↑

Properties

Label 1287.2.a.a.1.1
Level $1287$
Weight $2$
Character 1287.1
Self dual yes
Analytic conductor $10.277$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1287,2,Mod(1,1287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1287, 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("1287.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1287.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: \(10.2767467401\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1287.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} +3.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} -1.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} -4.00000 q^{17} +6.00000 q^{19} -2.00000 q^{20} +1.00000 q^{22} -8.00000 q^{23} -1.00000 q^{25} +1.00000 q^{26} +2.00000 q^{28} +10.0000 q^{31} -5.00000 q^{32} +4.00000 q^{34} -4.00000 q^{35} -6.00000 q^{37} -6.00000 q^{38} +6.00000 q^{40} +2.00000 q^{41} +1.00000 q^{44} +8.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} -2.00000 q^{55} -6.00000 q^{56} +4.00000 q^{59} -6.00000 q^{61} -10.0000 q^{62} +7.00000 q^{64} -2.00000 q^{65} -10.0000 q^{67} +4.00000 q^{68} +4.00000 q^{70} +10.0000 q^{73} +6.00000 q^{74} -6.00000 q^{76} +2.00000 q^{77} -4.00000 q^{79} -2.00000 q^{80} -2.00000 q^{82} -12.0000 q^{83} -8.00000 q^{85} -3.00000 q^{88} -18.0000 q^{89} +2.00000 q^{91} +8.00000 q^{92} +8.00000 q^{94} +12.0000 q^{95} -10.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −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 \(0.623376\pi\)
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 − 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) 6.00000 0.948683
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) −6.00000 −0.801784
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −10.0000 −1.27000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 0 0
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) −18.0000 −1.90800 −0.953998 − 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −8.00000 −0.796030 −0.398015 − 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) −12.0000 −1.12887 −0.564433 − 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) −16.0000 −1.49201
\(116\) 0 0
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −12.0000 −1.04053
\(134\) 10.0000 0.863868
\(135\) 0 0
\(136\) −12.0000 −1.02899
\(137\) −14.0000 −1.19610 −0.598050 − 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 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
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) −2.00000 −0.163846 −0.0819232 − 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 − 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 18.0000 1.45999
\(153\) 0 0
\(154\) −2.00000 −0.161165
\(155\) 20.0000 1.60644
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) −10.0000 −0.790569
\(161\) 16.0000 1.26098
\(162\) 0 0
\(163\) 22.0000 1.72317 0.861586 − 0.507611i \(-0.169471\pi\)
0.861586 + 0.507611i \(0.169471\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 8.00000 0.613572
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 18.0000 1.34916
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −26.0000 −1.93256 −0.966282 − 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) −24.0000 −1.76930
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) −12.0000 −0.870572
\(191\) 16.0000 1.15772 0.578860 − 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) 22.0000 1.58359 0.791797 − 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −22.0000 −1.56744 −0.783718 − 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 − 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) 8.00000 0.562878
\(203\) 0 0
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −20.0000 −1.35769
\(218\) −10.0000 −0.677285
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 − 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 10.0000 0.668153
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 16.0000 1.05501
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) −8.00000 −0.518563
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 − 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 30.0000 1.90500
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) −16.0000 −1.00991 −0.504956 − 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 20.0000 1.24757 0.623783 − 0.781598i \(-0.285595\pi\)
0.623783 + 0.781598i \(0.285595\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 12.0000 0.735767
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) 20.0000 1.21942 0.609711 − 0.792624i \(-0.291286\pi\)
0.609711 + 0.792624i \(0.291286\pi\)
\(270\) 0 0
\(271\) 14.0000 0.850439 0.425220 − 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 14.0000 0.845771
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) −12.0000 −0.717137
\(281\) −30.0000 −1.78965 −0.894825 − 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 − 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −1.00000 −0.0591312
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) −10.0000 −0.585206
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) −18.0000 −1.04623
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) −2.00000 −0.115087
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) −10.0000 −0.570730 −0.285365 − 0.958419i \(-0.592115\pi\)
