Properties

Label 4650.2.a.r.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
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] = [4650,2,Mod(1,4650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4650, 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("4650.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.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: \(37.1304369399\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4650.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -6.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +6.00000 q^{22} -2.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} -8.00000 q^{29} +1.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} +2.00000 q^{39} -2.00000 q^{41} -4.00000 q^{43} -6.00000 q^{44} +2.00000 q^{46} -4.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +4.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{58} +4.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} +4.00000 q^{67} +4.00000 q^{68} -2.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} +4.00000 q^{73} -6.00000 q^{74} -2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{86} -8.00000 q^{87} +6.00000 q^{88} -2.00000 q^{89} -2.00000 q^{92} +1.00000 q^{93} +4.00000 q^{94} -1.00000 q^{96} -14.0000 q^{97} +7.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) −6.00000 −1.04447
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 8.00000 1.05045
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 4.00000 0.485071
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −8.00000 −0.857690
\(88\) 6.00000 0.639602
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) 1.00000 0.103695
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 7.00000 0.707107
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −4.00000 −0.396059
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) −4.00000 −0.362143
\(123\) −2.00000 −0.180334
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) 0 0
\(127\) −14.0000 −1.24230 −0.621150 0.783692i \(-0.713334\pi\)
−0.621150 + 0.783692i \(0.713334\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −6.00000 −0.522233
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 2.00000 0.170251
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 8.00000 0.671345
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) −7.00000 −0.577350
\(148\) 6.00000 0.493197
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 4.00000 0.318223
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 0 0
\(167\) 10.0000 0.773823 0.386912 0.922117i \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 8.00000 0.606478
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 4.00000 0.295689
\(184\) 2.00000 0.147442
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) −24.0000 −1.75505
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000 0.0721688
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 6.00000 0.426401
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) −2.00000 −0.139010
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) −8.00000 −0.548151
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) −6.00000 −0.402694
\(223\) −18.0000 −1.20537 −0.602685 0.797980i \(-0.705902\pi\)
−0.602685 + 0.797980i \(0.705902\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 8.00000 0.525226
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −25.0000 −1.60706
\(243\) 1.00000 0.0641500
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) −1.00000 −0.0635001
\(249\) 0 0
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 14.0000 0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) 0 0
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) −2.00000 −0.122398
\(268\) 4.00000 0.244339
\(269\) −16.0000 −0.975537 −0.487769 0.872973i \(-0.662189\pi\)
−0.487769 + 0.872973i \(0.662189\pi\)
\(270\) 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
\(274\) 8.00000 0.483298
\(275\) 0 0
\(276\) −2.00000 −0.120386
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 2.00000 0.119952
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 4.00000 0.238197
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 4.00000 0.234082
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) −6.00000 −0.348155
\(298\) 14.0000 0.810998
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) 0 0
\(302\) −12.0000 −0.690522
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −2.00000 −0.113228
\(313\) 12.0000 0.678280 0.339140 0.940736i \(-0.389864\pi\)
0.339140 + 0.940736i \(0.389864\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 26.0000 1.46031 0.730153 0.683284i \(-0.239449\pi\)
