Properties

Label 418.2.a.a.1.1
Level $418$
Weight $2$
Character 418.1
Self dual yes
Analytic conductor $3.338$
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] = [418,2,Mod(1,418)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(418, 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("418.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.33774680449\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 418.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -3.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -7.00000 q^{17} -2.00000 q^{18} +1.00000 q^{19} -2.00000 q^{20} +3.00000 q^{21} -1.00000 q^{22} -5.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} +5.00000 q^{27} -3.00000 q^{28} +1.00000 q^{29} +2.00000 q^{30} +10.0000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -7.00000 q^{34} +6.00000 q^{35} -2.00000 q^{36} -6.00000 q^{37} +1.00000 q^{38} -1.00000 q^{39} -2.00000 q^{40} +6.00000 q^{41} +3.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +4.00000 q^{45} -5.00000 q^{46} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +7.00000 q^{51} +1.00000 q^{52} -1.00000 q^{53} +5.00000 q^{54} +2.00000 q^{55} -3.00000 q^{56} -1.00000 q^{57} +1.00000 q^{58} +3.00000 q^{59} +2.00000 q^{60} -12.0000 q^{61} +10.0000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} +3.00000 q^{67} -7.00000 q^{68} +5.00000 q^{69} +6.00000 q^{70} -10.0000 q^{71} -2.00000 q^{72} +3.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} +1.00000 q^{76} +3.00000 q^{77} -1.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +8.00000 q^{83} +3.00000 q^{84} +14.0000 q^{85} -4.00000 q^{86} -1.00000 q^{87} -1.00000 q^{88} -8.00000 q^{89} +4.00000 q^{90} -3.00000 q^{91} -5.00000 q^{92} -10.0000 q^{93} -2.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} +2.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −3.00000 −0.801784
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) −2.00000 −0.471405
\(19\) 1.00000 0.229416
\(20\) −2.00000 −0.447214
\(21\) 3.00000 0.654654
\(22\) −1.00000 −0.213201
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) 1.00000 0.196116
\(27\) 5.00000 0.962250
\(28\) −3.00000 −0.566947
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 2.00000 0.365148
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) −7.00000 −1.20049
\(35\) 6.00000 1.01419
\(36\) −2.00000 −0.333333
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 1.00000 0.162221
\(39\) −1.00000 −0.160128
\(40\) −2.00000 −0.316228
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 3.00000 0.462910
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 4.00000 0.596285
\(46\) −5.00000 −0.737210
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) −1.00000 −0.141421
\(51\) 7.00000 0.980196
\(52\) 1.00000 0.138675
\(53\) −1.00000 −0.137361 −0.0686803 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) 5.00000 0.680414
\(55\) 2.00000 0.269680
\(56\) −3.00000 −0.400892
\(57\) −1.00000 −0.132453
\(58\) 1.00000 0.131306
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 2.00000 0.258199
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 10.0000 1.27000
\(63\) 6.00000 0.755929
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 1.00000 0.123091
\(67\) 3.00000 0.366508 0.183254 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) −7.00000 −0.848875
\(69\) 5.00000 0.601929
\(70\) 6.00000 0.717137
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) −2.00000 −0.235702
\(73\) 3.00000 0.351123 0.175562 0.984468i \(-0.443826\pi\)
0.175562 + 0.984468i \(0.443826\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) 1.00000 0.114708
\(77\) 3.00000 0.341882
\(78\) −1.00000 −0.113228
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 3.00000 0.327327
\(85\) 14.0000 1.51851
\(86\) −4.00000 −0.431331
\(87\) −1.00000 −0.107211
\(88\) −1.00000 −0.106600
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 4.00000 0.421637
\(91\) −3.00000 −0.314485
\(92\) −5.00000 −0.521286
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 2.00000 0.202031
\(99\) 2.00000 0.201008
\(100\) −1.00000 −0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 7.00000 0.693103
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 1.00000 0.0980581
