Properties

Label 141.2.a.a.1.1
Level $141$
Weight $2$
Character 141.1
Self dual yes
Analytic conductor $1.126$
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] = [141,2,Mod(1,141)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(141, 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("141.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 141.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: \(1.12589066850\)
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 \(\chi\) \(=\) 141.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} -3.00000 q^{7} +1.00000 q^{9} +6.00000 q^{10} -5.00000 q^{11} +2.00000 q^{12} +2.00000 q^{13} +6.00000 q^{14} -3.00000 q^{15} -4.00000 q^{16} -6.00000 q^{17} -2.00000 q^{18} -6.00000 q^{19} -6.00000 q^{20} -3.00000 q^{21} +10.0000 q^{22} +9.00000 q^{23} +4.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -6.00000 q^{28} +1.00000 q^{29} +6.00000 q^{30} -2.00000 q^{31} +8.00000 q^{32} -5.00000 q^{33} +12.0000 q^{34} +9.00000 q^{35} +2.00000 q^{36} +1.00000 q^{37} +12.0000 q^{38} +2.00000 q^{39} +6.00000 q^{41} +6.00000 q^{42} +2.00000 q^{43} -10.0000 q^{44} -3.00000 q^{45} -18.0000 q^{46} +1.00000 q^{47} -4.00000 q^{48} +2.00000 q^{49} -8.00000 q^{50} -6.00000 q^{51} +4.00000 q^{52} -2.00000 q^{54} +15.0000 q^{55} -6.00000 q^{57} -2.00000 q^{58} -12.0000 q^{59} -6.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} -3.00000 q^{63} -8.00000 q^{64} -6.00000 q^{65} +10.0000 q^{66} +2.00000 q^{67} -12.0000 q^{68} +9.00000 q^{69} -18.0000 q^{70} -2.00000 q^{71} -2.00000 q^{73} -2.00000 q^{74} +4.00000 q^{75} -12.0000 q^{76} +15.0000 q^{77} -4.00000 q^{78} -15.0000 q^{79} +12.0000 q^{80} +1.00000 q^{81} -12.0000 q^{82} -4.00000 q^{83} -6.00000 q^{84} +18.0000 q^{85} -4.00000 q^{86} +1.00000 q^{87} +10.0000 q^{89} +6.00000 q^{90} -6.00000 q^{91} +18.0000 q^{92} -2.00000 q^{93} -2.00000 q^{94} +18.0000 q^{95} +8.00000 q^{96} +1.00000 q^{97} -4.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) −2.00000 −0.816497
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 6.00000 1.89737
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 2.00000 0.577350
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 6.00000 1.60357
\(15\) −3.00000 −0.774597
\(16\) −4.00000 −1.00000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −2.00000 −0.471405
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −6.00000 −1.34164
\(21\) −3.00000 −0.654654
\(22\) 10.0000 2.13201
\(23\) 9.00000 1.87663 0.938315 0.345782i \(-0.112386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) −6.00000 −1.13389
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 6.00000 1.09545
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 8.00000 1.41421
\(33\) −5.00000 −0.870388
\(34\) 12.0000 2.05798
\(35\) 9.00000 1.52128
\(36\) 2.00000 0.333333
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 12.0000 1.94666
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 6.00000 0.925820
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −10.0000 −1.50756
\(45\) −3.00000 −0.447214
\(46\) −18.0000 −2.65396
\(47\) 1.00000 0.145865
\(48\) −4.00000 −0.577350
\(49\) 2.00000 0.285714
\(50\) −8.00000 −1.13137
\(51\) −6.00000 −0.840168
\(52\) 4.00000 0.554700
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −2.00000 −0.272166
\(55\) 15.0000 2.02260
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) −2.00000 −0.262613
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −6.00000 −0.774597
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.00000 0.508001
\(63\) −3.00000 −0.377964
\(64\) −8.00000 −1.00000
