Properties

Label 3330.2.a.h.1.1
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
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] = [3330,2,Mod(1,3330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3330, 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("3330.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.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: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3330.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -3.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{19} +1.00000 q^{20} +3.00000 q^{22} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{28} +3.00000 q^{29} -1.00000 q^{31} -1.00000 q^{32} +3.00000 q^{34} -1.00000 q^{35} +1.00000 q^{37} -2.00000 q^{38} -1.00000 q^{40} -9.00000 q^{41} +11.0000 q^{43} -3.00000 q^{44} -6.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} +9.00000 q^{53} -3.00000 q^{55} +1.00000 q^{56} -3.00000 q^{58} +6.00000 q^{59} -1.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +8.00000 q^{67} -3.00000 q^{68} +1.00000 q^{70} +12.0000 q^{71} +8.00000 q^{73} -1.00000 q^{74} +2.00000 q^{76} +3.00000 q^{77} -4.00000 q^{79} +1.00000 q^{80} +9.00000 q^{82} +6.00000 q^{83} -3.00000 q^{85} -11.0000 q^{86} +3.00000 q^{88} +6.00000 q^{89} -2.00000 q^{91} +2.00000 q^{95} -13.0000 q^{97} +6.00000 q^{98} +O(q^{100})\)

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 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 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
\(22\) 3.00000 0.639602
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) 11.0000 1.67748 0.838742 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −3.00000 −0.393919
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 9.00000 0.993884
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) −11.0000 −1.18616
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 6.00000 0.606092
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 3.00000 0.286039
\(111\) 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
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 1.00000 0.0905357
\(123\) 0 0
\(124\) −1.00000 −0.0898027
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) −8.00000 −0.662085
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) −3.00000 −0.241747
\(155\) −1.00000 −0.0803219
\(156\) 0 0
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 3.00000 0.230089
\(171\) 0 0
\(172\) 11.0000 0.838742
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) 0 0
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 13.0000 0.933346
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 0 0
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 11.0000 0.757271 0.378636 0.925546i \(-0.376393\pi\)
0.378636 + 0.925546i \(0.376393\pi\)
\(212\) 9.00000 0.618123
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) 11.0000 0.750194
\(216\) 0 0
\(217\) 1.00000 0.0678844
\(218\) 1.00000 0.0677285
\(219\) 0 0
\(220\) −3.00000 −0.202260
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) 5.00000 0.334825 0.167412 0.985887i \(-0.446459\pi\)
0.167412 + 0.985887i \(0.446459\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 9.00000 0.598671
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) 0 0
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) −3.00000 −0.194461
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 2.00000 0.128565
\(243\) 0 0
\(244\) −1.00000 −0.0640184
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 1.00000 0.0635001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) −1.00000 −0.0621370
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) −5.00000 −0.299880
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 9.00000 0.531253
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) −3.00000 −0.176166
\(291\) 0 0
\(292\) 8.00000 0.468165
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) −11.0000 −0.634029
\(302\) −2.00000 −0.115087
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 3.00000 0.170941
\(309\) 0 0
\(310\) 1.00000 0.0567962
\(311\) −3.00000 −0.170114 −0.0850572 0.996376i \(-0.527107\pi\)
−0.0850572 + 0.996376i \(0.527107\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 7.00000 0.395033
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 33.0000 1.85346 0.926732 0.375722i \(-0.122605\pi\)
0.926732 + 0.375722i \(0.122605\pi\)
\(318\) 0 0
\(319\) −9.00000 −0.503903
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) −11.0000 −0.609234
\(327\) 0 0
\(328\) 9.00000 0.496942