\(105\) −6.00000 −0.585540
\(106\) −1.00000 −0.0971286
\(107\) −13.0000 −1.25676 −0.628379 0.777908i \(-0.716281\pi\)
−0.628379 + 0.777908i \(0.716281\pi\)
\(108\) 5.00000 0.481125
\(109\) −13.0000 −1.24517 −0.622587 0.782551i \(-0.713918\pi\)
−0.622587 + 0.782551i \(0.713918\pi\)
\(110\) 2.00000 0.190693
\(111\) 6.00000 0.569495
\(112\) −3.00000 −0.283473
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 10.0000 0.932505
\(116\) 1.00000 0.0928477
\(117\) −2.00000 −0.184900
\(118\) 3.00000 0.276172
\(119\) 21.0000 1.92507
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) −12.0000 −1.08643
\(123\) −6.00000 −0.541002
\(124\) 10.0000 0.898027
\(125\) 12.0000 1.07331
\(126\) 6.00000 0.534522
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) −2.00000 −0.175412
\(131\) −22.0000 −1.92215 −0.961074 0.276289i \(-0.910895\pi\)
−0.961074 + 0.276289i \(0.910895\pi\)
\(132\) 1.00000 0.0870388
\(133\) −3.00000 −0.260133
\(134\) 3.00000 0.259161
\(135\) −10.0000 −0.860663
\(136\) −7.00000 −0.600245
\(137\) −17.0000 −1.45241 −0.726204 0.687479i \(-0.758717\pi\)
−0.726204 + 0.687479i \(0.758717\pi\)
\(138\) 5.00000 0.425628
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 6.00000 0.507093
\(141\) 0 0
\(142\) −10.0000 −0.839181
\(143\) −1.00000 −0.0836242
\(144\) −2.00000 −0.166667
\(145\) −2.00000 −0.166091
\(146\) 3.00000 0.248282
\(147\) −2.00000 −0.164957
\(148\) −6.00000 −0.493197
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 1.00000 0.0816497
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 1.00000 0.0811107
\(153\) 14.0000 1.13183
\(154\) 3.00000 0.241747
\(155\) −20.0000 −1.60644
\(156\) −1.00000 −0.0800641
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 8.00000 0.636446
\(159\) 1.00000 0.0793052
\(160\) −2.00000 −0.158114
\(161\) 15.0000 1.18217
\(162\) 1.00000 0.0785674
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 6.00000 0.468521
\(165\) −2.00000 −0.155700
\(166\) 8.00000 0.620920
\(167\) −22.0000 −1.70241 −0.851206 0.524832i \(-0.824128\pi\)
−0.851206 + 0.524832i \(0.824128\pi\)
\(168\) 3.00000 0.231455
\(169\) −12.0000 −0.923077
\(170\) 14.0000 1.07375
\(171\) −2.00000 −0.152944
\(172\) −4.00000 −0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 3.00000 0.226779
\(176\) −1.00000 −0.0753778
\(177\) −3.00000 −0.225494
\(178\) −8.00000 −0.599625
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 4.00000 0.298142
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −3.00000 −0.222375
\(183\) 12.0000 0.887066
\(184\) −5.00000 −0.368605
\(185\) 12.0000 0.882258
\(186\) −10.0000 −0.733236
\(187\) 7.00000 0.511891
\(188\) 0 0
\(189\) −15.0000 −1.09109
\(190\) −2.00000 −0.145095
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 8.00000 0.574367
\(195\) 2.00000 0.143223
\(196\) 2.00000 0.142857
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 2.00000 0.142134
\(199\) 23.0000 1.63043 0.815213 0.579161i \(-0.196620\pi\)
0.815213 + 0.579161i \(0.196620\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −3.00000 −0.211604
\(202\) 6.00000 0.422159
\(203\) −3.00000 −0.210559
\(204\) 7.00000 0.490098
\(205\) −12.0000 −0.838116
\(206\) 12.0000 0.836080
\(207\) 10.0000 0.695048
\(208\) 1.00000 0.0693375
\(209\) −1.00000 −0.0691714
\(210\) −6.00000 −0.414039
\(211\) 1.00000 0.0688428 0.0344214 0.999407i \(-0.489041\pi\)
0.0344214 + 0.999407i \(0.489041\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 10.0000 0.685189
\(214\) −13.0000 −0.888662
\(215\) 8.00000 0.545595
\(216\) 5.00000 0.340207
\(217\) −30.0000 −2.03653
\(218\) −13.0000 −0.880471
\(219\) −3.00000 −0.202721
\(220\) 2.00000 0.134840
\(221\) −7.00000 −0.470871
\(222\) 6.00000 0.402694
\(223\) 10.0000 0.669650 0.334825 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) −3.00000 −0.200446
\(225\) 2.00000 0.133333
\(226\) −12.0000 −0.798228
\(227\) −15.0000 −0.995585 −0.497792 0.867296i \(-0.665856\pi\)
−0.497792 + 0.867296i \(0.665856\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 10.0000 0.659380
\(231\) −3.00000 −0.197386
\(232\) 1.00000 0.0656532
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 3.00000 0.195283
\(237\) −8.00000 −0.519656
\(238\) 21.0000 1.36123
\(239\) 25.0000 1.61712 0.808558 0.588417i \(-0.200249\pi\)
