Properties

Label 4022.2.a.a.1.1
Level $4022$
Weight $2$
Character 4022.1
Self dual yes
Analytic conductor $32.116$
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] = [4022,2,Mod(1,4022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4022, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4022 = 2 \cdot 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4022.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: \(32.1158316930\)
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 \(\chi\) \(=\) 4022.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} -6.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +3.00000 q^{18} -1.00000 q^{19} +2.00000 q^{20} -1.00000 q^{22} +5.00000 q^{23} -1.00000 q^{25} +6.00000 q^{26} -2.00000 q^{28} +6.00000 q^{29} -5.00000 q^{31} -1.00000 q^{32} -4.00000 q^{35} -3.00000 q^{36} -7.00000 q^{37} +1.00000 q^{38} -2.00000 q^{40} -1.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -6.00000 q^{45} -5.00000 q^{46} +8.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} +13.0000 q^{53} +2.00000 q^{55} +2.00000 q^{56} -6.00000 q^{58} -3.00000 q^{59} +5.00000 q^{61} +5.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -12.0000 q^{65} +5.00000 q^{67} +4.00000 q^{70} +5.00000 q^{71} +3.00000 q^{72} +2.00000 q^{73} +7.00000 q^{74} -1.00000 q^{76} -2.00000 q^{77} +2.00000 q^{79} +2.00000 q^{80} +9.00000 q^{81} +1.00000 q^{82} -4.00000 q^{86} -1.00000 q^{88} +18.0000 q^{89} +6.00000 q^{90} +12.0000 q^{91} +5.00000 q^{92} -8.00000 q^{94} -2.00000 q^{95} +12.0000 q^{97} +3.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) −2.00000 −0.632456
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 3.00000 0.707107
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 5.00000 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) −3.00000 −0.500000
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) −1.00000 −0.156174 −0.0780869 0.996947i \(-0.524881\pi\)
−0.0780869 + 0.996947i \(0.524881\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) −6.00000 −0.894427
\(46\) −5.00000 −0.737210
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) 13.0000 1.78569 0.892844 0.450367i \(-0.148707\pi\)
0.892844 + 0.450367i \(0.148707\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 5.00000 0.635001
\(63\) 6.00000 0.755929
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 5.00000 0.593391 0.296695 0.954972i \(-0.404115\pi\)
0.296695 + 0.954972i \(0.404115\pi\)
\(72\) 3.00000 0.353553
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 7.00000 0.813733
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 2.00000 0.223607
\(81\) 9.00000 1.00000
\(82\) 1.00000 0.110432
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 6.00000 0.632456
\(91\) 12.0000 1.25794
\(92\) 5.00000 0.521286
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 3.00000 0.303046
\(99\) −3.00000 −0.301511
\(100\) −1.00000 −0.100000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) −5.00000 −0.492665 −0.246332 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −13.0000 −1.26267
\(107\) 5.00000 0.483368 0.241684 0.970355i \(-0.422300\pi\)
0.241684 + 0.970355i \(0.422300\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 10.0000 0.932505
\(116\) 6.00000 0.557086
\(117\) 18.0000 1.66410
\(118\) 3.00000 0.276172
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −5.00000 −0.452679
\(123\) 0 0
\(124\) −5.00000 −0.449013
\(125\) −12.0000 −1.07331
\(126\) −6.00000 −0.534522
\(127\) 17.0000 1.50851 0.754253 0.656584i \(-0.227999\pi\)
0.754253 + 0.656584i \(0.227999\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 12.0000 1.05247
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) −5.00000 −0.431934
\(135\) 0 0
\(136\) 0 0
\(137\) 19.0000 1.62328 0.811640 0.584158i \(-0.198575\pi\)
