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Properties

Label 3042.2.a.o.1.1
Level $3042$
Weight $2$
Character 3042.1
Self dual yes
Analytic conductor $24.290$
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] = [3042,2,Mod(1,3042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.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: \(24.2904922949\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 234)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} +2.00000 q^{10} +4.00000 q^{11} +2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{19} +2.00000 q^{20} +4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{25} +2.00000 q^{28} -8.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} +4.00000 q^{35} -6.00000 q^{37} +6.00000 q^{38} +2.00000 q^{40} -6.00000 q^{41} -8.00000 q^{43} +4.00000 q^{44} +4.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} -1.00000 q^{50} +12.0000 q^{53} +8.00000 q^{55} +2.00000 q^{56} -8.00000 q^{58} -4.00000 q^{59} +10.0000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +2.00000 q^{67} +4.00000 q^{70} +16.0000 q^{71} -14.0000 q^{73} -6.00000 q^{74} +6.00000 q^{76} +8.00000 q^{77} -4.00000 q^{79} +2.00000 q^{80} -6.00000 q^{82} +12.0000 q^{83} -8.00000 q^{86} +4.00000 q^{88} +6.00000 q^{89} +4.00000 q^{92} -8.00000 q^{94} +12.0000 q^{95} +10.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
\(3\) 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 \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 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
\(22\) 4.00000 0.852803
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 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
\(40\) 2.00000 0.316228
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −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
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) −8.00000 −1.05045
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 16.0000 1.89885 0.949425 − 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 8.00000 0.911685
\(78\) 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
\(82\) −6.00000 −0.662589
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −16.0000 −1.59206 −0.796030 − 0.605257i \(-0.793070\pi\)
−0.796030 + 0.605257i \(0.793070\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −20.0000 −1.93347 −0.966736 − 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 8.00000 0.762770
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) 4.00000 0.376288 0.188144 − 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) −8.00000 −0.742781
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 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
\(142\) 16.0000 1.34269
\(143\) 0 0
\(144\) 0 0
\(145\) −16.0000 −1.32873
\(146\) −14.0000 −1.15865
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −18.0000 −1.46482 −0.732410 − 0.680864i \(-0.761604\pi\)
−0.732410 + 0.680864i \(0.761604\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 8.00000 0.644658
\(155\) 4.00000 0.321288
\(156\) 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
\(160\) 2.00000 0.158114
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 − 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 16.0000 1.21646 0.608229 − 0.793762i \(-0.291880\pi\)
0.608229 + 0.793762i \(0.291880\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 4.00000 0.301511
\(177\) 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
\(181\) −18.0000 −1.33793 −0.668965 − 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 12.0000 0.870572
\(191\) −4.00000 −0.289430 −0.144715 − 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 − 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −16.0000 −1.12576
\(203\) −16.0000 −1.12298
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 0 0
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) −20.0000 −1.36717
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) −10.0000 −0.677285
\(219\) 0 0
\(220\) 8.00000 0.539360
\(221\) 0 0
\(222\) 0 0
\(223\) −6.00000 −0.401790 −0.200895 − 0.979613i \(-0.564385\pi\)
−0.200895 + 0.979613i \(0.564385\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 4.00000 0.266076
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 30.0000 1.98246 0.991228 − 0.132164i \(-0.0421925\pi\)
0.991228 + 0.132164i \(0.0421925\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) −8.00000 −0.525226
\(233\) 20.0000 1.31024 0.655122 − 0.755523i \(-0.272617\pi\)
0.655122 + 0.755523i \(0.272617\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 − 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 0 0
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) −12.0000 −0.745644
\(260\) 0 0
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 24.0000 1.47431
\(266\) 12.0000 0.735767
\(267\) 0 0
\(268\) 2.00000 0.122169
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −22.0000 −1.33640 −0.668202 − 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 − 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) −16.0000 −0.939552
\(291\) 0 0
\(292\) −14.0000 −0.819288
\(293\) −2.00000 −0.116841 −0.0584206 − 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) −18.0000 −1.03578
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 20.0000 1.14520
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 8.00000 0.455842
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) −12.0000 −0.680458 −0.340229 − 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 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
