Properties

Label 1890.2.a.i.1.1
Level $1890$
Weight $2$
Character 1890.1
Self dual yes
Analytic conductor $15.092$
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] = [1890,2,Mod(1,1890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1890, 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("1890.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1890.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: \(15.0917259820\)
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 \(\chi\) \(=\) 1890.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -6.00000 q^{11} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +2.00000 q^{19} +1.00000 q^{20} +6.00000 q^{22} +6.00000 q^{23} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{28} +9.00000 q^{29} +5.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{35} +11.0000 q^{37} -2.00000 q^{38} -1.00000 q^{40} +9.00000 q^{41} +8.00000 q^{43} -6.00000 q^{44} -6.00000 q^{46} -3.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -4.00000 q^{52} -6.00000 q^{53} -6.00000 q^{55} -1.00000 q^{56} -9.00000 q^{58} +3.00000 q^{59} -1.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +14.0000 q^{67} -6.00000 q^{68} -1.00000 q^{70} -9.00000 q^{71} +11.0000 q^{73} -11.0000 q^{74} +2.00000 q^{76} -6.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} -9.00000 q^{82} -6.00000 q^{85} -8.00000 q^{86} +6.00000 q^{88} +6.00000 q^{89} -4.00000 q^{91} +6.00000 q^{92} +3.00000 q^{94} +2.00000 q^{95} -10.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 11.0000 1.80839 0.904194 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −9.00000 −1.18176
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) −5.00000 −0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −9.00000 −1.06810 −0.534052 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) −11.0000 −1.27872
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −9.00000 −0.993884
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) 3.00000 0.309426
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 6.00000 0.572078
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 9.00000 0.835629
\(117\) 0 0
\(118\) −3.00000 −0.276172
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 1.00000 0.0905357
\(123\) 0 0
\(124\) 5.00000 0.449013
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) 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
\(134\) −14.0000 −1.20942
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 9.00000 0.755263
\(143\) 24.0000 2.00698
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) −11.0000 −0.910366
\(147\) 0 0
\(148\) 11.0000 0.904194
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) 5.00000 0.401610
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 9.00000 0.702782
\(165\) 0 0
\(166\) 0 0
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −21.0000 −1.59660 −0.798300 0.602260i \(-0.794267\pi\)
−0.798300 + 0.602260i \(0.794267\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) 11.0000 0.808736
\(186\) 0 0
\(187\) 36.0000 2.63258
\(188\) −3.00000 −0.218797
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −12.0000 −0.844317
\(203\) 9.00000 0.631676
\(204\) 0 0
\(205\) 9.00000 0.628587
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 3.00000 0.205076
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 5.00000 0.339422
\(218\) 16.0000 1.08366
\(219\) 0 0
\(220\) −6.00000 −0.404520
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 9.00000 0.598671
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) −9.00000 −0.590879
\(233\) 15.0000 0.982683 0.491341 0.870967i \(-0.336507\pi\)
0.491341 + 0.870967i \(0.336507\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) 3.00000 0.195283
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) −25.0000 −1.60706
\(243\) 0 0
\(244\) −1.00000 −0.0640184
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) −5.00000 −0.317500
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −36.0000 −2.26330
\(254\) 7.00000 0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 0 0
\(259\) 11.0000 0.683507
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) −15.0000 −0.926703
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) 14.0000 0.855186
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −3.00000 −0.181237
\(275\) −6.00000 −0.361814
\(276\) 0 0
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −14.0000 −0.839664
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 5.00000 0.297219 0.148610 0.988896i \(-0.452520\pi\)
0.148610 + 0.988896i \(0.452520\pi\)
\(284\) −9.00000 −0.534052
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 9.00000 0.531253
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −9.00000 −0.528498
\(291\) 0 0
\(292\) 11.0000 0.643726
\(293\) −27.0000 −1.57736 −0.788678 0.614806i \(-0.789234\pi\)
−0.788678 + 0.614806i \(0.789234\pi\)
\(294\) 0 0
\(295\) 3.00000 0.174667
\(296\) −11.0000 −0.639362
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) −6.00000 −0.341882