0.808558 + 0.588417i \(0.200249\pi\)
\(240\) 2.00000 0.129099
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 1.00000 0.0642824
\(243\) −16.0000 −1.02640
\(244\) −12.0000 −0.768221
\(245\) −4.00000 −0.255551
\(246\) −6.00000 −0.382546
\(247\) 1.00000 0.0636285
\(248\) 10.0000 0.635001
\(249\) −8.00000 −0.506979
\(250\) 12.0000 0.758947
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 6.00000 0.377964
\(253\) 5.00000 0.314347
\(254\) 16.0000 1.00393
\(255\) −14.0000 −0.876714
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 4.00000 0.249029
\(259\) 18.0000 1.11847
\(260\) −2.00000 −0.124035
\(261\) −2.00000 −0.123797
\(262\) −22.0000 −1.35916
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 1.00000 0.0615457
\(265\) 2.00000 0.122859
\(266\) −3.00000 −0.183942
\(267\) 8.00000 0.489592
\(268\) 3.00000 0.183254
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −10.0000 −0.608581
\(271\) −15.0000 −0.911185 −0.455593 0.890188i \(-0.650573\pi\)
−0.455593 + 0.890188i \(0.650573\pi\)
\(272\) −7.00000 −0.424437
\(273\) 3.00000 0.181568
\(274\) −17.0000 −1.02701
\(275\) 1.00000 0.0603023
\(276\) 5.00000 0.300965
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) −14.0000 −0.839664
\(279\) −20.0000 −1.19737
\(280\) 6.00000 0.358569
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −10.0000 −0.593391
\(285\) 2.00000 0.118470
\(286\) −1.00000 −0.0591312
\(287\) −18.0000 −1.06251
\(288\) −2.00000 −0.117851
\(289\) 32.0000 1.88235
\(290\) −2.00000 −0.117444
\(291\) −8.00000 −0.468968
\(292\) 3.00000 0.175562
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) −2.00000 −0.116642
\(295\) −6.00000 −0.349334
\(296\) −6.00000 −0.348743
\(297\) −5.00000 −0.290129
\(298\) −18.0000 −1.04271
\(299\) −5.00000 −0.289157
\(300\) 1.00000 0.0577350
\(301\) 12.0000 0.691669
\(302\) 12.0000 0.690522
\(303\) −6.00000 −0.344691
\(304\) 1.00000 0.0573539
\(305\) 24.0000 1.37424
\(306\) 14.0000 0.800327
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 3.00000 0.170941
\(309\) −12.0000 −0.682656
\(310\) −20.0000 −1.13592
\(311\) 3.00000 0.170114 0.0850572 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 21.0000 1.18699 0.593495 0.804838i \(-0.297748\pi\)
0.593495 + 0.804838i \(0.297748\pi\)
\(314\) −22.0000 −1.24153
\(315\) −12.0000 −0.676123
\(316\) 8.00000 0.450035
\(317\) 21.0000 1.17948 0.589739 0.807594i \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) 1.00000 0.0560772
\(319\) −1.00000 −0.0559893
\(320\) −2.00000 −0.111803
\(321\) 13.0000 0.725589
\(322\) 15.0000 0.835917
\(323\) −7.00000 −0.389490
\(324\) 1.00000 0.0555556
\(325\) −1.00000 −0.0554700
\(326\) −2.00000 −0.110770
\(327\) 13.0000 0.718902
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) −2.00000 −0.110096
\(331\) 29.0000 1.59398 0.796992 0.603990i \(-0.206423\pi\)
0.796992 + 0.603990i \(0.206423\pi\)
\(332\) 8.00000 0.439057
\(333\) 12.0000 0.657596
\(334\) −22.0000 −1.20379
\(335\) −6.00000 −0.327815
\(336\) 3.00000 0.163663
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) −12.0000 −0.652714
\(339\) 12.0000 0.651751
\(340\) 14.0000 0.759257
\(341\) −10.0000 −0.541530
\(342\) −2.00000 −0.108148
\(343\) 15.0000 0.809924
\(344\) −4.00000 −0.215666
\(345\) −10.0000 −0.538382
\(346\) 14.0000 0.752645
\(347\) −26.0000 −1.39575 −0.697877 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) −1.00000 −0.0536056
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 3.00000 0.160357
\(351\) 5.00000 0.266880
\(352\) −1.00000 −0.0533002
\(353\) 1.00000 0.0532246 0.0266123 0.999646i \(-0.491528\pi\)
0.0266123 + 0.999646i \(0.491528\pi\)
\(354\) −3.00000 −0.159448
\(355\) 20.0000 1.06149
\(356\) −8.00000 −0.423999
\(357\) −21.0000 −1.11144
\(358\) 4.00000 0.211407
\(359\) −17.0000 −0.897226 −0.448613 0.893726i \(-0.648082\pi\)
−0.448613 + 0.893726i \(0.648082\pi\)
\(360\) 4.00000 0.210819
\(361\) 1.00000 0.0526316
\(362\) −2.00000 −0.105118
\(363\) −1.00000 −0.0524864
\(364\) −3.00000 −0.157243
\(365\) −6.00000 −0.314054
\(366\) 12.0000 0.627250
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −5.00000 −0.260643
\(369\) −12.0000 −0.624695
\(370\) 12.0000 0.623850
\(371\) 3.00000 0.155752
\(372\) −10.0000 −0.518476