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) −32.0000 −1.79166
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 10.0000 0.553849
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 − 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 − 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 16.0000 0.860165
\(347\) −24.0000 −1.28839 −0.644194 − 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 − 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) −14.0000 −0.745145 −0.372572 − 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 32.0000 1.69838
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −18.0000 −0.946059
\(363\) 0 0
\(364\) 0 0
\(365\) −28.0000 −1.46559
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 − 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) −12.0000 −0.623850
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 − 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 0 0
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 − 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 12.0000 0.615587
\(381\) 0 0
\(382\) −4.00000 −0.204658
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) −12.0000 −0.608424 −0.304212 − 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 10.0000 0.501886 0.250943 − 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 6.00000 0.299626 0.149813 − 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −16.0000 −0.796030
\(405\) 0 0
\(406\) −16.0000 −0.794067
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 − 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) −12.0000 −0.592638
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) 0 0
\(418\) 24.0000 1.17388
\(419\) 36.0000 1.75872 0.879358 − 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) 0 0
\(426\) 0 0
\(427\) 20.0000 0.967868
\(428\) −20.0000 −0.966736
\(429\) 0 0
\(430\) −16.0000 −0.771589
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 − 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 8.00000 0.381385
\(441\) 0 0
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 − 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) −6.00000 −0.284108
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) −26.0000 −1.22702 −0.613508 − 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 4.00000 0.188144
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 − 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 30.0000 1.40181
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −30.0000 −1.39724 −0.698620 − 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 − 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) 20.0000 0.926482
\(467\) −8.00000 −0.370196 −0.185098 − 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) −16.0000 −0.738025
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) −32.0000 −1.47136
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) 0 0
\(478\) 24.0000 1.09773
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −6.00000 −0.273293
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 20.0000 0.908153
\(486\) 0 0
\(487\) −38.0000 −1.72194 −0.860972 − 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) 10.0000 0.452679
\(489\) 0 0
\(490\) −6.00000 −0.271052
\(491\) −16.0000 −0.722070 −0.361035 − 0.932552i \(-0.617576\pi\)
−0.361035 + 0.932552i \(0.617576\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 32.0000 1.43540
\(498\) 0 0
\(499\) −22.0000 −0.984855 −0.492428 − 0.870353i \(-0.663890\pi\)
−0.492428 + 0.870353i \(0.663890\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) 0 0
\(503\) 32.0000 1.42681 0.713405 − 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) 0 0
\(505\) −32.0000 −1.42398
\(506\) 16.0000 0.711287
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 10.0000 0.443242 0.221621 − 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) −28.0000 −1.23865
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −32.0000 −1.40736
\(518\) −12.0000 −0.527250
\(519\) 0 0
\(520\) 0 0
\(521\) −36.0000 −1.57719 −0.788594 − 0.614914i \(-0.789191\pi\)
−0.788594 + 0.614914i \(0.789191\pi\)
\(522\) 0 0
\(523\) −12.0000 −0.524723 −0.262362 − 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 24.0000 1.04249
\(531\) 0 0
\(532\) 12.0000 0.520266
\(533\) 0 0
\(534\) 0 0
\(535\) −40.0000 −1.72935
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) 0 0
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −22.0000 −0.944981
\(543\) 0 0
\(544\) 0 0
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 − 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) −48.0000 −2.04487
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) −6.00000 −0.254916
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) −20.0000 −0.842900 −0.421450 − 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) 8.00000 0.336563
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) 16.0000 0.671345
\(569\) −4.00000 −0.167689 −0.0838444 − 0.996479i \(-0.526720\pi\)
−0.0838444 + 0.996479i \(0.526720\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 − 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 30.0000 1.24892 0.624458 − 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) −17.0000 −0.707107
\(579\) 0 0
\(580\) −16.0000 −0.664364
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 48.0000 1.98796
