Properties

Label 435.2.a.c.1.1
Level $435$
Weight $2$
Character 435.1
Self dual yes
Analytic conductor $3.473$
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] = [435,2,Mod(1,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, 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("435.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.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: \(3.47349248793\)
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 \(\chi\) \(=\) 435.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} +4.00000 q^{16} +2.00000 q^{19} +2.00000 q^{20} +2.00000 q^{21} +3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.00000 q^{28} -1.00000 q^{29} +8.00000 q^{31} +3.00000 q^{33} -2.00000 q^{35} -2.00000 q^{36} -1.00000 q^{37} +2.00000 q^{39} -3.00000 q^{41} -1.00000 q^{43} -6.00000 q^{44} -1.00000 q^{45} -6.00000 q^{47} +4.00000 q^{48} -3.00000 q^{49} -4.00000 q^{52} -3.00000 q^{53} -3.00000 q^{55} +2.00000 q^{57} -12.0000 q^{59} +2.00000 q^{60} +8.00000 q^{61} +2.00000 q^{63} -8.00000 q^{64} -2.00000 q^{65} +14.0000 q^{67} +3.00000 q^{69} -6.00000 q^{71} -7.00000 q^{73} +1.00000 q^{75} -4.00000 q^{76} +6.00000 q^{77} -4.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +9.00000 q^{83} -4.00000 q^{84} -1.00000 q^{87} -6.00000 q^{89} +4.00000 q^{91} -6.00000 q^{92} +8.00000 q^{93} -2.00000 q^{95} +11.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.00000 0.577350
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −2.00000 −0.577350
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 2.00000 0.447214
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) −2.00000 −0.333333
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −6.00000 −0.904534
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 4.00000 0.577350
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) −8.00000 −1.00000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −6.00000 −0.625543
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 11.0000 1.11688 0.558440 0.829545i \(-0.311400\pi\)
0.558440 + 0.829545i \(0.311400\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) −2.00000 −0.200000
\(101\) −9.00000 −0.895533 −0.447767 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −2.00000 −0.192450
\(109\) −13.0000 −1.24517 −0.622587 0.782551i \(-0.713918\pi\)
−0.622587 + 0.782551i \(0.713918\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 8.00000 0.755929
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −3.00000 −0.270501
\(124\) −16.0000 −1.43684
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −6.00000 −0.522233
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 4.00000 0.338062
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 4.00000 0.333333
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) 2.00000 0.164399
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) −4.00000 −0.320256
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) −19.0000 −1.48819 −0.744097 0.668071i \(-0.767120\pi\)
−0.744097 + 0.668071i \(0.767120\pi\)
\(164\) 6.00000 0.468521
\(165\) −3.00000 −0.233550
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 2.00000 0.152499
\(173\) 21.0000 1.59660 0.798300 0.602260i \(-0.205733\pi\)
0.798300 + 0.602260i \(0.205733\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 12.0000 0.904534
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 2.00000 0.149071
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 0 0
\(188\) 12.0000 0.875190
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 15.0000 1.08536 0.542681 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) −8.00000 −0.577350
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) −2.00000 −0.143223
\(196\) 6.00000 0.428571
\(197\) 21.0000 1.49619 0.748094 0.663593i \(-0.230969\pi\)