\(373\) 11.0000 0.569558 0.284779 0.958593i \(-0.408080\pi\)
0.284779 + 0.958593i \(0.408080\pi\)
\(374\) 7.00000 0.361961
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 1.00000 0.0515026
\(378\) −15.0000 −0.771517
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) −2.00000 −0.102598
\(381\) −16.0000 −0.819705
\(382\) 3.00000 0.153493
\(383\) 2.00000 0.102195 0.0510976 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −6.00000 −0.305788
\(386\) −4.00000 −0.203595
\(387\) 8.00000 0.406663
\(388\) 8.00000 0.406138
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 2.00000 0.101274
\(391\) 35.0000 1.77003
\(392\) 2.00000 0.101015
\(393\) 22.0000 1.10975
\(394\) −12.0000 −0.604551
\(395\) −16.0000 −0.805047
\(396\) 2.00000 0.100504
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 23.0000 1.15289
\(399\) 3.00000 0.150188
\(400\) −1.00000 −0.0500000
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) −3.00000 −0.149626
\(403\) 10.0000 0.498135
\(404\) 6.00000 0.298511
\(405\) −2.00000 −0.0993808
\(406\) −3.00000 −0.148888
\(407\) 6.00000 0.297409
\(408\) 7.00000 0.346552
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) −12.0000 −0.592638
\(411\) 17.0000 0.838548
\(412\) 12.0000 0.591198
\(413\) −9.00000 −0.442861
\(414\) 10.0000 0.491473
\(415\) −16.0000 −0.785409
\(416\) 1.00000 0.0490290
\(417\) 14.0000 0.685583
\(418\) −1.00000 −0.0489116
\(419\) 10.0000 0.488532 0.244266 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) −6.00000 −0.292770
\(421\) 15.0000 0.731055 0.365528 0.930800i \(-0.380889\pi\)
0.365528 + 0.930800i \(0.380889\pi\)
\(422\) 1.00000 0.0486792
\(423\) 0 0
\(424\) −1.00000 −0.0485643
\(425\) 7.00000 0.339550
\(426\) 10.0000 0.484502
\(427\) 36.0000 1.74216
\(428\) −13.0000 −0.628379
\(429\) 1.00000 0.0482805
\(430\) 8.00000 0.385794
\(431\) 34.0000 1.63772 0.818861 0.573992i \(-0.194606\pi\)
0.818861 + 0.573992i \(0.194606\pi\)
\(432\) 5.00000 0.240563
\(433\) 24.0000 1.15337 0.576683 0.816968i \(-0.304347\pi\)
0.576683 + 0.816968i \(0.304347\pi\)
\(434\) −30.0000 −1.44005
\(435\) 2.00000 0.0958927
\(436\) −13.0000 −0.622587
\(437\) −5.00000 −0.239182
\(438\) −3.00000 −0.143346
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 2.00000 0.0953463
\(441\) −4.00000 −0.190476
\(442\) −7.00000 −0.332956
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) 6.00000 0.284747
\(445\) 16.0000 0.758473
\(446\) 10.0000 0.473514
\(447\) 18.0000 0.851371
\(448\) −3.00000 −0.141737
\(449\) 24.0000 1.13263 0.566315 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) 2.00000 0.0942809
\(451\) −6.00000 −0.282529
\(452\) −12.0000 −0.564433
\(453\) −12.0000 −0.563809
\(454\) −15.0000 −0.703985
\(455\) 6.00000 0.281284
\(456\) −1.00000 −0.0468293
\(457\) −37.0000 −1.73079 −0.865393 0.501093i \(-0.832931\pi\)
−0.865393 + 0.501093i \(0.832931\pi\)
\(458\) 20.0000 0.934539
\(459\) −35.0000 −1.63366
\(460\) 10.0000 0.466252
\(461\) −8.00000 −0.372597 −0.186299 0.982493i \(-0.559649\pi\)
−0.186299 + 0.982493i \(0.559649\pi\)
\(462\) −3.00000 −0.139573
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 1.00000 0.0464238
\(465\) 20.0000 0.927478
\(466\) −10.0000 −0.463241
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −9.00000 −0.415581
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) 3.00000 0.138086
\(473\) 4.00000 0.183920
\(474\) −8.00000 −0.367452
\(475\) −1.00000 −0.0458831
\(476\) 21.0000 0.962533
\(477\) 2.00000 0.0915737
\(478\) 25.0000 1.14347
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 2.00000 0.0912871
\(481\) −6.00000 −0.273576
\(482\) −4.00000 −0.182195
\(483\) −15.0000 −0.682524
\(484\) 1.00000 0.0454545
\(485\) −16.0000 −0.726523
\(486\) −16.0000 −0.725775
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) −12.0000 −0.543214
\(489\) 2.00000 0.0904431
\(490\) −4.00000 −0.180702
\(491\) 10.0000 0.451294 0.225647 0.974209i \(-0.427550\pi\)
0.225647 + 0.974209i \(0.427550\pi\)
\(492\) −6.00000 −0.270501
\(493\) −7.00000 −0.315264
\(494\) 1.00000 0.0449921
\(495\) −4.00000 −0.179787
\(496\) 10.0000 0.449013
\(497\) 30.0000 1.34568
\(498\) −8.00000 −0.358489
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 12.0000 0.536656
\(501\) 22.0000 0.982888