\(65\) −6.00000 −0.744208
\(66\) 10.0000 1.23091
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −12.0000 −1.45521
\(69\) 9.00000 1.08347
\(70\) −18.0000 −2.15141
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) 4.00000 0.461880
\(76\) −12.0000 −1.37649
\(77\) 15.0000 1.70941
\(78\) −4.00000 −0.452911
\(79\) −15.0000 −1.68763 −0.843816 0.536633i \(-0.819696\pi\)
−0.843816 + 0.536633i \(0.819696\pi\)
\(80\) 12.0000 1.34164
\(81\) 1.00000 0.111111
\(82\) −12.0000 −1.32518
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −6.00000 −0.654654
\(85\) 18.0000 1.95237
\(86\) −4.00000 −0.431331
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 6.00000 0.632456
\(91\) −6.00000 −0.628971
\(92\) 18.0000 1.87663
\(93\) −2.00000 −0.207390
\(94\) −2.00000 −0.206284
\(95\) 18.0000 1.84676
\(96\) 8.00000 0.816497
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) −4.00000 −0.404061
\(99\) −5.00000 −0.502519
\(100\) 8.00000 0.800000
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 12.0000 1.18818
\(103\) −13.0000 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 0 0
\(105\) 9.00000 0.878310
\(106\) 0 0
\(107\) −17.0000 −1.64345 −0.821726 0.569883i \(-0.806989\pi\)
−0.821726 + 0.569883i \(0.806989\pi\)
\(108\) 2.00000 0.192450
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) −30.0000 −2.86039
\(111\) 1.00000 0.0949158
\(112\) 12.0000 1.13389
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 12.0000 1.12390
\(115\) −27.0000 −2.51776
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) 24.0000 2.20938
\(119\) 18.0000 1.65006
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 4.00000 0.362143
\(123\) 6.00000 0.541002
\(124\) −4.00000 −0.359211
\(125\) 3.00000 0.268328
\(126\) 6.00000 0.534522
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 12.0000 1.05247
\(131\) −22.0000 −1.92215 −0.961074 0.276289i \(-0.910895\pi\)
−0.961074 + 0.276289i \(0.910895\pi\)
\(132\) −10.0000 −0.870388
\(133\) 18.0000 1.56080
\(134\) −4.00000 −0.345547
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −18.0000 −1.53226
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 18.0000 1.52128
\(141\) 1.00000 0.0842152
\(142\) 4.00000 0.335673
\(143\) −10.0000 −0.836242
\(144\) −4.00000 −0.333333
\(145\) −3.00000 −0.249136
\(146\) 4.00000 0.331042
\(147\) 2.00000 0.164957
\(148\) 2.00000 0.164399
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −8.00000 −0.653197
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) −30.0000 −2.41747
\(155\) 6.00000 0.481932
\(156\) 4.00000 0.320256
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) 30.0000 2.38667
\(159\) 0 0
\(160\) −24.0000 −1.89737
\(161\) −27.0000 −2.12790
\(162\) −2.00000 −0.157135
\(163\) 18.0000 1.40987 0.704934 0.709273i \(-0.250976\pi\)
0.704934 + 0.709273i \(0.250976\pi\)
\(164\) 12.0000 0.937043
\(165\) 15.0000 1.16775
\(166\) 8.00000 0.620920
\(167\) −1.00000 −0.0773823 −0.0386912 0.999251i \(-0.512319\pi\)
−0.0386912 + 0.999251i \(0.512319\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −36.0000 −2.76107
\(171\) −6.00000 −0.458831
\(172\) 4.00000 0.304997
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) −2.00000 −0.151620
\(175\) −12.0000 −0.907115
\(176\) 20.0000 1.50756
\(177\) −12.0000 −0.901975
\(178\) −20.0000 −1.49906
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) −6.00000 −0.447214
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 12.0000 0.889499
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 4.00000 0.293294
\(187\) 30.0000 2.19382
\(188\) 2.00000 0.145865
\(189\) −3.00000 −0.218218
\(190\) −36.0000 −2.61171
