Properties

Label 182.2.a.a.1.1
Level $182$
Weight $2$
Character 182.1
Self dual yes
Analytic conductor $1.453$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [182,2,Mod(1,182)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(182, 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("182.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 182 = 2 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 182.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: \(1.45327731679\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 182.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -4.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +2.00000 q^{18} +2.00000 q^{19} +4.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} -7.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -1.00000 q^{26} -5.00000 q^{27} -1.00000 q^{28} -8.00000 q^{29} -4.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} -4.00000 q^{35} -2.00000 q^{36} +7.00000 q^{37} -2.00000 q^{38} +1.00000 q^{39} -4.00000 q^{40} -7.00000 q^{41} +1.00000 q^{42} -8.00000 q^{43} -1.00000 q^{44} -8.00000 q^{45} +7.00000 q^{46} +3.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -11.0000 q^{50} +4.00000 q^{51} +1.00000 q^{52} +5.00000 q^{54} -4.00000 q^{55} +1.00000 q^{56} +2.00000 q^{57} +8.00000 q^{58} -6.00000 q^{59} +4.00000 q^{60} -13.0000 q^{61} -3.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +1.00000 q^{66} +7.00000 q^{67} +4.00000 q^{68} -7.00000 q^{69} +4.00000 q^{70} +4.00000 q^{71} +2.00000 q^{72} +9.00000 q^{73} -7.00000 q^{74} +11.0000 q^{75} +2.00000 q^{76} +1.00000 q^{77} -1.00000 q^{78} -13.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +7.00000 q^{82} -16.0000 q^{83} -1.00000 q^{84} +16.0000 q^{85} +8.00000 q^{86} -8.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} +8.00000 q^{90} -1.00000 q^{91} -7.00000 q^{92} +3.00000 q^{93} -3.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} +11.0000 q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) −4.00000 −1.26491
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 1.00000 0.267261
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 2.00000 0.471405
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 4.00000 0.894427
\(21\) −1.00000 −0.218218
\(22\) 1.00000 0.213201
\(23\) −7.00000 −1.45960 −0.729800 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) −1.00000 −0.204124
\(25\) 11.0000 2.20000
\(26\) −1.00000 −0.196116
\(27\) −5.00000 −0.962250
\(28\) −1.00000 −0.188982
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) −4.00000 −0.730297
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −4.00000 −0.685994
\(35\) −4.00000 −0.676123
\(36\) −2.00000 −0.333333
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) −2.00000 −0.324443
\(39\) 1.00000 0.160128
\(40\) −4.00000 −0.632456
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 1.00000 0.154303
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −1.00000 −0.150756
\(45\) −8.00000 −1.19257
\(46\) 7.00000 1.03209
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −11.0000 −1.55563
\(51\) 4.00000 0.560112
\(52\) 1.00000 0.138675
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 5.00000 0.680414
\(55\) −4.00000 −0.539360
\(56\) 1.00000 0.133631
\(57\) 2.00000 0.264906
\(58\) 8.00000 1.05045
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 4.00000 0.516398
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) −3.00000 −0.381000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 1.00000 0.123091
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) 4.00000 0.485071
\(69\) −7.00000 −0.842701
\(70\) 4.00000 0.478091
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 2.00000 0.235702
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) −7.00000 −0.813733
\(75\) 11.0000 1.27017
\(76\) 2.00000 0.229416
\(77\) 1.00000 0.113961
\(78\) −1.00000 −0.113228
\(79\) −13.0000 −1.46261 −0.731307 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) 7.00000 0.773021
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) −1.00000 −0.109109
\(85\) 16.0000 1.73544
\(86\) 8.00000 0.862662
\(87\) −8.00000 −0.857690
\(88\) 1.00000 0.106600
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 8.00000 0.843274
\(91\) −1.00000 −0.104828
\(92\) −7.00000 −0.729800
\(93\) 3.00000 0.311086