\(502\) −6.00000 −0.267793
\(503\) −19.0000 −0.847168 −0.423584 0.905857i \(-0.639228\pi\)
−0.423584 + 0.905857i \(0.639228\pi\)
\(504\) 6.00000 0.267261
\(505\) −12.0000 −0.533993
\(506\) 5.00000 0.222277
\(507\) 12.0000 0.532939
\(508\) 16.0000 0.709885
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) −14.0000 −0.619930
\(511\) −9.00000 −0.398137
\(512\) 1.00000 0.0441942
\(513\) 5.00000 0.220755
\(514\) −22.0000 −0.970378
\(515\) −24.0000 −1.05757
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 18.0000 0.790875
\(519\) −14.0000 −0.614532
\(520\) −2.00000 −0.0877058
\(521\) −20.0000 −0.876216 −0.438108 0.898922i \(-0.644351\pi\)
−0.438108 + 0.898922i \(0.644351\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −21.0000 −0.918266 −0.459133 0.888368i \(-0.651840\pi\)
−0.459133 + 0.888368i \(0.651840\pi\)
\(524\) −22.0000 −0.961074
\(525\) −3.00000 −0.130931
\(526\) 4.00000 0.174408
\(527\) −70.0000 −3.04925
\(528\) 1.00000 0.0435194
\(529\) 2.00000 0.0869565
\(530\) 2.00000 0.0868744
\(531\) −6.00000 −0.260378
\(532\) −3.00000 −0.130066
\(533\) 6.00000 0.259889
\(534\) 8.00000 0.346194
\(535\) 26.0000 1.12408
\(536\) 3.00000 0.129580
\(537\) −4.00000 −0.172613
\(538\) 18.0000 0.776035
\(539\) −2.00000 −0.0861461
\(540\) −10.0000 −0.430331
\(541\) −16.0000 −0.687894 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) −15.0000 −0.644305
\(543\) 2.00000 0.0858282
\(544\) −7.00000 −0.300123
\(545\) 26.0000 1.11372
\(546\) 3.00000 0.128388
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −17.0000 −0.726204
\(549\) 24.0000 1.02430
\(550\) 1.00000 0.0426401
\(551\) 1.00000 0.0426014
\(552\) 5.00000 0.212814
\(553\) −24.0000 −1.02058
\(554\) −16.0000 −0.679775
\(555\) −12.0000 −0.509372
\(556\) −14.0000 −0.593732
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) −20.0000 −0.846668
\(559\) −4.00000 −0.169182
\(560\) 6.00000 0.253546
\(561\) −7.00000 −0.295540
\(562\) 16.0000 0.674919
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 24.0000 1.00969
\(566\) −14.0000 −0.588464
\(567\) −3.00000 −0.125988
\(568\) −10.0000 −0.419591
\(569\) −12.0000 −0.503066 −0.251533 0.967849i \(-0.580935\pi\)
−0.251533 + 0.967849i \(0.580935\pi\)
\(570\) 2.00000 0.0837708
\(571\) −14.0000 −0.585882 −0.292941 0.956131i \(-0.594634\pi\)
−0.292941 + 0.956131i \(0.594634\pi\)
\(572\) −1.00000 −0.0418121
\(573\) −3.00000 −0.125327
\(574\) −18.0000 −0.751305
\(575\) 5.00000 0.208514
\(576\) −2.00000 −0.0833333
\(577\) 23.0000 0.957503 0.478751 0.877951i \(-0.341090\pi\)
0.478751 + 0.877951i \(0.341090\pi\)
\(578\) 32.0000 1.33102
\(579\) 4.00000 0.166234
\(580\) −2.00000 −0.0830455
\(581\) −24.0000 −0.995688
\(582\) −8.00000 −0.331611
\(583\) 1.00000 0.0414158
\(584\) 3.00000 0.124141
\(585\) 4.00000 0.165380
\(586\) −9.00000 −0.371787
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 10.0000 0.412043
\(590\) −6.00000 −0.247016
\(591\) 12.0000 0.493614
\(592\) −6.00000 −0.246598
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) −5.00000 −0.205152
\(595\) −42.0000 −1.72183
\(596\) −18.0000 −0.737309
\(597\) −23.0000 −0.941327
\(598\) −5.00000 −0.204465
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 1.00000 0.0408248
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 12.0000 0.489083
\(603\) −6.00000 −0.244339
\(604\) 12.0000 0.488273
\(605\) −2.00000 −0.0813116
\(606\) −6.00000 −0.243733
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 1.00000 0.0405554
\(609\) 3.00000 0.121566
\(610\) 24.0000 0.971732
\(611\) 0 0
\(612\) 14.0000 0.565916
\(613\) −28.0000 −1.13091 −0.565455 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(614\) −16.0000 −0.645707
\(615\) 12.0000 0.483887
\(616\) 3.00000 0.120873
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) −12.0000 −0.482711
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) −20.0000 −0.803219
\(621\) −25.0000 −1.00322
\(622\) 3.00000 0.120289
\(623\) 24.0000 0.961540
\(624\) −1.00000 −0.0400320
\(625\) −19.0000 −0.760000
\(626\) 21.0000 0.839329
\(627\) 1.00000 0.0399362
\(628\) −22.0000 −0.877896
\(629\) 42.0000 1.67465
\(630\) −12.0000 −0.478091
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 8.00000 0.318223
\(633\) −1.00000 −0.0397464
\(634\) 21.0000 0.834017