\(191\) 26.0000 1.88129 0.940647 0.339387i \(-0.110219\pi\)
0.940647 + 0.339387i \(0.110219\pi\)
\(192\) −8.00000 −0.577350
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) −2.00000 −0.143592
\(195\) −6.00000 −0.429669
\(196\) 4.00000 0.285714
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 10.0000 0.710669
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) 8.00000 0.562878
\(203\) −3.00000 −0.210559
\(204\) −12.0000 −0.840168
\(205\) −18.0000 −1.25717
\(206\) 26.0000 1.81151
\(207\) 9.00000 0.625543
\(208\) −8.00000 −0.554700
\(209\) 30.0000 2.07514
\(210\) −18.0000 −1.24212
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) −2.00000 −0.137038
\(214\) 34.0000 2.32419
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) −12.0000 −0.812743
\(219\) −2.00000 −0.135147
\(220\) 30.0000 2.02260
\(221\) −12.0000 −0.807207
\(222\) −2.00000 −0.134231
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) −24.0000 −1.60357
\(225\) 4.00000 0.266667
\(226\) 28.0000 1.86253
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) −12.0000 −0.794719
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 54.0000 3.56065
\(231\) 15.0000 0.986928
\(232\) 0 0
\(233\) 21.0000 1.37576 0.687878 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) −4.00000 −0.261488
\(235\) −3.00000 −0.195698
\(236\) −24.0000 −1.56227
\(237\) −15.0000 −0.974355
\(238\) −36.0000 −2.33353
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 12.0000 0.774597
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) −28.0000 −1.79991
\(243\) 1.00000 0.0641500
\(244\) −4.00000 −0.256074
\(245\) −6.00000 −0.383326
\(246\) −12.0000 −0.765092
\(247\) −12.0000 −0.763542
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) −6.00000 −0.379473
\(251\) 14.0000 0.883672 0.441836 0.897096i \(-0.354327\pi\)
0.441836 + 0.897096i \(0.354327\pi\)
\(252\) −6.00000 −0.377964
\(253\) −45.0000 −2.82913
\(254\) −40.0000 −2.50982
\(255\) 18.0000 1.12720
\(256\) 16.0000 1.00000
\(257\) 11.0000 0.686161 0.343081 0.939306i \(-0.388530\pi\)
0.343081 + 0.939306i \(0.388530\pi\)
\(258\) −4.00000 −0.249029
\(259\) −3.00000 −0.186411
\(260\) −12.0000 −0.744208
\(261\) 1.00000 0.0618984
\(262\) 44.0000 2.71833
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −36.0000 −2.20730
\(267\) 10.0000 0.611990
\(268\) 4.00000 0.244339
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 6.00000 0.365148
\(271\) 9.00000 0.546711 0.273356 0.961913i \(-0.411866\pi\)
0.273356 + 0.961913i \(0.411866\pi\)
\(272\) 24.0000 1.45521
\(273\) −6.00000 −0.363137
\(274\) −12.0000 −0.724947
\(275\) −20.0000 −1.20605
\(276\) 18.0000 1.08347
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 20.0000 1.19952
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) −2.00000 −0.119098
\(283\) −5.00000 −0.297219 −0.148610 0.988896i \(-0.547480\pi\)
−0.148610 + 0.988896i \(0.547480\pi\)
\(284\) −4.00000 −0.237356
\(285\) 18.0000 1.06623
\(286\) 20.0000 1.18262
\(287\) −18.0000 −1.06251
\(288\) 8.00000 0.471405
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) 1.00000 0.0586210
\(292\) −4.00000 −0.234082
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) −4.00000 −0.233285
\(295\) 36.0000 2.09600
\(296\) 0 0
\(297\) −5.00000 −0.290129
\(298\) 0 0
\(299\) 18.0000 1.04097
\(300\) 8.00000 0.461880
\(301\) −6.00000 −0.345834
\(302\) 20.0000 1.15087
\(303\) −4.00000 −0.229794
\(304\) 24.0000 1.37649
\(305\) 6.00000 0.343559
\(306\) 12.0000 0.685994
\(307\) −17.0000 −0.970241 −0.485121 0.874447i \(-0.661224\pi\)
−0.485121 + 0.874447i \(0.661224\pi\)
\(308\) 30.0000 1.70941
\(309\) −13.0000 −0.739544