−0.285365 + 0.958419i \(0.592115\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) −20.0000 −1.13592
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 − 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 2.00000 0.112331 0.0561656 − 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 14.0000 0.782624
\(321\) 0 0
\(322\) −16.0000 −0.891645
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) −22.0000 −1.21847
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) −2.00000 −0.109930 −0.0549650 − 0.998488i \(-0.517505\pi\)
−0.0549650 + 0.998488i \(0.517505\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 0 0
\(335\) −20.0000 −1.09272
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) 8.00000 0.433861
\(341\) −10.0000 −0.541530
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.0000 1.07366 0.536828 − 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 − 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 32.0000 1.68890 0.844448 − 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 26.0000 1.36653
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 20.0000 1.04685
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) 12.0000 0.623850
\(371\) 0 0
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 − 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) −24.0000 −1.23771
\(377\) 0 0
\(378\) 0 0
\(379\) 34.0000 1.74646 0.873231 − 0.487306i \(-0.162020\pi\)
0.873231 + 0.487306i \(0.162020\pi\)
\(380\) −12.0000 −0.615587
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) −32.0000 −1.63512 −0.817562 − 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) 24.0000 1.21685 0.608424 − 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 0 0
\(391\) 32.0000 1.61831
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) 22.0000 1.10834
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 − 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −24.0000 −1.20301
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 22.0000 1.09863 0.549314 − 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) −10.0000 −0.498135
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 − 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −4.00000 −0.197546
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 5.00000 0.245145
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) 28.0000 1.36789 0.683945 − 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 − 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) 12.0000 0.580721
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 − 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 20.0000 0.960031
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) −48.0000 −2.29615
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 − 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) −6.00000 −0.286039
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −28.0000 −1.33032 −0.665160 − 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) 0 0
\(445\) −36.0000 −1.70656
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) −14.0000 −0.661438
\(449\) −26.0000 −1.22702 −0.613508 − 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 12.0000 0.564433
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 − 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) 16.0000 0.746004
\(461\) −30.0000 −1.39724 −0.698620 − 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −38.0000 −1.76601 −0.883005 − 0.469364i \(-0.844483\pi\)
−0.883005 + 0.469364i \(0.844483\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 20.0000 0.923514
\(470\) 16.0000 0.738025
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) 0 0
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) −8.00000 −0.366679
\(477\) 0 0
\(478\) −24.0000 −1.09773
\(479\) 40.0000 1.82765 0.913823 − 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) −20.0000 −0.908153
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 − 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) −18.0000 −0.814822
\(489\) 0 0
\(490\) 6.00000 0.271052
\(491\) −28.0000 −1.26362 −0.631811 − 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 0 0
\(498\) 0 0
\(499\) 14.0000 0.626726 0.313363 − 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 16.0000 0.714115
\(503\) 32.0000 1.42681 0.713405 − 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) 0 0
\(505\) −16.0000 −0.711991
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 18.0000 0.797836 0.398918 − 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) −20.0000 −0.884748
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −20.0000 −0.882162
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) −12.0000 −0.527250
\(519\) 0 0
\(520\) −6.00000 −0.263117
\(521\) −12.0000 −0.525730 −0.262865 − 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −40.0000 −1.74243
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 0 0
\(532\) 12.0000 0.520266
\(533\) −2.00000 −0.0866296
\(534\) 0 0
\(535\) 0 0
\(536\) −30.0000 −1.29580
\(537\) 0 0
\(538\) −20.0000 −0.862261
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 − 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) −14.0000 −0.601351
\(543\) 0 0
\(544\) 20.0000 0.857493
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 − 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 14.0000 0.598050