\(635\) −32.0000 −1.26988
\(636\) 1.00000 0.0396526
\(637\) 2.00000 0.0792429
\(638\) −1.00000 −0.0395904
\(639\) 20.0000 0.791188
\(640\) −2.00000 −0.0790569
\(641\) 40.0000 1.57991 0.789953 0.613168i \(-0.210105\pi\)
0.789953 + 0.613168i \(0.210105\pi\)
\(642\) 13.0000 0.513069
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) 15.0000 0.591083
\(645\) −8.00000 −0.315000
\(646\) −7.00000 −0.275411
\(647\) 3.00000 0.117942 0.0589711 0.998260i \(-0.481218\pi\)
0.0589711 + 0.998260i \(0.481218\pi\)
\(648\) 1.00000 0.0392837
\(649\) −3.00000 −0.117760
\(650\) −1.00000 −0.0392232
\(651\) 30.0000 1.17579
\(652\) −2.00000 −0.0783260
\(653\) 4.00000 0.156532 0.0782660 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) 13.0000 0.508340
\(655\) 44.0000 1.71922
\(656\) 6.00000 0.234261
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 11.0000 0.428499 0.214250 0.976779i \(-0.431269\pi\)
0.214250 + 0.976779i \(0.431269\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 29.0000 1.12797 0.563985 0.825785i \(-0.309268\pi\)
0.563985 + 0.825785i \(0.309268\pi\)
\(662\) 29.0000 1.12712
\(663\) 7.00000 0.271857
\(664\) 8.00000 0.310460
\(665\) 6.00000 0.232670
\(666\) 12.0000 0.464991
\(667\) −5.00000 −0.193601
\(668\) −22.0000 −0.851206
\(669\) −10.0000 −0.386622
\(670\) −6.00000 −0.231800
\(671\) 12.0000 0.463255
\(672\) 3.00000 0.115728
\(673\) −24.0000 −0.925132 −0.462566 0.886585i \(-0.653071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) −20.0000 −0.770371
\(675\) −5.00000 −0.192450
\(676\) −12.0000 −0.461538
\(677\) −29.0000 −1.11456 −0.557280 0.830324i \(-0.688155\pi\)
−0.557280 + 0.830324i \(0.688155\pi\)
\(678\) 12.0000 0.460857
\(679\) −24.0000 −0.921035
\(680\) 14.0000 0.536875
\(681\) 15.0000 0.574801
\(682\) −10.0000 −0.382920
\(683\) −8.00000 −0.306111 −0.153056 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 34.0000 1.29907
\(686\) 15.0000 0.572703
\(687\) −20.0000 −0.763048
\(688\) −4.00000 −0.152499
\(689\) −1.00000 −0.0380970
\(690\) −10.0000 −0.380693
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) 14.0000 0.532200
\(693\) −6.00000 −0.227921
\(694\) −26.0000 −0.986947
\(695\) 28.0000 1.06210
\(696\) −1.00000 −0.0379049
\(697\) −42.0000 −1.59086
\(698\) 10.0000 0.378506
\(699\) 10.0000 0.378235
\(700\) 3.00000 0.113389
\(701\) 24.0000 0.906467 0.453234 0.891392i \(-0.350270\pi\)
0.453234 + 0.891392i \(0.350270\pi\)
\(702\) 5.00000 0.188713
\(703\) −6.00000 −0.226294
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 1.00000 0.0376355
\(707\) −18.0000 −0.676960
\(708\) −3.00000 −0.112747
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 20.0000 0.750587
\(711\) −16.0000 −0.600047
\(712\) −8.00000 −0.299813
\(713\) −50.0000 −1.87251
\(714\) −21.0000 −0.785905
\(715\) 2.00000 0.0747958
\(716\) 4.00000 0.149487
\(717\) −25.0000 −0.933642
\(718\) −17.0000 −0.634434
\(719\) −33.0000 −1.23069 −0.615346 0.788257i \(-0.710984\pi\)
−0.615346 + 0.788257i \(0.710984\pi\)
\(720\) 4.00000 0.149071
\(721\) −36.0000 −1.34071
\(722\) 1.00000 0.0372161
\(723\) 4.00000 0.148762
\(724\) −2.00000 −0.0743294
\(725\) −1.00000 −0.0371391
\(726\) −1.00000 −0.0371135
\(727\) 47.0000 1.74313 0.871567 0.490277i \(-0.163104\pi\)
0.871567 + 0.490277i \(0.163104\pi\)
\(728\) −3.00000 −0.111187
\(729\) 13.0000 0.481481
\(730\) −6.00000 −0.222070
\(731\) 28.0000 1.03562
\(732\) 12.0000 0.443533
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 16.0000 0.590571
\(735\) 4.00000 0.147542
\(736\) −5.00000 −0.184302
\(737\) −3.00000 −0.110506
\(738\) −12.0000 −0.441726
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 12.0000 0.441129
\(741\) −1.00000 −0.0367359
\(742\) 3.00000 0.110133
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −10.0000 −0.366618
\(745\) 36.0000 1.31894
\(746\) 11.0000 0.402739
\(747\) −16.0000 −0.585409
\(748\) 7.00000 0.255945
\(749\) 39.0000 1.42503
\(750\) −12.0000 −0.438178
\(751\) −50.0000 −1.82453 −0.912263 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) 0 0
\(753\) 6.00000 0.218652
\(754\) 1.00000 0.0364179
\(755\) −24.0000 −0.873449
\(756\) −15.0000 −0.545545
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −5.00000 −0.181608
\(759\) −5.00000 −0.181489