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) −3.00000 −0.162698
\(341\) 3.00000 0.162459
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) −11.0000 −0.593080
\(345\) 0 0
\(346\) 15.0000 0.806405
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) −3.00000 −0.159674 −0.0798369 0.996808i \(-0.525440\pi\)
−0.0798369 + 0.996808i \(0.525440\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 8.00000 0.418739
\(366\) 0 0
\(367\) 23.0000 1.20059 0.600295 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −9.00000 −0.465379
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) 15.0000 0.767467
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 3.00000 0.152894
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) −13.0000 −0.659975
\(389\) 33.0000 1.67317 0.836583 0.547840i \(-0.184550\pi\)
0.836583 + 0.547840i \(0.184550\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 6.00000 0.303046
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 4.00000 0.200502
\(399\) 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
\(403\) −2.00000 −0.0996271
\(404\) 0 0
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) −3.00000 −0.148704
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 9.00000 0.444478
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −11.0000 −0.535472
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) 1.00000 0.0483934
\(428\) 18.0000 0.870063
\(429\) 0 0
\(430\) −11.0000 −0.530467
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 0 0
\(433\) 20.0000 0.961139 0.480569 0.876957i \(-0.340430\pi\)
0.480569 + 0.876957i \(0.340430\pi\)
\(434\) −1.00000 −0.0480015
\(435\) 0 0
\(436\) −1.00000 −0.0478913
\(437\) 0 0
\(438\) 0 0
\(439\) 17.0000 0.811366 0.405683 0.914014i \(-0.367034\pi\)
0.405683 + 0.914014i \(0.367034\pi\)
\(440\) 3.00000 0.143019
\(441\) 0 0
\(442\) 6.00000 0.285391
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) −5.00000 −0.236757
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 27.0000 1.27138
\(452\) −9.00000 −0.423324
\(453\) 0 0
\(454\) 3.00000 0.140797
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) 35.0000 1.63723 0.818615 0.574342i \(-0.194742\pi\)
0.818615 + 0.574342i \(0.194742\pi\)
\(458\) −8.00000 −0.373815
\(459\) 0 0
\(460\) 0 0
\(461\) −39.0000 −1.81641 −0.908206 0.418524i \(-0.862547\pi\)
−0.908206 + 0.418524i \(0.862547\pi\)
\(462\) 0 0
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) 15.0000 0.694117 0.347059 0.937843i \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) −33.0000 −1.51734
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 3.00000 0.137505
\(477\) 0 0
\(478\) −9.00000 −0.411650
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −13.0000 −0.590300
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 1.00000 0.0452679
\(489\) 0 0
\(490\) 6.00000 0.271052
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 36.0000 1.59567 0.797836 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 0 0
\(518\) 1.00000 0.0439375
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) −27.0000 −1.18289 −0.591446 0.806345i \(-0.701443\pi\)
−0.591446 + 0.806345i \(0.701443\pi\)
\(522\) 0 0
\(523\) −40.0000 −1.74908 −0.874539 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) 3.00000 0.130682
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −9.00000 −0.390935
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) −18.0000 −0.779667
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) −24.0000 −1.03471
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 4.00000 0.171815
\(543\) 0 0
\(544\) 3.00000 0.128624
\(545\) −1.00000 −0.0428353
\(546\) 0 0
\(547\) 11.0000 0.470326 0.235163 0.971956i \(-0.424438\pi\)
0.235163 + 0.971956i \(0.424438\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) 3.00000 0.127920
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) 5.00000 0.212047
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 0 0
\(559\) 22.0000 0.930501
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −39.0000 −1.64365 −0.821827 0.569737i \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) 0 0
\(565\) −9.00000 −0.378633
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −13.0000 −0.544033 −0.272017 0.962293i \(-0.587691\pi\)
−0.272017 + 0.962293i \(0.587691\pi\)
\(572\) −6.00000 −0.250873
\(573\) 0 0
\(574\) −9.00000 −0.375653
\(575\) 0 0
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 8.00000 0.332756
\(579\) 0 0
\(580\) 3.00000 0.124568