0.811640 + 0.584158i \(0.198575\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) −5.00000 −0.419591
\(143\) −6.00000 −0.501745
\(144\) −3.00000 −0.250000
\(145\) 12.0000 0.996546
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −17.0000 −1.38344 −0.691720 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 2.00000 0.161165
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −2.00000 −0.159111
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) −10.0000 −0.788110
\(162\) −9.00000 −0.707107
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) −1.00000 −0.0780869
\(165\) 0 0
\(166\) 0 0
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 3.00000 0.229416
\(172\) 4.00000 0.304997
\(173\) −11.0000 −0.836315 −0.418157 0.908375i \(-0.637324\pi\)
−0.418157 + 0.908375i \(0.637324\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −18.0000 −1.34916
\(179\) 21.0000 1.56961 0.784807 0.619740i \(-0.212762\pi\)
0.784807 + 0.619740i \(0.212762\pi\)
\(180\) −6.00000 −0.447214
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) −12.0000 −0.889499
\(183\) 0 0
\(184\) −5.00000 −0.368605
\(185\) −14.0000 −1.02930
\(186\) 0 0
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 3.00000 0.213201
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −14.0000 −0.985037
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 5.00000 0.348367
\(207\) −15.0000 −1.04257
\(208\) −6.00000 −0.416025
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 13.0000 0.892844
\(213\) 0 0
\(214\) −5.00000 −0.341793
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 2.00000 0.133631
\(225\) 3.00000 0.200000
\(226\) 6.00000 0.399114
\(227\) 16.0000 1.06196 0.530979 0.847385i \(-0.321824\pi\)
0.530979 + 0.847385i \(0.321824\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) −10.0000 −0.659380
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) −18.0000 −1.17670
\(235\) 16.0000 1.04372
\(236\) −3.00000 −0.195283
\(237\) 0 0
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 25.0000 1.61039 0.805196 0.593009i \(-0.202060\pi\)
0.805196 + 0.593009i \(0.202060\pi\)
\(242\) 10.0000 0.642824
\(243\) 0 0
\(244\) 5.00000 0.320092
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 5.00000 0.317500
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) −19.0000 −1.19927 −0.599635 0.800274i \(-0.704687\pi\)
−0.599635 + 0.800274i \(0.704687\pi\)
\(252\) 6.00000 0.377964
\(253\) 5.00000 0.314347
\(254\) −17.0000 −1.06667
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 14.0000 0.869918
\(260\) −12.0000 −0.744208
\(261\) −18.0000 −1.11417
\(262\) −15.0000 −0.926703
\(263\) −11.0000 −0.678289 −0.339145 0.940734i \(-0.610138\pi\)
−0.339145 + 0.940734i \(0.610138\pi\)
\(264\) 0 0
\(265\) 26.0000 1.59717
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) 5.00000 0.305424
\(269\) 1.00000 0.0609711 0.0304855 0.999535i \(-0.490295\pi\)
0.0304855 + 0.999535i \(0.490295\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −19.0000 −1.14783
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −7.00000 −0.420589 −0.210295 0.977638i \(-0.567442\pi\)
−0.210295 + 0.977638i \(0.567442\pi\)
\(278\) 20.0000 1.19952
\(279\) 15.0000 0.898027
\(280\) 4.00000 0.239046
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 5.00000 0.296695
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 2.00000 0.118056
\(288\) 3.00000 0.176777
\(289\) −17.0000 −1.00000
\(290\) −12.0000 −0.704664
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) −8.00000 −0.467365 −0.233682 0.972313i \(-0.575078\pi\)
−0.233682 + 0.972313i \(0.575078\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 7.00000 0.406867
\(297\) 0 0
\(298\) 22.0000 1.27443
\(299\) −30.0000 −1.73494
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 17.0000 0.978240