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) 14.0000 0.593199 0.296600 − 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) 8.00000 0.337160 0.168580 − 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(564\) 0 0
\(565\) −24.0000 −1.00969
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 − 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −1.00000 −0.0418121
\(573\) 0 0
\(574\) 4.00000 0.166957
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) −6.00000 −0.249783 −0.124892 − 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 0 0
\(584\) 30.0000 1.24141
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −28.0000 −1.15568 −0.577842 − 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) 60.0000 2.47226
\(590\) −8.00000 −0.329355
\(591\) 0 0
\(592\) 6.00000 0.246598
\(593\) −26.0000 −1.06769 −0.533846 − 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) 2.00000 0.0819232
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) −4.00000 −0.163436 −0.0817178 − 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 − 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −2.00000 −0.0813788
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 − 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −30.0000 −1.21666
\(609\) 0 0
\(610\) 12.0000 0.485866
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 − 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 10.0000 0.403567
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −46.0000 −1.84890 −0.924448 − 0.381308i \(-0.875474\pi\)
−0.924448 + 0.381308i \(0.875474\pi\)
\(620\) −20.0000 −0.803219
\(621\) 0 0
\(622\) 0 0
\(623\) 36.0000 1.44231
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) −2.00000 −0.0796187 −0.0398094 − 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) −12.0000 −0.477334
\(633\) 0 0
\(634\) −2.00000 −0.0794301
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) 0 0
\(640\) 6.00000 0.237171
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −34.0000 −1.34083 −0.670415 − 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 4.00000 0.157256 0.0786281 − 0.996904i \(-0.474946\pi\)
0.0786281 + 0.996904i \(0.474946\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) −1.00000 −0.0392232
\(651\) 0 0
\(652\) −22.0000 −0.861586
\(653\) −4.00000 −0.156532 −0.0782660 − 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) −16.0000 −0.623745
\(659\) −28.0000 −1.09073 −0.545363 − 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 2.00000 0.0777322
\(663\) 0 0
\(664\) −36.0000 −1.39707
\(665\) −24.0000 −0.930680
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 20.0000 0.772667
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 − 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) 12.0000 0.461197 0.230599 − 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) −24.0000 −0.920358
\(681\) 0 0
\(682\) 10.0000 0.382920
\(683\) −20.0000 −0.765279 −0.382639 − 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) −28.0000 −1.06983
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 − 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) −40.0000 −1.51729
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) 18.0000 0.681310
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) −20.0000 −0.755390 −0.377695 − 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) 0 0
\(703\) −36.0000 −1.35777
\(704\) −7.00000 −0.263822
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 16.0000 0.601742
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 − 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −54.0000 −2.02374
\(713\) −80.0000 −2.99602
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −32.0000 −1.19423
\(719\) −24.0000 −0.895049 −0.447524 − 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) 26.0000 0.966282
\(725\) 0 0
\(726\) 0 0
\(727\) −24.0000 −0.890111 −0.445055 − 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(728\) 6.00000 0.222375
\(729\) 0 0
\(730\) −20.0000 −0.740233
\(731\) 0 0
\(732\) 0 0
\(733\) 42.0000 1.55131 0.775653 − 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 40.0000 1.47442
\(737\) 10.0000 0.368355
\(738\) 0 0
\(739\) 14.0000 0.514998 0.257499 − 0.966279i \(-0.417102\pi\)
0.257499 + 0.966279i \(0.417102\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) 0 0
\(743\) −40.0000 −1.46746 −0.733729 − 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) −4.00000 −0.146549
\(746\) 26.0000 0.951928
\(747\) 0 0
\(748\) −4.00000 −0.146254
\(749\) 0 0
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 − 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 0 0
\(755\) 4.00000 0.145575
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −34.0000 −1.23494
\(759\) 0 0
\(760\) 36.0000 1.30586
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) −20.0000 −0.724049
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) 32.0000 1.15621
\(767\) −4.00000 −0.144432
\(768\) 0 0
\(769\) 38.0000 1.37032 0.685158 − 0.728395i \(-0.259733\pi\)
0.685158 + 0.728395i \(0.259733\pi\)
\(770\) −4.00000 −0.144150
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) 30.0000 1.07903 0.539513 − 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 0 0
\(775\) −10.0000 −0.359211
\(776\) −30.0000 −1.07694
\(777\) 0 0
\(778\) −24.0000 −0.860442
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) −32.0000 −1.14432
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) −26.0000 −0.926800 −0.463400 − 0.886149i \(-0.653371\pi\)
