Properties

Label 3822.2.a.f.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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +1.00000 q^{20} +3.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +9.00000 q^{29} +1.00000 q^{30} +5.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} +1.00000 q^{39} -1.00000 q^{40} -4.00000 q^{41} -3.00000 q^{44} +1.00000 q^{45} -6.00000 q^{46} -1.00000 q^{48} +4.00000 q^{50} +2.00000 q^{51} -1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} -3.00000 q^{55} -9.00000 q^{58} -7.00000 q^{59} -1.00000 q^{60} -4.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -3.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} -6.00000 q^{69} +6.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +8.00000 q^{74} +4.00000 q^{75} -1.00000 q^{78} -13.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} -3.00000 q^{83} -2.00000 q^{85} -9.00000 q^{87} +3.00000 q^{88} -8.00000 q^{89} -1.00000 q^{90} +6.00000 q^{92} -5.00000 q^{93} +1.00000 q^{96} -15.0000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) 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 \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 1.00000 0.182574
\(31\) 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
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) −1.00000 −0.158114
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −3.00000 −0.452267
\(45\) 1.00000 0.149071
\(46\) −6.00000 −0.884652
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) 2.00000 0.280056
\(52\) −1.00000 −0.138675
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) −9.00000 −1.18176
\(59\) −7.00000 −0.911322 −0.455661 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) −1.00000 −0.129099
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) −5.00000 −0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) −3.00000 −0.369274
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.00000 −0.242536
\(69\) −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
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 8.00000 0.929981
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) −13.0000 −1.46261 −0.731307 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) −9.00000 −0.964901
\(88\) 3.00000 0.319801
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) −5.00000 −0.518476
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −15.0000 −1.52302 −0.761510 0.648154i \(-0.775541\pi\)
−0.761510 + 0.648154i \(0.775541\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) −4.00000 −0.400000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −2.00000 −0.198030
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) 17.0000 1.64345 0.821726 0.569883i \(-0.193011\pi\)
0.821726 + 0.569883i \(0.193011\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 3.00000 0.286039
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 9.00000 0.835629
\(117\) −1.00000 −0.0924500
\(118\) 7.00000 0.644402
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −2.00000 −0.181818
\(122\) 4.00000 0.362143
\(123\) 4.00000 0.360668
\(124\) 5.00000 0.449013
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 1.00000 0.0877058
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 3.00000 0.261116
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) −1.00000 −0.0860663
\(136\) 2.00000 0.171499
\(137\) −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 \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 3.00000 0.250873
\(144\) 1.00000 0.0833333
\(145\) 9.00000 0.747409