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) −11.0000 −0.627803 −0.313902 0.949456i \(-0.601636\pi\)
−0.313902 + 0.949456i \(0.601636\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) 10.0000 0.567962
\(311\) 33.0000 1.87126 0.935629 0.352985i \(-0.114833\pi\)
0.935629 + 0.352985i \(0.114833\pi\)
\(312\) 0 0
\(313\) −16.0000 −0.904373 −0.452187 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) 2.00000 0.112867
\(315\) 12.0000 0.676123
\(316\) 2.00000 0.112509
\(317\) 33.0000 1.85346 0.926732 0.375722i \(-0.122605\pi\)
0.926732 + 0.375722i \(0.122605\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) 10.0000 0.557278
\(323\) 0 0
\(324\) 9.00000 0.500000
\(325\) 6.00000 0.332820
\(326\) −11.0000 −0.609234
\(327\) 0 0
\(328\) 1.00000 0.0552158
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 0 0
\(333\) 21.0000 1.15079
\(334\) 2.00000 0.109435
\(335\) 10.0000 0.546358
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) 0 0
\(341\) −5.00000 −0.270765
\(342\) −3.00000 −0.162221
\(343\) 20.0000 1.07990
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 11.0000 0.591364
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 10.0000 0.530745
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) −21.0000 −1.10988
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 6.00000 0.316228
\(361\) −18.0000 −0.947368
\(362\) −6.00000 −0.315353
\(363\) 0 0
\(364\) 12.0000 0.628971
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 5.00000 0.260643
\(369\) 3.00000 0.156174
\(370\) 14.0000 0.727825
\(371\) −26.0000 −1.34985
\(372\) 0 0
\(373\) 7.00000 0.362446 0.181223 0.983442i \(-0.441994\pi\)
0.181223 + 0.983442i \(0.441994\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) −2.00000 −0.102598
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) 36.0000 1.83951 0.919757 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 2.00000 0.101797
\(387\) −12.0000 −0.609994
\(388\) 12.0000 0.609208
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) −12.0000 −0.604551
\(395\) 4.00000 0.201262
\(396\) −3.00000 −0.150756
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 30.0000 1.49441
\(404\) 14.0000 0.696526
\(405\) 18.0000 0.894427
\(406\) 12.0000 0.595550
\(407\) −7.00000 −0.346977
\(408\) 0 0
\(409\) 23.0000 1.13728 0.568638 0.822588i \(-0.307470\pi\)
0.568638 + 0.822588i \(0.307470\pi\)
\(410\) 2.00000 0.0987730
\(411\) 0 0
\(412\) −5.00000 −0.246332
\(413\) 6.00000 0.295241
\(414\) 15.0000 0.737210
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) 1.00000 0.0489116
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) −24.0000 −1.16692
\(424\) −13.0000 −0.631336
\(425\) 0 0
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 5.00000 0.241684
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −21.0000 −1.01153 −0.505767 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) 0 0
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) −10.0000 −0.480015
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −5.00000 −0.239182
\(438\) 0 0
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 36.0000 1.70656
\(446\) 0 0
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) −3.00000 −0.141421
\(451\) −1.00000 −0.0470882
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) −16.0000 −0.750917
\(455\) 24.0000 1.12514
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −22.0000 −1.02799
\(459\) 0 0
\(460\) 10.0000 0.466252
\(461\) 40.0000 1.86299 0.931493 0.363760i \(-0.118507\pi\)
0.931493 + 0.363760i \(0.118507\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 18.0000 0.832050
\(469\) −10.0000 −0.461757
\(470\) −16.0000 −0.738025
\(471\) 0 0
\(472\) 3.00000 0.138086
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −39.0000 −1.78569