\(94\) −3.00000 −0.309426
\(95\) 8.00000 0.820783
\(96\) −1.00000 −0.102062
\(97\) 11.0000 1.11688 0.558440 0.829545i \(-0.311400\pi\)
0.558440 + 0.829545i \(0.311400\pi\)
\(98\) −1.00000 −0.101015
\(99\) 2.00000 0.201008
\(100\) 11.0000 1.10000
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) −4.00000 −0.396059
\(103\) 10.0000 0.985329 0.492665 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) −1.00000 −0.0980581
\(105\) −4.00000 −0.390360
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −5.00000 −0.481125
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 4.00000 0.381385
\(111\) 7.00000 0.664411
\(112\) −1.00000 −0.0944911
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) −2.00000 −0.187317
\(115\) −28.0000 −2.61101
\(116\) −8.00000 −0.742781
\(117\) −2.00000 −0.184900
\(118\) 6.00000 0.552345
\(119\) −4.00000 −0.366679
\(120\) −4.00000 −0.365148
\(121\) −10.0000 −0.909091
\(122\) 13.0000 1.17696
\(123\) −7.00000 −0.631169
\(124\) 3.00000 0.269408
\(125\) 24.0000 2.14663
\(126\) −2.00000 −0.178174
\(127\) 13.0000 1.15356 0.576782 0.816898i \(-0.304308\pi\)
0.576782 + 0.816898i \(0.304308\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.00000 −0.704361
\(130\) −4.00000 −0.350823
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −2.00000 −0.173422
\(134\) −7.00000 −0.604708
\(135\) −20.0000 −1.72133
\(136\) −4.00000 −0.342997
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 7.00000 0.595880
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −4.00000 −0.338062
\(141\) 3.00000 0.252646
\(142\) −4.00000 −0.335673
\(143\) −1.00000 −0.0836242
\(144\) −2.00000 −0.166667
\(145\) −32.0000 −2.65746
\(146\) −9.00000 −0.744845
\(147\) 1.00000 0.0824786
\(148\) 7.00000 0.575396
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) −11.0000 −0.898146
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −2.00000 −0.162221
\(153\) −8.00000 −0.646762
\(154\) −1.00000 −0.0805823
\(155\) 12.0000 0.963863
\(156\) 1.00000 0.0800641
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 13.0000 1.03422
\(159\) 0 0
\(160\) −4.00000 −0.316228
\(161\) 7.00000 0.551677
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −7.00000 −0.546608
\(165\) −4.00000 −0.311400
\(166\) 16.0000 1.24184
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.00000 0.0771517
\(169\) 1.00000 0.0769231
\(170\) −16.0000 −1.22714
\(171\) −4.00000 −0.305888
\(172\) −8.00000 −0.609994
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 8.00000 0.606478
\(175\) −11.0000 −0.831522
\(176\) −1.00000 −0.0753778
\(177\) −6.00000 −0.450988
\(178\) 6.00000 0.449719
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) −8.00000 −0.596285
\(181\) 13.0000 0.966282 0.483141 0.875542i \(-0.339496\pi\)
0.483141 + 0.875542i \(0.339496\pi\)
\(182\) 1.00000 0.0741249
\(183\) −13.0000 −0.960988
\(184\) 7.00000 0.516047
\(185\) 28.0000 2.05860
\(186\) −3.00000 −0.219971
\(187\) −4.00000 −0.292509
\(188\) 3.00000 0.218797
\(189\) 5.00000 0.363696
\(190\) −8.00000 −0.580381
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 1.00000 0.0721688
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −11.0000 −0.789754
\(195\) 4.00000 0.286446
\(196\) 1.00000 0.0714286
\(197\) 3.00000 0.213741 0.106871 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) −2.00000 −0.142134
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −11.0000 −0.777817
\(201\) 7.00000 0.493742
\(202\) −9.00000 −0.633238
\(203\) 8.00000 0.561490
\(204\) 4.00000 0.280056
\(205\) −28.0000 −1.95560
\(206\) −10.0000 −0.696733
\(207\) 14.0000 0.973067
\(208\) 1.00000 0.0693375
\(209\) −2.00000 −0.138343
\(210\) 4.00000 0.276026
\(211\) 22.0000 1.51454 0.757271 0.653101i \(-0.226532\pi\)
0.757271 + 0.653101i \(0.226532\pi\)
\(212\) 0 0
\(213\) 4.00000 0.274075
\(214\) −12.0000 −0.820303
\(215\) −32.0000 −2.18238
\(216\) 5.00000 0.340207
\(217\) −3.00000 −0.203653
\(218\) −14.0000 −0.948200
\(219\) 9.00000 0.608164
\(220\) −4.00000 −0.269680
\(221\) 4.00000 0.269069
\(222\) −7.00000 −0.469809
\(223\) 9.00000 0.602685 0.301342 0.953516i \(-0.402565\pi\)
0.301342 + 0.953516i \(0.402565\pi\)
\(224\) 1.00000 0.0668153
\(225\) −22.0000 −1.46667
\(226\) −1.00000 −0.0665190
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) 2.00000 0.132453
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 28.0000 1.84627