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −4.00000 −0.326599
\(151\) −11.0000 −0.895167 −0.447584 0.894242i \(-0.647715\pi\)
−0.447584 + 0.894242i \(0.647715\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 5.00000 0.401610
\(156\) 1.00000 0.0800641
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 13.0000 1.03422
\(159\) −1.00000 −0.0793052
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −4.00000 −0.312348
\(165\) 3.00000 0.233550
\(166\) 3.00000 0.232845
\(167\) −4.00000 −0.309529 −0.154765 0.987951i \(-0.549462\pi\)
−0.154765 + 0.987951i \(0.549462\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) 0 0
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 9.00000 0.682288
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 7.00000 0.526152
\(178\) 8.00000 0.599625
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 1.00000 0.0745356
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 0 0
\(183\) 4.00000 0.295689
\(184\) −6.00000 −0.442326
\(185\) −8.00000 −0.588172
\(186\) 5.00000 0.366618
\(187\) 6.00000 0.438763
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 19.0000 1.36765 0.683825 0.729646i \(-0.260315\pi\)
0.683825 + 0.729646i \(0.260315\pi\)
\(194\) 15.0000 1.07694
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 3.00000 0.213201
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 4.00000 0.282843
\(201\) 4.00000 0.282138
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) −4.00000 −0.279372
\(206\) 16.0000 1.11477
\(207\) 6.00000 0.417029
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 1.00000 0.0686803
\(213\) −6.00000 −0.411113
\(214\) −17.0000 −1.16210
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −16.0000 −1.08366
\(219\) −6.00000 −0.405442
\(220\) −3.00000 −0.202260
\(221\) 2.00000 0.134535
\(222\) −8.00000 −0.536925
\(223\) −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
\(227\) −13.0000 −0.862840 −0.431420 0.902151i \(-0.641987\pi\)
−0.431420 + 0.902151i \(0.641987\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) −9.00000 −0.590879
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −7.00000 −0.455661
\(237\) 13.0000 0.844441
\(238\) 0 0
\(239\) 22.0000 1.42306 0.711531 0.702655i \(-0.248002\pi\)
0.711531 + 0.702655i \(0.248002\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −27.0000 −1.73922 −0.869611 0.493737i \(-0.835631\pi\)
−0.869611 + 0.493737i \(0.835631\pi\)
\(242\) 2.00000 0.128565
\(243\) −1.00000 −0.0641500
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) −4.00000 −0.255031
\(247\) 0 0
\(248\) −5.00000 −0.317500
\(249\) 3.00000 0.190117
\(250\) 9.00000 0.569210
\(251\) 23.0000 1.45175 0.725874 0.687828i \(-0.241436\pi\)
0.725874 + 0.687828i \(0.241436\pi\)
\(252\) 0 0
\(253\) −18.0000 −1.13165
\(254\) −5.00000 −0.313728
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −26.0000 −1.62184 −0.810918 0.585160i \(-0.801032\pi\)
−0.810918 + 0.585160i \(0.801032\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.00000 −0.0620174
\(261\) 9.00000 0.557086
\(262\) −15.0000 −0.926703
\(263\) −26.0000 −1.60323 −0.801614 0.597841i \(-0.796025\pi\)
−0.801614 + 0.597841i \(0.796025\pi\)
\(264\) −3.00000 −0.184637
\(265\) 1.00000 0.0614295
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) −4.00000 −0.244339
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 1.00000 0.0608581
\(271\) 9.00000 0.546711 0.273356 0.961913i \(-0.411866\pi\)
0.273356 + 0.961913i \(0.411866\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 12.0000 0.723627
\(276\) −6.00000 −0.361158
