0.0474579
\(445\) 0 0
\(446\) 2.00000 0.0947027
\(447\) −4.00000 −0.189194
\(448\) 0 0
\(449\) −39.0000 −1.84052 −0.920262 0.391303i \(-0.872024\pi\)
−0.920262 + 0.391303i \(0.872024\pi\)
\(450\) 4.00000 0.188562
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) 9.00000 0.422857
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −4.00000 −0.186908
\(459\) 3.00000 0.140028
\(460\) 3.00000 0.139876
\(461\) −33.0000 −1.53696 −0.768482 0.639872i \(-0.778987\pi\)
−0.768482 + 0.639872i \(0.778987\pi\)
\(462\) 0 0
\(463\) −21.0000 −0.975953 −0.487976 0.872857i \(-0.662265\pi\)
−0.487976 + 0.872857i \(0.662265\pi\)
\(464\) 5.00000 0.232119
\(465\) −18.0000 −0.834730
\(466\) −8.00000 −0.370593
\(467\) 33.0000 1.52706 0.763529 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) −12.0000 −0.553519
\(471\) −13.0000 −0.599008
\(472\) −2.00000 −0.0920575
\(473\) −3.00000 −0.137940
\(474\) −12.0000 −0.551178
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −6.00000 −0.274434
\(479\) −5.00000 −0.228456 −0.114228 0.993455i \(-0.536439\pi\)
−0.114228 + 0.993455i \(0.536439\pi\)
\(480\) 3.00000 0.136931
\(481\) −1.00000 −0.0455961
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) −54.0000 −2.45201
\(486\) −1.00000 −0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 1.00000 0.0452679
\(489\) −6.00000 −0.271329
\(490\) 0 0
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) 0 0
\(493\) −15.0000 −0.675566
\(494\) 1.00000 0.0449921
\(495\) 3.00000 0.134840
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) 6.00000 0.268866
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 3.00000 0.134164
\(501\) 3.00000 0.134030
\(502\) −27.0000 −1.20507
\(503\) 4.00000 0.178351 0.0891756 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) 0 0
\(505\) −42.0000 −1.86898
\(506\) 1.00000 0.0444554
\(507\) −1.00000 −0.0444116
\(508\) 16.0000 0.709885
\(509\) −33.0000 −1.46270 −0.731350 0.682003i \(-0.761109\pi\)
−0.731350 + 0.682003i \(0.761109\pi\)
\(510\) −9.00000 −0.398527
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 6.00000 0.264649
\(515\) −3.00000 −0.132196
\(516\) −3.00000 −0.132068
\(517\) −4.00000 −0.175920
\(518\) 0 0
\(519\) −4.00000 −0.175581
\(520\) −3.00000 −0.131559
\(521\) −17.0000 −0.744784 −0.372392 0.928076i \(-0.621462\pi\)
−0.372392 + 0.928076i \(0.621462\pi\)
\(522\) 5.00000 0.218844
\(523\) 42.0000 1.83653 0.918266 0.395964i \(-0.129590\pi\)
0.918266 + 0.395964i \(0.129590\pi\)
\(524\) 9.00000 0.393167
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) 18.0000 0.784092
\(528\) 1.00000 0.0435194
\(529\) −22.0000 −0.956522
\(530\) 18.0000 0.781870
\(531\) −2.00000 −0.0867926
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) 16.0000 0.691095
\(537\) 18.0000 0.776757
\(538\) 16.0000 0.689809
\(539\) 0 0
\(540\) 3.00000 0.129099
\(541\) −35.0000 −1.50477 −0.752384 0.658725i \(-0.771096\pi\)
−0.752384 + 0.658725i \(0.771096\pi\)
\(542\) 8.00000 0.343629
\(543\) −14.0000 −0.600798
\(544\) −3.00000 −0.128624
\(545\) 21.0000 0.899541
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −15.0000 −0.640768
\(549\) 1.00000 0.0426790
\(550\) −4.00000 −0.170561
\(551\) 5.00000 0.213007
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −8.00000 −0.339887
\(555\) −3.00000 −0.127343
\(556\) 16.0000 0.678551
\(557\) 20.0000 0.847427 0.423714 0.905796i \(-0.360726\pi\)
0.423714 + 0.905796i \(0.360726\pi\)
\(558\) −6.00000 −0.254000
\(559\) 3.00000 0.126886