0.730153 + 0.683284i \(0.239449\pi\)
\(318\) −6.00000 −0.336463
\(319\) 48.0000 2.68748
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) −10.0000 −0.553001
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) −18.0000 −0.989369 −0.494685 0.869072i \(-0.664716\pi\)
−0.494685 + 0.869072i \(0.664716\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) −10.0000 −0.547176
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 9.00000 0.489535
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) −8.00000 −0.428845
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 6.00000 0.319801
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) 2.00000 0.105703
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −8.00000 −0.420471
\(363\) 25.0000 1.31216
\(364\) 0 0
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) 10.0000 0.521996 0.260998 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) −2.00000 −0.104257
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 1.00000 0.0518476
\(373\) −18.0000 −0.932005 −0.466002 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) −16.0000 −0.824042
\(378\) 0 0
\(379\) −24.0000 −1.23280 −0.616399 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 0 0
\(381\) −14.0000 −0.717242
\(382\) 8.00000 0.409316
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) −4.00000 −0.203331
\(388\) −14.0000 −0.710742
\(389\) −20.0000 −1.01404 −0.507020 0.861934i \(-0.669253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 7.00000 0.353553
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) −38.0000 −1.90717 −0.953583 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) 0 0
\(401\) −26.0000 −1.29838 −0.649189 0.760627i \(-0.724892\pi\)
−0.649189 + 0.760627i \(0.724892\pi\)
\(402\) −4.00000 −0.199502
\(403\) 2.00000 0.0996271
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) −36.0000 −1.78445
\(408\) −4.00000 −0.198030
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −8.00000 −0.394611
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −2.00000 −0.0979404
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 4.00000 0.194717
\(423\) −4.00000 −0.194487
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) 1.00000 0.0481125
\(433\) 28.0000 1.34559 0.672797 0.739827i \(-0.265093\pi\)
0.672797 + 0.739827i \(0.265093\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) −4.00000 −0.191127
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) −8.00000 −0.380521
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) 18.0000 0.852325
\(447\) −14.0000 −0.662177
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) −6.00000 −0.282216
\(453\) 12.0000 0.563809
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) −16.0000 −0.747631
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 14.0000 0.650635 0.325318 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 24.0000 1.10352
\(474\) 4.00000 0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 24.0000 1.09773
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 34.0000 1.54069 0.770344 0.637629i \(-0.220085\pi\)
0.770344 + 0.637629i \(0.220085\pi\)
\(488\) −4.00000 −0.181071
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −34.0000 −1.53440 −0.767199 0.641409i \(-0.778350\pi\)
−0.767199 + 0.641409i \(0.778350\pi\)
\(492\) −2.00000 −0.0901670
\(493\) −32.0000 −1.44121
\(494\) 0 0
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 0 0
\(498\) 0 0
\(499\) −38.0000 −1.70111 −0.850557 0.525883i \(-0.823735\pi\)
−0.850557 + 0.525883i \(0.823735\pi\)
\(500\) 0 0
\(501\) 10.0000 0.446767
\(502\) −18.0000 −0.803379
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −12.0000 −0.533465
\(507\) −9.00000 −0.399704
\(508\) −14.0000 −0.621150
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −26.0000 −1.14681
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 8.00000 0.350150
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 18.0000 0.784837
\(527\) 4.00000 0.174243
\(528\) −6.00000 −0.261116
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 2.00000 0.0865485
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) −2.00000 −0.0863064
\(538\) 16.0000 0.689809
\(539\) 42.0000 1.80907
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −16.0000 −0.687259
\(543\) 8.00000 0.343313
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) −8.00000 −0.341743
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) 0 0
\(552\) 2.00000 0.0851257
\(553\) 0 0
\(554\) 18.0000 0.764747
\(555\) 0 0
\(556\) −2.00000 −0.0848189
\(557\) −38.0000 −1.61011 −0.805056 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) −1.00000 −0.0423334