0.748094 + 0.663593i \(0.230969\pi\)
\(198\) 0 0
\(199\) −1.00000 −0.0708881 −0.0354441 0.999372i \(-0.511285\pi\)
−0.0354441 + 0.999372i \(0.511285\pi\)
\(200\) 0 0
\(201\) 14.0000 0.987484
\(202\) 0 0
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 3.00000 0.209529
\(206\) 0 0
\(207\) 3.00000 0.208514
\(208\) 8.00000 0.554700
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 6.00000 0.412082
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 0 0
\(219\) −7.00000 −0.473016
\(220\) 6.00000 0.404520
\(221\) 0 0
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −21.0000 −1.39382 −0.696909 0.717159i \(-0.745442\pi\)
−0.696909 + 0.717159i \(0.745442\pi\)
\(228\) −4.00000 −0.264906
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) 15.0000 0.982683 0.491341 0.870967i \(-0.336507\pi\)
0.491341 + 0.870967i \(0.336507\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 24.0000 1.56227
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) −4.00000 −0.258199
\(241\) −7.00000 −0.450910 −0.225455 0.974254i \(-0.572387\pi\)
−0.225455 + 0.974254i \(0.572387\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −16.0000 −1.02430
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −4.00000 −0.251976
\(253\) 9.00000 0.565825
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 4.00000 0.248069
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 3.00000 0.184289
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) −28.0000 −1.71037
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) 3.00000 0.180907
\(276\) −6.00000 −0.361158
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −10.0000 −0.594438 −0.297219 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) 12.0000 0.712069
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 11.0000 0.644831
\(292\) 14.0000 0.819288
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) 0 0
\(299\) 6.00000 0.346989
\(300\) −2.00000 −0.115470
\(301\) −2.00000 −0.115278
\(302\) 0 0
\(303\) −9.00000 −0.517036
\(304\) 8.00000 0.458831
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) −12.0000 −0.683763
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) 0 0
\(313\) −4.00000 −0.226093 −0.113047 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) 0 0
\(315\) −2.00000 −0.112687
\(316\) 8.00000 0.450035
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 0 0
\(319\) −3.00000 −0.167968
\(320\) 8.00000 0.447214
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) −2.00000 −0.111111
\(325\) 2.00000 0.110940
\(326\) 0 0
\(327\) −13.0000 −0.718902
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −18.0000 −0.987878
\(333\) −1.00000 −0.0547997
\(334\) 0 0
\(335\) −14.0000 −0.764902
\(336\) 8.00000 0.436436
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) −3.00000 −0.161515
\(346\) 0 0
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 2.00000 0.107211
\(349\) −25.0000 −1.33822 −0.669110 0.743164i \(-0.733324\pi\)
−0.669110 + 0.743164i \(0.733324\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 12.0000 0.635999
\(357\) 0 0
\(358\) 0 0
\(359\) 3.00000 0.158334 0.0791670 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) −8.00000 −0.419314
\(365\) 7.00000 0.366397
\(366\) 0 0
\(367\) −19.0000 −0.991792 −0.495896 0.868382i \(-0.665160\pi\)
−0.495896 + 0.868382i \(0.665160\pi\)
\(368\) 12.0000 0.625543
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) −16.0000 −0.829561
\(373\) 32.0000 1.65690 0.828449 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 4.00000 0.205196
\(381\) 5.00000 0.256158
\(382\) 0 0
\(383\) −21.0000 −1.07305 −0.536525 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 0 0
\(385\) −6.00000 −0.305788
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) −22.0000 −1.11688