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −4.00000 −0.239904
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) −3.00000 −0.177394
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −9.00000 −0.528498
\(291\) 15.0000 0.879316
\(292\) 6.00000 0.351123
\(293\) 15.0000 0.876309 0.438155 0.898900i \(-0.355632\pi\)
0.438155 + 0.898900i \(0.355632\pi\)
\(294\) 0 0
\(295\) −7.00000 −0.407556
\(296\) 8.00000 0.464991
\(297\) 3.00000 0.174078
\(298\) 6.00000 0.347571
\(299\) −6.00000 −0.346989
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) 11.0000 0.632979
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) −4.00000 −0.229039
\(306\) 2.00000 0.114332
\(307\) −18.0000 −1.02731 −0.513657 0.857996i \(-0.671710\pi\)
−0.513657 + 0.857996i \(0.671710\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) −5.00000 −0.283981
\(311\) 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 \(0.380247\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) −13.0000 −0.731307
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 1.00000 0.0560772
\(319\) −27.0000 −1.51171
\(320\) 1.00000 0.0559017
\(321\) −17.0000 −0.948847
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) 4.00000 0.221540
\(327\) −16.0000 −0.884802
\(328\) 4.00000 0.220863
\(329\) 0 0
\(330\) −3.00000 −0.165145
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −3.00000 −0.164646
\(333\) −8.00000 −0.438397
\(334\) 4.00000 0.218870
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 1.00000 0.0544735 0.0272367 0.999629i \(-0.491329\pi\)
0.0272367 + 0.999629i \(0.491329\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −12.0000 −0.651751
\(340\) −2.00000 −0.108465
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −6.00000 −0.323029
\(346\) 14.0000 0.752645
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) −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
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −7.00000 −0.372046
\(355\) 6.00000 0.318447
\(356\) −8.00000 −0.423999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −2.00000 −0.105556 −0.0527780 0.998606i \(-0.516808\pi\)
−0.0527780 + 0.998606i \(0.516808\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −19.0000 −1.00000
\(362\) 8.00000 0.420471
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) −4.00000 −0.209083
\(367\) −3.00000 −0.156599 −0.0782994 0.996930i \(-0.524949\pi\)
−0.0782994 + 0.996930i \(0.524949\pi\)
\(368\) 6.00000 0.312772
\(369\) −4.00000 −0.208232
\(370\) 8.00000 0.415900
\(371\) 0 0
\(372\) −5.00000 −0.259238
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −6.00000 −0.310253
\(375\) 9.00000 0.464758
\(376\) 0 0
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) −24.0000 −1.23280 −0.616399 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 0 0
\(381\) −5.00000 −0.256158
\(382\) 6.00000 0.306987
\(383\) −26.0000 −1.32854 −0.664269 0.747494i \(-0.731257\pi\)
−0.664269 + 0.747494i \(0.731257\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −19.0000 −0.967075
\(387\) 0 0
\(388\) −15.0000 −0.761510
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) −1.00000 −0.0506370
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) −15.0000 −0.756650
\(394\) 22.0000 1.10834
\(395\) −13.0000 −0.654101
\(396\) −3.00000 −0.150756
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) −4.00000 −0.199502
\(403\) −5.00000 −0.249068
\(404\) −2.00000 −0.0995037
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) −2.00000 −0.0990148
\(409\) −11.0000 −0.543915 −0.271957 0.962309i \(-0.587671\pi\)