\(478\) −8.00000 −0.365911
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) 42.0000 1.91504
\(482\) −25.0000 −1.13872
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 24.0000 1.08978
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) −5.00000 −0.226339
\(489\) 0 0
\(490\) 6.00000 0.271052
\(491\) −38.0000 −1.71492 −0.857458 0.514554i \(-0.827958\pi\)
−0.857458 + 0.514554i \(0.827958\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −6.00000 −0.269953
\(495\) −6.00000 −0.269680
\(496\) −5.00000 −0.224507
\(497\) −10.0000 −0.448561
\(498\) 0 0
\(499\) −34.0000 −1.52205 −0.761025 0.648723i \(-0.775303\pi\)
−0.761025 + 0.648723i \(0.775303\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) 19.0000 0.848012
\(503\) 15.0000 0.668817 0.334408 0.942428i \(-0.391463\pi\)
0.334408 + 0.942428i \(0.391463\pi\)
\(504\) −6.00000 −0.267261
\(505\) 28.0000 1.24598
\(506\) −5.00000 −0.222277
\(507\) 0 0
\(508\) 17.0000 0.754253
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) −10.0000 −0.440653
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) −14.0000 −0.615125
\(519\) 0 0
\(520\) 12.0000 0.526235
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 18.0000 0.787839
\(523\) −14.0000 −0.612177 −0.306089 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) 15.0000 0.655278
\(525\) 0 0
\(526\) 11.0000 0.479623
\(527\) 0 0
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) −26.0000 −1.12937
\(531\) 9.00000 0.390567
\(532\) 2.00000 0.0867110
\(533\) 6.00000 0.259889
\(534\) 0 0
\(535\) 10.0000 0.432338
\(536\) −5.00000 −0.215967
\(537\) 0 0
\(538\) −1.00000 −0.0431131
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 7.00000 0.300954 0.150477 0.988614i \(-0.451919\pi\)
0.150477 + 0.988614i \(0.451919\pi\)
\(542\) 2.00000 0.0859074
\(543\) 0 0
\(544\) 0 0
\(545\) −8.00000 −0.342682
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 19.0000 0.811640
\(549\) −15.0000 −0.640184
\(550\) 1.00000 0.0426401
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 7.00000 0.297402
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 11.0000 0.466085 0.233042 0.972467i \(-0.425132\pi\)
0.233042 + 0.972467i \(0.425132\pi\)
\(558\) −15.0000 −0.635001
\(559\) −24.0000 −1.01509
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) 22.0000 0.928014
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) −12.0000 −0.504844
\(566\) 0 0
\(567\) −18.0000 −0.755929
\(568\) −5.00000 −0.209795
\(569\) −28.0000 −1.17382 −0.586911 0.809652i \(-0.699656\pi\)
−0.586911 + 0.809652i \(0.699656\pi\)
\(570\) 0 0
\(571\) −2.00000 −0.0836974 −0.0418487 0.999124i \(-0.513325\pi\)
−0.0418487 + 0.999124i \(0.513325\pi\)
\(572\) −6.00000 −0.250873
\(573\) 0 0
\(574\) −2.00000 −0.0834784
\(575\) −5.00000 −0.208514
\(576\) −3.00000 −0.125000
\(577\) −11.0000 −0.457936 −0.228968 0.973434i \(-0.573535\pi\)
−0.228968 + 0.973434i \(0.573535\pi\)
\(578\) 17.0000 0.707107
\(579\) 0 0
\(580\) 12.0000 0.498273
\(581\) 0 0
\(582\) 0 0
\(583\) 13.0000 0.538405
\(584\) −2.00000 −0.0827606
\(585\) 36.0000 1.48842
\(586\) 8.00000 0.330477
\(587\) −7.00000 −0.288921 −0.144460 0.989511i \(-0.546145\pi\)
−0.144460 + 0.989511i \(0.546145\pi\)
\(588\) 0 0
\(589\) 5.00000 0.206021
\(590\) 6.00000 0.247016
\(591\) 0 0
\(592\) −7.00000 −0.287698
\(593\) 39.0000 1.60154 0.800769 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −22.0000 −0.901155
\(597\) 0 0
\(598\) 30.0000 1.22679
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −3.00000 −0.122373 −0.0611863 0.998126i \(-0.519488\pi\)
−0.0611863 + 0.998126i \(0.519488\pi\)
\(602\) 8.00000 0.326056
\(603\) −15.0000 −0.610847
\(604\) −17.0000 −0.691720
\(605\) −20.0000 −0.813116
\(606\) 0 0
\(607\) 1.00000 0.0405887 0.0202944 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) −12.0000 −0.484675 −0.242338 0.970192i \(-0.577914\pi\)