\(231\) 1.00000 0.0657952
\(232\) 8.00000 0.525226
\(233\) −13.0000 −0.851658 −0.425829 0.904804i \(-0.640018\pi\)
−0.425829 + 0.904804i \(0.640018\pi\)
\(234\) 2.00000 0.130744
\(235\) 12.0000 0.782794
\(236\) −6.00000 −0.390567
\(237\) −13.0000 −0.844441
\(238\) 4.00000 0.259281
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 4.00000 0.258199
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 10.0000 0.642824
\(243\) 16.0000 1.02640
\(244\) −13.0000 −0.832240
\(245\) 4.00000 0.255551
\(246\) 7.00000 0.446304
\(247\) 2.00000 0.127257
\(248\) −3.00000 −0.190500
\(249\) −16.0000 −1.01396
\(250\) −24.0000 −1.51789
\(251\) 17.0000 1.07303 0.536515 0.843891i \(-0.319740\pi\)
0.536515 + 0.843891i \(0.319740\pi\)
\(252\) 2.00000 0.125988
\(253\) 7.00000 0.440086
\(254\) −13.0000 −0.815693
\(255\) 16.0000 1.00196
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 8.00000 0.498058
\(259\) −7.00000 −0.434959
\(260\) 4.00000 0.248069
\(261\) 16.0000 0.990375
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 1.00000 0.0615457
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) −6.00000 −0.367194
\(268\) 7.00000 0.427593
\(269\) 3.00000 0.182913 0.0914566 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 20.0000 1.21716
\(271\) −3.00000 −0.182237 −0.0911185 0.995840i \(-0.529044\pi\)
−0.0911185 + 0.995840i \(0.529044\pi\)
\(272\) 4.00000 0.242536
\(273\) −1.00000 −0.0605228
\(274\) 14.0000 0.845771
\(275\) −11.0000 −0.663325
\(276\) −7.00000 −0.421350
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 20.0000 1.19952
\(279\) −6.00000 −0.359211
\(280\) 4.00000 0.239046
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) −3.00000 −0.178647
\(283\) 17.0000 1.01055 0.505273 0.862960i \(-0.331392\pi\)
0.505273 + 0.862960i \(0.331392\pi\)
\(284\) 4.00000 0.237356
\(285\) 8.00000 0.473879
\(286\) 1.00000 0.0591312
\(287\) 7.00000 0.413197
\(288\) 2.00000 0.117851
\(289\) −1.00000 −0.0588235
\(290\) 32.0000 1.87910
\(291\) 11.0000 0.644831
\(292\) 9.00000 0.526685
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −24.0000 −1.39733
\(296\) −7.00000 −0.406867
\(297\) 5.00000 0.290129
\(298\) −15.0000 −0.868927
\(299\) −7.00000 −0.404820
\(300\) 11.0000 0.635085
\(301\) 8.00000 0.461112
\(302\) 4.00000 0.230174
\(303\) 9.00000 0.517036
\(304\) 2.00000 0.114708
\(305\) −52.0000 −2.97751
\(306\) 8.00000 0.457330
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 1.00000 0.0569803
\(309\) 10.0000 0.568880
\(310\) −12.0000 −0.681554
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −7.00000 −0.395033
\(315\) 8.00000 0.450749
\(316\) −13.0000 −0.731307
\(317\) −13.0000 −0.730153 −0.365076 0.930978i \(-0.618957\pi\)
−0.365076 + 0.930978i \(0.618957\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 4.00000 0.223607
\(321\) 12.0000 0.669775
\(322\) −7.00000 −0.390095
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 11.0000 0.610170
\(326\) −12.0000 −0.664619
\(327\) 14.0000 0.774202
\(328\) 7.00000 0.386510
\(329\) −3.00000 −0.165395
\(330\) 4.00000 0.220193
\(331\) 15.0000 0.824475 0.412237 0.911077i \(-0.364747\pi\)
0.412237 + 0.911077i \(0.364747\pi\)
\(332\) −16.0000 −0.878114
\(333\) −14.0000 −0.767195
\(334\) 0 0
\(335\) 28.0000 1.52980
\(336\) −1.00000 −0.0545545
\(337\) 9.00000 0.490261 0.245131 0.969490i \(-0.421169\pi\)
0.245131 + 0.969490i \(0.421169\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 1.00000 0.0543125
\(340\) 16.0000 0.867722
\(341\) −3.00000 −0.162459
\(342\) 4.00000 0.216295
\(343\) −1.00000 −0.0539949
\(344\) 8.00000 0.431331
\(345\) −28.0000 −1.50747
\(346\) −14.0000 −0.752645
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) −8.00000 −0.428845
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 11.0000 0.587975
\(351\) −5.00000 −0.266880
\(352\) 1.00000 0.0533002
\(353\) −11.0000 −0.585471 −0.292735 0.956193i \(-0.594566\pi\)
−0.292735 + 0.956193i \(0.594566\pi\)
\(354\) 6.00000 0.318896
\(355\) 16.0000 0.849192
\(356\) −6.00000 −0.317999
\(357\) −4.00000 −0.211702
\(358\) 18.0000 0.951330
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 8.00000 0.421637
\(361\) −15.0000 −0.789474
\(362\) −13.0000 −0.683265
\(363\) −10.0000 −0.524864
\(364\) −1.00000 −0.0524142
\(365\) 36.0000 1.88433
\(366\) 13.0000 0.679521
\(367\) 12.0000 0.626395 0.313197 0.949688i \(-0.398600\pi\)