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 26.0000 1.09674
\(563\) −44.0000 −1.85438 −0.927189 0.374593i \(-0.877783\pi\)
−0.927189 + 0.374593i \(0.877783\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −2.00000 −0.0838444 −0.0419222 0.999121i \(-0.513348\pi\)
−0.0419222 + 0.999121i \(0.513348\pi\)
\(570\) 0 0
\(571\) −38.0000 −1.59025 −0.795125 0.606445i \(-0.792595\pi\)
−0.795125 + 0.606445i \(0.792595\pi\)
\(572\) −12.0000 −0.501745
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 1.00000 0.0415945
\(579\) 18.0000 0.748054
\(580\) 0 0
\(581\) 0 0
\(582\) 14.0000 0.580319
\(583\) −36.0000 −1.49097
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) −30.0000 −1.23929
\(587\) −32.0000 −1.32078 −0.660391 0.750922i \(-0.729609\pi\)
−0.660391 + 0.750922i \(0.729609\pi\)
\(588\) −7.00000 −0.288675
\(589\) 0 0
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 6.00000 0.246598
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) −20.0000 −0.818546
\(598\) 4.00000 0.163572
\(599\) 48.0000 1.96123 0.980613 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 12.0000 0.488273
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 4.00000 0.161690
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 16.0000 0.643614
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) −2.00000 −0.0802572
\(622\) 24.0000 0.962312
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −12.0000 −0.479616
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 4.00000 0.159111
\(633\) −4.00000 −0.158986
\(634\) −26.0000 −1.03259
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) −14.0000 −0.554700
\(638\) −48.0000 −1.90034
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −22.0000 −0.868948 −0.434474 0.900684i \(-0.643066\pi\)
−0.434474 + 0.900684i \(0.643066\pi\)
\(642\) 12.0000 0.473602
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 18.0000 0.699590
\(663\) 8.00000 0.310694
\(664\) 0 0
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 16.0000 0.619522
\(668\) 10.0000 0.386912
\(669\) −18.0000 −0.695920
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) −32.0000 −1.23351 −0.616755 0.787155i \(-0.711553\pi\)
−0.616755 + 0.787155i \(0.711553\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 6.00000 0.229752
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 16.0000 0.610438
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) 16.0000 0.607352
\(695\) 0 0
\(696\) 8.00000 0.303239
\(697\) −8.00000 −0.303022
\(698\) 30.0000 1.13552
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 2.00000 0.0749532
\(713\) −2.00000 −0.0749006
\(714\) 0 0
\(715\) 0 0
\(716\) −2.00000 −0.0747435
\(717\) −24.0000 −0.896296
\(718\) −24.0000 −0.895672
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) 18.0000 0.669427
\(724\) 8.00000 0.297318
\(725\) 0 0
\(726\) −25.0000 −0.927837
\(727\) 12.0000 0.445055 0.222528 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 4.00000 0.147844
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) −24.0000 −0.884051
\(738\) 2.00000 0.0736210
\(739\) −2.00000 −0.0735712 −0.0367856 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 22.0000 0.807102 0.403551 0.914957i \(-0.367776\pi\)
0.403551 + 0.914957i \(0.367776\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) 18.0000 0.659027
\(747\) 0 0
\(748\) −24.0000 −0.877527
\(749\) 0 0
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) −4.00000 −0.145865
\(753\) 18.0000 0.655956
\(754\) 16.0000 0.582686
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 24.0000 0.871719
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 14.0000 0.507166
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 26.0000 0.936367
\(772\) 18.0000 0.647834
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 20.0000 0.717035
\(779\) 0 0
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) 8.00000 0.286079
\(783\) −8.00000 −0.285897
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) 18.0000 0.641223
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) 0 0
\(792\) 6.00000 0.213201
\(793\) 8.00000 0.284088
\(794\) 38.0000 1.34857
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 26.0000 0.918092
\(803\) −24.0000 −0.846942
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −2.00000 −0.0704470
\(807\) −16.0000 −0.563227
\(808\) −2.00000 −0.0703598
\(809\) −46.0000 −1.61727 −0.808637 0.588308i \(-0.799794\pi\)
−0.808637 + 0.588308i \(0.799794\pi\)
\(810\) 0 0
\(811\) 24.0000 0.842754 0.421377 0.906886i \(-0.361547\pi\)