\(389\) 33.0000 1.67317 0.836583 0.547840i \(-0.184550\pi\)
0.836583 + 0.547840i \(0.184550\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 4.00000 0.201262
\(396\) −6.00000 −0.301511
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 4.00000 0.200000
\(401\) −36.0000 −1.79775 −0.898877 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 18.0000 0.895533
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −3.00000 −0.148704
\(408\) 0 0
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) −28.0000 −1.37946
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) −9.00000 −0.441793
\(416\) 0 0
\(417\) 5.00000 0.244851
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 4.00000 0.195180
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 16.0000 0.774294
\(428\) 24.0000 1.16008
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 4.00000 0.192450
\(433\) 5.00000 0.240285 0.120142 0.992757i \(-0.461665\pi\)
0.120142 + 0.992757i \(0.461665\pi\)
\(434\) 0 0
\(435\) 1.00000 0.0479463
\(436\) 26.0000 1.24517
\(437\) 6.00000 0.287019
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 2.00000 0.0949158
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) −16.0000 −0.755929
\(449\) 27.0000 1.27421 0.637104 0.770778i \(-0.280132\pi\)
0.637104 + 0.770778i \(0.280132\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) 0 0
\(453\) −19.0000 −0.892698
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −16.0000 −0.748448 −0.374224 0.927338i \(-0.622091\pi\)
−0.374224 + 0.927338i \(0.622091\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 6.00000 0.279751
\(461\) 33.0000 1.53696 0.768482 0.639872i \(-0.221013\pi\)
0.768482 + 0.639872i \(0.221013\pi\)
\(462\) 0 0
\(463\) −34.0000 −1.58011 −0.790057 0.613033i \(-0.789949\pi\)
−0.790057 + 0.613033i \(0.789949\pi\)
\(464\) −4.00000 −0.185695
\(465\) −8.00000 −0.370991
\(466\) 0 0
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) −4.00000 −0.184900
\(469\) 28.0000 1.29292
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) −3.00000 −0.137940
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) −3.00000 −0.137361
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 6.00000 0.273009
\(484\) 4.00000 0.181818
\(485\) −11.0000 −0.499484
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 0 0
\(489\) −19.0000 −0.859210
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 6.00000 0.270501
\(493\) 0 0
\(494\) 0 0
\(495\) −3.00000 −0.134840
\(496\) 32.0000 1.43684
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 2.00000 0.0894427
\(501\) 0 0
\(502\) 0 0
\(503\) −18.0000 −0.802580 −0.401290 0.915951i \(-0.631438\pi\)
−0.401290 + 0.915951i \(0.631438\pi\)
\(504\) 0 0
\(505\) 9.00000 0.400495
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) −10.0000 −0.443678
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) −14.0000 −0.616914
\(516\) 2.00000 0.0880451
\(517\) −18.0000 −0.791639
\(518\) 0 0
\(519\) 21.0000 0.921798
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 24.0000 1.04844
\(525\) 2.00000 0.0872872
\(526\) 0 0
\(527\) 0 0
\(528\) 12.0000 0.522233
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) −8.00000 −0.346844
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 6.00000 0.258919
\(538\) 0 0
\(539\) −9.00000 −0.387657
\(540\) 2.00000 0.0860663
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 0 0
\(543\) −7.00000 −0.300399
\(544\) 0 0
\(545\) 13.0000 0.556859
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 24.0000 1.02523
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) −2.00000 −0.0852029
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) 1.00000 0.0424476
\(556\) −10.0000 −0.424094
\(557\) −27.0000 −1.14403 −0.572013 0.820244i \(-0.693837\pi\)
−0.572013 + 0.820244i \(0.693837\pi\)