−0.271957 + 0.962309i \(0.587671\pi\)
\(410\) 4.00000 0.197546
\(411\) 12.0000 0.591916
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) −3.00000 −0.147264
\(416\) 1.00000 0.0490290
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −12.0000 −0.584844 −0.292422 0.956289i \(-0.594461\pi\)
−0.292422 + 0.956289i \(0.594461\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) −1.00000 −0.0485643
\(425\) 8.00000 0.388057
\(426\) 6.00000 0.290701
\(427\) 0 0
\(428\) 17.0000 0.821726
\(429\) −3.00000 −0.144841
\(430\) 0 0
\(431\) 26.0000 1.25238 0.626188 0.779672i \(-0.284614\pi\)
0.626188 + 0.779672i \(0.284614\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −38.0000 −1.82616 −0.913082 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 16.0000 0.766261
\(437\) 0 0
\(438\) 6.00000 0.286691
\(439\) −29.0000 −1.38409 −0.692047 0.721852i \(-0.743291\pi\)
−0.692047 + 0.721852i \(0.743291\pi\)
\(440\) 3.00000 0.143019
\(441\) 0 0
\(442\) −2.00000 −0.0951303
\(443\) 39.0000 1.85295 0.926473 0.376361i \(-0.122825\pi\)
0.926473 + 0.376361i \(0.122825\pi\)
\(444\) 8.00000 0.379663
\(445\) −8.00000 −0.379236
\(446\) 1.00000 0.0473514
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 8.00000 0.377543 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 4.00000 0.188562
\(451\) 12.0000 0.565058
\(452\) 12.0000 0.564433
\(453\) 11.0000 0.516825
\(454\) 13.0000 0.610120
\(455\) 0 0
\(456\) 0 0
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) 14.0000 0.654177
\(459\) 2.00000 0.0933520
\(460\) 6.00000 0.279751
\(461\) −26.0000 −1.21094 −0.605470 0.795868i \(-0.707015\pi\)
−0.605470 + 0.795868i \(0.707015\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 9.00000 0.417815
\(465\) −5.00000 −0.231869
\(466\) 14.0000 0.648537
\(467\) 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
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 7.00000 0.322201
\(473\) 0 0
\(474\) −13.0000 −0.597110
\(475\) 0 0
\(476\) 0 0
\(477\) 1.00000 0.0457869
\(478\) −22.0000 −1.00626
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 1.00000 0.0456435
\(481\) 8.00000 0.364769
\(482\) 27.0000 1.22982
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −15.0000 −0.681115
\(486\) 1.00000 0.0453609
\(487\) −9.00000 −0.407829 −0.203914 0.978989i \(-0.565366\pi\)
−0.203914 + 0.978989i \(0.565366\pi\)
\(488\) 4.00000 0.181071
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −29.0000 −1.30875 −0.654376 0.756169i \(-0.727069\pi\)
−0.654376 + 0.756169i \(0.727069\pi\)
\(492\) 4.00000 0.180334
\(493\) −18.0000 −0.810679
\(494\) 0 0
\(495\) −3.00000 −0.134840
\(496\) 5.00000 0.224507
\(497\) 0 0
\(498\) −3.00000 −0.134433
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) −9.00000 −0.402492
\(501\) 4.00000 0.178707
\(502\) −23.0000 −1.02654
\(503\) −26.0000 −1.15928 −0.579641 0.814872i \(-0.696807\pi\)
−0.579641 + 0.814872i \(0.696807\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 18.0000 0.800198
\(507\) −1.00000 −0.0444116
\(508\) 5.00000 0.221839
\(509\) −21.0000 −0.930809 −0.465404 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 26.0000 1.14681
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 1.00000 0.0438529
\(521\) 28.0000 1.22670 0.613351 0.789810i \(-0.289821\pi\)
0.613351 + 0.789810i \(0.289821\pi\)
\(522\) −9.00000 −0.393919
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 15.0000 0.655278
\(525\) 0 0
\(526\) 26.0000 1.13365
\(527\) −10.0000 −0.435607
\(528\) 3.00000 0.130558
\(529\) 13.0000 0.565217
\(530\) −1.00000 −0.0434372
