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Properties

Label 8046.2.a.a.1.1
Level $8046$
Weight $2$
Character 8046.1
Self dual yes
Analytic conductor $64.248$
Analytic rank $1$
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] = [8046,2,Mod(1,8046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8046, 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("8046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8046.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: \(64.2476334663\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{10} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +4.00000 q^{19} +2.00000 q^{20} -5.00000 q^{23} -1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} +1.00000 q^{29} -5.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} +2.00000 q^{35} -6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{40} -9.00000 q^{41} -4.00000 q^{43} +5.00000 q^{46} -6.00000 q^{47} -6.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} +1.00000 q^{53} -1.00000 q^{56} -1.00000 q^{58} +2.00000 q^{59} -6.00000 q^{61} +5.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +4.00000 q^{68} -2.00000 q^{70} -3.00000 q^{71} -6.00000 q^{73} +6.00000 q^{74} +4.00000 q^{76} -8.00000 q^{79} +2.00000 q^{80} +9.00000 q^{82} +6.00000 q^{83} +8.00000 q^{85} +4.00000 q^{86} +1.00000 q^{89} -2.00000 q^{91} -5.00000 q^{92} +6.00000 q^{94} +8.00000 q^{95} +10.0000 q^{97} +6.00000 q^{98} +O(q^{100})\)

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
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) −5.00000 −1.04257 −0.521286 − 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 1.00000 0.185695 0.0928477 − 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 − 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 5.00000 0.737210
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 1.00000 0.137361 0.0686803 − 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) 2.00000 0.260378 0.130189 − 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 5.00000 0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −3.00000 −0.356034 −0.178017 − 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) 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
\(85\) 8.00000 0.867722
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 1.00000 0.106000 0.0529999 − 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −5.00000 −0.521286
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 6.00000 0.606092
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 14.0000 1.39305 0.696526 − 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 7.00000 0.689730 0.344865 − 0.938652i \(-0.387925\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) −16.0000 −1.54678 −0.773389 − 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 0 0
\(109\) −17.0000 −1.62830 −0.814152 − 0.580651i \(-0.802798\pi\)
−0.814152 + 0.580651i \(0.802798\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −4.00000 −0.376288 −0.188144 − 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) −10.0000 −0.932505
\(116\) 1.00000 0.0928477
\(117\) 0 0
\(118\) −2.00000 −0.184115
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −5.00000 −0.449013
\(125\) −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
\(130\) 4.00000 0.350823
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 21.0000 1.79415 0.897076 − 0.441877i \(-0.145687\pi\)
0.897076 + 0.441877i \(0.145687\pi\)
\(138\) 0 0
\(139\) −9.00000 −0.763370 −0.381685 − 0.924292i \(-0.624656\pi\)
−0.381685 + 0.924292i \(0.624656\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 3.00000 0.251754
\(143\) 0 0
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) −1.00000 −0.0819232
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 0 0
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) −21.0000 −1.67598 −0.837991 − 0.545684i \(-0.816270\pi\)
−0.837991 + 0.545684i \(0.816270\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) −5.00000 −0.394055
\(162\) 0 0
\(163\) −11.0000 −0.861586 −0.430793 − 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) −16.0000 −1.23812 −0.619059 − 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −8.00000 −0.613572
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 13.0000 0.988372 0.494186 − 0.869356i \(-0.335466\pi\)
0.494186 + 0.869356i \(0.335466\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) −1.00000 −0.0749532
