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Properties

Label 1638.2.a.h.1.1
Level $1638$
Weight $2$
Character 1638.1
Self dual yes
Analytic conductor $13.079$
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] = [1638,2,Mod(1,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, 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("1638.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.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: \(13.0794958511\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1638.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -5.00000 q^{11} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{19} +1.00000 q^{20} +5.00000 q^{22} -3.00000 q^{23} -4.00000 q^{25} +1.00000 q^{26} +1.00000 q^{28} -9.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{34} +1.00000 q^{35} -11.0000 q^{37} +1.00000 q^{38} -1.00000 q^{40} -5.00000 q^{43} -5.00000 q^{44} +3.00000 q^{46} +8.00000 q^{47} +1.00000 q^{49} +4.00000 q^{50} -1.00000 q^{52} +2.00000 q^{53} -5.00000 q^{55} -1.00000 q^{56} +9.00000 q^{58} -4.00000 q^{59} -15.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -2.00000 q^{67} +3.00000 q^{68} -1.00000 q^{70} +12.0000 q^{71} +11.0000 q^{73} +11.0000 q^{74} -1.00000 q^{76} -5.00000 q^{77} +10.0000 q^{79} +1.00000 q^{80} +14.0000 q^{83} +3.00000 q^{85} +5.00000 q^{86} +5.00000 q^{88} -6.00000 q^{89} -1.00000 q^{91} -3.00000 q^{92} -8.00000 q^{94} -1.00000 q^{95} -14.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 5.00000 1.06600
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −11.0000 −1.80839 −0.904194 − 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 − 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 9.00000 1.18176
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −15.0000 −1.92055 −0.960277 − 0.279050i \(-0.909981\pi\)
−0.960277 + 0.279050i \(0.909981\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 3.00000 0.363803
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 − 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 11.0000 1.27872
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 0 0
\(83\) 14.0000 1.53670 0.768350 − 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) 5.00000 0.539164
\(87\) 0 0
\(88\) 5.00000 0.533002
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) −3.00000 −0.312772
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) −4.00000 −0.398015 −0.199007 − 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 0 0
\(103\) −15.0000 −1.47799 −0.738997 − 0.673709i \(-0.764700\pi\)
−0.738997 + 0.673709i \(0.764700\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −10.0000 −0.966736 −0.483368 − 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) 0 0
\(109\) −9.00000 −0.862044 −0.431022 − 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) 5.00000 0.476731
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) −9.00000 −0.835629
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 15.0000 1.35804
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 1.00000 0.0877058
\(131\) −13.0000 −1.13582 −0.567908 − 0.823092i \(-0.692247\pi\)
−0.567908 + 0.823092i \(0.692247\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) −19.0000 −1.62328 −0.811640 − 0.584158i \(-0.801425\pi\)
−0.811640 + 0.584158i \(0.801425\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 5.00000 0.418121
\(144\) 0 0
\(145\) −9.00000 −0.747409
\(146\) −11.0000 −0.910366
\(147\) 0 0
\(148\) −11.0000 −0.904194
\(149\) 4.00000 0.327693 0.163846 − 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) −23.0000 −1.87171 −0.935857 − 0.352381i \(-0.885372\pi\)
−0.935857 + 0.352381i \(0.885372\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 5.00000 0.402911
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −3.00000 −0.239426 −0.119713 − 0.992809i \(-0.538197\pi\)
−0.119713 + 0.992809i \(0.538197\pi\)
\(158\) −10.0000 −0.795557
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 − 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −14.0000 −1.08661
\(167\) 21.0000 1.62503 0.812514 − 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −3.00000 −0.230089
\(171\) 0 0
\(172\) −5.00000 −0.381246
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) −5.00000 −0.376889
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 − 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 1.00000 0.0741249
\(183\) 0 0
\(184\) 3.00000 0.221163
\(185\) −11.0000 −0.808736
\(186\) 0 0