\(531\) −7.00000 −0.303774
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) −8.00000 −0.346194
\(535\) 17.0000 0.734974
\(536\) 4.00000 0.172774
\(537\) −12.0000 −0.517838
\(538\) 3.00000 0.129339
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −32.0000 −1.37579 −0.687894 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) −9.00000 −0.386583
\(543\) 8.00000 0.343313
\(544\) 2.00000 0.0857493
\(545\) 16.0000 0.685365
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −12.0000 −0.512615
\(549\) −4.00000 −0.170716
\(550\) −12.0000 −0.511682
\(551\) 0 0
\(552\) 6.00000 0.255377
\(553\) 0 0
\(554\) 22.0000 0.934690
\(555\) 8.00000 0.339581
\(556\) 4.00000 0.169638
\(557\) 17.0000 0.720313 0.360157 0.932892i \(-0.382723\pi\)
0.360157 + 0.932892i \(0.382723\pi\)
\(558\) −5.00000 −0.211667
\(559\) 0 0
\(560\) 0 0
\(561\) −6.00000 −0.253320
\(562\) 0 0
\(563\) 3.00000 0.126435 0.0632175 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 3.00000 0.125436
\(573\) 6.00000 0.250654
\(574\) 0 0
\(575\) −24.0000 −1.00087
\(576\) 1.00000 0.0416667
\(577\) −3.00000 −0.124892 −0.0624458 0.998048i \(-0.519890\pi\)
−0.0624458 + 0.998048i \(0.519890\pi\)
\(578\) 13.0000 0.540729
\(579\) −19.0000 −0.789613
\(580\) 9.00000 0.373705
\(581\) 0 0
\(582\) −15.0000 −0.621770
\(583\) −3.00000 −0.124247
\(584\) −6.00000 −0.248282
\(585\) −1.00000 −0.0413449
\(586\) −15.0000 −0.619644
\(587\) −9.00000 −0.371470 −0.185735 0.982600i \(-0.559467\pi\)
−0.185735 + 0.982600i \(0.559467\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 7.00000 0.288185
\(591\) 22.0000 0.904959
\(592\) −8.00000 −0.328798
\(593\) −40.0000 −1.64260 −0.821302 0.570494i \(-0.806752\pi\)
−0.821302 + 0.570494i \(0.806752\pi\)
\(594\) −3.00000 −0.123091
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 24.0000 0.982255
\(598\) 6.00000 0.245358
\(599\) 18.0000 0.735460 0.367730 0.929933i \(-0.380135\pi\)
0.367730 + 0.929933i \(0.380135\pi\)
\(600\) −4.00000 −0.163299
\(601\) 19.0000 0.775026 0.387513 0.921864i \(-0.373334\pi\)
0.387513 + 0.921864i \(0.373334\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −11.0000 −0.447584
\(605\) −2.00000 −0.0813116
\(606\) −2.00000 −0.0812444
\(607\) 1.00000 0.0405887 0.0202944 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) −28.0000 −1.13091 −0.565455 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(614\) 18.0000 0.726421
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) 4.00000 0.161034 0.0805170 0.996753i \(-0.474343\pi\)
0.0805170 + 0.996753i \(0.474343\pi\)
\(618\) −16.0000 −0.643614
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 5.00000 0.200805
\(621\) −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
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −13.0000 −0.517522 −0.258761 0.965941i \(-0.583314\pi\)
−0.258761 + 0.965941i \(0.583314\pi\)
\(632\) 13.0000 0.517112
\(633\) −20.0000 −0.794929
\(634\) −3.00000 −0.119145
\(635\) 5.00000 0.198419
\(636\) −1.00000 −0.0396526
\(637\) 0 0
\(638\) 27.0000 1.06894
\(639\) 6.00000 0.237356
\(640\) −1.00000 −0.0395285
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 17.0000 0.670936
\(643\) −32.0000 −1.26196 −0.630978 0.775800i \(-0.717346\pi\)
−0.630978 + 0.775800i \(0.717346\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 36.0000 1.41531 0.707653 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 21.0000 0.824322
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 39.0000 1.52619 0.763094 0.646288i \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) 16.0000 0.625650