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) −27.0000 −1.11823
\(584\) −8.00000 −0.331042
\(585\) 0 0
\(586\) 21.0000 0.867502
\(587\) −33.0000 −1.36206 −0.681028 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) −6.00000 −0.247016
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −31.0000 −1.26452 −0.632258 0.774758i \(-0.717872\pi\)
−0.632258 + 0.774758i \(0.717872\pi\)
\(602\) 11.0000 0.448327
\(603\) 0 0
\(604\) 2.00000 0.0813788
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 1.00000 0.0404888
\(611\) 0 0
\(612\) 0 0
\(613\) −25.0000 −1.00974 −0.504870 0.863195i \(-0.668460\pi\)
−0.504870 + 0.863195i \(0.668460\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 5.00000 0.200967 0.100483 0.994939i \(-0.467961\pi\)
0.100483 + 0.994939i \(0.467961\pi\)
\(620\) −1.00000 −0.0401610
\(621\) 0 0
\(622\) 3.00000 0.120289
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) −7.00000 −0.279330
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) 47.0000 1.87104 0.935520 0.353273i \(-0.114931\pi\)
0.935520 + 0.353273i \(0.114931\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) −33.0000 −1.31060
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 9.00000 0.356313
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 0 0
\(643\) 23.0000 0.907031 0.453516 0.891248i \(-0.350170\pi\)
0.453516 + 0.891248i \(0.350170\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) −18.0000 −0.706562
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 11.0000 0.430793
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) −9.00000 −0.351391
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 17.0000 0.661223 0.330612 0.943767i \(-0.392745\pi\)
0.330612 + 0.943767i \(0.392745\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 3.00000 0.115814
\(672\) 0 0
\(673\) −28.0000 −1.07932 −0.539660 0.841883i \(-0.681447\pi\)
−0.539660 + 0.841883i \(0.681447\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 13.0000 0.498894
\(680\) 3.00000 0.115045
\(681\) 0 0
\(682\) −3.00000 −0.114876
\(683\) −27.0000 −1.03313 −0.516563 0.856249i \(-0.672789\pi\)
−0.516563 + 0.856249i \(0.672789\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) −13.0000 −0.496342
\(687\) 0 0
\(688\) 11.0000 0.419371
\(689\) 18.0000 0.685745
\(690\) 0 0
\(691\) −25.0000 −0.951045 −0.475522 0.879704i \(-0.657741\pi\)
−0.475522 + 0.879704i \(0.657741\pi\)
\(692\) −15.0000 −0.570214
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 5.00000 0.189661
\(696\) 0 0
\(697\) 27.0000 1.02270
\(698\) 28.0000 1.05982
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 3.00000 0.112906
\(707\) 0 0
\(708\) 0 0
\(709\) −13.0000 −0.488225 −0.244113 0.969747i \(-0.578497\pi\)
−0.244113 + 0.969747i \(0.578497\pi\)
\(710\) −12.0000 −0.450352
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 6.00000 0.223918
\(719\) −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
\(724\) 2.00000 0.0743294
\(725\) 3.00000 0.111417
\(726\) 0 0
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) −8.00000 −0.296093
\(731\) −33.0000 −1.22055
\(732\) 0 0
\(733\) −1.00000 −0.0369358 −0.0184679 0.999829i \(-0.505879\pi\)
−0.0184679 + 0.999829i \(0.505879\pi\)
\(734\) −23.0000 −0.848945
\(735\) 0 0
\(736\) 0 0
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) −7.00000 −0.257499 −0.128750 0.991677i \(-0.541096\pi\)
−0.128750 + 0.991677i \(0.541096\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) 9.00000 0.330400
\(743\) −39.0000 −1.43077 −0.715386 0.698730i \(-0.753749\pi\)
−0.715386 + 0.698730i \(0.753749\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) 9.00000 0.329073
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −6.00000 −0.218507
\(755\) 2.00000 0.0727875
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 16.0000 0.581146
\(759\) 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
\(763\) 1.00000 0.0362024
\(764\) −15.0000 −0.542681
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) −3.00000 −0.108112
\(771\) 0 0
\(772\) 2.00000 0.0719816
\(773\) 51.0000 1.83434 0.917171 0.398493i \(-0.130467\pi\)
0.917171 + 0.398493i \(0.130467\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 13.0000 0.466673
\(777\) 0 0
\(778\) −33.0000 −1.18311
\(779\) −18.0000 −0.644917
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) −7.00000 −0.249841
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 4.00000 0.142314
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −30.0000 −1.05934
\(803\) −24.0000 −0.846942
\(804\) 0 0