\(187\) −15.0000 −1.09691
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 1.00000 0.0725476
\(191\) −3.00000 −0.217072 −0.108536 − 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) 12.0000 0.863779 0.431889 − 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 12.0000 0.854965 0.427482 − 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 11.0000 0.779769 0.389885 − 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 4.00000 0.282843
\(201\) 0 0
\(202\) 4.00000 0.281439
\(203\) −9.00000 −0.631676
\(204\) 0 0
\(205\) 0 0
\(206\) 15.0000 1.04510
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) 9.00000 0.619586 0.309793 − 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 10.0000 0.683586
\(215\) −5.00000 −0.340997
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 9.00000 0.609557
\(219\) 0 0
\(220\) −5.00000 −0.337100
\(221\) −3.00000 −0.201802
\(222\) 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
\(226\) 2.00000 0.133038
\(227\) −18.0000 −1.19470 −0.597351 − 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 3.00000 0.197814
\(231\) 0 0
\(232\) 9.00000 0.590879
\(233\) −24.0000 −1.57229 −0.786146 − 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) −3.00000 −0.194461
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 − 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −14.0000 −0.899954
\(243\) 0 0
\(244\) −15.0000 −0.960277
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 1.00000 0.0636285
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) 9.00000 0.569210
\(251\) 7.00000 0.441836 0.220918 − 0.975292i \(-0.429095\pi\)
0.220918 + 0.975292i \(0.429095\pi\)
\(252\) 0 0
\(253\) 15.0000 0.943042
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) −11.0000 −0.683507
\(260\) −1.00000 −0.0620174
\(261\) 0 0
\(262\) 13.0000 0.803143
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 1.00000 0.0613139
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −22.0000 −1.33640 −0.668202 − 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) 19.0000 1.14783
\(275\) 20.0000 1.20605
\(276\) 0 0
\(277\) 26.0000 1.56219 0.781094 − 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 26.0000 1.55103 0.775515 − 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) −5.00000 −0.295656
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 9.00000 0.528498
\(291\) 0 0
\(292\) 11.0000 0.643726
\(293\) −14.0000 −0.817889 −0.408944 − 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 11.0000 0.639362
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) −5.00000 −0.288195
\(302\) 23.0000 1.32350
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) −15.0000 −0.858898
\(306\) 0 0
\(307\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −5.00000 −0.284901
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) 10.0000 0.567048 0.283524 − 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 − 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 3.00000 0.169300
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) −24.0000 −1.34797 −0.673987 − 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) 45.0000 2.51952
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 3.00000 0.167183
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) −10.0000 −0.553849
\(327\) 0 0
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 − 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 14.0000 0.768350
\(333\) 0 0
\(334\) −21.0000 −1.14907
\(335\) −2.00000 −0.109272
\(336\) 0 0
\(337\) −5.00000 −0.272367 −0.136184 − 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) 3.00000 0.162698
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 5.00000 0.269582
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 16.0000 0.858925 0.429463 − 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 − 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 4.00000 0.211112 0.105556 − 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −14.0000 −0.735824
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) 11.0000 0.575766
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 − 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) −3.00000 −0.156386
\(369\) 0 0
\(370\) 11.0000 0.571863
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 15.0000 0.775632
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 9.00000 0.463524
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 − 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 0 0