−0.242338 + 0.970192i \(0.577914\pi\)
\(614\) 11.0000 0.443924
\(615\) 0 0
\(616\) 2.00000 0.0805823
\(617\) 32.0000 1.28827 0.644136 0.764911i \(-0.277217\pi\)
0.644136 + 0.764911i \(0.277217\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) −10.0000 −0.401610
\(621\) 0 0
\(622\) −33.0000 −1.32318
\(623\) −36.0000 −1.44231
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 16.0000 0.639489
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 0 0
\(630\) −12.0000 −0.478091
\(631\) −48.0000 −1.91085 −0.955425 0.295234i \(-0.904602\pi\)
−0.955425 + 0.295234i \(0.904602\pi\)
\(632\) −2.00000 −0.0795557
\(633\) 0 0
\(634\) −33.0000 −1.31060
\(635\) 34.0000 1.34925
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) −6.00000 −0.237542
\(639\) −15.0000 −0.593391
\(640\) −2.00000 −0.0790569
\(641\) 3.00000 0.118493 0.0592464 0.998243i \(-0.481130\pi\)
0.0592464 + 0.998243i \(0.481130\pi\)
\(642\) 0 0
\(643\) −1.00000 −0.0394362 −0.0197181 0.999806i \(-0.506277\pi\)
−0.0197181 + 0.999806i \(0.506277\pi\)
\(644\) −10.0000 −0.394055
\(645\) 0 0
\(646\) 0 0
\(647\) 45.0000 1.76913 0.884566 0.466415i \(-0.154454\pi\)
0.884566 + 0.466415i \(0.154454\pi\)
\(648\) −9.00000 −0.353553
\(649\) −3.00000 −0.117760
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) 11.0000 0.430793
\(653\) 38.0000 1.48705 0.743527 0.668705i \(-0.233151\pi\)
0.743527 + 0.668705i \(0.233151\pi\)
\(654\) 0 0
\(655\) 30.0000 1.17220
\(656\) −1.00000 −0.0390434
\(657\) −6.00000 −0.234082
\(658\) 16.0000 0.623745
\(659\) −18.0000 −0.701180 −0.350590 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) 0 0
\(661\) −15.0000 −0.583432 −0.291716 0.956505i \(-0.594226\pi\)
−0.291716 + 0.956505i \(0.594226\pi\)
\(662\) 16.0000 0.621858
\(663\) 0 0
\(664\) 0 0
\(665\) 4.00000 0.155113
\(666\) −21.0000 −0.813733
\(667\) 30.0000 1.16160
\(668\) −2.00000 −0.0773823
\(669\) 0 0
\(670\) −10.0000 −0.386334
\(671\) 5.00000 0.193023
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −45.0000 −1.72949 −0.864745 0.502211i \(-0.832520\pi\)
−0.864745 + 0.502211i \(0.832520\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 0 0
\(682\) 5.00000 0.191460
\(683\) −51.0000 −1.95146 −0.975730 0.218975i \(-0.929729\pi\)
−0.975730 + 0.218975i \(0.929729\pi\)
\(684\) 3.00000 0.114708
\(685\) 38.0000 1.45191
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −78.0000 −2.97156
\(690\) 0 0
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) −11.0000 −0.418157
\(693\) 6.00000 0.227921
\(694\) 18.0000 0.683271
\(695\) −40.0000 −1.51729
\(696\) 0 0
\(697\) 0 0
\(698\) −14.0000 −0.529908
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) −5.00000 −0.188847 −0.0944237 0.995532i \(-0.530101\pi\)
−0.0944237 + 0.995532i \(0.530101\pi\)
\(702\) 0 0
\(703\) 7.00000 0.264010
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) −28.0000 −1.05305
\(708\) 0 0
\(709\) 32.0000 1.20179 0.600893 0.799330i \(-0.294812\pi\)
0.600893 + 0.799330i \(0.294812\pi\)
\(710\) −10.0000 −0.375293
\(711\) −6.00000 −0.225018
\(712\) −18.0000 −0.674579
\(713\) −25.0000 −0.936257
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 21.0000 0.784807
\(717\) 0 0
\(718\) 36.0000 1.34351
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) −6.00000 −0.223607
\(721\) 10.0000 0.372419
\(722\) 18.0000 0.669891
\(723\) 0 0
\(724\) 6.00000 0.222988
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 22.0000 0.815935 0.407967 0.912996i \(-0.366238\pi\)
0.407967 + 0.912996i \(0.366238\pi\)
\(728\) −12.0000 −0.444750
\(729\) −27.0000 −1.00000
\(730\) −4.00000 −0.148047
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) −17.0000 −0.627481
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) 5.00000 0.184177
\(738\) −3.00000 −0.110432