\(309\) 0 0
\(310\) −5.00000 −0.283981
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) −54.0000 −3.02342
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −6.00000 −0.334367
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 16.0000 0.886158
\(327\) 0 0
\(328\) −9.00000 −0.496942
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −9.00000 −0.492458
\(335\) 14.0000 0.764902
\(336\) 0 0
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) −30.0000 −1.62459
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 21.0000 1.12897
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 6.00000 0.319801
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −9.00000 −0.477670
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 6.00000 0.317110
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 7.00000 0.367912
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) 11.0000 0.575766
\(366\) 0 0
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) −11.0000 −0.571863
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 23.0000 1.19089 0.595447 0.803394i \(-0.296975\pi\)
0.595447 + 0.803394i \(0.296975\pi\)
\(374\) −36.0000 −1.86152
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) 11.0000 0.565032 0.282516 0.959263i \(-0.408831\pi\)
0.282516 + 0.959263i \(0.408831\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) −24.0000 −1.22795
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) −6.00000 −0.305788
\(386\) 22.0000 1.11977
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) −15.0000 −0.760530 −0.380265 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) −9.00000 −0.446663
\(407\) −66.0000 −3.27150
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −9.00000 −0.444478
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 3.00000 0.147620
\(414\) 0 0
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) 12.0000 0.586939
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −5.00000 −0.243396
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −1.00000 −0.0483934
\(428\) −3.00000 −0.145010
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) −5.00000 −0.240008
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 12.0000 0.574038
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 6.00000 0.286039
\(441\) 0 0
\(442\) −24.0000 −1.14156
\(443\) −3.00000 −0.142534 −0.0712672 0.997457i \(-0.522704\pi\)
−0.0712672 + 0.997457i \(0.522704\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 10.0000 0.473514
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) −54.0000 −2.54276
\(452\) −9.00000 −0.423324
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 22.0000 1.02799
\(459\) 0 0
\(460\) 6.00000 0.279751
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 23.0000 1.06890 0.534450 0.845200i \(-0.320519\pi\)
0.534450 + 0.845200i \(0.320519\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) −15.0000 −0.694862
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) 14.0000 0.646460
\(470\) 3.00000 0.138380
\(471\) 0 0
\(472\) −3.00000 −0.138086
\(473\) −48.0000 −2.20704
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) −9.00000 −0.411650
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −44.0000 −2.00623
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 11.0000 0.498458 0.249229 0.968445i \(-0.419823\pi\)
0.249229 + 0.968445i \(0.419823\pi\)
\(488\) 1.00000 0.0452679
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −54.0000 −2.43204
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) −9.00000 −0.403705
\(498\) 0 0
\(499\) −31.0000 −1.38775 −0.693875 0.720095i \(-0.744098\pi\)
−0.693875 + 0.720095i \(0.744098\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 36.0000 1.60040
\(507\) 0 0
\(508\) −7.00000 −0.310575
\(509\) −12.0000 −0.531891 −0.265945 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(510\) 0 0
\(511\) 11.0000 0.486611
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 18.0000 0.791639
\(518\) −11.0000 −0.483312
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 15.0000 0.655278
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) −30.0000 −1.30682
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 2.00000 0.0867110
\(533\) −36.0000 −1.55933
\(534\) 0 0
\(535\) −3.00000 −0.129701
\(536\) −14.0000 −0.604708
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −11.0000 −0.472490
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) −16.0000 −0.685365
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 3.00000 0.128154
\(549\) 0 0
\(550\) 6.00000 0.255841
\(551\) 18.0000 0.766826
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) −32.0000 −1.35346
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) −9.00000 −0.378633
\(566\) −5.00000 −0.210166
\(567\) 0 0
\(568\) 9.00000 0.377632
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −13.0000 −0.544033 −0.272017 0.962293i \(-0.587691\pi\)
−0.272017 + 0.962293i \(0.587691\pi\)
\(572\) 24.0000 1.00349
\(573\) 0 0
\(574\) −9.00000 −0.375653