\(558\) 0 0
\(559\) −2.00000 −0.0845910
\(560\) −8.00000 −0.338062
\(561\) 0 0
\(562\) 0 0
\(563\) −30.0000 −1.26435 −0.632175 0.774826i \(-0.717837\pi\)
−0.632175 + 0.774826i \(0.717837\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) 0 0
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −43.0000 −1.79949 −0.899747 0.436412i \(-0.856249\pi\)
−0.899747 + 0.436412i \(0.856249\pi\)
\(572\) −12.0000 −0.501745
\(573\) 15.0000 0.626634
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) −8.00000 −0.333333
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 0 0
\(579\) 14.0000 0.581820
\(580\) −2.00000 −0.0830455
\(581\) 18.0000 0.746766
\(582\) 0 0
\(583\) −9.00000 −0.372742
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 6.00000 0.247436
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 21.0000 0.863825
\(592\) −4.00000 −0.164399
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −36.0000 −1.47462
\(597\) −1.00000 −0.0409273
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 20.0000 0.815817 0.407909 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(602\) 0 0
\(603\) 14.0000 0.570124
\(604\) 38.0000 1.54620
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 3.00000 0.120972
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 16.0000 0.642575
\(621\) 3.00000 0.120386
\(622\) 0 0
\(623\) −12.0000 −0.480770
\(624\) 8.00000 0.320256
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.00000 0.239617
\(628\) −4.00000 −0.159617
\(629\) 0 0
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) −10.0000 −0.397464
\(634\) 0 0
\(635\) −5.00000 −0.198419
\(636\) 6.00000 0.237915
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 3.00000 0.118493 0.0592464 0.998243i \(-0.481130\pi\)
0.0592464 + 0.998243i \(0.481130\pi\)
\(642\) 0 0
\(643\) 32.0000 1.26196 0.630978 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) −12.0000 −0.472866
\(645\) 1.00000 0.0393750
\(646\) 0 0
\(647\) −27.0000 −1.06148 −0.530740 0.847535i \(-0.678086\pi\)
−0.530740 + 0.847535i \(0.678086\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 38.0000 1.48819
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) −12.0000 −0.468521
\(657\) −7.00000 −0.273096
\(658\) 0 0
\(659\) 27.0000 1.05177 0.525885 0.850555i \(-0.323734\pi\)
0.525885 + 0.850555i \(0.323734\pi\)
\(660\) 6.00000 0.233550
\(661\) 23.0000 0.894596 0.447298 0.894385i \(-0.352386\pi\)
0.447298 + 0.894385i \(0.352386\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.00000 −0.155113
\(666\) 0 0
\(667\) −3.00000 −0.116160
\(668\) 0 0
\(669\) −4.00000 −0.154649
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 18.0000 0.692308
\(677\) 36.0000 1.38359 0.691796 0.722093i \(-0.256820\pi\)
0.691796 + 0.722093i \(0.256820\pi\)
\(678\) 0 0
\(679\) 22.0000 0.844283
\(680\) 0 0
\(681\) −21.0000 −0.804722
\(682\) 0 0
\(683\) 15.0000 0.573959 0.286980 0.957937i \(-0.407349\pi\)
0.286980 + 0.957937i \(0.407349\pi\)
\(684\) −4.00000 −0.152944
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) −4.00000 −0.152499
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −42.0000 −1.59660
\(693\) 6.00000 0.227921
\(694\) 0 0
\(695\) −5.00000 −0.189661
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 15.0000 0.567352
\(700\) −4.00000 −0.151186
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) −24.0000 −0.904534
\(705\) 6.00000 0.225973
\(706\) 0 0
\(707\) −18.0000 −0.676960
\(708\) 24.0000 0.901975
\(709\) −1.00000 −0.0375558 −0.0187779 0.999824i \(-0.505978\pi\)
−0.0187779 + 0.999824i \(0.505978\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) −12.0000 −0.448461
\(717\) 6.00000 0.224074
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) −4.00000 −0.149071
\(721\) 28.0000 1.04277
\(722\) 0 0
\(723\) −7.00000 −0.260333