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) −14.0000 −0.514650
\(741\) 0 0
\(742\) 26.0000 0.954490
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) −44.0000 −1.61204
\(746\) −7.00000 −0.256288
\(747\) 0 0
\(748\) 0 0
\(749\) −10.0000 −0.365392
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 36.0000 1.31104
\(755\) −34.0000 −1.23739
\(756\) 0 0
\(757\) 6.00000 0.218074 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 6.00000 0.217930
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) 39.0000 1.41375 0.706874 0.707339i \(-0.250105\pi\)
0.706874 + 0.707339i \(0.250105\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) 18.0000 0.649942
\(768\) 0 0
\(769\) 54.0000 1.94729 0.973645 0.228069i \(-0.0732413\pi\)
0.973645 + 0.228069i \(0.0732413\pi\)
\(770\) 4.00000 0.144150
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) 33.0000 1.18693 0.593464 0.804861i \(-0.297760\pi\)
0.593464 + 0.804861i \(0.297760\pi\)
\(774\) 12.0000 0.431331
\(775\) 5.00000 0.179605
\(776\) −12.0000 −0.430775
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 1.00000 0.0358287
\(780\) 0 0
\(781\) 5.00000 0.178914
\(782\) 0 0
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) −13.0000 −0.463400 −0.231700 0.972787i \(-0.574429\pi\)
−0.231700 + 0.972787i \(0.574429\pi\)
\(788\) 12.0000 0.427482
\(789\) 0 0
\(790\) −4.00000 −0.142314
\(791\) 12.0000 0.426671
\(792\) 3.00000 0.106600
\(793\) −30.0000 −1.06533
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −54.0000 −1.90800
\(802\) −30.0000 −1.05934
\(803\) 2.00000 0.0705785
\(804\) 0 0
\(805\) −20.0000 −0.704907
\(806\) −30.0000 −1.05670
\(807\) 0 0
\(808\) −14.0000 −0.492518
\(809\) 25.0000 0.878953 0.439477 0.898254i \(-0.355164\pi\)
0.439477 + 0.898254i \(0.355164\pi\)
\(810\) −18.0000 −0.632456
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) −12.0000 −0.421117
\(813\) 0 0
\(814\) 7.00000 0.245350
\(815\) 22.0000 0.770626
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) −23.0000 −0.804176
\(819\) −36.0000 −1.25794
\(820\) −2.00000 −0.0698430
\(821\) −14.0000 −0.488603 −0.244302 0.969699i \(-0.578559\pi\)
−0.244302 + 0.969699i \(0.578559\pi\)
\(822\) 0 0
\(823\) 42.0000 1.46403 0.732014 0.681290i \(-0.238581\pi\)
0.732014 + 0.681290i \(0.238581\pi\)
\(824\) 5.00000 0.174183
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) −15.0000 −0.521286
\(829\) −49.0000 −1.70184 −0.850920 0.525295i \(-0.823955\pi\)
−0.850920 + 0.525295i \(0.823955\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −6.00000 −0.208013
\(833\) 0 0
\(834\) 0 0
\(835\) −4.00000 −0.138426
\(836\) −1.00000 −0.0345857
\(837\) 0 0
\(838\) 6.00000 0.207267
\(839\) −28.0000 −0.966667 −0.483334 0.875436i \(-0.660574\pi\)
−0.483334 + 0.875436i \(0.660574\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −20.0000 −0.689246
\(843\) 0 0
\(844\) 0 0
\(845\) 46.0000 1.58245
\(846\) 24.0000 0.825137
\(847\) 20.0000 0.687208
\(848\) 13.0000 0.446422
\(849\) 0 0
\(850\) 0 0
\(851\) −35.0000 −1.19978
\(852\) 0 0
\(853\) −54.0000 −1.84892 −0.924462 0.381273i \(-0.875486\pi\)
−0.924462 + 0.381273i \(0.875486\pi\)
\(854\) 10.0000 0.342193
\(855\) 6.00000 0.205196
\(856\) −5.00000 −0.170896
\(857\) −17.0000 −0.580709 −0.290354 0.956919i \(-0.593773\pi\)
−0.290354 + 0.956919i \(0.593773\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 21.0000 0.715263
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 0 0
\(865\) −22.0000 −0.748022
\(866\) −18.0000 −0.611665
\(867\) 0 0
\(868\) 10.0000 0.339422
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) −30.0000 −1.01651
\(872\) 4.00000 0.135457
\(873\) −36.0000 −1.21842
\(874\) 5.00000 0.169128
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) −25.0000 −0.844190 −0.422095 0.906552i \(-0.638705\pi\)