\(382\) 3.00000 0.153493
\(383\) 29.0000 1.48183 0.740915 − 0.671598i \(-0.234392\pi\)
0.740915 + 0.671598i \(0.234392\pi\)
\(384\) 0 0
\(385\) −5.00000 −0.254824
\(386\) −12.0000 −0.610784
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) 2.00000 0.101404 0.0507020 − 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −9.00000 −0.455150
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −12.0000 −0.604551
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 − 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) −11.0000 −0.551380
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) 9.00000 0.446663
\(407\) 55.0000 2.72625
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 − 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −15.0000 −0.738997
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 14.0000 0.687233
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) −5.00000 −0.244558
\(419\) −7.00000 −0.341972 −0.170986 − 0.985273i \(-0.554695\pi\)
−0.170986 + 0.985273i \(0.554695\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 − 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) −9.00000 −0.438113
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) −15.0000 −0.725901
\(428\) −10.0000 −0.483368
\(429\) 0 0
\(430\) 5.00000 0.241121
\(431\) 2.00000 0.0963366 0.0481683 − 0.998839i \(-0.484662\pi\)
0.0481683 + 0.998839i \(0.484662\pi\)
\(432\) 0 0
\(433\) 28.0000 1.34559 0.672797 − 0.739827i \(-0.265093\pi\)
0.672797 + 0.739827i \(0.265093\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −9.00000 −0.431022
\(437\) 3.00000 0.143509
\(438\) 0 0
\(439\) 13.0000 0.620456 0.310228 − 0.950662i \(-0.399595\pi\)
0.310228 + 0.950662i \(0.399595\pi\)
\(440\) 5.00000 0.238366
\(441\) 0 0
\(442\) 3.00000 0.142695
\(443\) 18.0000 0.855206 0.427603 − 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −23.0000 −1.08544 −0.542719 − 0.839915i \(-0.682605\pi\)
−0.542719 + 0.839915i \(0.682605\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) 18.0000 0.844782
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) −32.0000 −1.49690 −0.748448 − 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) −3.00000 −0.139876
\(461\) 35.0000 1.63011 0.815056 − 0.579382i \(-0.196706\pi\)
0.815056 + 0.579382i \(0.196706\pi\)
\(462\) 0 0
\(463\) 9.00000 0.418265 0.209133 − 0.977887i \(-0.432936\pi\)
0.209133 + 0.977887i \(0.432936\pi\)
\(464\) −9.00000 −0.417815
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) 15.0000 0.694117 0.347059 − 0.937843i \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) 0 0
\(469\) −2.00000 −0.0923514
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) 25.0000 1.14950
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 3.00000 0.137505
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) 31.0000 1.41643 0.708213 − 0.705999i \(-0.249502\pi\)
0.708213 + 0.705999i \(0.249502\pi\)
\(480\) 0 0
\(481\) 11.0000 0.501557
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 − 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 15.0000 0.679018
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 26.0000 1.17336 0.586682 − 0.809818i \(-0.300434\pi\)
0.586682 + 0.809818i \(0.300434\pi\)
\(492\) 0 0
\(493\) −27.0000 −1.21602
\(494\) −1.00000 −0.0449921
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −9.00000 −0.402492
\(501\) 0 0
\(502\) −7.00000 −0.312425
\(503\) −42.0000 −1.87269 −0.936344 − 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 0 0
\(505\) −4.00000 −0.177998
\(506\) −15.0000 −0.666831
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) −13.0000 −0.576215 −0.288107 − 0.957598i \(-0.593026\pi\)
−0.288107 + 0.957598i \(0.593026\pi\)
\(510\) 0 0
\(511\) 11.0000 0.486611
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) −15.0000 −0.660979
\(516\) 0 0
\(517\) −40.0000 −1.75920
\(518\) 11.0000 0.483312
\(519\) 0 0
\(520\) 1.00000 0.0438529
\(521\) 17.0000 0.744784 0.372392 − 0.928076i \(-0.378538\pi\)
0.372392 + 0.928076i \(0.378538\pi\)
\(522\) 0 0
\(523\) −8.00000 −0.349816 −0.174908 − 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) −13.0000 −0.567908
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −2.00000 −0.0868744
\(531\) 0 0
\(532\) −1.00000 −0.0433555
\(533\) 0 0
\(534\) 0 0
\(535\) −10.0000 −0.432338
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) 18.0000 0.776035
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 − 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 22.0000 0.944981