\(724\) 14.0000 0.520306
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) 44.0000 1.63187 0.815935 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −16.0000 −0.591377
\(733\) 50.0000 1.84679 0.923396 0.383849i \(-0.125402\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) 0 0
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) 42.0000 1.54709
\(738\) 0 0
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) −42.0000 −1.54083 −0.770415 0.637542i \(-0.779951\pi\)
−0.770415 + 0.637542i \(0.779951\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) 9.00000 0.329293
\(748\) 0 0
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) 44.0000 1.60558 0.802791 0.596260i \(-0.203347\pi\)
0.802791 + 0.596260i \(0.203347\pi\)
\(752\) −24.0000 −0.875190
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 19.0000 0.691481
\(756\) −4.00000 −0.145479
\(757\) −43.0000 −1.56286 −0.781431 0.623992i \(-0.785510\pi\)
−0.781431 + 0.623992i \(0.785510\pi\)
\(758\) 0 0
\(759\) 9.00000 0.326679
\(760\) 0 0
\(761\) −24.0000 −0.869999 −0.435000 0.900431i \(-0.643252\pi\)
−0.435000 + 0.900431i \(0.643252\pi\)
\(762\) 0 0
\(763\) −26.0000 −0.941263
\(764\) −30.0000 −1.08536
\(765\) 0 0
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 16.0000 0.577350
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 3.00000 0.108042
\(772\) −28.0000 −1.00774
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) −2.00000 −0.0717496
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 4.00000 0.143223
\(781\) −18.0000 −0.644091
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) −12.0000 −0.428571
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −42.0000 −1.49619
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 0 0
\(795\) 3.00000 0.106399
\(796\) 2.00000 0.0708881
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) −21.0000 −0.741074
\(804\) −28.0000 −0.987484
\(805\) −6.00000 −0.211472
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) 0 0
\(809\) −27.0000 −0.949269 −0.474635 0.880183i \(-0.657420\pi\)
−0.474635 + 0.880183i \(0.657420\pi\)
\(810\) 0 0
\(811\) −49.0000 −1.72062 −0.860311 0.509769i \(-0.829731\pi\)
−0.860311 + 0.509769i \(0.829731\pi\)
\(812\) 4.00000 0.140372
\(813\) 2.00000 0.0701431
\(814\) 0 0
\(815\) 19.0000 0.665541
\(816\) 0 0
\(817\) −2.00000 −0.0699711
\(818\) 0 0
\(819\) 4.00000 0.139771
\(820\) −6.00000 −0.209529
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) 3.00000 0.104447
\(826\) 0 0
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) −6.00000 −0.208514
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) −16.0000 −0.554700
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −4.00000 −0.137442
\(848\) −12.0000 −0.412082
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) 12.0000 0.411113
\(853\) 17.0000 0.582069 0.291034 0.956713i \(-0.406001\pi\)
0.291034 + 0.956713i \(0.406001\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) 0 0
\(857\) −39.0000 −1.33221 −0.666107 0.745856i \(-0.732041\pi\)
−0.666107 + 0.745856i \(0.732041\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) −2.00000 −0.0681994
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −21.0000 −0.714021
\(866\) 0 0
\(867\) −17.0000 −0.577350
\(868\) −32.0000 −1.08615
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) 28.0000 0.948744
\(872\) 0 0
\(873\) 11.0000 0.372294
\(874\) 0 0
\(875\) −2.00000 −0.0676123
\(876\) 14.0000 0.473016
\(877\) −28.0000 −0.945493 −0.472746 0.881199i \(-0.656737\pi\)
−0.472746 + 0.881199i \(0.656737\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −12.0000 −0.404520
\(881\) 3.00000 0.101073 0.0505363 0.998722i \(-0.483907\pi\)
0.0505363 + 0.998722i \(0.483907\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) 0 0