0.313197 + 0.949688i \(0.398600\pi\)
\(368\) −7.00000 −0.364900
\(369\) 14.0000 0.728811
\(370\) −28.0000 −1.45565
\(371\) 0 0
\(372\) 3.00000 0.155543
\(373\) −32.0000 −1.65690 −0.828449 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) 4.00000 0.206835
\(375\) 24.0000 1.23935
\(376\) −3.00000 −0.154713
\(377\) −8.00000 −0.412021
\(378\) −5.00000 −0.257172
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 8.00000 0.410391
\(381\) 13.0000 0.666010
\(382\) −24.0000 −1.22795
\(383\) 21.0000 1.07305 0.536525 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 4.00000 0.203859
\(386\) 4.00000 0.203595
\(387\) 16.0000 0.813326
\(388\) 11.0000 0.558440
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) −4.00000 −0.202548
\(391\) −28.0000 −1.41602
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −3.00000 −0.151138
\(395\) −52.0000 −2.61640
\(396\) 2.00000 0.100504
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) −16.0000 −0.802008
\(399\) −2.00000 −0.100125
\(400\) 11.0000 0.550000
\(401\) 8.00000 0.399501 0.199750 0.979847i \(-0.435987\pi\)
0.199750 + 0.979847i \(0.435987\pi\)
\(402\) −7.00000 −0.349128
\(403\) 3.00000 0.149441
\(404\) 9.00000 0.447767
\(405\) 4.00000 0.198762
\(406\) −8.00000 −0.397033
\(407\) −7.00000 −0.346977
\(408\) −4.00000 −0.198030
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 28.0000 1.38282
\(411\) −14.0000 −0.690569
\(412\) 10.0000 0.492665
\(413\) 6.00000 0.295241
\(414\) −14.0000 −0.688062
\(415\) −64.0000 −3.14164
\(416\) −1.00000 −0.0490290
\(417\) −20.0000 −0.979404
\(418\) 2.00000 0.0978232
\(419\) −19.0000 −0.928211 −0.464105 0.885780i \(-0.653624\pi\)
−0.464105 + 0.885780i \(0.653624\pi\)
\(420\) −4.00000 −0.195180
\(421\) −11.0000 −0.536107 −0.268054 0.963404i \(-0.586380\pi\)
−0.268054 + 0.963404i \(0.586380\pi\)
\(422\) −22.0000 −1.07094
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 44.0000 2.13431
\(426\) −4.00000 −0.193801
\(427\) 13.0000 0.629114
\(428\) 12.0000 0.580042
\(429\) −1.00000 −0.0482805
\(430\) 32.0000 1.54318
\(431\) 2.00000 0.0963366 0.0481683 0.998839i \(-0.484662\pi\)
0.0481683 + 0.998839i \(0.484662\pi\)
\(432\) −5.00000 −0.240563
\(433\) −18.0000 −0.865025 −0.432512 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) 3.00000 0.144005
\(435\) −32.0000 −1.53428
\(436\) 14.0000 0.670478
\(437\) −14.0000 −0.669711
\(438\) −9.00000 −0.430037
\(439\) −2.00000 −0.0954548 −0.0477274 0.998860i \(-0.515198\pi\)
−0.0477274 + 0.998860i \(0.515198\pi\)
\(440\) 4.00000 0.190693
\(441\) −2.00000 −0.0952381
\(442\) −4.00000 −0.190261
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 7.00000 0.332205
\(445\) −24.0000 −1.13771
\(446\) −9.00000 −0.426162
\(447\) 15.0000 0.709476
\(448\) −1.00000 −0.0472456
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 22.0000 1.03709
\(451\) 7.00000 0.329617
\(452\) 1.00000 0.0470360
\(453\) −4.00000 −0.187936
\(454\) −28.0000 −1.31411
\(455\) −4.00000 −0.187523
\(456\) −2.00000 −0.0936586
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 20.0000 0.934539
\(459\) −20.0000 −0.933520
\(460\) −28.0000 −1.30551
\(461\) 16.0000 0.745194 0.372597 0.927993i \(-0.378467\pi\)
0.372597 + 0.927993i \(0.378467\pi\)
\(462\) −1.00000 −0.0465242
\(463\) −12.0000 −0.557687 −0.278844 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) −8.00000 −0.371391
\(465\) 12.0000 0.556487
\(466\) 13.0000 0.602213
\(467\) −20.0000 −0.925490 −0.462745 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −7.00000 −0.323230
\(470\) −12.0000 −0.553519
\(471\) 7.00000 0.322543
\(472\) 6.00000 0.276172
\(473\) 8.00000 0.367840
\(474\) 13.0000 0.597110
\(475\) 22.0000 1.00943
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) 6.00000 0.274434
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −4.00000 −0.182574
\(481\) 7.00000 0.319173
\(482\) 10.0000 0.455488
\(483\) 7.00000 0.318511
\(484\) −10.0000 −0.454545
\(485\) 44.0000 1.99794
\(486\) −16.0000 −0.725775
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) 13.0000 0.588482
\(489\) 12.0000 0.542659
\(490\) −4.00000 −0.180702
\(491\) 40.0000 1.80517 0.902587 0.430507i \(-0.141665\pi\)
0.902587 + 0.430507i \(0.141665\pi\)
\(492\) −7.00000 −0.315584
\(493\) −32.0000 −1.44121
\(494\) −2.00000 −0.0899843
\(495\) 8.00000 0.359573