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −3.00000 −0.105279
\(813\) 0 0
\(814\) 3.00000 0.105150
\(815\) 11.0000 0.385313
\(816\) 0 0
\(817\) 22.0000 0.769683
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 9.00000 0.312961 0.156480 0.987681i \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(828\) 0 0
\(829\) −13.0000 −0.451509 −0.225754 0.974184i \(-0.572485\pi\)
−0.225754 + 0.974184i \(0.572485\pi\)
\(830\) −6.00000 −0.208263
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 10.0000 0.344623
\(843\) 0 0
\(844\) 11.0000 0.378636
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 9.00000 0.309061
\(849\) 0 0
\(850\) 3.00000 0.102899
\(851\) 0 0
\(852\) 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
\(856\) −18.0000 −0.615227
\(857\) −45.0000 −1.53717 −0.768585 0.639747i \(-0.779039\pi\)
−0.768585 + 0.639747i \(0.779039\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 11.0000 0.375097
\(861\) 0 0
\(862\) 3.00000 0.102180
\(863\) −9.00000 −0.306364 −0.153182 0.988198i \(-0.548952\pi\)
−0.153182 + 0.988198i \(0.548952\pi\)
\(864\) 0 0
\(865\) −15.0000 −0.510015
\(866\) −20.0000 −0.679628
\(867\) 0 0
\(868\) 1.00000 0.0339422
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 1.00000 0.0338643
\(873\) 0 0
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 41.0000 1.38447 0.692236 0.721671i \(-0.256626\pi\)
0.692236 + 0.721671i \(0.256626\pi\)
\(878\) −17.0000 −0.573722
\(879\) 0 0
\(880\) −3.00000 −0.101130
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) 0 0
\(883\) −43.0000 −1.44707 −0.723533 0.690290i \(-0.757483\pi\)
−0.723533 + 0.690290i \(0.757483\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) −3.00000 −0.100730 −0.0503651 0.998731i \(-0.516038\pi\)
−0.0503651 + 0.998731i \(0.516038\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) 5.00000 0.167412
\(893\) 0 0
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 0 0
\(899\) −3.00000 −0.100056
\(900\) 0 0
\(901\) −27.0000 −0.899500
\(902\) −27.0000 −0.899002
\(903\) 0 0
\(904\) 9.00000 0.299336
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) −3.00000 −0.0995585
\(909\) 0 0
\(910\) 2.00000 0.0662994
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −18.0000 −0.595713
\(914\) −35.0000 −1.15770
\(915\) 0 0
\(916\) 8.00000 0.264327
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 39.0000 1.28440
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) −26.0000 −0.854413
\(927\) 0 0
\(928\) −3.00000 −0.0984798
\(929\) −15.0000 −0.492134 −0.246067 0.969253i \(-0.579138\pi\)
−0.246067 + 0.969253i \(0.579138\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 12.0000 0.393073
\(933\) 0 0
\(934\) −15.0000 −0.490815
\(935\) 9.00000 0.294331
\(936\) 0 0
\(937\) −16.0000 −0.522697 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) 8.00000 0.261209
\(939\) 0 0
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 33.0000 1.07292
\(947\) −27.0000 −0.877382 −0.438691 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 0 0
\(949\) 16.0000 0.519382
\(950\) −2.00000 −0.0648886
\(951\) 0 0
\(952\) −3.00000 −0.0972306
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 0 0
\(955\) −15.0000 −0.485389
\(956\) 9.00000 0.291081
\(957\) 0 0
\(958\) 0 0
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) −2.00000 −0.0644826
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) 13.0000 0.417405
\(971\) 51.0000 1.63667 0.818334 0.574743i \(-0.194898\pi\)
0.818334 + 0.574743i \(0.194898\pi\)
\(972\) 0 0
\(973\) −5.00000 −0.160293
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) −15.0000 −0.479893 −0.239946 0.970786i \(-0.577130\pi\)
−0.239946 + 0.970786i \(0.577130\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) −6.00000 −0.191663
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) 39.0000 1.24391 0.621953 0.783054i \(-0.286339\pi\)
0.621953 + 0.783054i \(0.286339\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 9.00000 0.286618
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) 29.0000 0.921215 0.460608 0.887604i \(-0.347632\pi\)
0.460608 + 0.887604i \(0.347632\pi\)
\(992\) 1.00000 0.0317500
\(993\) 0 0
\(994\) 12.0000 0.380617
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −32.0000 −1.01294
\(999\) 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 3330.2.a.h.1.1 1
3.2 odd 2 1110.2.a.n.1.1 1
12.11 even 2 8880.2.a.g.1.1 1
15.14 odd 2 5550.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.n.1.1 1 3.2 odd 2
3330.2.a.h.1.1 1 1.1 even 1 trivial
5550.2.a.g.1.1 1 15.14 odd 2
8880.2.a.g.1.1 1 12.11 even 2