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) −28.0000 −1.15568 −0.577842 − 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) −8.00000 −0.329355
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) 38.0000 1.56047 0.780236 − 0.625485i \(-0.215099\pi\)
0.780236 + 0.625485i \(0.215099\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) −40.0000 −1.63436 −0.817178 − 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 − 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −16.0000 −0.652111
\(603\) 0 0
\(604\) −18.0000 −0.732410
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) 24.0000 0.974130 0.487065 − 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) 20.0000 0.809776
\(611\) 0 0
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 − 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) −14.0000 −0.563619 −0.281809 − 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 0 0
\(619\) −34.0000 −1.36658 −0.683288 − 0.730149i \(-0.739451\pi\)
−0.683288 + 0.730149i \(0.739451\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 0 0
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 − 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) 0 0
\(638\) −32.0000 −1.26689
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) −40.0000 −1.57991 −0.789953 − 0.613168i \(-0.789895\pi\)
−0.789953 + 0.613168i \(0.789895\pi\)
\(642\) 0 0
\(643\) 10.0000 0.394362 0.197181 − 0.980367i \(-0.436821\pi\)
0.197181 + 0.980367i \(0.436821\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 − 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 8.00000 0.312586
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) −16.0000 −0.623745
\(659\) −48.0000 −1.86981 −0.934907 − 0.354892i \(-0.884518\pi\)
−0.934907 + 0.354892i \(0.884518\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 10.0000 0.388661
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 24.0000 0.930680
\(666\) 0 0
\(667\) −32.0000 −1.23904
\(668\) 0 0
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 − 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 0 0
\(677\) −12.0000 −0.461197 −0.230599 − 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 0 0
\(682\) 8.00000 0.306336
\(683\) 4.00000 0.153056 0.0765279 − 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 36.0000 1.37549
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) 0 0
\(691\) 26.0000 0.989087 0.494543 − 0.869153i \(-0.335335\pi\)
0.494543 + 0.869153i \(0.335335\pi\)
\(692\) 16.0000 0.608229
\(693\) 0 0
\(694\) −24.0000 −0.911028
\(695\) −40.0000 −1.51729
\(696\) 0 0
\(697\) 0 0
\(698\) −22.0000 −0.832712
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) −12.0000 −0.453234 −0.226617 − 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 0 0
\(703\) −36.0000 −1.35777
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) −32.0000 −1.20348
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 − 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 32.0000 1.20094
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) 28.0000 1.04422 0.522112 − 0.852877i \(-0.325144\pi\)
0.522112 + 0.852877i \(0.325144\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) −18.0000 −0.668965
\(725\) 8.00000 0.297113
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −28.0000 −1.03633
\(731\) 0 0
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 − 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 54.0000 1.98642 0.993211 − 0.116326i \(-0.0371118\pi\)
0.993211 + 0.116326i \(0.0371118\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) 40.0000 1.46746 0.733729 − 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) 0 0
\(749\) −40.0000 −1.46157
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 − 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 0 0
\(755\) −36.0000 −1.31017
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 − 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) −10.0000 −0.363216
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) −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
\(764\) −4.00000 −0.144715
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 − 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 16.0000 0.576600
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) 14.0000 0.503545 0.251773 − 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) 64.0000 2.29010
\(782\) 0 0
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 − 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) 8.00000 0.284447
\(792\) 0 0
\(793\) 0 0
\(794\) 10.0000 0.354887
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) −12.0000 −0.425062 −0.212531 − 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 6.00000 0.211867
\(803\) −56.0000 −1.97620
\(804\) 0 0
\(805\) 16.0000 0.563926
\(806\) 0 0
\(807\) 0 0
\(808\) −16.0000 −0.562878
\(809\) 36.0000 1.26569 0.632846 − 0.774277i \(-0.281886\pi\)
0.632846 + 0.774277i \(0.281886\pi\)
\(810\) 0 0
\(811\) −50.0000 −1.75574 −0.877869 − 0.478901i \(-0.841035\pi\)
−0.877869 + 0.478901i \(0.841035\pi\)
\(812\) −16.0000 −0.561490
\(813\) 0 0
\(814\) −24.0000 −0.841200
\(815\) 20.0000 0.700569
\(816\) 0 0
\(817\) −48.0000 −1.67931
\(818\) 26.0000 0.909069
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) −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 \(-0.567069\pi\)
−0.209147 + 0.977884i \(0.567069\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 46.0000 1.59765 0.798823 − 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 24.0000 0.833052