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) 9.00000 0.373705
\(581\) 0 0
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) −11.0000 −0.455183
\(585\) 0 0
\(586\) 27.0000 1.11536
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 10.0000 0.412043
\(590\) −3.00000 −0.123508
\(591\) 0 0
\(592\) 11.0000 0.452097
\(593\) 24.0000 0.985562 0.492781 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) 0 0
\(595\) −6.00000 −0.245976
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 24.0000 0.981433
\(599\) 3.00000 0.122577 0.0612883 0.998120i \(-0.480479\pi\)
0.0612883 + 0.998120i \(0.480479\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) −8.00000 −0.326056
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) 25.0000 1.01639
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 1.00000 0.0404888
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) 27.0000 1.08698 0.543490 0.839416i \(-0.317103\pi\)
0.543490 + 0.839416i \(0.317103\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 5.00000 0.200805
\(621\) 0 0
\(622\) −30.0000 −1.20289
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) −66.0000 −2.63159
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) −7.00000 −0.277787
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 54.0000 2.13788
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) 5.00000 0.197181 0.0985904 0.995128i \(-0.468567\pi\)
0.0985904 + 0.995128i \(0.468567\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 21.0000 0.825595 0.412798 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) 0 0
\(649\) −18.0000 −0.706562
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 15.0000 0.586098
\(656\) 9.00000 0.351391
\(657\) 0 0
\(658\) 3.00000 0.116952
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 23.0000 0.894596 0.447298 0.894385i \(-0.352386\pi\)
0.447298 + 0.894385i \(0.352386\pi\)
\(662\) 19.0000 0.738456
\(663\) 0 0
\(664\) 0 0
\(665\) 2.00000 0.0775567
\(666\) 0 0
\(667\) 54.0000 2.09089
\(668\) 9.00000 0.348220
\(669\) 0 0
\(670\) −14.0000 −0.540867
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −32.0000 −1.23259
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 33.0000 1.26829 0.634147 0.773213i \(-0.281352\pi\)
0.634147 + 0.773213i \(0.281352\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) 30.0000 1.14876
\(683\) 21.0000 0.803543 0.401771 0.915740i \(-0.368395\pi\)
0.401771 + 0.915740i \(0.368395\pi\)
\(684\) 0 0
\(685\) 3.00000 0.114624
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 26.0000 0.989087 0.494543 0.869153i \(-0.335335\pi\)
0.494543 + 0.869153i \(0.335335\pi\)
\(692\) −21.0000 −0.798300
\(693\) 0 0
\(694\) −3.00000 −0.113878
\(695\) 14.0000 0.531050
\(696\) 0 0
\(697\) −54.0000 −2.04540
\(698\) 10.0000 0.378506
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −27.0000 −1.01978 −0.509888 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(702\) 0 0
\(703\) 22.0000 0.829746
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 9.00000 0.337764
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 30.0000 1.12351
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) −6.00000 −0.224231
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 15.0000 0.558242
\(723\) 0 0
\(724\) −7.00000 −0.260153
\(725\) 9.00000 0.334252
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 4.00000 0.148250
\(729\) 0 0
\(730\) −11.0000 −0.407128
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) 10.0000 0.369107
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −84.0000 −3.09418
\(738\) 0 0
\(739\) −7.00000 −0.257499 −0.128750 0.991677i \(-0.541096\pi\)
−0.128750 + 0.991677i \(0.541096\pi\)
\(740\) 11.0000 0.404368
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) 18.0000 0.660356 0.330178 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) −23.0000 −0.842090
\(747\) 0 0
\(748\) 36.0000 1.31629
\(749\) −3.00000 −0.109618
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) −3.00000 −0.109399
\(753\) 0 0
\(754\) 36.0000 1.31104
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −7.00000 −0.254419 −0.127210 0.991876i \(-0.540602\pi\)
−0.127210 + 0.991876i \(0.540602\pi\)
\(758\) −11.0000 −0.399538
\(759\) 0 0
\(760\) −2.00000 −0.0725476
\(761\) 15.0000 0.543750 0.271875 0.962333i \(-0.412356\pi\)
0.271875 + 0.962333i \(0.412356\pi\)
\(762\) 0 0
\(763\) −16.0000 −0.579239
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 6.00000 0.216225
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) 3.00000 0.107903 0.0539513 0.998544i \(-0.482818\pi\)
0.0539513 + 0.998544i \(0.482818\pi\)
\(774\) 0 0
\(775\) 5.00000 0.179605
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) 15.0000 0.537776
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) 54.0000 1.93227
\(782\) 36.0000 1.28736
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −22.0000 −0.785214
\(786\) 0 0
\(787\) 23.0000 0.819861 0.409931 0.912117i \(-0.365553\pi\)
0.409931 + 0.912117i \(0.365553\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −9.00000 −0.320003