\(179\) −3.00000 −0.224231 −0.112115 − 0.993695i \(-0.535763\pi\)
−0.112115 + 0.993695i \(0.535763\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 − 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) 5.00000 0.368605
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 0 0
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 10.0000 0.723575 0.361787 − 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 − 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 24.0000 1.70993 0.854965 − 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −14.0000 −0.985037
\(203\) 1.00000 0.0701862
\(204\) 0 0
\(205\) −18.0000 −1.25717
\(206\) −7.00000 −0.487713
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 19.0000 1.30801 0.654007 − 0.756489i \(-0.273087\pi\)
0.654007 + 0.756489i \(0.273087\pi\)
\(212\) 1.00000 0.0686803
\(213\) 0 0
\(214\) 16.0000 1.09374
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −5.00000 −0.339422
\(218\) 17.0000 1.15139
\(219\) 0 0
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 − 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 4.00000 0.266076
\(227\) 18.0000 1.19470 0.597351 − 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 10.0000 0.659380
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 17.0000 1.11371 0.556854 − 0.830611i \(-0.312008\pi\)
0.556854 + 0.830611i \(0.312008\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 2.00000 0.130189
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) −9.00000 −0.582162 −0.291081 − 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 − 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) −12.0000 −0.766652
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 5.00000 0.317500
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) −25.0000 −1.57799 −0.788993 − 0.614402i \(-0.789397\pi\)
−0.788993 + 0.614402i \(0.789397\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 5.00000 0.311891 0.155946 − 0.987766i \(-0.450158\pi\)
0.155946 + 0.987766i \(0.450158\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) 0 0
\(263\) 30.0000 1.84988 0.924940 − 0.380114i \(-0.124115\pi\)
0.924940 + 0.380114i \(0.124115\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) 0 0
\(269\) −23.0000 −1.40233 −0.701167 − 0.712997i \(-0.747337\pi\)
−0.701167 + 0.712997i \(0.747337\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −21.0000 −1.26866
\(275\) 0 0
\(276\) 0 0
\(277\) −27.0000 −1.62227 −0.811136 − 0.584857i \(-0.801151\pi\)
−0.811136 + 0.584857i \(0.801151\pi\)
\(278\) 9.00000 0.539784
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) −2.00000 −0.119310 −0.0596550 − 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) −3.00000 −0.178331 −0.0891657 − 0.996017i \(-0.528420\pi\)
−0.0891657 + 0.996017i \(0.528420\pi\)
\(284\) −3.00000 −0.178017
\(285\) 0 0
\(286\) 0 0
\(287\) −9.00000 −0.531253
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) −2.00000 −0.117444
\(291\) 0 0
\(292\) −6.00000 −0.351123
\(293\) 7.00000 0.408944 0.204472 − 0.978872i \(-0.434452\pi\)
0.204472 + 0.978872i \(0.434452\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 1.00000 0.0579284
\(299\) 10.0000 0.578315
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) 6.00000 0.342438 0.171219 − 0.985233i \(-0.445229\pi\)
0.171219 + 0.985233i \(0.445229\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 10.0000 0.567962
\(311\) 31.0000 1.75785 0.878924 − 0.476961i \(-0.158262\pi\)
0.878924 + 0.476961i \(0.158262\pi\)
\(312\) 0 0
\(313\) 20.0000 1.13047 0.565233 − 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) 21.0000 1.18510
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −2.00000 −0.112331 −0.0561656 − 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) 5.00000 0.278639
\(323\) 16.0000 0.890264
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 11.0000 0.609234
\(327\) 0 0
\(328\) 9.00000 0.496942
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) 0 0
\(337\) 31.0000 1.68868 0.844339 − 0.535810i \(-0.179994\pi\)
0.844339 + 0.535810i \(0.179994\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 8.00000 0.433861
\(341\) 0 0
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −13.0000 −0.698884
\(347\) −5.00000 −0.268414 −0.134207 − 0.990953i \(-0.542849\pi\)
−0.134207 + 0.990953i \(0.542849\pi\)
\(348\) 0 0
\(349\) 19.0000 1.01705 0.508523 − 0.861048i \(-0.330192\pi\)