\(760\) −2.00000 −0.0725476
\(761\) −11.0000 −0.398750 −0.199375 0.979923i \(-0.563891\pi\)
−0.199375 + 0.979923i \(0.563891\pi\)
\(762\) −16.0000 −0.579619
\(763\) 39.0000 1.41189
\(764\) 3.00000 0.108536
\(765\) −28.0000 −1.01234
\(766\) 2.00000 0.0722629
\(767\) 3.00000 0.108324
\(768\) −1.00000 −0.0360844
\(769\) −49.0000 −1.76699 −0.883493 0.468445i \(-0.844814\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) −6.00000 −0.216225
\(771\) 22.0000 0.792311
\(772\) −4.00000 −0.143963
\(773\) −23.0000 −0.827253 −0.413626 0.910447i \(-0.635738\pi\)
−0.413626 + 0.910447i \(0.635738\pi\)
\(774\) 8.00000 0.287554
\(775\) −10.0000 −0.359211
\(776\) 8.00000 0.287183
\(777\) −18.0000 −0.645746
\(778\) 18.0000 0.645331
\(779\) 6.00000 0.214972
\(780\) 2.00000 0.0716115
\(781\) 10.0000 0.357828
\(782\) 35.0000 1.25160
\(783\) 5.00000 0.178685
\(784\) 2.00000 0.0714286
\(785\) 44.0000 1.57043
\(786\) 22.0000 0.784714
\(787\) 49.0000 1.74666 0.873331 0.487128i \(-0.161955\pi\)
0.873331 + 0.487128i \(0.161955\pi\)
\(788\) −12.0000 −0.427482
\(789\) −4.00000 −0.142404
\(790\) −16.0000 −0.569254
\(791\) 36.0000 1.28001
\(792\) 2.00000 0.0710669
\(793\) −12.0000 −0.426132
\(794\) 34.0000 1.20661
\(795\) −2.00000 −0.0709327
\(796\) 23.0000 0.815213
\(797\) 19.0000 0.673015 0.336507 0.941681i \(-0.390754\pi\)
0.336507 + 0.941681i \(0.390754\pi\)
\(798\) 3.00000 0.106199
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 16.0000 0.565332
\(802\) −2.00000 −0.0706225
\(803\) −3.00000 −0.105868
\(804\) −3.00000 −0.105802
\(805\) −30.0000 −1.05736
\(806\) 10.0000 0.352235
\(807\) −18.0000 −0.633630
\(808\) 6.00000 0.211079
\(809\) −13.0000 −0.457056 −0.228528 0.973537i \(-0.573391\pi\)
−0.228528 + 0.973537i \(0.573391\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −25.0000 −0.877869 −0.438934 0.898519i \(-0.644644\pi\)
−0.438934 + 0.898519i \(0.644644\pi\)
\(812\) −3.00000 −0.105279
\(813\) 15.0000 0.526073
\(814\) 6.00000 0.210300
\(815\) 4.00000 0.140114
\(816\) 7.00000 0.245049
\(817\) −4.00000 −0.139942
\(818\) 34.0000 1.18878
\(819\) 6.00000 0.209657
\(820\) −12.0000 −0.419058
\(821\) −16.0000 −0.558404 −0.279202 0.960232i \(-0.590070\pi\)
−0.279202 + 0.960232i \(0.590070\pi\)
\(822\) 17.0000 0.592943
\(823\) −43.0000 −1.49889 −0.749443 0.662069i \(-0.769679\pi\)
−0.749443 + 0.662069i \(0.769679\pi\)
\(824\) 12.0000 0.418040
\(825\) −1.00000 −0.0348155
\(826\) −9.00000 −0.313150
\(827\) 37.0000 1.28662 0.643308 0.765607i \(-0.277561\pi\)
0.643308 + 0.765607i \(0.277561\pi\)
\(828\) 10.0000 0.347524
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) −16.0000 −0.555368
\(831\) 16.0000 0.555034
\(832\) 1.00000 0.0346688
\(833\) −14.0000 −0.485071
\(834\) 14.0000 0.484780
\(835\) 44.0000 1.52268
\(836\) −1.00000 −0.0345857
\(837\) 50.0000 1.72825
\(838\) 10.0000 0.345444
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) −6.00000 −0.207020
\(841\) −28.0000 −0.965517
\(842\) 15.0000 0.516934
\(843\) −16.0000 −0.551069
\(844\) 1.00000 0.0344214
\(845\) 24.0000 0.825625
\(846\) 0 0
\(847\) −3.00000 −0.103081
\(848\) −1.00000 −0.0343401
\(849\) 14.0000 0.480479
\(850\) 7.00000 0.240098
\(851\) 30.0000 1.02839
\(852\) 10.0000 0.342594
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 36.0000 1.23189
\(855\) 4.00000 0.136797
\(856\) −13.0000 −0.444331
\(857\) −30.0000 −1.02478 −0.512390 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 1.00000 0.0341394
\(859\) −30.0000 −1.02359 −0.511793 0.859109i \(-0.671019\pi\)
−0.511793 + 0.859109i \(0.671019\pi\)
\(860\) 8.00000 0.272798
\(861\) 18.0000 0.613438
\(862\) 34.0000 1.15804
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 5.00000 0.170103
\(865\) −28.0000 −0.952029
\(866\) 24.0000 0.815553
\(867\) −32.0000 −1.08678
\(868\) −30.0000 −1.01827
\(869\) −8.00000 −0.271381
\(870\) 2.00000 0.0678064
\(871\) 3.00000 0.101651
\(872\) −13.0000 −0.440236
\(873\) −16.0000 −0.541518
\(874\) −5.00000 −0.169128
\(875\) −36.0000 −1.21702
\(876\) −3.00000 −0.101361
\(877\) 27.0000 0.911725 0.455863 0.890050i \(-0.349331\pi\)
0.455863 + 0.890050i \(0.349331\pi\)
\(878\) −14.0000 −0.472477
\(879\) 9.00000 0.303562
\(880\) 2.00000 0.0674200
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) −4.00000 −0.134687