\(655\) 15.0000 0.586098
\(656\) −4.00000 −0.156174
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 3.00000 0.116775
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) −8.00000 −0.310929
\(663\) −2.00000 −0.0776736
\(664\) 3.00000 0.116423
\(665\) 0 0
\(666\) 8.00000 0.309994
\(667\) 54.0000 2.09089
\(668\) −4.00000 −0.154765
\(669\) 1.00000 0.0386622
\(670\) 4.00000 0.154533
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) 17.0000 0.655302 0.327651 0.944799i \(-0.393743\pi\)
0.327651 + 0.944799i \(0.393743\pi\)
\(674\) −1.00000 −0.0385186
\(675\) 4.00000 0.153960
\(676\) 1.00000 0.0384615
\(677\) 31.0000 1.19143 0.595713 0.803197i \(-0.296869\pi\)
0.595713 + 0.803197i \(0.296869\pi\)
\(678\) 12.0000 0.460857
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) 13.0000 0.498161
\(682\) 15.0000 0.574380
\(683\) −41.0000 −1.56882 −0.784411 0.620242i \(-0.787034\pi\)
−0.784411 + 0.620242i \(0.787034\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 0 0
\(689\) −1.00000 −0.0380970
\(690\) 6.00000 0.228416
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) 4.00000 0.151729
\(696\) 9.00000 0.341144
\(697\) 8.00000 0.303022
\(698\) 34.0000 1.28692
\(699\) 14.0000 0.529529
\(700\) 0 0
\(701\) −39.0000 −1.47301 −0.736505 0.676432i \(-0.763525\pi\)
−0.736505 + 0.676432i \(0.763525\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 0 0
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 7.00000 0.263076
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) −6.00000 −0.225176
\(711\) −13.0000 −0.487538
\(712\) 8.00000 0.299813
\(713\) 30.0000 1.12351
\(714\) 0 0
\(715\) 3.00000 0.112194
\(716\) 12.0000 0.448461
\(717\) −22.0000 −0.821605
\(718\) 2.00000 0.0746393
\(719\) 2.00000 0.0745874 0.0372937 0.999304i \(-0.488126\pi\)
0.0372937 + 0.999304i \(0.488126\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) 27.0000 1.00414
\(724\) −8.00000 −0.297318
\(725\) −36.0000 −1.33701
\(726\) −2.00000 −0.0742270
\(727\) −9.00000 −0.333792 −0.166896 0.985975i \(-0.553374\pi\)
−0.166896 + 0.985975i \(0.553374\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) 0 0
\(732\) 4.00000 0.147844
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 3.00000 0.110732
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 12.0000 0.442026
\(738\) 4.00000 0.147242
\(739\) −18.0000 −0.662141 −0.331070 0.943606i \(-0.607410\pi\)
−0.331070 + 0.943606i \(0.607410\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) 0 0
\(743\) −20.0000 −0.733729 −0.366864 0.930274i \(-0.619569\pi\)
−0.366864 + 0.930274i \(0.619569\pi\)
\(744\) 5.00000 0.183309
\(745\) −6.00000 −0.219823
\(746\) 26.0000 0.951928
\(747\) −3.00000 −0.109764
\(748\) 6.00000 0.219382
\(749\) 0 0
\(750\) −9.00000 −0.328634
\(751\) 13.0000 0.474377 0.237188 0.971464i \(-0.423774\pi\)
0.237188 + 0.971464i \(0.423774\pi\)
\(752\) 0 0
\(753\) −23.0000 −0.838167
\(754\) 9.00000 0.327761
\(755\) −11.0000 −0.400331
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 24.0000 0.871719
\(759\) 18.0000 0.653359
\(760\) 0 0
\(761\) −32.0000 −1.16000 −0.580000 0.814617i \(-0.696947\pi\)
−0.580000 + 0.814617i \(0.696947\pi\)
\(762\) 5.00000 0.181131
\(763\) 0 0
\(764\) −6.00000 −0.217072
\(765\) −2.00000 −0.0723102
\(766\) 26.0000 0.939418
\(767\) 7.00000 0.252755
\(768\) −1.00000 −0.0360844
\(769\) −13.0000 −0.468792 −0.234396 0.972141i \(-0.575311\pi\)
−0.234396 + 0.972141i \(0.575311\pi\)
\(770\) 0 0
\(771\) 26.0000 0.936367
\(772\) 19.0000 0.683825
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) −20.0000 −0.718421