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −45.0000 −1.59398 −0.796991 0.603991i \(-0.793576\pi\)
−0.796991 + 0.603991i \(0.793576\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −24.0000 −0.847469
\(803\) −66.0000 −2.32909
\(804\) 0 0
\(805\) 6.00000 0.211472
\(806\) 20.0000 0.704470
\(807\) 0 0
\(808\) −12.0000 −0.422159
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 9.00000 0.315838
\(813\) 0 0
\(814\) 66.0000 2.31330
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 9.00000 0.314294
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) 5.00000 0.174289 0.0871445 0.996196i \(-0.472226\pi\)
0.0871445 + 0.996196i \(0.472226\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −3.00000 −0.104383
\(827\) −27.0000 −0.938882 −0.469441 0.882964i \(-0.655545\pi\)
−0.469441 + 0.882964i \(0.655545\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 9.00000 0.311458
\(836\) −12.0000 −0.415029
\(837\) 0 0
\(838\) 9.00000 0.310900
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −26.0000 −0.896019
\(843\) 0 0
\(844\) 5.00000 0.172107
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 25.0000 0.859010
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 66.0000 2.26245
\(852\) 0 0
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 1.00000 0.0342193
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) −24.0000 −0.819824 −0.409912 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) −21.0000 −0.715263
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) −21.0000 −0.714021
\(866\) 25.0000 0.849535
\(867\) 0 0
\(868\) 5.00000 0.169711
\(869\) −48.0000 −1.62829
\(870\) 0 0
\(871\) −56.0000 −1.89749
\(872\) 16.0000 0.541828
\(873\) 0 0
\(874\) −12.0000 −0.405906
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 41.0000 1.38447 0.692236 0.721671i \(-0.256626\pi\)
0.692236 + 0.721671i \(0.256626\pi\)
\(878\) −8.00000 −0.269987
\(879\) 0 0
\(880\) −6.00000 −0.202260
\(881\) 9.00000 0.303218 0.151609 0.988441i \(-0.451555\pi\)
0.151609 + 0.988441i \(0.451555\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 3.00000 0.100787
\(887\) −39.0000 −1.30949 −0.654746 0.755849i \(-0.727224\pi\)
−0.654746 + 0.755849i \(0.727224\pi\)
\(888\) 0 0
\(889\) −7.00000 −0.234772
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) −10.0000 −0.334825
\(893\) −6.00000 −0.200782
\(894\) 0 0
\(895\) −6.00000 −0.200558
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 0 0
\(899\) 45.0000 1.50083
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 54.0000 1.79800
\(903\) 0 0
\(904\) 9.00000 0.299336
\(905\) −7.00000 −0.232688
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) 4.00000 0.132599
\(911\) −9.00000 −0.298183 −0.149092 0.988823i \(-0.547635\pi\)
−0.149092 + 0.988823i \(0.547635\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) 15.0000 0.495344
\(918\) 0 0
\(919\) 38.0000 1.25350 0.626752 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) −6.00000 −0.197814
\(921\) 0 0
\(922\) 0 0
\(923\) 36.0000 1.18495
\(924\) 0 0
\(925\) 11.0000 0.361678
\(926\) −23.0000 −0.755827
\(927\) 0 0
\(928\) −9.00000 −0.295439
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 15.0000 0.491341
\(933\) 0 0
\(934\) −6.00000 −0.196326
\(935\) 36.0000 1.17733
\(936\) 0 0
\(937\) 47.0000 1.53542 0.767712 0.640796i \(-0.221395\pi\)
0.767712 + 0.640796i \(0.221395\pi\)
\(938\) −14.0000 −0.457116
\(939\) 0 0
\(940\) −3.00000 −0.0978492
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) 54.0000 1.75848
\(944\) 3.00000 0.0976417
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −44.0000 −1.42830
\(950\) −2.00000 −0.0648886
\(951\) 0 0
\(952\) 6.00000 0.194461
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) 24.0000 0.776622
\(956\) 9.00000 0.291081
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 3.00000 0.0968751
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 44.0000 1.41862
\(963\) 0 0
\(964\) 26.0000 0.837404
\(965\) −22.0000 −0.708205
\(966\) 0 0
\(967\) −13.0000 −0.418052 −0.209026 0.977910i \(-0.567029\pi\)
−0.209026 + 0.977910i \(0.567029\pi\)
\(968\) −25.0000 −0.803530
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) 33.0000 1.05902 0.529510 0.848304i \(-0.322376\pi\)
0.529510 + 0.848304i \(0.322376\pi\)
\(972\) 0 0
\(973\) 14.0000 0.448819
\(974\) −11.0000 −0.352463
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) −36.0000 −1.15056
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 54.0000 1.71971
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) −5.00000 −0.158750
\(993\) 0 0
\(994\) 9.00000 0.285463
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 31.0000 0.981288
\(999\) 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 1890.2.a.i.1.1 1
3.2 odd 2 1890.2.a.t.1.1 yes 1
5.4 even 2 9450.2.a.cb.1.1 1
15.14 odd 2 9450.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1890.2.a.i.1.1 1 1.1 even 1 trivial
1890.2.a.t.1.1 yes 1 3.2 odd 2
9450.2.a.z.1.1 1 15.14 odd 2
9450.2.a.cb.1.1 1 5.4 even 2