\(310\) −12.0000 −0.681554
\(311\) 3.00000 0.170114 0.0850572 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 0 0
\(313\) −30.0000 −1.69570 −0.847850 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) −26.0000 −1.46726
\(315\) 9.00000 0.507093
\(316\) −30.0000 −1.68763
\(317\) −17.0000 −0.954815 −0.477408 0.878682i \(-0.658423\pi\)
−0.477408 + 0.878682i \(0.658423\pi\)
\(318\) 0 0
\(319\) −5.00000 −0.279946
\(320\) 24.0000 1.34164
\(321\) −17.0000 −0.948847
\(322\) 54.0000 3.00930
\(323\) 36.0000 2.00309
\(324\) 2.00000 0.111111
\(325\) 8.00000 0.443760
\(326\) −36.0000 −1.99386
\(327\) 6.00000 0.331801
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) −30.0000 −1.65145
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −8.00000 −0.439057
\(333\) 1.00000 0.0547997
\(334\) 2.00000 0.109435
\(335\) −6.00000 −0.327815
\(336\) 12.0000 0.654654
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) 18.0000 0.979071
\(339\) −14.0000 −0.760376
\(340\) 36.0000 1.95237
\(341\) 10.0000 0.541530
\(342\) 12.0000 0.648886
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) −27.0000 −1.45363
\(346\) 0 0
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 2.00000 0.107211
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 24.0000 1.28285
\(351\) 2.00000 0.106752
\(352\) −40.0000 −2.13201
\(353\) 36.0000 1.91609 0.958043 0.286623i \(-0.0925328\pi\)
0.958043 + 0.286623i \(0.0925328\pi\)
\(354\) 24.0000 1.27559
\(355\) 6.00000 0.318447
\(356\) 20.0000 1.06000
\(357\) 18.0000 0.952661
\(358\) 18.0000 0.951330
\(359\) 3.00000 0.158334 0.0791670 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −24.0000 −1.26141
\(363\) 14.0000 0.734809
\(364\) −12.0000 −0.628971
\(365\) 6.00000 0.314054
\(366\) 4.00000 0.209083
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) −36.0000 −1.87663
\(369\) 6.00000 0.312348
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) −4.00000 −0.207390
\(373\) 20.0000 1.03556 0.517780 0.855514i \(-0.326758\pi\)
0.517780 + 0.855514i \(0.326758\pi\)
\(374\) −60.0000 −3.10253
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) 2.00000 0.103005
\(378\) 6.00000 0.308607
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 36.0000 1.84676
\(381\) 20.0000 1.02463
\(382\) −52.0000 −2.66055
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) −45.0000 −2.29341
\(386\) −32.0000 −1.62876
\(387\) 2.00000 0.101666
\(388\) 2.00000 0.101535
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 12.0000 0.607644
\(391\) −54.0000 −2.73090
\(392\) 0 0
\(393\) −22.0000 −1.10975
\(394\) 48.0000 2.41821
\(395\) 45.0000 2.26420
\(396\) −10.0000 −0.502519
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) −20.0000 −1.00251
\(399\) 18.0000 0.901127
\(400\) −16.0000 −0.800000
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) −4.00000 −0.199502
\(403\) −4.00000 −0.199254
\(404\) −8.00000 −0.398015
\(405\) −3.00000 −0.149071
\(406\) 6.00000 0.297775
\(407\) −5.00000 −0.247841
\(408\) 0 0
\(409\) 16.0000 0.791149 0.395575 0.918434i \(-0.370545\pi\)
0.395575 + 0.918434i \(0.370545\pi\)
\(410\) 36.0000 1.77791
\(411\) 6.00000 0.295958
\(412\) −26.0000 −1.28093
\(413\) 36.0000 1.77144
\(414\) −18.0000 −0.884652
\(415\) 12.0000 0.589057
\(416\) 16.0000 0.784465
\(417\) −10.0000 −0.489702
\(418\) −60.0000 −2.93470
\(419\) 3.00000 0.146560 0.0732798 0.997311i \(-0.476653\pi\)
0.0732798 + 0.997311i \(0.476653\pi\)
\(420\) 18.0000 0.878310
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 32.0000 1.55774
\(423\) 1.00000 0.0486217
\(424\) 0 0
\(425\) −24.0000 −1.16417
\(426\) 4.00000 0.193801