\(776\) 15.0000 0.538469
\(777\) 0 0
\(778\) −34.0000 −1.21896
\(779\) 0 0
\(780\) 1.00000 0.0358057
\(781\) −18.0000 −0.644091
\(782\) 12.0000 0.429119
\(783\) −9.00000 −0.321634
\(784\) 0 0
\(785\) −14.0000 −0.499681
\(786\) 15.0000 0.535032
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −22.0000 −0.783718
\(789\) 26.0000 0.925625
\(790\) 13.0000 0.462519
\(791\) 0 0
\(792\) 3.00000 0.106600
\(793\) 4.00000 0.142044
\(794\) −34.0000 −1.20661
\(795\) −1.00000 −0.0354663
\(796\) −24.0000 −0.850657
\(797\) 7.00000 0.247953 0.123976 0.992285i \(-0.460435\pi\)
0.123976 + 0.992285i \(0.460435\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.00000 0.141421
\(801\) −8.00000 −0.282666
\(802\) −6.00000 −0.211867
\(803\) −18.0000 −0.635206
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 5.00000 0.176117
\(807\) 3.00000 0.105605
\(808\) 2.00000 0.0703598
\(809\) 4.00000 0.140633 0.0703163 0.997525i \(-0.477599\pi\)
0.0703163 + 0.997525i \(0.477599\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) 0 0
\(813\) −9.00000 −0.315644
\(814\) −24.0000 −0.841200
\(815\) −4.00000 −0.140114
\(816\) 2.00000 0.0700140
\(817\) 0 0
\(818\) 11.0000 0.384606
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 3.00000 0.104701 0.0523504 0.998629i \(-0.483329\pi\)
0.0523504 + 0.998629i \(0.483329\pi\)
\(822\) −12.0000 −0.418548
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) 16.0000 0.557386
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) −27.0000 −0.938882 −0.469441 0.882964i \(-0.655545\pi\)
−0.469441 + 0.882964i \(0.655545\pi\)
\(828\) 6.00000 0.208514
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 3.00000 0.104132
\(831\) 22.0000 0.763172
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) 4.00000 0.138509
\(835\) −4.00000 −0.138426
\(836\) 0 0
\(837\) −5.00000 −0.172825
\(838\) 0 0
\(839\) 44.0000 1.51905 0.759524 0.650479i \(-0.225432\pi\)
0.759524 + 0.650479i \(0.225432\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 12.0000 0.413547
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 0 0
\(848\) 1.00000 0.0343401
\(849\) −14.0000 −0.480479
\(850\) −8.00000 −0.274398
\(851\) −48.0000 −1.64542
\(852\) −6.00000 −0.205557
\(853\) 58.0000 1.98588 0.992941 0.118609i \(-0.0378434\pi\)
0.992941 + 0.118609i \(0.0378434\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −17.0000 −0.581048
\(857\) −50.0000 −1.70797 −0.853984 0.520300i \(-0.825820\pi\)
−0.853984 + 0.520300i \(0.825820\pi\)
\(858\) 3.00000 0.102418
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −26.0000 −0.885564
\(863\) 40.0000 1.36162 0.680808 0.732462i \(-0.261629\pi\)
0.680808 + 0.732462i \(0.261629\pi\)
\(864\) 1.00000 0.0340207
\(865\) −14.0000 −0.476014
\(866\) 38.0000 1.29129
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 39.0000 1.32298
\(870\) 9.00000 0.305129
\(871\) 4.00000 0.135535
\(872\) −16.0000 −0.541828
\(873\) −15.0000 −0.507673
\(874\) 0 0
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 29.0000 0.978703
\(879\) −15.0000 −0.505937
\(880\) −3.00000 −0.101130
\(881\) 50.0000 1.68454 0.842271 0.539054i \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) 0 0
\(883\) 10.0000 0.336527 0.168263 0.985742i \(-0.446184\pi\)
0.168263 + 0.985742i \(0.446184\pi\)
\(884\) 2.00000 0.0672673
\(885\) 7.00000 0.235302
\(886\) −39.0000 −1.31023
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) −8.00000 −0.268462
\(889\) 0 0
\(890\) 8.00000 0.268161
\(891\) −3.00000 −0.100504
\(892\) −1.00000 −0.0334825
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) −8.00000 −0.266963