\(496\) 3.00000 0.134704
\(497\) −4.00000 −0.179425
\(498\) 16.0000 0.716977
\(499\) 37.0000 1.65635 0.828174 0.560471i \(-0.189380\pi\)
0.828174 + 0.560471i \(0.189380\pi\)
\(500\) 24.0000 1.07331
\(501\) 0 0
\(502\) −17.0000 −0.758747
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 36.0000 1.60198
\(506\) −7.00000 −0.311188
\(507\) 1.00000 0.0444116
\(508\) 13.0000 0.576782
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) −16.0000 −0.708492
\(511\) −9.00000 −0.398137
\(512\) −1.00000 −0.0441942
\(513\) −10.0000 −0.441511
\(514\) 12.0000 0.529297
\(515\) 40.0000 1.76261
\(516\) −8.00000 −0.352180
\(517\) −3.00000 −0.131940
\(518\) 7.00000 0.307562
\(519\) 14.0000 0.614532
\(520\) −4.00000 −0.175412
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −16.0000 −0.700301
\(523\) −15.0000 −0.655904 −0.327952 0.944694i \(-0.606358\pi\)
−0.327952 + 0.944694i \(0.606358\pi\)
\(524\) 0 0
\(525\) −11.0000 −0.480079
\(526\) −16.0000 −0.697633
\(527\) 12.0000 0.522728
\(528\) −1.00000 −0.0435194
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) −2.00000 −0.0867110
\(533\) −7.00000 −0.303204
\(534\) 6.00000 0.259645
\(535\) 48.0000 2.07522
\(536\) −7.00000 −0.302354
\(537\) −18.0000 −0.776757
\(538\) −3.00000 −0.129339
\(539\) −1.00000 −0.0430730
\(540\) −20.0000 −0.860663
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 3.00000 0.128861
\(543\) 13.0000 0.557883
\(544\) −4.00000 −0.171499
\(545\) 56.0000 2.39878
\(546\) 1.00000 0.0427960
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −14.0000 −0.598050
\(549\) 26.0000 1.10965
\(550\) 11.0000 0.469042
\(551\) −16.0000 −0.681623
\(552\) 7.00000 0.297940
\(553\) 13.0000 0.552816
\(554\) 14.0000 0.594803
\(555\) 28.0000 1.18853
\(556\) −20.0000 −0.848189
\(557\) 31.0000 1.31351 0.656756 0.754103i \(-0.271928\pi\)
0.656756 + 0.754103i \(0.271928\pi\)
\(558\) 6.00000 0.254000
\(559\) −8.00000 −0.338364
\(560\) −4.00000 −0.169031
\(561\) −4.00000 −0.168880
\(562\) 20.0000 0.843649
\(563\) −31.0000 −1.30649 −0.653247 0.757145i \(-0.726594\pi\)
−0.653247 + 0.757145i \(0.726594\pi\)
\(564\) 3.00000 0.126323
\(565\) 4.00000 0.168281
\(566\) −17.0000 −0.714563
\(567\) −1.00000 −0.0419961
\(568\) −4.00000 −0.167836
\(569\) 29.0000 1.21574 0.607872 0.794035i \(-0.292024\pi\)
0.607872 + 0.794035i \(0.292024\pi\)
\(570\) −8.00000 −0.335083
\(571\) 26.0000 1.08807 0.544033 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) −1.00000 −0.0418121
\(573\) 24.0000 1.00261
\(574\) −7.00000 −0.292174
\(575\) −77.0000 −3.21112
\(576\) −2.00000 −0.0833333
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 1.00000 0.0415945
\(579\) −4.00000 −0.166234
\(580\) −32.0000 −1.32873
\(581\) 16.0000 0.663792
\(582\) −11.0000 −0.455965
\(583\) 0 0
\(584\) −9.00000 −0.372423
\(585\) −8.00000 −0.330759
\(586\) −14.0000 −0.578335
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 1.00000 0.0412393
\(589\) 6.00000 0.247226
\(590\) 24.0000 0.988064
\(591\) 3.00000 0.123404
\(592\) 7.00000 0.287698
\(593\) −10.0000 −0.410651 −0.205325 0.978694i \(-0.565825\pi\)
−0.205325 + 0.978694i \(0.565825\pi\)
\(594\) −5.00000 −0.205152
\(595\) −16.0000 −0.655936
\(596\) 15.0000 0.614424
\(597\) 16.0000 0.654836
\(598\) 7.00000 0.286251
\(599\) 27.0000 1.10319 0.551595 0.834112i \(-0.314019\pi\)
0.551595 + 0.834112i \(0.314019\pi\)
\(600\) −11.0000 −0.449073
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) −8.00000 −0.326056
\(603\) −14.0000 −0.570124
\(604\) −4.00000 −0.162758
\(605\) −40.0000 −1.62623
\(606\) −9.00000 −0.365600
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 8.00000 0.324176
\(610\) 52.0000 2.10542
\(611\) 3.00000 0.121367
\(612\) −8.00000 −0.323381
\(613\) 17.0000 0.686624 0.343312 0.939222i \(-0.388451\pi\)
0.343312 + 0.939222i \(0.388451\pi\)
\(614\) 12.0000 0.484281
\(615\) −28.0000 −1.12907
\(616\) −1.00000 −0.0402911
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) −10.0000 −0.402259
\(619\) 14.0000 0.562708 0.281354 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) 12.0000 0.481932
\(621\) 35.0000 1.40450
\(622\) 30.0000 1.20289
\(623\) 6.00000 0.240385
\(624\) 1.00000 0.0400320
\(625\) 41.0000 1.64000
\(626\) −6.00000 −0.239808
\(627\) −2.00000 −0.0798723
\(628\) 7.00000 0.279330
\(629\) 28.0000 1.11643
\(630\) −8.00000 −0.318728