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) −18.0000 −0.759284
\(563\) −19.0000 −0.800755 −0.400377 0.916350i \(-0.631121\pi\)
−0.400377 + 0.916350i \(0.631121\pi\)
\(564\) −4.00000 −0.168430
\(565\) −36.0000 −1.51453
\(566\) 0 0
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 3.00000 0.125656
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) −1.00000 −0.0418121
\(573\) 5.00000 0.208878
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −42.0000 −1.74848 −0.874241 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) −8.00000 −0.332756
\(579\) −2.00000 −0.0831172
\(580\) −15.0000 −0.622841
\(581\) 0 0
\(582\) −18.0000 −0.746124
\(583\) 6.00000 0.248495
\(584\) 5.00000 0.206901
\(585\) −3.00000 −0.124035
\(586\) −14.0000 −0.578335
\(587\) 2.00000 0.0825488 0.0412744 0.999148i \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) 6.00000 0.247016
\(591\) −18.0000 −0.740421
\(592\) −1.00000 −0.0410997
\(593\) −44.0000 −1.80686 −0.903432 0.428732i \(-0.858960\pi\)
−0.903432 + 0.428732i \(0.858960\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) −3.00000 −0.122782
\(598\) −1.00000 −0.0408930
\(599\) −39.0000 −1.59350 −0.796748 0.604311i \(-0.793448\pi\)
−0.796748 + 0.604311i \(0.793448\pi\)
\(600\) −4.00000 −0.163299
\(601\) 32.0000 1.30531 0.652654 0.757656i \(-0.273656\pi\)
0.652654 + 0.757656i \(0.273656\pi\)
\(602\) 0 0
\(603\) 16.0000 0.651570
\(604\) −9.00000 −0.366205
\(605\) 30.0000 1.21967
\(606\) −14.0000 −0.568711
\(607\) −35.0000 −1.42061 −0.710303 0.703896i \(-0.751442\pi\)
−0.710303 + 0.703896i \(0.751442\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) −3.00000 −0.121466
\(611\) 4.00000 0.161823
\(612\) −3.00000 −0.121268
\(613\) 25.0000 1.00974 0.504870 0.863195i \(-0.331540\pi\)
0.504870 + 0.863195i \(0.331540\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) −9.00000 −0.362326 −0.181163 0.983453i \(-0.557986\pi\)
−0.181163 + 0.983453i \(0.557986\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 37.0000 1.48716 0.743578 0.668649i \(-0.233127\pi\)
0.743578 + 0.668649i \(0.233127\pi\)
\(620\) 18.0000 0.722897
\(621\) 1.00000 0.0401286
\(622\) 6.00000 0.240578
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) −29.0000 −1.16000
\(626\) −14.0000 −0.559553
\(627\) 1.00000 0.0399362
\(628\) 13.0000 0.518756
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) −1.00000 −0.0398094 −0.0199047 0.999802i \(-0.506336\pi\)
−0.0199047 + 0.999802i \(0.506336\pi\)
\(632\) 12.0000 0.477334
\(633\) −25.0000 −0.993661
\(634\) 22.0000 0.873732
\(635\) −48.0000 −1.90482
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) −5.00000 −0.197952
\(639\) 6.00000 0.237356
\(640\) −3.00000 −0.118585
\(641\) 40.0000 1.57991 0.789953 0.613168i \(-0.210105\pi\)
0.789953 + 0.613168i \(0.210105\pi\)
\(642\) 8.00000 0.315735
\(643\) 35.0000 1.38027 0.690133 0.723683i \(-0.257552\pi\)
0.690133 + 0.723683i \(0.257552\pi\)
\(644\) 0 0
\(645\) 9.00000 0.354375
\(646\) −3.00000 −0.118033
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 1.00000 0.0392837
\(649\) 2.00000 0.0785069
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 6.00000 0.234978
\(653\) −1.00000 −0.0391330 −0.0195665 0.999809i \(-0.506229\pi\)
−0.0195665 + 0.999809i \(0.506229\pi\)
\(654\) 7.00000 0.273722
\(655\) −27.0000 −1.05498
\(656\) 0 0
\(657\) 5.00000 0.195069
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) −3.00000 −0.116775