\(899\) 45.0000 1.50083
\(900\) −4.00000 −0.133333
\(901\) −2.00000 −0.0666297
\(902\) −12.0000 −0.399556
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) −8.00000 −0.265929
\(906\) −11.0000 −0.365451
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) −13.0000 −0.431420
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −50.0000 −1.65657 −0.828287 0.560304i \(-0.810684\pi\)
−0.828287 + 0.560304i \(0.810684\pi\)
\(912\) 0 0
\(913\) 9.00000 0.297857
\(914\) −1.00000 −0.0330771
\(915\) 4.00000 0.132236
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) −6.00000 −0.197814
\(921\) 18.0000 0.593120
\(922\) 26.0000 0.856264
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) 32.0000 1.05215
\(926\) 8.00000 0.262896
\(927\) −16.0000 −0.525509
\(928\) −9.00000 −0.295439
\(929\) 28.0000 0.918650 0.459325 0.888268i \(-0.348091\pi\)
0.459325 + 0.888268i \(0.348091\pi\)
\(930\) 5.00000 0.163956
\(931\) 0 0
\(932\) −14.0000 −0.458585
\(933\) −2.00000 −0.0654771
\(934\) −12.0000 −0.392652
\(935\) 6.00000 0.196221
\(936\) 1.00000 0.0326860
\(937\) −37.0000 −1.20874 −0.604369 0.796705i \(-0.706575\pi\)
−0.604369 + 0.796705i \(0.706575\pi\)
\(938\) 0 0
\(939\) −13.0000 −0.424239
\(940\) 0 0
\(941\) −39.0000 −1.27136 −0.635682 0.771951i \(-0.719281\pi\)
−0.635682 + 0.771951i \(0.719281\pi\)
\(942\) −14.0000 −0.456145
\(943\) −24.0000 −0.781548
\(944\) −7.00000 −0.227831
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 13.0000 0.422220
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) −3.00000 −0.0972817
\(952\) 0 0
\(953\) 12.0000 0.388718 0.194359 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) −1.00000 −0.0323762
\(955\) −6.00000 −0.194155
\(956\) 22.0000 0.711531
\(957\) 27.0000 0.872786
\(958\) −30.0000 −0.969256
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) −6.00000 −0.193548
\(962\) −8.00000 −0.257930
\(963\) 17.0000 0.547817
\(964\) −27.0000 −0.869611
\(965\) 19.0000 0.611632
\(966\) 0 0
\(967\) 21.0000 0.675314 0.337657 0.941269i \(-0.390366\pi\)
0.337657 + 0.941269i \(0.390366\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) 15.0000 0.481621
\(971\) 13.0000 0.417190 0.208595 0.978002i \(-0.433111\pi\)
0.208595 + 0.978002i \(0.433111\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 9.00000 0.288379
\(975\) −4.00000 −0.128103
\(976\) −4.00000 −0.128037
\(977\) −22.0000 −0.703842 −0.351921 0.936030i \(-0.614471\pi\)
−0.351921 + 0.936030i \(0.614471\pi\)
\(978\) −4.00000 −0.127906
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 16.0000 0.510841
\(982\) 29.0000 0.925427
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) −4.00000 −0.127515
\(985\) −22.0000 −0.700978
\(986\) 18.0000 0.573237
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 3.00000 0.0953463
\(991\) 15.0000 0.476491 0.238245 0.971205i \(-0.423428\pi\)
0.238245 + 0.971205i \(0.423428\pi\)
\(992\) −5.00000 −0.158750
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) 3.00000 0.0950586
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) −16.0000 −0.506471
\(999\) 8.00000 0.253109
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.f.1.1 1
7.2 even 3 546.2.i.f.235.1 yes 2
7.4 even 3 546.2.i.f.79.1 2
7.6 odd 2 3822.2.a.l.1.1 1
21.2 odd 6 1638.2.j.e.235.1 2
21.11 odd 6 1638.2.j.e.1171.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.f.79.1 2 7.4 even 3
546.2.i.f.235.1 yes 2 7.2 even 3
1638.2.j.e.235.1 2 21.2 odd 6
1638.2.j.e.1171.1 2 21.11 odd 6
3822.2.a.f.1.1 1 1.1 even 1 trivial
3822.2.a.l.1.1 1 7.6 odd 2