\(631\) 22.0000 0.875806 0.437903 0.899022i \(-0.355721\pi\)
0.437903 + 0.899022i \(0.355721\pi\)
\(632\) 13.0000 0.517112
\(633\) 22.0000 0.874421
\(634\) 13.0000 0.516296
\(635\) 52.0000 2.06356
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) −8.00000 −0.316723
\(639\) −8.00000 −0.316475
\(640\) −4.00000 −0.158114
\(641\) −25.0000 −0.987441 −0.493720 0.869621i \(-0.664363\pi\)
−0.493720 + 0.869621i \(0.664363\pi\)
\(642\) −12.0000 −0.473602
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 7.00000 0.275839
\(645\) −32.0000 −1.26000
\(646\) −8.00000 −0.314756
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 6.00000 0.235521
\(650\) −11.0000 −0.431455
\(651\) −3.00000 −0.117579
\(652\) 12.0000 0.469956
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) −14.0000 −0.547443
\(655\) 0 0
\(656\) −7.00000 −0.273304
\(657\) −18.0000 −0.702247
\(658\) 3.00000 0.116952
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) −4.00000 −0.155700
\(661\) −8.00000 −0.311164 −0.155582 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) −15.0000 −0.582992
\(663\) 4.00000 0.155347
\(664\) 16.0000 0.620920
\(665\) −8.00000 −0.310227
\(666\) 14.0000 0.542489
\(667\) 56.0000 2.16833
\(668\) 0 0
\(669\) 9.00000 0.347960
\(670\) −28.0000 −1.08173
\(671\) 13.0000 0.501859
\(672\) 1.00000 0.0385758
\(673\) −33.0000 −1.27206 −0.636028 0.771666i \(-0.719424\pi\)
−0.636028 + 0.771666i \(0.719424\pi\)
\(674\) −9.00000 −0.346667
\(675\) −55.0000 −2.11695
\(676\) 1.00000 0.0384615
\(677\) 13.0000 0.499631 0.249815 0.968294i \(-0.419630\pi\)
0.249815 + 0.968294i \(0.419630\pi\)
\(678\) −1.00000 −0.0384048
\(679\) −11.0000 −0.422141
\(680\) −16.0000 −0.613572
\(681\) 28.0000 1.07296
\(682\) 3.00000 0.114876
\(683\) 31.0000 1.18618 0.593091 0.805135i \(-0.297907\pi\)
0.593091 + 0.805135i \(0.297907\pi\)
\(684\) −4.00000 −0.152944
\(685\) −56.0000 −2.13965
\(686\) 1.00000 0.0381802
\(687\) −20.0000 −0.763048
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) 28.0000 1.06594
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 14.0000 0.532200
\(693\) −2.00000 −0.0759737
\(694\) 32.0000 1.21470
\(695\) −80.0000 −3.03457
\(696\) 8.00000 0.303239
\(697\) −28.0000 −1.06058
\(698\) −2.00000 −0.0757011
\(699\) −13.0000 −0.491705
\(700\) −11.0000 −0.415761
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 5.00000 0.188713
\(703\) 14.0000 0.528020
\(704\) −1.00000 −0.0376889
\(705\) 12.0000 0.451946
\(706\) 11.0000 0.413990
\(707\) −9.00000 −0.338480
\(708\) −6.00000 −0.225494
\(709\) −51.0000 −1.91535 −0.957673 0.287860i \(-0.907056\pi\)
−0.957673 + 0.287860i \(0.907056\pi\)
\(710\) −16.0000 −0.600469
\(711\) 26.0000 0.975076
\(712\) 6.00000 0.224860
\(713\) −21.0000 −0.786456
\(714\) 4.00000 0.149696
\(715\) −4.00000 −0.149592
\(716\) −18.0000 −0.672692
\(717\) −6.00000 −0.224074
\(718\) 0 0
\(719\) −22.0000 −0.820462 −0.410231 0.911982i \(-0.634552\pi\)
−0.410231 + 0.911982i \(0.634552\pi\)
\(720\) −8.00000 −0.298142
\(721\) −10.0000 −0.372419
\(722\) 15.0000 0.558242
\(723\) −10.0000 −0.371904
\(724\) 13.0000 0.483141
\(725\) −88.0000 −3.26824
\(726\) 10.0000 0.371135
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 1.00000 0.0370625
\(729\) 13.0000 0.481481
\(730\) −36.0000 −1.33242
\(731\) −32.0000 −1.18356
\(732\) −13.0000 −0.480494
\(733\) −12.0000 −0.443230 −0.221615 0.975134i \(-0.571133\pi\)
−0.221615 + 0.975134i \(0.571133\pi\)
\(734\) −12.0000 −0.442928
\(735\) 4.00000 0.147542
\(736\) 7.00000 0.258023
\(737\) −7.00000 −0.257848
\(738\) −14.0000 −0.515347
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 28.0000 1.02930
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) −44.0000 −1.61420 −0.807102 0.590412i \(-0.798965\pi\)
−0.807102 + 0.590412i \(0.798965\pi\)
\(744\) −3.00000 −0.109985
\(745\) 60.0000 2.19823
\(746\) 32.0000 1.17160
\(747\) 32.0000 1.17082
\(748\) −4.00000 −0.146254
\(749\) −12.0000 −0.438470
\(750\) −24.0000 −0.876356
\(751\) 37.0000 1.35015 0.675075 0.737749i \(-0.264111\pi\)
0.675075 + 0.737749i \(0.264111\pi\)
\(752\) 3.00000 0.109399
\(753\) 17.0000 0.619514
\(754\) 8.00000 0.291343
\(755\) −16.0000 −0.582300
\(756\) 5.00000 0.181848
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 8.00000 0.290573
\(759\) 7.00000 0.254084
\(760\) −8.00000 −0.290191
\(761\) 1.00000 0.0362500 0.0181250 0.999836i \(-0.494230\pi\)