0.421377 + 0.906886i \(0.361547\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 36.0000 1.26180
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 0 0
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 0 0
\(821\) 16.0000 0.558404 0.279202 0.960232i \(-0.409930\pi\)
0.279202 + 0.960232i \(0.409930\pi\)
\(822\) 8.00000 0.279032
\(823\) −6.00000 −0.209147 −0.104573 0.994517i \(-0.533348\pi\)
−0.104573 + 0.994517i \(0.533348\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) −32.0000 −1.11275 −0.556375 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) −2.00000 −0.0695048
\(829\) −32.0000 −1.11141 −0.555703 0.831381i \(-0.687551\pi\)
−0.555703 + 0.831381i \(0.687551\pi\)
\(830\) 0 0
\(831\) −18.0000 −0.624413
\(832\) 2.00000 0.0693375
\(833\) −28.0000 −0.970143
\(834\) 2.00000 0.0692543
\(835\) 0 0
\(836\) 0 0
\(837\) 1.00000 0.0345651
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) −26.0000 −0.896019
\(843\) −26.0000 −0.895488
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) −8.00000 −0.274075
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 12.0000 0.409673
\(859\) −22.0000 −0.750630 −0.375315 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −40.0000 −1.36241
\(863\) −22.0000 −0.748889 −0.374444 0.927249i \(-0.622167\pi\)
−0.374444 + 0.927249i \(0.622167\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −28.0000 −0.951479
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 10.0000 0.338643
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) 50.0000 1.68838 0.844190 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) 0 0
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) 38.0000 1.28025 0.640126 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) 7.00000 0.235702
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 4.00000 0.134307 0.0671534 0.997743i \(-0.478608\pi\)
0.0671534 + 0.997743i \(0.478608\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) −18.0000 −0.602685
\(893\) 0 0
\(894\) 14.0000 0.468230
\(895\) 0 0
\(896\) 0 0
\(897\) −4.00000 −0.133556
\(898\) −14.0000 −0.467186
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) −12.0000 −0.399556
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −12.0000 −0.398673
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) −12.0000 −0.398234
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) 16.0000 0.528655
\(917\) 0 0
\(918\) −4.00000 −0.132020
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 0 0
\(926\) −14.0000 −0.460069
\(927\) −16.0000 −0.525509
\(928\) 8.00000 0.262613
\(929\) 46.0000 1.50921 0.754606 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) −24.0000 −0.785725
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) 12.0000 0.391605
\(940\) 0 0
\(941\) 52.0000 1.69515 0.847576 0.530674i \(-0.178061\pi\)
0.847576 + 0.530674i \(0.178061\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 4.00000 0.130258
\(944\) 0 0
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −4.00000 −0.129914
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 26.0000 0.843108
\(952\) 0 0
\(953\) −8.00000 −0.259145 −0.129573 0.991570i \(-0.541361\pi\)
−0.129573 + 0.991570i \(0.541361\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 48.0000 1.55162
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −12.0000 −0.386896
\(963\) −12.0000 −0.386695
\(964\) 18.0000 0.579741
\(965\) 0 0
\(966\) 0 0
\(967\) −6.00000 −0.192947 −0.0964735 0.995336i \(-0.530756\pi\)
−0.0964735 + 0.995336i \(0.530756\pi\)
\(968\) −25.0000 −0.803530
\(969\) 0 0
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −34.0000 −1.08943
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) −26.0000 −0.831814 −0.415907 0.909407i \(-0.636536\pi\)
−0.415907 + 0.909407i \(0.636536\pi\)
\(978\) −4.00000 −0.127906
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 34.0000 1.08498
\(983\) −10.0000 −0.318950 −0.159475 0.987202i \(-0.550980\pi\)
−0.159475 + 0.987202i \(0.550980\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) 32.0000 1.01909
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −1.00000 −0.0317500
\(993\) −18.0000 −0.571213
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) 38.0000 1.20287
\(999\) 6.00000 0.189832
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 4650.2.a.r.1.1 1
5.2 odd 4 4650.2.d.a.3349.1 2
5.3 odd 4 4650.2.d.a.3349.2 2
5.4 even 2 930.2.a.l.1.1 1
15.14 odd 2 2790.2.a.j.1.1 1
20.19 odd 2 7440.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.l.1.1 1 5.4 even 2
2790.2.a.j.1.1 1 15.14 odd 2
4650.2.a.r.1.1 1 1.1 even 1 trivial
4650.2.d.a.3349.1 2 5.2 odd 4
4650.2.d.a.3349.2 2 5.3 odd 4
7440.2.a.r.1.1 1 20.19 odd 2