0.508523 + 0.861048i \(0.330192\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 − 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 1.00000 0.0529999
\(357\) 0 0
\(358\) 3.00000 0.158555
\(359\) 16.0000 0.844448 0.422224 − 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 5.00000 0.262794
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) −12.0000 −0.628109
\(366\) 0 0
\(367\) 29.0000 1.51379 0.756894 − 0.653538i \(-0.226716\pi\)
0.756894 + 0.653538i \(0.226716\pi\)
\(368\) −5.00000 −0.260643
\(369\) 0 0
\(370\) 12.0000 0.623850
\(371\) 1.00000 0.0519174
\(372\) 0 0
\(373\) 23.0000 1.19089 0.595447 − 0.803394i \(-0.296975\pi\)
0.595447 + 0.803394i \(0.296975\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 − 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) −10.0000 −0.511645
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) 26.0000 1.31825 0.659126 − 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) −20.0000 −1.01144
\(392\) 6.00000 0.303046
\(393\) 0 0
\(394\) −24.0000 −1.20910
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 − 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 24.0000 1.19850 0.599251 − 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 10.0000 0.498135
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) −1.00000 −0.0496292
\(407\) 0 0
\(408\) 0 0
\(409\) −38.0000 −1.87898 −0.939490 − 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 18.0000 0.888957
\(411\) 0 0
\(412\) 7.00000 0.344865
\(413\) 2.00000 0.0984136
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 0 0
\(419\) −23.0000 −1.12362 −0.561812 − 0.827265i \(-0.689895\pi\)
−0.561812 + 0.827265i \(0.689895\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −19.0000 −0.924906
\(423\) 0 0
\(424\) −1.00000 −0.0485643
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) −16.0000 −0.773389
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) 26.0000 1.25238 0.626188 − 0.779672i \(-0.284614\pi\)
0.626188 + 0.779672i \(0.284614\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 5.00000 0.240008
\(435\) 0 0
\(436\) −17.0000 −0.814152
\(437\) −20.0000 −0.956730
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 − 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 8.00000 0.380521
\(443\) −27.0000 −1.28281 −0.641404 − 0.767203i \(-0.721648\pi\)
−0.641404 + 0.767203i \(0.721648\pi\)
\(444\) 0 0
\(445\) 2.00000 0.0948091
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −27.0000 −1.27421 −0.637104 − 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −4.00000 −0.188144
\(453\) 0 0
\(454\) −18.0000 −0.844782
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −34.0000 −1.59045 −0.795226 − 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) 10.0000 0.467269
\(459\) 0 0
\(460\) −10.0000 −0.466252
\(461\) 2.00000 0.0931493 0.0465746 − 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 13.0000 0.604161 0.302081 − 0.953282i \(-0.402319\pi\)
0.302081 + 0.953282i \(0.402319\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −17.0000 −0.787510
\(467\) −20.0000 −0.925490 −0.462745 − 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 12.0000 0.553519
\(471\) 0 0
\(472\) −2.00000 −0.0920575
\(473\) 0 0
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) 9.00000 0.411650
\(479\) 13.0000 0.593985 0.296993 − 0.954880i \(-0.404016\pi\)
0.296993 + 0.954880i \(0.404016\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 20.0000 0.908153
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 − 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) 12.0000 0.542105
\(491\) 24.0000 1.08310 0.541552 − 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 4.00000 0.180151
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) −5.00000 −0.224507
\(497\) −3.00000 −0.134568
\(498\) 0 0
\(499\) −9.00000 −0.402895 −0.201448 − 0.979499i \(-0.564565\pi\)
−0.201448 + 0.979499i \(0.564565\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) 25.0000 1.11580
\(503\) 12.0000 0.535054 0.267527 − 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 28.0000 1.24598
\(506\) 0 0
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 6.00000 0.265945 0.132973 − 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −5.00000 −0.220541
\(515\) 14.0000 0.616914
\(516\) 0 0
\(517\) 0 0
\(518\) 6.00000 0.263625