0.0181250 + 0.999836i \(0.494230\pi\)
\(762\) −13.0000 −0.470940
\(763\) −14.0000 −0.506834
\(764\) 24.0000 0.868290
\(765\) −32.0000 −1.15696
\(766\) −21.0000 −0.758761
\(767\) −6.00000 −0.216647
\(768\) 1.00000 0.0360844
\(769\) 25.0000 0.901523 0.450762 0.892644i \(-0.351152\pi\)
0.450762 + 0.892644i \(0.351152\pi\)
\(770\) −4.00000 −0.144150
\(771\) −12.0000 −0.432169
\(772\) −4.00000 −0.143963
\(773\) 38.0000 1.36677 0.683383 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) −16.0000 −0.575108
\(775\) 33.0000 1.18539
\(776\) −11.0000 −0.394877
\(777\) −7.00000 −0.251124
\(778\) −8.00000 −0.286814
\(779\) −14.0000 −0.501602
\(780\) 4.00000 0.143223
\(781\) −4.00000 −0.143131
\(782\) 28.0000 1.00128
\(783\) 40.0000 1.42948
\(784\) 1.00000 0.0357143
\(785\) 28.0000 0.999363
\(786\) 0 0
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) 3.00000 0.106871
\(789\) 16.0000 0.569615
\(790\) 52.0000 1.85008
\(791\) −1.00000 −0.0355559
\(792\) −2.00000 −0.0710669
\(793\) −13.0000 −0.461644
\(794\) −8.00000 −0.283909
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) 21.0000 0.743858 0.371929 0.928261i \(-0.378696\pi\)
0.371929 + 0.928261i \(0.378696\pi\)
\(798\) 2.00000 0.0707992
\(799\) 12.0000 0.424529
\(800\) −11.0000 −0.388909
\(801\) 12.0000 0.423999
\(802\) −8.00000 −0.282490
\(803\) −9.00000 −0.317603
\(804\) 7.00000 0.246871
\(805\) 28.0000 0.986870
\(806\) −3.00000 −0.105670
\(807\) 3.00000 0.105605
\(808\) −9.00000 −0.316619
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) −4.00000 −0.140546
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 8.00000 0.280745
\(813\) −3.00000 −0.105215
\(814\) 7.00000 0.245350
\(815\) 48.0000 1.68137
\(816\) 4.00000 0.140028
\(817\) −16.0000 −0.559769
\(818\) 10.0000 0.349642
\(819\) 2.00000 0.0698857
\(820\) −28.0000 −0.977802
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 14.0000 0.488306
\(823\) −35.0000 −1.22002 −0.610012 0.792392i \(-0.708835\pi\)
−0.610012 + 0.792392i \(0.708835\pi\)
\(824\) −10.0000 −0.348367
\(825\) −11.0000 −0.382971
\(826\) −6.00000 −0.208767
\(827\) 4.00000 0.139094 0.0695468 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 14.0000 0.486534
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 64.0000 2.22147
\(831\) −14.0000 −0.485655
\(832\) 1.00000 0.0346688
\(833\) 4.00000 0.138592
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) −15.0000 −0.518476
\(838\) 19.0000 0.656344
\(839\) 21.0000 0.725001 0.362500 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(840\) 4.00000 0.138013
\(841\) 35.0000 1.20690
\(842\) 11.0000 0.379085
\(843\) −20.0000 −0.688837
\(844\) 22.0000 0.757271
\(845\) 4.00000 0.137604
\(846\) 6.00000 0.206284
\(847\) 10.0000 0.343604
\(848\) 0 0
\(849\) 17.0000 0.583438
\(850\) −44.0000 −1.50919
\(851\) −49.0000 −1.67970
\(852\) 4.00000 0.137038
\(853\) −44.0000 −1.50653 −0.753266 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) −13.0000 −0.444851
\(855\) −16.0000 −0.547188
\(856\) −12.0000 −0.410152
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 1.00000 0.0341394
\(859\) 31.0000 1.05771 0.528853 0.848713i \(-0.322622\pi\)
0.528853 + 0.848713i \(0.322622\pi\)
\(860\) −32.0000 −1.09119
\(861\) 7.00000 0.238559
\(862\) −2.00000 −0.0681203
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 5.00000 0.170103
\(865\) 56.0000 1.90406
\(866\) 18.0000 0.611665
\(867\) −1.00000 −0.0339618
\(868\) −3.00000 −0.101827
\(869\) 13.0000 0.440995
\(870\) 32.0000 1.08490
\(871\) 7.00000 0.237186
\(872\) −14.0000 −0.474100
\(873\) −22.0000 −0.744587
\(874\) 14.0000 0.473557
\(875\) −24.0000 −0.811348
\(876\) 9.00000 0.304082
\(877\) 29.0000 0.979260 0.489630 0.871930i \(-0.337132\pi\)
0.489630 + 0.871930i \(0.337132\pi\)
\(878\) 2.00000 0.0674967
\(879\) 14.0000 0.472208
\(880\) −4.00000 −0.134840
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 2.00000 0.0673435
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 4.00000 0.134535
\(885\) −24.0000 −0.806751
\(886\) 0 0
\(887\) 14.0000 0.470074 0.235037 0.971986i \(-0.424479\pi\)
0.235037 + 0.971986i \(0.424479\pi\)
\(888\) −7.00000 −0.234905
\(889\) −13.0000 −0.436006
\(890\) 24.0000 0.804482
\(891\) −1.00000 −0.0335013
\(892\) 9.00000 0.301342
\(893\) 6.00000 0.200782
\(894\) −15.0000 −0.501675
\(895\) −72.0000 −2.40669