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 0 0
\(663\) 3.00000 0.116510
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) −5.00000 −0.193601
\(668\) −3.00000 −0.116073
\(669\) −2.00000 −0.0773245
\(670\) −48.0000 −1.85440
\(671\) −1.00000 −0.0386046
\(672\) 0 0
\(673\) 3.00000 0.115642 0.0578208 0.998327i \(-0.481585\pi\)
0.0578208 + 0.998327i \(0.481585\pi\)
\(674\) 31.0000 1.19408
\(675\) −4.00000 −0.153960
\(676\) 1.00000 0.0384615
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) −12.0000 −0.460857
\(679\) 0 0
\(680\) 9.00000 0.345134
\(681\) −12.0000 −0.459841
\(682\) 6.00000 0.229752
\(683\) 21.0000 0.803543 0.401771 0.915740i \(-0.368395\pi\)
0.401771 + 0.915740i \(0.368395\pi\)
\(684\) 1.00000 0.0382360
\(685\) 45.0000 1.71936
\(686\) 0 0
\(687\) 4.00000 0.152610
\(688\) 3.00000 0.114374
\(689\) −6.00000 −0.228582
\(690\) −3.00000 −0.114208
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 4.00000 0.152057
\(693\) 0 0
\(694\) −18.0000 −0.683271
\(695\) −48.0000 −1.82074
\(696\) −5.00000 −0.189525
\(697\) 0 0
\(698\) −4.00000 −0.151402
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −1.00000 −0.0377157
\(704\) −1.00000 −0.0376889
\(705\) 12.0000 0.451946
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) 2.00000 0.0751646
\(709\) 2.00000 0.0751116 0.0375558 0.999295i \(-0.488043\pi\)
0.0375558 + 0.999295i \(0.488043\pi\)
\(710\) −18.0000 −0.675528
\(711\) 12.0000 0.450035
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 3.00000 0.112194
\(716\) −18.0000 −0.672692
\(717\) 6.00000 0.224074
\(718\) 8.00000 0.298557
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) −3.00000 −0.111803
\(721\) 0 0
\(722\) −18.0000 −0.669891
\(723\) −14.0000 −0.520666
\(724\) 14.0000 0.520306
\(725\) 20.0000 0.742781
\(726\) 10.0000 0.371135
\(727\) 1.00000 0.0370879 0.0185440 0.999828i \(-0.494097\pi\)
0.0185440 + 0.999828i \(0.494097\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −15.0000 −0.555175
\(731\) −9.00000 −0.332877
\(732\) −1.00000 −0.0369611
\(733\) 54.0000 1.99454 0.997268 0.0738717i \(-0.0235355\pi\)
0.997268 + 0.0738717i \(0.0235355\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 3.00000 0.110282
\(741\) −1.00000 −0.0367359
\(742\) 0 0
\(743\) 30.0000 1.10059 0.550297 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) 6.00000 0.219971
\(745\) −12.0000 −0.439646
\(746\) −22.0000 −0.805477
\(747\) −6.00000 −0.219529
\(748\) 3.00000 0.109691
\(749\) 0 0
\(750\) −3.00000 −0.109545
\(751\) 10.0000 0.364905 0.182453 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(752\) 4.00000 0.145865
\(753\) 27.0000 0.983935
\(754\) 5.00000 0.182089
\(755\) 27.0000 0.982631
\(756\) 0 0
\(757\) −12.0000 −0.436147 −0.218074 0.975932i \(-0.569977\pi\)
−0.218074 + 0.975932i \(0.569977\pi\)
\(758\) 22.0000 0.799076
\(759\) −1.00000 −0.0362977
\(760\) −3.00000 −0.108821
\(761\) −40.0000 −1.45000 −0.724999 0.688749i \(-0.758160\pi\)
−0.724999 + 0.688749i \(0.758160\pi\)
\(762\) −16.0000 −0.579619
\(763\) 0 0
\(764\) −5.00000 −0.180894
\(765\) 9.00000 0.325396
\(766\) 33.0000 1.19234
\(767\) −2.00000 −0.0722158
\(768\) −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
\(771\) −6.00000 −0.216085
\(772\) 2.00000 0.0719816
\(773\) 39.0000 1.40273 0.701366 0.712801i \(-0.252574\pi\)
0.701366 + 0.712801i \(0.252574\pi\)
\(774\) 3.00000 0.107833
\(775\) −24.0000 −0.862105
\(776\) 18.0000 0.646162
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) 0 0