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) 18.0000 0.788594 0.394297 − 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) −22.0000 −0.961993 −0.480996 − 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −30.0000 −1.30806
\(527\) −20.0000 −0.871214
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) −2.00000 −0.0868744
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) 18.0000 0.779667
\(534\) 0 0
\(535\) −32.0000 −1.38348
\(536\) 0 0
\(537\) 0 0
\(538\) 23.0000 0.991600
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 − 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −16.0000 −0.687259
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) −34.0000 −1.45640
\(546\) 0 0
\(547\) −18.0000 −0.769624 −0.384812 − 0.922995i \(-0.625734\pi\)
−0.384812 + 0.922995i \(0.625734\pi\)
\(548\) 21.0000 0.897076
\(549\) 0 0
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 27.0000 1.14712
\(555\) 0 0
\(556\) −9.00000 −0.381685
\(557\) −13.0000 −0.550828 −0.275414 − 0.961326i \(-0.588815\pi\)
−0.275414 + 0.961326i \(0.588815\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) −13.0000 −0.547885 −0.273942 − 0.961746i \(-0.588328\pi\)
−0.273942 + 0.961746i \(0.588328\pi\)
\(564\) 0 0
\(565\) −8.00000 −0.336563
\(566\) 3.00000 0.126099
\(567\) 0 0
\(568\) 3.00000 0.125877
\(569\) −21.0000 −0.880366 −0.440183 − 0.897908i \(-0.645086\pi\)
−0.440183 + 0.897908i \(0.645086\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 − 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 9.00000 0.375653
\(575\) 5.00000 0.208514
\(576\) 0 0
\(577\) 7.00000 0.291414 0.145707 − 0.989328i \(-0.453454\pi\)
0.145707 + 0.989328i \(0.453454\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 0 0
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) −7.00000 −0.289167
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) −4.00000 −0.164677
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) −17.0000 −0.698106 −0.349053 − 0.937103i \(-0.613497\pi\)
−0.349053 + 0.937103i \(0.613497\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) −1.00000 −0.0409616
\(597\) 0 0
\(598\) −10.0000 −0.408930
\(599\) −16.0000 −0.653742 −0.326871 − 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −23.0000 −0.938190 −0.469095 − 0.883148i \(-0.655420\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) −22.0000 −0.894427
\(606\) 0 0
\(607\) −34.0000 −1.38002 −0.690009 − 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 12.0000 0.485866
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) 32.0000 1.29247 0.646234 − 0.763139i \(-0.276343\pi\)
0.646234 + 0.763139i \(0.276343\pi\)
\(614\) −6.00000 −0.242140
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) −10.0000 −0.401610
\(621\) 0 0
\(622\) −31.0000 −1.24299
\(623\) 1.00000 0.0400642
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −20.0000 −0.799361
\(627\) 0 0
\(628\) −21.0000 −0.837991
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 − 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 2.00000 0.0794301
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) −34.0000 −1.34292 −0.671460 − 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) 2.00000 0.0788723 0.0394362 − 0.999222i \(-0.487444\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(644\) −5.00000 −0.197028
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) 31.0000 1.21874 0.609368 − 0.792888i \(-0.291423\pi\)
0.609368 + 0.792888i \(0.291423\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −11.0000 −0.430793
\(653\) −4.00000 −0.156532 −0.0782660 − 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −9.00000 −0.351391
\(657\) 0 0
\(658\) 6.00000 0.233904
\(659\) −19.0000 −0.740135 −0.370067 − 0.929005i \(-0.620665\pi\)
−0.370067 + 0.929005i \(0.620665\pi\)
\(660\) 0 0
\(661\) −3.00000 −0.116686 −0.0583432 − 0.998297i \(-0.518582\pi\)
−0.0583432 + 0.998297i \(0.518582\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) −5.00000 −0.193601
\(668\) −16.0000 −0.619059
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 46.0000 1.77317 0.886585 − 0.462566i \(-0.153071\pi\)
0.886585 + 0.462566i \(0.153071\pi\)
\(674\) −31.0000 −1.19408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 14.0000 0.538064 0.269032 − 0.963131i \(-0.413296\pi\)
0.269032 + 0.963131i \(0.413296\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) −8.00000 −0.306786
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 42.0000 1.60474
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) −6.00000 −0.228251 −0.114125 − 0.993466i \(-0.536407\pi\)