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) −9.00000 −0.385518
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 − 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) −19.0000 −0.811640
\(549\) 0 0
\(550\) −20.0000 −0.852803
\(551\) 9.00000 0.383413
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) 0 0
\(557\) 44.0000 1.86434 0.932170 − 0.362021i \(-0.117913\pi\)
0.932170 + 0.362021i \(0.117913\pi\)
\(558\) 0 0
\(559\) 5.00000 0.211477
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −26.0000 −1.09674
\(563\) 3.00000 0.126435 0.0632175 − 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) 46.0000 1.92842 0.964210 − 0.265139i \(-0.0854179\pi\)
0.964210 + 0.265139i \(0.0854179\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 − 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 5.00000 0.209061
\(573\) 0 0
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 − 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 8.00000 0.332756
\(579\) 0 0
\(580\) −9.00000 −0.373705
\(581\) 14.0000 0.580818
\(582\) 0 0
\(583\) −10.0000 −0.414158
\(584\) −11.0000 −0.455183
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) −14.0000 −0.577842 −0.288921 − 0.957353i \(-0.593296\pi\)
−0.288921 + 0.957353i \(0.593296\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) −11.0000 −0.452097
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) −3.00000 −0.122679
\(599\) −5.00000 −0.204294 −0.102147 − 0.994769i \(-0.532571\pi\)
−0.102147 + 0.994769i \(0.532571\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 − 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 5.00000 0.203785
\(603\) 0 0
\(604\) −23.0000 −0.935857
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) −43.0000 −1.74532 −0.872658 − 0.488332i \(-0.837606\pi\)
−0.872658 + 0.488332i \(0.837606\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 15.0000 0.607332
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) −9.00000 −0.363507 −0.181753 − 0.983344i \(-0.558177\pi\)
−0.181753 + 0.983344i \(0.558177\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 5.00000 0.201456
\(617\) −9.00000 −0.362326 −0.181163 − 0.983453i \(-0.557986\pi\)
−0.181163 + 0.983453i \(0.557986\pi\)
\(618\) 0 0
\(619\) −17.0000 −0.683288 −0.341644 − 0.939829i \(-0.610984\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) −10.0000 −0.400963
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) −3.00000 −0.119713
\(629\) −33.0000 −1.31580
\(630\) 0 0
\(631\) 37.0000 1.47295 0.736473 − 0.676467i \(-0.236490\pi\)
0.736473 + 0.676467i \(0.236490\pi\)
\(632\) −10.0000 −0.397779
\(633\) 0 0
\(634\) 24.0000 0.953162
\(635\) 2.00000 0.0793676
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) −45.0000 −1.78157
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 30.0000 1.18493 0.592464 − 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) −7.00000 −0.276053 −0.138027 − 0.990429i \(-0.544076\pi\)
−0.138027 + 0.990429i \(0.544076\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) 3.00000 0.118033
\(647\) −32.0000 −1.25805 −0.629025 − 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) −31.0000 −1.21312 −0.606562 − 0.795036i \(-0.707452\pi\)
−0.606562 + 0.795036i \(0.707452\pi\)
\(654\) 0 0
\(655\) −13.0000 −0.507952
\(656\) 0 0
\(657\) 0 0
\(658\) −8.00000 −0.311872
\(659\) −30.0000 −1.16863 −0.584317 − 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) 46.0000 1.78919 0.894596 − 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) −10.0000 −0.388661
\(663\) 0 0
\(664\) −14.0000 −0.543305
\(665\) −1.00000 −0.0387783
\(666\) 0 0
\(667\) 27.0000 1.04544
\(668\) 21.0000 0.812514
\(669\) 0 0
\(670\) 2.00000 0.0772667
\(671\) 75.0000 2.89534
\(672\) 0 0
\(673\) −33.0000 −1.27206 −0.636028 − 0.771666i \(-0.719424\pi\)
−0.636028 + 0.771666i \(0.719424\pi\)
\(674\) 5.00000 0.192593
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −34.0000 −1.30673 −0.653363 − 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) −3.00000 −0.115045
\(681\) 0 0
\(682\) 20.0000 0.765840
\(683\) 9.00000 0.344375 0.172188 − 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) 0 0
\(685\) −19.0000 −0.725953
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −5.00000 −0.190623
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −16.0000 −0.607352
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −14.0000 −0.529908