\(780\) 3.00000 0.107417
\(781\) −6.00000 −0.214697
\(782\) 3.00000 0.107280
\(783\) −5.00000 −0.178685
\(784\) 0 0
\(785\) −39.0000 −1.39197
\(786\) −9.00000 −0.321019
\(787\) 29.0000 1.03374 0.516869 0.856064i \(-0.327097\pi\)
0.516869 + 0.856064i \(0.327097\pi\)
\(788\) 18.0000 0.641223
\(789\) 4.00000 0.142404
\(790\) −36.0000 −1.28082
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) 1.00000 0.0355110
\(794\) 2.00000 0.0709773
\(795\) −18.0000 −0.638394
\(796\) 3.00000 0.106332
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) 2.00000 0.0706225
\(803\) −5.00000 −0.176446
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) −6.00000 −0.211341
\(807\) −16.0000 −0.563227
\(808\) 14.0000 0.492518
\(809\) −24.0000 −0.843795 −0.421898 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) −3.00000 −0.105409
\(811\) 29.0000 1.01833 0.509164 0.860670i \(-0.329955\pi\)
0.509164 + 0.860670i \(0.329955\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 1.00000 0.0350500
\(815\) −18.0000 −0.630512
\(816\) 3.00000 0.105021
\(817\) 3.00000 0.104957
\(818\) 15.0000 0.524463
\(819\) 0 0
\(820\) 0 0
\(821\) −16.0000 −0.558404 −0.279202 0.960232i \(-0.590070\pi\)
−0.279202 + 0.960232i \(0.590070\pi\)
\(822\) 15.0000 0.523185
\(823\) 10.0000 0.348578 0.174289 0.984695i \(-0.444237\pi\)
0.174289 + 0.984695i \(0.444237\pi\)
\(824\) 1.00000 0.0348367
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −15.0000 −0.520972 −0.260486 0.965478i \(-0.583883\pi\)
−0.260486 + 0.965478i \(0.583883\pi\)
\(830\) 18.0000 0.624789
\(831\) 8.00000 0.277517
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) −16.0000 −0.554035
\(835\) 9.00000 0.311458
\(836\) −1.00000 −0.0345857
\(837\) 6.00000 0.207390
\(838\) −21.0000 −0.725433
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −30.0000 −1.03387
\(843\) 18.0000 0.619953
\(844\) 25.0000 0.860535
\(845\) −3.00000 −0.103203
\(846\) 4.00000 0.137523
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −12.0000 −0.411597
\(851\) 1.00000 0.0342796
\(852\) −6.00000 −0.205557
\(853\) −16.0000 −0.547830 −0.273915 0.961754i \(-0.588319\pi\)
−0.273915 + 0.961754i \(0.588319\pi\)
\(854\) 0 0
\(855\) −3.00000 −0.102598
\(856\) −8.00000 −0.273434
\(857\) 26.0000 0.888143 0.444072 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) 1.00000 0.0341394
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) −9.00000 −0.306897
\(861\) 0 0
\(862\) 6.00000 0.204361
\(863\) −18.0000 −0.612727 −0.306364 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −12.0000 −0.408012
\(866\) 4.00000 0.135926
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −12.0000 −0.407072
\(870\) 15.0000 0.508548
\(871\) 16.0000 0.542139
\(872\) −7.00000 −0.237050
\(873\) 18.0000 0.609208
\(874\) −1.00000 −0.0338255
\(875\) 0 0
\(876\) −5.00000 −0.168934
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) −19.0000 −0.641219
\(879\) 14.0000 0.472208
\(880\) 3.00000 0.101130
\(881\) −35.0000 −1.17918 −0.589590 0.807703i \(-0.700711\pi\)
−0.589590 + 0.807703i \(0.700711\pi\)
\(882\) 0 0
\(883\) −25.0000 −0.841317 −0.420658 0.907219i \(-0.638201\pi\)
−0.420658 + 0.907219i \(0.638201\pi\)
\(884\) −3.00000 −0.100901
\(885\) −6.00000 −0.201688
\(886\) 42.0000 1.41102
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 1.00000 0.0335578
\(889\) 0 0
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 2.00000 0.0669650
\(893\) 4.00000 0.133855
\(894\) −4.00000 −0.133780