−0.114125 + 0.993466i \(0.536407\pi\)
\(692\) 13.0000 0.494186
\(693\) 0 0
\(694\) 5.00000 0.189797
\(695\) −18.0000 −0.682779
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) −19.0000 −0.719161
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) 10.0000 0.377695 0.188847 − 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) 14.0000 0.526524
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 − 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 6.00000 0.225176
\(711\) 0 0
\(712\) −1.00000 −0.0374766
\(713\) 25.0000 0.936257
\(714\) 0 0
\(715\) 0 0
\(716\) −3.00000 −0.112115
\(717\) 0 0
\(718\) −16.0000 −0.597115
\(719\) −28.0000 −1.04422 −0.522112 − 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) 0 0
\(721\) 7.00000 0.260694
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) −5.00000 −0.185824
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) 22.0000 0.815935 0.407967 − 0.912996i \(-0.366238\pi\)
0.407967 + 0.912996i \(0.366238\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) 12.0000 0.444140
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) −17.0000 −0.627909 −0.313955 − 0.949438i \(-0.601654\pi\)
−0.313955 + 0.949438i \(0.601654\pi\)
\(734\) −29.0000 −1.07041
\(735\) 0 0
\(736\) 5.00000 0.184302
\(737\) 0 0
\(738\) 0 0
\(739\) −8.00000 −0.294285 −0.147142 − 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) −1.00000 −0.0367112
\(743\) −33.0000 −1.21065 −0.605326 − 0.795977i \(-0.706957\pi\)
−0.605326 + 0.795977i \(0.706957\pi\)
\(744\) 0 0
\(745\) −2.00000 −0.0732743
\(746\) −23.0000 −0.842090
\(747\) 0 0
\(748\) 0 0
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 29.0000 1.05823 0.529113 − 0.848552i \(-0.322525\pi\)
0.529113 + 0.848552i \(0.322525\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) 2.00000 0.0728357
\(755\) 40.0000 1.45575
\(756\) 0 0
\(757\) 25.0000 0.908640 0.454320 − 0.890838i \(-0.349882\pi\)
0.454320 + 0.890838i \(0.349882\pi\)
\(758\) 10.0000 0.363216
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) −22.0000 −0.797499 −0.398750 − 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) −17.0000 −0.615441
\(764\) 10.0000 0.361787
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −4.00000 −0.144432
\(768\) 0 0
\(769\) 10.0000 0.360609 0.180305 − 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.00000 −0.143963
\(773\) −19.0000 −0.683383 −0.341691 − 0.939812i \(-0.611000\pi\)
−0.341691 + 0.939812i \(0.611000\pi\)
\(774\) 0 0
\(775\) 5.00000 0.179605
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) −26.0000 −0.932145
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) 0 0
\(782\) 20.0000 0.715199
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) −42.0000 −1.49904
\(786\) 0 0
\(787\) 46.0000 1.63972 0.819861 − 0.572562i \(-0.194050\pi\)
0.819861 + 0.572562i \(0.194050\pi\)
\(788\) 24.0000 0.854965
\(789\) 0 0
\(790\) 16.0000 0.569254
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −34.0000 −1.20434 −0.602171 − 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −24.0000 −0.847469
\(803\) 0 0
\(804\) 0 0
\(805\) −10.0000 −0.352454
\(806\) −10.0000 −0.352235
\(807\) 0 0
\(808\) −14.0000 −0.492518
\(809\) −54.0000 −1.89854 −0.949269 − 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 − 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 1.00000 0.0350931
\(813\) 0 0
\(814\) 0 0
\(815\) −22.0000 −0.770626
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 38.0000 1.32864
\(819\) 0 0
\(820\) −18.0000 −0.628587
\(821\) 33.0000 1.15171 0.575854 − 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) 0 0
\(823\) −44.0000 −1.53374 −0.766872 − 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) −7.00000 −0.243857
\(825\) 0 0
\(826\) −2.00000 −0.0695889
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 13.0000 0.451509 0.225754 − 0.974184i \(-0.427515\pi\)
0.225754 + 0.974184i \(0.427515\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) −24.0000 −0.831551
\(834\) 0 0
\(835\) −32.0000 −1.10741
\(836\) 0 0
\(837\) 0 0
\(838\) 23.0000 0.794522
\(839\) −9.00000 −0.310715 −0.155357 − 0.987858i \(-0.549653\pi\)
−0.155357 + 0.987858i \(0.549653\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) 19.0000 0.654007
\(845\) −18.0000 −0.619219
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 1.00000 0.0343401
\(849\) 0 0
\(850\) 4.00000 0.137199