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) −2.00000 −0.0755390 −0.0377695 − 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 11.0000 0.414873
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) −4.00000 −0.150435
\(708\) 0 0
\(709\) −18.0000 −0.676004 −0.338002 − 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) −12.0000 −0.450352
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) 5.00000 0.186989
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −4.00000 −0.149279
\(719\) −34.0000 −1.26799 −0.633993 − 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 0 0
\(721\) −15.0000 −0.558629
\(722\) 18.0000 0.669891
\(723\) 0 0
\(724\) 14.0000 0.520306
\(725\) 36.0000 1.33701
\(726\) 0 0
\(727\) −35.0000 −1.29808 −0.649039 − 0.760755i \(-0.724829\pi\)
−0.649039 + 0.760755i \(0.724829\pi\)
\(728\) 1.00000 0.0370625
\(729\) 0 0
\(730\) −11.0000 −0.407128
\(731\) −15.0000 −0.554795
\(732\) 0 0
\(733\) 20.0000 0.738717 0.369358 − 0.929287i \(-0.379577\pi\)
0.369358 + 0.929287i \(0.379577\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) 10.0000 0.368355
\(738\) 0 0
\(739\) −2.00000 −0.0735712 −0.0367856 − 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) −11.0000 −0.404368
\(741\) 0 0
\(742\) −2.00000 −0.0734223
\(743\) 26.0000 0.953847 0.476924 − 0.878945i \(-0.341752\pi\)
0.476924 + 0.878945i \(0.341752\pi\)
\(744\) 0 0
\(745\) 4.00000 0.146549
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) −15.0000 −0.548454
\(749\) −10.0000 −0.365392
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 − 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) −9.00000 −0.327761
\(755\) −23.0000 −0.837056
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 − 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 1.00000 0.0362738
\(761\) −20.0000 −0.724999 −0.362500 − 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(762\) 0 0
\(763\) −9.00000 −0.325822
\(764\) −3.00000 −0.108536
\(765\) 0 0
\(766\) −29.0000 −1.04781
\(767\) 4.00000 0.144432
\(768\) 0 0
\(769\) 49.0000 1.76699 0.883493 − 0.468445i \(-0.155186\pi\)
0.883493 + 0.468445i \(0.155186\pi\)
\(770\) 5.00000 0.180187
\(771\) 0 0
\(772\) 12.0000 0.431889
\(773\) 3.00000 0.107903 0.0539513 − 0.998544i \(-0.482818\pi\)
0.0539513 + 0.998544i \(0.482818\pi\)
\(774\) 0 0
\(775\) −16.0000 −0.574737
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) −2.00000 −0.0717035
\(779\) 0 0
\(780\) 0 0
\(781\) −60.0000 −2.14697
\(782\) 9.00000 0.321839
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −3.00000 −0.107075
\(786\) 0 0
\(787\) 39.0000 1.39020 0.695100 − 0.718913i \(-0.255360\pi\)
0.695100 + 0.718913i \(0.255360\pi\)
\(788\) 12.0000 0.427482
\(789\) 0 0
\(790\) −10.0000 −0.355784
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) 15.0000 0.532666
\(794\) −6.00000 −0.212932
\(795\) 0 0
\(796\) 11.0000 0.389885
\(797\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) −55.0000 −1.94091
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) 4.00000 0.140894
\(807\) 0 0
\(808\) 4.00000 0.140720
\(809\) −54.0000 −1.89854 −0.949269 − 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) 35.0000 1.22902 0.614508 − 0.788911i \(-0.289355\pi\)
0.614508 + 0.788911i \(0.289355\pi\)
\(812\) −9.00000 −0.315838
\(813\) 0 0
\(814\) −55.0000 −1.92775
\(815\) 10.0000 0.350285
\(816\) 0 0
\(817\) 5.00000 0.174928
\(818\) −25.0000 −0.874105
\(819\) 0 0
\(820\) 0 0
\(821\) 32.0000 1.11681 0.558404 − 0.829569i \(-0.311414\pi\)
0.558404 + 0.829569i \(0.311414\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 15.0000 0.522550
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) 19.0000 0.660695 0.330347 − 0.943859i \(-0.392834\pi\)
0.330347 + 0.943859i \(0.392834\pi\)
\(828\) 0 0
\(829\) 25.0000 0.868286 0.434143 − 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) −14.0000 −0.485947
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) 21.0000 0.726735
\(836\) 5.00000 0.172929
\(837\) 0 0
\(838\) 7.00000 0.241811
\(839\) 28.0000 0.966667 0.483334 − 0.875436i \(-0.339426\pi\)
0.483334 + 0.875436i \(0.339426\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −30.0000 −1.03387
\(843\) 0 0
\(844\) 9.00000 0.309793
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) 12.0000 0.411597
\(851\) 33.0000 1.13123
\(852\) 0 0
\(853\) −28.0000 −0.958702 −0.479351 − 0.877623i \(-0.659128\pi\)
−0.479351 + 0.877623i \(0.659128\pi\)
\(854\) 15.0000 0.513289
\(855\) 0 0
\(856\) 10.0000 0.341793