−0.422095 + 0.906552i \(0.638705\pi\)
\(878\) −14.0000 −0.472477
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) 12.0000 0.404290 0.202145 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) −9.00000 −0.303046
\(883\) −22.0000 −0.740359 −0.370179 0.928960i \(-0.620704\pi\)
−0.370179 + 0.928960i \(0.620704\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) −45.0000 −1.51095 −0.755476 0.655176i \(-0.772594\pi\)
−0.755476 + 0.655176i \(0.772594\pi\)
\(888\) 0 0
\(889\) −34.0000 −1.14032
\(890\) −36.0000 −1.20672
\(891\) 9.00000 0.301511
\(892\) 0 0
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) 42.0000 1.40391
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −12.0000 −0.400445
\(899\) −30.0000 −1.00056
\(900\) 3.00000 0.100000
\(901\) 0 0
\(902\) 1.00000 0.0332964
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 12.0000 0.398893
\(906\) 0 0
\(907\) −30.0000 −0.996134 −0.498067 0.867139i \(-0.665957\pi\)
−0.498067 + 0.867139i \(0.665957\pi\)
\(908\) 16.0000 0.530979
\(909\) −42.0000 −1.39305
\(910\) −24.0000 −0.795592
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) −30.0000 −0.990687
\(918\) 0 0
\(919\) 2.00000 0.0659739 0.0329870 0.999456i \(-0.489498\pi\)
0.0329870 + 0.999456i \(0.489498\pi\)
\(920\) −10.0000 −0.329690
\(921\) 0 0
\(922\) −40.0000 −1.31733
\(923\) −30.0000 −0.987462
\(924\) 0 0
\(925\) 7.00000 0.230159
\(926\) 16.0000 0.525793
\(927\) 15.0000 0.492665
\(928\) −6.00000 −0.196960
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) 12.0000 0.393073
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) −18.0000 −0.588348
\(937\) 50.0000 1.63343 0.816714 0.577042i \(-0.195793\pi\)
0.816714 + 0.577042i \(0.195793\pi\)
\(938\) 10.0000 0.326512
\(939\) 0 0
\(940\) 16.0000 0.521862
\(941\) −27.0000 −0.880175 −0.440087 0.897955i \(-0.645053\pi\)
−0.440087 + 0.897955i \(0.645053\pi\)
\(942\) 0 0
\(943\) −5.00000 −0.162822
\(944\) −3.00000 −0.0976417
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 58.0000 1.88475 0.942373 0.334563i \(-0.108589\pi\)
0.942373 + 0.334563i \(0.108589\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) −1.00000 −0.0324443
\(951\) 0 0
\(952\) 0 0
\(953\) 1.00000 0.0323932 0.0161966 0.999869i \(-0.494844\pi\)
0.0161966 + 0.999869i \(0.494844\pi\)
\(954\) 39.0000 1.26267
\(955\) −16.0000 −0.517748
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) −36.0000 −1.16311
\(959\) −38.0000 −1.22708
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) −42.0000 −1.35413
\(963\) −15.0000 −0.483368
\(964\) 25.0000 0.805196
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) 31.0000 0.996893 0.498446 0.866921i \(-0.333904\pi\)
0.498446 + 0.866921i \(0.333904\pi\)
\(968\) 10.0000 0.321412
\(969\) 0 0
\(970\) −24.0000 −0.770594
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 40.0000 1.28234
\(974\) 24.0000 0.769010
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) −31.0000 −0.991778 −0.495889 0.868386i \(-0.665158\pi\)
−0.495889 + 0.868386i \(0.665158\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) −6.00000 −0.191663
\(981\) 12.0000 0.383131
\(982\) 38.0000 1.21263
\(983\) 43.0000 1.37149 0.685744 0.727843i \(-0.259477\pi\)
0.685744 + 0.727843i \(0.259477\pi\)
\(984\) 0 0
\(985\) 24.0000 0.764704
\(986\) 0 0
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) 20.0000 0.635963
\(990\) 6.00000 0.190693
\(991\) −2.00000 −0.0635321 −0.0317660 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 5.00000 0.158750
\(993\) 0 0
\(994\) 10.0000 0.317181
\(995\) −48.0000 −1.52170
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 34.0000 1.07625
\(999\) 0 0
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 4022.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4022.2.a.a.1.1 1 1.1 even 1 trivial