\(851\) 30.0000 1.02839
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 − 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) 16.0000 0.546869
\(857\) −24.0000 −0.819824 −0.409912 − 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) −18.0000 −0.614152 −0.307076 − 0.951685i \(-0.599351\pi\)
−0.307076 + 0.951685i \(0.599351\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) −26.0000 −0.885564
\(863\) 34.0000 1.15737 0.578687 − 0.815550i \(-0.303565\pi\)
0.578687 + 0.815550i \(0.303565\pi\)
\(864\) 0 0
\(865\) 26.0000 0.884027
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) −5.00000 −0.169711
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 17.0000 0.575693
\(873\) 0 0
\(874\) 20.0000 0.676510
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 − 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) −10.0000 −0.337484
\(879\) 0 0
\(880\) 0 0
\(881\) 3.00000 0.101073 0.0505363 − 0.998722i \(-0.483907\pi\)
0.0505363 + 0.998722i \(0.483907\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) 27.0000 0.907083
\(887\) 10.0000 0.335767 0.167884 − 0.985807i \(-0.446307\pi\)
0.167884 + 0.985807i \(0.446307\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) −2.00000 −0.0670402
\(891\) 0 0
\(892\) −14.0000 −0.468755
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) −6.00000 −0.200558
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 27.0000 0.901002
\(899\) −5.00000 −0.166759
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 0 0
\(903\) 0 0
\(904\) 4.00000 0.133038
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) 48.0000 1.59381 0.796907 − 0.604102i \(-0.206468\pi\)
0.796907 + 0.604102i \(0.206468\pi\)
\(908\) 18.0000 0.597351
\(909\) 0 0
\(910\) 4.00000 0.132599
\(911\) 4.00000 0.132526 0.0662630 − 0.997802i \(-0.478892\pi\)
0.0662630 + 0.997802i \(0.478892\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 34.0000 1.12462
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 0 0
\(919\) 7.00000 0.230909 0.115454 − 0.993313i \(-0.463168\pi\)
0.115454 + 0.993313i \(0.463168\pi\)
\(920\) 10.0000 0.329690
\(921\) 0 0
\(922\) −2.00000 −0.0658665
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) −13.0000 −0.427207
\(927\) 0 0
\(928\) −1.00000 −0.0328266
\(929\) 10.0000 0.328089 0.164045 − 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) −24.0000 −0.786568
\(932\) 17.0000 0.556854
\(933\) 0 0
\(934\) 20.0000 0.654420
\(935\) 0 0
\(936\) 0 0
\(937\) 32.0000 1.04539 0.522697 − 0.852518i \(-0.324926\pi\)
0.522697 + 0.852518i \(0.324926\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −12.0000 −0.391397
\(941\) −25.0000 −0.814977 −0.407488 − 0.913210i \(-0.633595\pi\)
−0.407488 + 0.913210i \(0.633595\pi\)
\(942\) 0 0
\(943\) 45.0000 1.46540
\(944\) 2.00000 0.0650945
\(945\) 0 0
\(946\) 0 0
\(947\) 33.0000 1.07236 0.536178 − 0.844105i \(-0.319868\pi\)
0.536178 + 0.844105i \(0.319868\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) −4.00000 −0.129641
\(953\) −5.00000 −0.161966 −0.0809829 − 0.996715i \(-0.525806\pi\)
−0.0809829 + 0.996715i \(0.525806\pi\)
\(954\) 0 0
\(955\) 20.0000 0.647185
\(956\) −9.00000 −0.291081
\(957\) 0 0
\(958\) −13.0000 −0.420011
\(959\) 21.0000 0.678125
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) −8.00000 −0.257529
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 − 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) −20.0000 −0.642161
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −9.00000 −0.288527
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) −18.0000 −0.575871 −0.287936 − 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −12.0000 −0.383326
\(981\) 0 0
\(982\) −24.0000 −0.765871
\(983\) −39.0000 −1.24391 −0.621953 − 0.783054i \(-0.713661\pi\)
−0.621953 + 0.783054i \(0.713661\pi\)
\(984\) 0 0
\(985\) 48.0000 1.52941
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 20.0000 0.635963
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 − 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) 5.00000 0.158750
\(993\) 0 0
\(994\) 3.00000 0.0951542
\(995\) 40.0000 1.26809
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 9.00000 0.284890
\(999\) 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 8046.2.a.a.1.1 ✓ 1
3.2 odd 2 8046.2.a.b.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8046.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
8046.2.a.b.1.1 yes 1 3.2 odd 2