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) −7.00000 −0.235435
\(885\) 6.00000 0.201688
\(886\) 18.0000 0.604722
\(887\) −4.00000 −0.134307 −0.0671534 0.997743i \(-0.521392\pi\)
−0.0671534 + 0.997743i \(0.521392\pi\)
\(888\) 6.00000 0.201347
\(889\) −48.0000 −1.60987
\(890\) 16.0000 0.536321
\(891\) −1.00000 −0.0335013
\(892\) 10.0000 0.334825
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) −8.00000 −0.267411
\(896\) −3.00000 −0.100223
\(897\) 5.00000 0.166945
\(898\) 24.0000 0.800890
\(899\) 10.0000 0.333519
\(900\) 2.00000 0.0666667
\(901\) 7.00000 0.233204
\(902\) −6.00000 −0.199778
\(903\) −12.0000 −0.399335
\(904\) −12.0000 −0.399114
\(905\) 4.00000 0.132964
\(906\) −12.0000 −0.398673
\(907\) 13.0000 0.431658 0.215829 0.976431i \(-0.430755\pi\)
0.215829 + 0.976431i \(0.430755\pi\)
\(908\) −15.0000 −0.497792
\(909\) −12.0000 −0.398015
\(910\) 6.00000 0.198898
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) −1.00000 −0.0331133
\(913\) −8.00000 −0.264761
\(914\) −37.0000 −1.22385
\(915\) −24.0000 −0.793416
\(916\) 20.0000 0.660819
\(917\) 66.0000 2.17951
\(918\) −35.0000 −1.15517
\(919\) 3.00000 0.0989609 0.0494804 0.998775i \(-0.484243\pi\)
0.0494804 + 0.998775i \(0.484243\pi\)
\(920\) 10.0000 0.329690
\(921\) 16.0000 0.527218
\(922\) −8.00000 −0.263466
\(923\) −10.0000 −0.329154
\(924\) −3.00000 −0.0986928
\(925\) 6.00000 0.197279
\(926\) 16.0000 0.525793
\(927\) −24.0000 −0.788263
\(928\) 1.00000 0.0328266
\(929\) 29.0000 0.951459 0.475730 0.879592i \(-0.342184\pi\)
0.475730 + 0.879592i \(0.342184\pi\)
\(930\) 20.0000 0.655826
\(931\) 2.00000 0.0655474
\(932\) −10.0000 −0.327561
\(933\) −3.00000 −0.0982156
\(934\) 20.0000 0.654420
\(935\) −14.0000 −0.457849
\(936\) −2.00000 −0.0653720
\(937\) −5.00000 −0.163343 −0.0816714 0.996659i \(-0.526026\pi\)
−0.0816714 + 0.996659i \(0.526026\pi\)
\(938\) −9.00000 −0.293860
\(939\) −21.0000 −0.685309
\(940\) 0 0
\(941\) 33.0000 1.07577 0.537885 0.843018i \(-0.319224\pi\)
0.537885 + 0.843018i \(0.319224\pi\)
\(942\) 22.0000 0.716799
\(943\) −30.0000 −0.976934
\(944\) 3.00000 0.0976417
\(945\) 30.0000 0.975900
\(946\) 4.00000 0.130051
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) −8.00000 −0.259828
\(949\) 3.00000 0.0973841
\(950\) −1.00000 −0.0324443
\(951\) −21.0000 −0.680972
\(952\) 21.0000 0.680614
\(953\) 60.0000 1.94359 0.971795 0.235826i \(-0.0757795\pi\)
0.971795 + 0.235826i \(0.0757795\pi\)
\(954\) 2.00000 0.0647524
\(955\) −6.00000 −0.194155
\(956\) 25.0000 0.808558
\(957\) 1.00000 0.0323254
\(958\) −12.0000 −0.387702
\(959\) 51.0000 1.64688
\(960\) 2.00000 0.0645497
\(961\) 69.0000 2.22581
\(962\) −6.00000 −0.193448
\(963\) 26.0000 0.837838
\(964\) −4.00000 −0.128831
\(965\) 8.00000 0.257529
\(966\) −15.0000 −0.482617
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 1.00000 0.0321412
\(969\) 7.00000 0.224872
\(970\) −16.0000 −0.513729
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) −16.0000 −0.513200
\(973\) 42.0000 1.34646
\(974\) −12.0000 −0.384505
\(975\) 1.00000 0.0320256
\(976\) −12.0000 −0.384111
\(977\) −60.0000 −1.91957 −0.959785 0.280736i \(-0.909421\pi\)
−0.959785 + 0.280736i \(0.909421\pi\)
\(978\) 2.00000 0.0639529
\(979\) 8.00000 0.255681
\(980\) −4.00000 −0.127775
\(981\) 26.0000 0.830116
\(982\) 10.0000 0.319113
\(983\) −46.0000 −1.46717 −0.733586 0.679597i \(-0.762155\pi\)
−0.733586 + 0.679597i \(0.762155\pi\)
\(984\) −6.00000 −0.191273
\(985\) 24.0000 0.764704
\(986\) −7.00000 −0.222925
\(987\) 0 0
\(988\) 1.00000 0.0318142
\(989\) 20.0000 0.635963
\(990\) −4.00000 −0.127128
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) 10.0000 0.317500
\(993\) −29.0000 −0.920287
\(994\) 30.0000 0.951542
\(995\) −46.0000 −1.45830
\(996\) −8.00000 −0.253490
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −28.0000 −0.886325
\(999\) −30.0000 −0.949158
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 418.2.a.a.1.1 1
3.2 odd 2 3762.2.a.g.1.1 1
4.3 odd 2 3344.2.a.h.1.1 1
11.10 odd 2 4598.2.a.b.1.1 1
19.18 odd 2 7942.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.a.1.1 1 1.1 even 1 trivial
3344.2.a.h.1.1 1 4.3 odd 2
3762.2.a.g.1.1 1 3.2 odd 2
4598.2.a.b.1.1 1 11.10 odd 2
7942.2.a.i.1.1 1 19.18 odd 2