\(895\) 54.0000 1.80502
\(896\) 0 0
\(897\) 1.00000 0.0333890
\(898\) −39.0000 −1.30145
\(899\) −30.0000 −1.00056
\(900\) 4.00000 0.133333
\(901\) 18.0000 0.599667
\(902\) 0 0
\(903\) 0 0
\(904\) 12.0000 0.399114
\(905\) −42.0000 −1.39613
\(906\) 9.00000 0.299005
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) 12.0000 0.398234
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) 9.00000 0.298183 0.149092 0.988823i \(-0.452365\pi\)
0.149092 + 0.988823i \(0.452365\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 6.00000 0.198571
\(914\) −22.0000 −0.727695
\(915\) 3.00000 0.0991769
\(916\) −4.00000 −0.132164
\(917\) 0 0
\(918\) 3.00000 0.0990148
\(919\) 34.0000 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 3.00000 0.0989071
\(921\) 20.0000 0.659022
\(922\) −33.0000 −1.08680
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −21.0000 −0.690103
\(927\) 1.00000 0.0328443
\(928\) 5.00000 0.164133
\(929\) 4.00000 0.131236 0.0656179 0.997845i \(-0.479098\pi\)
0.0656179 + 0.997845i \(0.479098\pi\)
\(930\) −18.0000 −0.590243
\(931\) 0 0
\(932\) −8.00000 −0.262049
\(933\) −6.00000 −0.196431
\(934\) 33.0000 1.07979
\(935\) −9.00000 −0.294331
\(936\) 1.00000 0.0326860
\(937\) −28.0000 −0.914720 −0.457360 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) −12.0000 −0.391397
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) −13.0000 −0.423563
\(943\) 0 0
\(944\) −2.00000 −0.0650945
\(945\) 0 0
\(946\) −3.00000 −0.0975384
\(947\) −33.0000 −1.07236 −0.536178 0.844105i \(-0.680132\pi\)
−0.536178 + 0.844105i \(0.680132\pi\)
\(948\) −12.0000 −0.389742
\(949\) 5.00000 0.162307
\(950\) 4.00000 0.129777
\(951\) −22.0000 −0.713399
\(952\) 0 0
\(953\) −8.00000 −0.259145 −0.129573 0.991570i \(-0.541361\pi\)
−0.129573 + 0.991570i \(0.541361\pi\)
\(954\) −6.00000 −0.194257
\(955\) 15.0000 0.485389
\(956\) −6.00000 −0.194054
\(957\) 5.00000 0.161627
\(958\) −5.00000 −0.161543
\(959\) 0 0
\(960\) 3.00000 0.0968246
\(961\) 5.00000 0.161290
\(962\) −1.00000 −0.0322413
\(963\) −8.00000 −0.257796
\(964\) 14.0000 0.450910
\(965\) −6.00000 −0.193147
\(966\) 0 0
\(967\) 41.0000 1.31847 0.659236 0.751936i \(-0.270880\pi\)
0.659236 + 0.751936i \(0.270880\pi\)
\(968\) −10.0000 −0.321412
\(969\) 3.00000 0.0963739
\(970\) −54.0000 −1.73384
\(971\) 44.0000 1.41203 0.706014 0.708198i \(-0.250492\pi\)
0.706014 + 0.708198i \(0.250492\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) −4.00000 −0.128103
\(976\) 1.00000 0.0320092
\(977\) −13.0000 −0.415907 −0.207953 0.978139i \(-0.566680\pi\)
−0.207953 + 0.978139i \(0.566680\pi\)
\(978\) −6.00000 −0.191859
\(979\) 0 0
\(980\) 0 0
\(981\) −7.00000 −0.223493
\(982\) 18.0000 0.574403
\(983\) −9.00000 −0.287055 −0.143528 0.989646i \(-0.545845\pi\)
−0.143528 + 0.989646i \(0.545845\pi\)
\(984\) 0 0
\(985\) −54.0000 −1.72058
\(986\) −15.0000 −0.477697
\(987\) 0 0
\(988\) 1.00000 0.0318142
\(989\) −3.00000 −0.0953945
\(990\) 3.00000 0.0953463
\(991\) 42.0000 1.33417 0.667087 0.744980i \(-0.267541\pi\)
0.667087 + 0.744980i \(0.267541\pi\)
\(992\) −6.00000 −0.190500
\(993\) 0 0
\(994\) 0 0
\(995\) −9.00000 −0.285319
\(996\) 6.00000 0.190117
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) −6.00000 −0.189927
\(999\) 1.00000 0.0316386
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.r.1.1 1
7.6 odd 2 3822.2.a.bg.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3822.2.a.r.1.1 1 1.1 even 1 trivial
3822.2.a.bg.1.1 yes 1 7.6 odd 2