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 − 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) −5.00000 −0.170499
\(861\) 0 0
\(862\) −2.00000 −0.0681203
\(863\) −38.0000 −1.29354 −0.646768 − 0.762687i \(-0.723880\pi\)
−0.646768 + 0.762687i \(0.723880\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) −28.0000 −0.951479
\(867\) 0 0
\(868\) 4.00000 0.135769
\(869\) −50.0000 −1.69613
\(870\) 0 0
\(871\) 2.00000 0.0677674
\(872\) 9.00000 0.304778
\(873\) 0 0
\(874\) −3.00000 −0.101477
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 − 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) −13.0000 −0.438729
\(879\) 0 0
\(880\) −5.00000 −0.168550
\(881\) −21.0000 −0.707508 −0.353754 − 0.935339i \(-0.615095\pi\)
−0.353754 + 0.935339i \(0.615095\pi\)
\(882\) 0 0
\(883\) 51.0000 1.71629 0.858143 − 0.513410i \(-0.171618\pi\)
0.858143 + 0.513410i \(0.171618\pi\)
\(884\) −3.00000 −0.100901
\(885\) 0 0
\(886\) −18.0000 −0.604722
\(887\) −48.0000 −1.61168 −0.805841 − 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 2.00000 0.0670778
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −14.0000 −0.468755
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 23.0000 0.767520
\(899\) −36.0000 −1.20067
\(900\) 0 0
\(901\) 6.00000 0.199889
\(902\) 0 0
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 − 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −18.0000 −0.597351
\(909\) 0 0
\(910\) 1.00000 0.0331497
\(911\) 19.0000 0.629498 0.314749 − 0.949175i \(-0.398080\pi\)
0.314749 + 0.949175i \(0.398080\pi\)
\(912\) 0 0
\(913\) −70.0000 −2.31666
\(914\) 32.0000 1.05847
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) −13.0000 −0.429298
\(918\) 0 0
\(919\) 38.0000 1.25350 0.626752 − 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) 3.00000 0.0989071
\(921\) 0 0
\(922\) −35.0000 −1.15266
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) 44.0000 1.44671
\(926\) −9.00000 −0.295758
\(927\) 0 0
\(928\) 9.00000 0.295439
\(929\) 8.00000 0.262471 0.131236 − 0.991351i \(-0.458106\pi\)
0.131236 + 0.991351i \(0.458106\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) −24.0000 −0.786146
\(933\) 0 0
\(934\) −15.0000 −0.490815
\(935\) −15.0000 −0.490552
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 − 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 2.00000 0.0653023
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) −18.0000 −0.586783 −0.293392 − 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −25.0000 −0.812820
\(947\) 39.0000 1.26733 0.633665 − 0.773608i \(-0.281550\pi\)
0.633665 + 0.773608i \(0.281550\pi\)
\(948\) 0 0
\(949\) −11.0000 −0.357075
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) −3.00000 −0.0972306
\(953\) −16.0000 −0.518291 −0.259145 − 0.965838i \(-0.583441\pi\)
−0.259145 + 0.965838i \(0.583441\pi\)
\(954\) 0 0
\(955\) −3.00000 −0.0970777
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −31.0000 −1.00156
\(959\) −19.0000 −0.613542
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −11.0000 −0.354654
\(963\) 0 0
\(964\) 18.0000 0.579741
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) −5.00000 −0.160789 −0.0803946 − 0.996763i \(-0.525618\pi\)
−0.0803946 + 0.996763i \(0.525618\pi\)
\(968\) −14.0000 −0.449977
\(969\) 0 0
\(970\) 14.0000 0.449513
\(971\) 8.00000 0.256732 0.128366 − 0.991727i \(-0.459027\pi\)
0.128366 + 0.991727i \(0.459027\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) −15.0000 −0.480138
\(977\) 51.0000 1.63163 0.815817 − 0.578310i \(-0.196287\pi\)
0.815817 + 0.578310i \(0.196287\pi\)
\(978\) 0 0
\(979\) 30.0000 0.958804
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) −26.0000 −0.829693
\(983\) −25.0000 −0.797376 −0.398688 − 0.917087i \(-0.630534\pi\)
−0.398688 + 0.917087i \(0.630534\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 27.0000 0.859855
\(987\) 0 0
\(988\) 1.00000 0.0318142
\(989\) 15.0000 0.476972
\(990\) 0 0
\(991\) −2.00000 −0.0635321 −0.0317660 − 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 11.0000 0.348723
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 4.00000 0.126618
\(999\) 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 1638.2.a.h.1.1 1
3.2 odd 2 546.2.a.f.1.1 ✓ 1
12.11 even 2 4368.2.a.e.1.1 1
21.20 even 2 3822.2.a.w.1.1 1
39.38 odd 2 7098.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.f.1.1 ✓ 1 3.2 odd 2
1638.2.a.h.1.1 1 1.1 even 1 trivial
3822.2.a.w.1.1 1 21.20 even 2
4368.2.a.e.1.1 1 12.11 even 2
7098.2.a.n.1.1 1 39.38 odd 2