−0.463400 + 0.886149i \(0.653371\pi\)
\(788\) 22.0000 0.783718
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) 40.0000 1.41687 0.708436 − 0.705775i \(-0.249401\pi\)
0.708436 + 0.705775i \(0.249401\pi\)
\(798\) 0 0
\(799\) 32.0000 1.13208
\(800\) 5.00000 0.176777
\(801\) 0 0
\(802\) −22.0000 −0.776847
\(803\) −10.0000 −0.352892
\(804\) 0 0
\(805\) 32.0000 1.12785
\(806\) 10.0000 0.352235
\(807\) 0 0
\(808\) −24.0000 −0.844317
\(809\) −48.0000 −1.68759 −0.843795 − 0.536666i \(-0.819684\pi\)
−0.843795 + 0.536666i \(0.819684\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 − 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −6.00000 −0.210300
\(815\) 44.0000 1.54125
\(816\) 0 0
\(817\) 0 0
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) −34.0000 −1.18661 −0.593304 − 0.804978i \(-0.702177\pi\)
−0.593304 + 0.804978i \(0.702177\pi\)
\(822\) 0 0
\(823\) 12.0000 0.418294 0.209147 − 0.977884i \(-0.432931\pi\)
0.209147 + 0.977884i \(0.432931\pi\)
\(824\) −12.0000 −0.418040
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) −44.0000 −1.53003 −0.765015 − 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 24.0000 0.833052
\(831\) 0 0
\(832\) −7.00000 −0.242681
\(833\) 12.0000 0.415775
\(834\) 0 0
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) −28.0000 −0.967244
\(839\) 8.00000 0.276191 0.138095 − 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 2.00000 0.0689246
\(843\) 0 0
\(844\) 0 0
\(845\) 2.00000 0.0688021
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) 0 0
\(850\) −4.00000 −0.137199
\(851\) 48.0000 1.64542
\(852\) 0 0
\(853\) 38.0000 1.30110 0.650548 − 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) −12.0000 −0.410632
\(855\) 0 0
\(856\) 0 0
\(857\) 16.0000 0.546550 0.273275 − 0.961936i \(-0.411893\pi\)
0.273275 + 0.961936i \(0.411893\pi\)
\(858\) 0 0
\(859\) −24.0000 −0.818869 −0.409435 − 0.912339i \(-0.634274\pi\)
−0.409435 + 0.912339i \(0.634274\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) 20.0000 0.678844
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 10.0000 0.338837
\(872\) 30.0000 1.01593
\(873\) 0 0
\(874\) 48.0000 1.62362
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 − 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) 4.00000 0.134763 0.0673817 − 0.997727i \(-0.478535\pi\)
0.0673817 + 0.997727i \(0.478535\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 − 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) −20.0000 −0.671534 −0.335767 − 0.941945i \(-0.608996\pi\)
−0.335767 + 0.941945i \(0.608996\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 36.0000 1.20672
\(891\) 0 0
\(892\) 14.0000 0.468755
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) −6.00000 −0.200446
\(897\) 0 0
\(898\) 26.0000 0.867631
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) −36.0000 −1.19734
\(905\) −52.0000 −1.72854
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 − 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) −4.00000 −0.132599
\(911\) 48.0000 1.59031 0.795155 − 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 0 0
\(918\) 0 0
\(919\) 28.0000 0.923635 0.461817 − 0.886975i \(-0.347198\pi\)
0.461817 + 0.886975i \(0.347198\pi\)
\(920\) −48.0000 −1.58251
\(921\) 0 0
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 38.0000 1.24876
\(927\) 0 0
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 − 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) −20.0000 −0.653023
\(939\) 0 0
\(940\) 16.0000 0.521862
\(941\) −42.0000 −1.36916 −0.684580 − 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 − 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) 6.00000 0.194666
\(951\) 0 0
\(952\) 24.0000 0.777844
\(953\) 56.0000 1.81402 0.907009 − 0.421111i \(-0.138360\pi\)
0.907009 + 0.421111i \(0.138360\pi\)
\(954\) 0 0
\(955\) 32.0000 1.03550
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) −40.0000 −1.29234
\(959\) 28.0000 0.904167
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) −6.00000 −0.193448
\(963\) 0 0
\(964\) 14.0000 0.450910
\(965\) 44.0000 1.41641
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 − 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 3.00000 0.0964237
\(969\) 0 0
\(970\) 20.0000 0.642161
\(971\) 40.0000 1.28366 0.641831 − 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 0 0
\(973\) 40.0000 1.28234
\(974\) 22.0000 0.704925
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 54.0000 1.72761 0.863807 − 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) 6.00000 0.191663
\(981\) 0 0
\(982\) 28.0000 0.893516
\(983\) −48.0000 −1.53096 −0.765481 − 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) 0 0
\(985\) −44.0000 −1.40196
\(986\) 0 0
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) 0 0
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 − 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) −50.0000 −1.58750
\(993\) 0 0
\(994\) 0 0
\(995\) 48.0000 1.52170
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 − 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −14.0000 −0.443162
\(999\) 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 1287.2.a.a.1.1 ✓ 1
3.2 odd 2 1287.2.a.c.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1287.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
1287.2.a.c.1.1 yes 1 3.2 odd 2