\(896\) 1.00000 0.0334077
\(897\) −7.00000 −0.233723
\(898\) −6.00000 −0.200223
\(899\) −24.0000 −0.800445
\(900\) −22.0000 −0.733333
\(901\) 0 0
\(902\) −7.00000 −0.233075
\(903\) 8.00000 0.266223
\(904\) −1.00000 −0.0332595
\(905\) 52.0000 1.72854
\(906\) 4.00000 0.132891
\(907\) 54.0000 1.79304 0.896520 0.443003i \(-0.146087\pi\)
0.896520 + 0.443003i \(0.146087\pi\)
\(908\) 28.0000 0.929213
\(909\) −18.0000 −0.597022
\(910\) 4.00000 0.132599
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 2.00000 0.0662266
\(913\) 16.0000 0.529523
\(914\) −18.0000 −0.595387
\(915\) −52.0000 −1.71907
\(916\) −20.0000 −0.660819
\(917\) 0 0
\(918\) 20.0000 0.660098
\(919\) −37.0000 −1.22052 −0.610259 0.792202i \(-0.708935\pi\)
−0.610259 + 0.792202i \(0.708935\pi\)
\(920\) 28.0000 0.923133
\(921\) −12.0000 −0.395413
\(922\) −16.0000 −0.526932
\(923\) 4.00000 0.131662
\(924\) 1.00000 0.0328976
\(925\) 77.0000 2.53174
\(926\) 12.0000 0.394344
\(927\) −20.0000 −0.656886
\(928\) 8.00000 0.262613
\(929\) −27.0000 −0.885841 −0.442921 0.896561i \(-0.646058\pi\)
−0.442921 + 0.896561i \(0.646058\pi\)
\(930\) −12.0000 −0.393496
\(931\) 2.00000 0.0655474
\(932\) −13.0000 −0.425829
\(933\) −30.0000 −0.982156
\(934\) 20.0000 0.654420
\(935\) −16.0000 −0.523256
\(936\) 2.00000 0.0653720
\(937\) −48.0000 −1.56809 −0.784046 0.620703i \(-0.786847\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) 7.00000 0.228558
\(939\) 6.00000 0.195803
\(940\) 12.0000 0.391397
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) −7.00000 −0.228072
\(943\) 49.0000 1.59566
\(944\) −6.00000 −0.195283
\(945\) 20.0000 0.650600
\(946\) −8.00000 −0.260102
\(947\) −8.00000 −0.259965 −0.129983 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) −13.0000 −0.422220
\(949\) 9.00000 0.292152
\(950\) −22.0000 −0.713774
\(951\) −13.0000 −0.421554
\(952\) 4.00000 0.129641
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) 96.0000 3.10649
\(956\) −6.00000 −0.194054
\(957\) 8.00000 0.258603
\(958\) 0 0
\(959\) 14.0000 0.452084
\(960\) 4.00000 0.129099
\(961\) −22.0000 −0.709677
\(962\) −7.00000 −0.225689
\(963\) −24.0000 −0.773389
\(964\) −10.0000 −0.322078
\(965\) −16.0000 −0.515058
\(966\) −7.00000 −0.225221
\(967\) 18.0000 0.578841 0.289420 0.957202i \(-0.406537\pi\)
0.289420 + 0.957202i \(0.406537\pi\)
\(968\) 10.0000 0.321412
\(969\) 8.00000 0.256997
\(970\) −44.0000 −1.41275
\(971\) −45.0000 −1.44412 −0.722059 0.691831i \(-0.756804\pi\)
−0.722059 + 0.691831i \(0.756804\pi\)
\(972\) 16.0000 0.513200
\(973\) 20.0000 0.641171
\(974\) −26.0000 −0.833094
\(975\) 11.0000 0.352282
\(976\) −13.0000 −0.416120
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) −12.0000 −0.383718
\(979\) 6.00000 0.191761
\(980\) 4.00000 0.127775
\(981\) −28.0000 −0.893971
\(982\) −40.0000 −1.27645
\(983\) 32.0000 1.02064 0.510321 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) 7.00000 0.223152
\(985\) 12.0000 0.382352
\(986\) 32.0000 1.01909
\(987\) −3.00000 −0.0954911
\(988\) 2.00000 0.0636285
\(989\) 56.0000 1.78070
\(990\) −8.00000 −0.254257
\(991\) −33.0000 −1.04828 −0.524140 0.851632i \(-0.675613\pi\)
−0.524140 + 0.851632i \(0.675613\pi\)
\(992\) −3.00000 −0.0952501
\(993\) 15.0000 0.476011
\(994\) 4.00000 0.126872
\(995\) 64.0000 2.02894
\(996\) −16.0000 −0.506979
\(997\) 31.0000 0.981780 0.490890 0.871222i \(-0.336672\pi\)
0.490890 + 0.871222i \(0.336672\pi\)
\(998\) −37.0000 −1.17121
\(999\) −35.0000 −1.10735
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 182.2.a.a.1.1 1
3.2 odd 2 1638.2.a.k.1.1 1
4.3 odd 2 1456.2.a.e.1.1 1
5.4 even 2 4550.2.a.t.1.1 1
7.2 even 3 1274.2.f.n.1145.1 2
7.3 odd 6 1274.2.f.t.79.1 2
7.4 even 3 1274.2.f.n.79.1 2
7.5 odd 6 1274.2.f.t.1145.1 2
7.6 odd 2 1274.2.a.b.1.1 1
8.3 odd 2 5824.2.a.w.1.1 1
8.5 even 2 5824.2.a.g.1.1 1
13.5 odd 4 2366.2.d.g.337.2 2
13.8 odd 4 2366.2.d.g.337.1 2
13.12 even 2 2366.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.a.a.1.1 1 1.1 even 1 trivial
1274.2.a.b.1.1 1 7.6 odd 2
1274.2.f.n.79.1 2 7.4 even 3
1274.2.f.n.1145.1 2 7.2 even 3
1274.2.f.t.79.1 2 7.3 odd 6
1274.2.f.t.1145.1 2 7.5 odd 6
1456.2.a.e.1.1 1 4.3 odd 2
1638.2.a.k.1.1 1 3.2 odd 2
2366.2.a.m.1.1 1 13.12 even 2
2366.2.d.g.337.1 2 13.8 odd 4
2366.2.d.g.337.2 2 13.5 odd 4
4550.2.a.t.1.1 1 5.4 even 2
5824.2.a.g.1.1 1 8.5 even 2
5824.2.a.w.1.1 1 8.3 odd 2