\(887\) 42.0000 1.41022 0.705111 0.709097i \(-0.250897\pi\)
0.705111 + 0.709097i \(0.250897\pi\)
\(888\) 0 0
\(889\) 10.0000 0.335389
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 8.00000 0.267860
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) −6.00000 −0.200558
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) 0 0
\(899\) −8.00000 −0.266815
\(900\) −2.00000 −0.0666667
\(901\) 0 0
\(902\) 0 0
\(903\) −2.00000 −0.0665558
\(904\) 0 0
\(905\) 7.00000 0.232688
\(906\) 0 0
\(907\) −19.0000 −0.630885 −0.315442 0.948945i \(-0.602153\pi\)
−0.315442 + 0.948945i \(0.602153\pi\)
\(908\) 42.0000 1.39382
\(909\) −9.00000 −0.298511
\(910\) 0 0
\(911\) 33.0000 1.09334 0.546669 0.837349i \(-0.315895\pi\)
0.546669 + 0.837349i \(0.315895\pi\)
\(912\) 8.00000 0.264906
\(913\) 27.0000 0.893570
\(914\) 0 0
\(915\) −8.00000 −0.264472
\(916\) −28.0000 −0.925146
\(917\) −24.0000 −0.792550
\(918\) 0 0
\(919\) −52.0000 −1.71532 −0.857661 0.514216i \(-0.828083\pi\)
−0.857661 + 0.514216i \(0.828083\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) −12.0000 −0.394771
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) 14.0000 0.459820
\(928\) 0 0
\(929\) −36.0000 −1.18112 −0.590561 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −30.0000 −0.982683
\(933\) −21.0000 −0.687509
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −52.0000 −1.69877 −0.849383 0.527777i \(-0.823026\pi\)
−0.849383 + 0.527777i \(0.823026\pi\)
\(938\) 0 0
\(939\) −4.00000 −0.130535
\(940\) −12.0000 −0.391397
\(941\) 54.0000 1.76035 0.880175 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) 0 0
\(943\) −9.00000 −0.293080
\(944\) −48.0000 −1.56227
\(945\) −2.00000 −0.0650600
\(946\) 0 0
\(947\) 30.0000 0.974869 0.487435 0.873160i \(-0.337933\pi\)
0.487435 + 0.873160i \(0.337933\pi\)
\(948\) 8.00000 0.259828
\(949\) −14.0000 −0.454459
\(950\) 0 0
\(951\) 24.0000 0.778253
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) −15.0000 −0.485389
\(956\) −12.0000 −0.388108
\(957\) −3.00000 −0.0969762
\(958\) 0 0
\(959\) −24.0000 −0.775000
\(960\) 8.00000 0.258199
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 14.0000 0.450910
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 41.0000 1.31847 0.659236 0.751936i \(-0.270880\pi\)
0.659236 + 0.751936i \(0.270880\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −57.0000 −1.82922 −0.914609 0.404341i \(-0.867501\pi\)
−0.914609 + 0.404341i \(0.867501\pi\)
\(972\) −2.00000 −0.0641500
\(973\) 10.0000 0.320585
\(974\) 0 0
\(975\) 2.00000 0.0640513
\(976\) 32.0000 1.02430
\(977\) 57.0000 1.82359 0.911796 0.410644i \(-0.134696\pi\)
0.911796 + 0.410644i \(0.134696\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) −6.00000 −0.191663
\(981\) −13.0000 −0.415058
\(982\) 0 0
\(983\) 30.0000 0.956851 0.478426 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 0 0
\(985\) −21.0000 −0.669116
\(986\) 0 0
\(987\) −12.0000 −0.381964
\(988\) −8.00000 −0.254514
\(989\) −3.00000 −0.0953945
\(990\) 0 0
\(991\) −25.0000 −0.794151 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) 0 0
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) 1.00000 0.0317021
\(996\) −18.0000 −0.570352
\(997\) −37.0000 −1.17180 −0.585901 0.810383i \(-0.699259\pi\)
−0.585901 + 0.810383i \(0.699259\pi\)
\(998\) 0 0
\(999\) −1.00000 −0.0316386
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 435.2.a.c.1.1 1
3.2 odd 2 1305.2.a.d.1.1 1
4.3 odd 2 6960.2.a.b.1.1 1
5.2 odd 4 2175.2.c.e.349.1 2
5.3 odd 4 2175.2.c.e.349.2 2
5.4 even 2 2175.2.a.d.1.1 1
15.14 odd 2 6525.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.c.1.1 1 1.1 even 1 trivial
1305.2.a.d.1.1 1 3.2 odd 2
2175.2.a.d.1.1 1 5.4 even 2
2175.2.c.e.349.1 2 5.2 odd 4
2175.2.c.e.349.2 2 5.3 odd 4
6525.2.a.f.1.1 1 15.14 odd 2
6960.2.a.b.1.1 1 4.3 odd 2