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 24.0000 0.830057
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) −16.0000 −0.552381 −0.276191 − 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) −10.0000 −0.344623
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) 12.0000 0.412082
\(849\) 0 0
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 20.0000 0.684386
\(855\) 0 0
\(856\) −20.0000 −0.683586
\(857\) −28.0000 −0.956462 −0.478231 − 0.878234i \(-0.658722\pi\)
−0.478231 + 0.878234i \(0.658722\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 − 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) −16.0000 −0.545595
\(861\) 0 0
\(862\) 0 0
\(863\) −8.00000 −0.272323 −0.136162 − 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) 0 0
\(865\) 32.0000 1.08803
\(866\) −2.00000 −0.0679628
\(867\) 0 0
\(868\) 4.00000 0.135769
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) −10.0000 −0.338643
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) −6.00000 −0.202606 −0.101303 − 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) −4.00000 −0.134993
\(879\) 0 0
\(880\) 8.00000 0.269680
\(881\) 36.0000 1.21287 0.606435 − 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 − 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 44.0000 1.47738 0.738688 − 0.674048i \(-0.235446\pi\)
0.738688 + 0.674048i \(0.235446\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 12.0000 0.402241
\(891\) 0 0
\(892\) −6.00000 −0.200895
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −26.0000 −0.867631
\(899\) −16.0000 −0.533630
\(900\) 0 0
\(901\) 0 0
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 4.00000 0.133038
\(905\) −36.0000 −1.19668
\(906\) 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
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 − 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 48.0000 1.58857
\(914\) 2.00000 0.0661541
\(915\) 0 0
\(916\) 30.0000 0.991228
\(917\) 8.00000 0.264183
\(918\) 0 0
\(919\) 12.0000 0.395843 0.197922 − 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) 8.00000 0.263752
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) −22.0000 −0.722965
\(927\) 0 0
\(928\) −8.00000 −0.262613
\(929\) −30.0000 −0.984268 −0.492134 − 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 20.0000 0.655122
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 − 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) −16.0000 −0.521862
\(941\) −26.0000 −0.847576 −0.423788 − 0.905761i \(-0.639300\pi\)
−0.423788 + 0.905761i \(0.639300\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −32.0000 −1.04041
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −6.00000 −0.194666
\(951\) 0 0
\(952\) 0 0
\(953\) 28.0000 0.907009 0.453504 − 0.891254i \(-0.350174\pi\)
0.453504 + 0.891254i \(0.350174\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 0 0
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) −6.00000 −0.193247
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) 58.0000 1.86515 0.932577 − 0.360971i \(-0.117555\pi\)
0.932577 + 0.360971i \(0.117555\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 20.0000 0.642161
\(971\) 48.0000 1.54039 0.770197 − 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 0 0
\(973\) −40.0000 −1.28234
\(974\) −38.0000 −1.21760
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −18.0000 −0.575871 −0.287936 − 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) −6.00000 −0.191663
\(981\) 0 0
\(982\) −16.0000 −0.510581
\(983\) 32.0000 1.02064 0.510321 − 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) −48.0000 −1.52477 −0.762385 − 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) 32.0000 1.01498
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 − 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −22.0000 −0.696398
\(999\) 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 3042.2.a.o.1.1 1
3.2 odd 2 3042.2.a.b.1.1 1
13.5 odd 4 3042.2.b.c.1351.1 2
13.8 odd 4 3042.2.b.c.1351.2 2
13.12 even 2 234.2.a.a.1.1 ✓ 1
39.5 even 4 3042.2.b.b.1351.2 2
39.8 even 4 3042.2.b.b.1351.1 2
39.38 odd 2 234.2.a.d.1.1 yes 1
52.51 odd 2 1872.2.a.g.1.1 1
65.12 odd 4 5850.2.e.d.5149.1 2
65.38 odd 4 5850.2.e.d.5149.2 2
65.64 even 2 5850.2.a.bv.1.1 1
104.51 odd 2 7488.2.a.bu.1.1 1
104.77 even 2 7488.2.a.bp.1.1 1
117.25 even 6 2106.2.e.z.1405.1 2
117.38 odd 6 2106.2.e.e.1405.1 2
117.77 odd 6 2106.2.e.e.703.1 2
117.103 even 6 2106.2.e.z.703.1 2
156.155 even 2 1872.2.a.p.1.1 1
195.38 even 4 5850.2.e.bd.5149.1 2
195.77 even 4 5850.2.e.bd.5149.2 2
195.194 odd 2 5850.2.a.v.1.1 1
312.77 odd 2 7488.2.a.j.1.1 1
312.155 even 2 7488.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
234.2.a.a.1.1 ✓ 1 13.12 even 2
234.2.a.d.1.1 yes 1 39.38 odd 2
1872.2.a.g.1.1 1 52.51 odd 2
1872.2.a.p.1.1 1 156.155 even 2
2106.2.e.e.703.1 2 117.77 odd 6
2106.2.e.e.1405.1 2 117.38 odd 6
2106.2.e.z.703.1 2 117.103 even 6
2106.2.e.z.1405.1 2 117.25 even 6
3042.2.a.b.1.1 1 3.2 odd 2
3042.2.a.o.1.1 1 1.1 even 1 trivial
3042.2.b.b.1351.1 2 39.8 even 4
3042.2.b.b.1351.2 2 39.5 even 4
3042.2.b.c.1351.1 2 13.5 odd 4
3042.2.b.c.1351.2 2 13.8 odd 4
5850.2.a.v.1.1 1 195.194 odd 2
5850.2.a.bv.1.1 1 65.64 even 2
5850.2.e.d.5149.1 2 65.12 odd 4
5850.2.e.d.5149.2 2 65.38 odd 4
5850.2.e.bd.5149.1 2 195.38 even 4
5850.2.e.bd.5149.2 2 195.77 even 4
7488.2.a.j.1.1 1 312.77 odd 2
7488.2.a.s.1.1 1 312.155 even 2
7488.2.a.bp.1.1 1 104.77 even 2
7488.2.a.bu.1.1 1 104.51 odd 2