\(427\) 6.00000 0.290360
\(428\) −34.0000 −1.64345
\(429\) −10.0000 −0.482805
\(430\) 12.0000 0.578691
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) −4.00000 −0.192450
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) −12.0000 −0.576018
\(435\) −3.00000 −0.143839
\(436\) 12.0000 0.574696
\(437\) −54.0000 −2.58317
\(438\) 4.00000 0.191127
\(439\) −15.0000 −0.715911 −0.357955 0.933739i \(-0.616526\pi\)
−0.357955 + 0.933739i \(0.616526\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 24.0000 1.14156
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 2.00000 0.0949158
\(445\) −30.0000 −1.42214
\(446\) 24.0000 1.13643
\(447\) 0 0
\(448\) 24.0000 1.13389
\(449\) 5.00000 0.235965 0.117982 0.993016i \(-0.462357\pi\)
0.117982 + 0.993016i \(0.462357\pi\)
\(450\) −8.00000 −0.377124
\(451\) −30.0000 −1.41264
\(452\) −28.0000 −1.31701
\(453\) −10.0000 −0.469841
\(454\) 6.00000 0.281594
\(455\) 18.0000 0.843853
\(456\) 0 0
\(457\) 11.0000 0.514558 0.257279 0.966337i \(-0.417174\pi\)
0.257279 + 0.966337i \(0.417174\pi\)
\(458\) 0 0
\(459\) −6.00000 −0.280056
\(460\) −54.0000 −2.51776
\(461\) −27.0000 −1.25752 −0.628758 0.777601i \(-0.716436\pi\)
−0.628758 + 0.777601i \(0.716436\pi\)
\(462\) −30.0000 −1.39573
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −4.00000 −0.185695
\(465\) 6.00000 0.278243
\(466\) −42.0000 −1.94561
\(467\) −23.0000 −1.06431 −0.532157 0.846646i \(-0.678618\pi\)
−0.532157 + 0.846646i \(0.678618\pi\)
\(468\) 4.00000 0.184900
\(469\) −6.00000 −0.277054
\(470\) 6.00000 0.276759
\(471\) 13.0000 0.599008
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 30.0000 1.37795
\(475\) −24.0000 −1.10120
\(476\) 36.0000 1.65006
\(477\) 0 0
\(478\) 8.00000 0.365911
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −24.0000 −1.09545
\(481\) 2.00000 0.0911922
\(482\) 10.0000 0.455488
\(483\) −27.0000 −1.22854
\(484\) 28.0000 1.27273
\(485\) −3.00000 −0.136223
\(486\) −2.00000 −0.0907218
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) 18.0000 0.813988
\(490\) 12.0000 0.542105
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 12.0000 0.541002
\(493\) −6.00000 −0.270226
\(494\) 24.0000 1.07981
\(495\) 15.0000 0.674200
\(496\) 8.00000 0.359211
\(497\) 6.00000 0.269137
\(498\) 8.00000 0.358489
\(499\) 2.00000 0.0895323 0.0447661 0.998997i \(-0.485746\pi\)
0.0447661 + 0.998997i \(0.485746\pi\)
\(500\) 6.00000 0.268328
\(501\) −1.00000 −0.0446767
\(502\) −28.0000 −1.24970
\(503\) 9.00000 0.401290 0.200645 0.979664i \(-0.435696\pi\)
0.200645 + 0.979664i \(0.435696\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 90.0000 4.00099
\(507\) −9.00000 −0.399704
\(508\) 40.0000 1.77471
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) −36.0000 −1.59411
\(511\) 6.00000 0.265424
\(512\) −32.0000 −1.41421
\(513\) −6.00000 −0.264906
\(514\) −22.0000 −0.970378
\(515\) 39.0000 1.71855
\(516\) 4.00000 0.176090
\(517\) −5.00000 −0.219900
\(518\) 6.00000 0.263625
\(519\) 0 0
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 11.0000 0.480996 0.240498 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) −44.0000 −1.92215
\(525\) −12.0000 −0.523723
\(526\) 24.0000 1.04645
\(527\) 12.0000 0.522728
\(528\) 20.0000 0.870388
\(529\) 58.0000 2.52174
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 36.0000 1.56080
\(533\) 12.0000 0.519778
\(534\) −20.0000 −0.865485
\(535\) 51.0000 2.20492
\(536\) 0 0
\(537\) −9.00000 −0.388379
\(538\) 8.00000 0.344904
\(539\) −10.0000 −0.430730
\(540\) −6.00000 −0.258199
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) −18.0000 −0.773166
\(543\) 12.0000 0.514969
\(544\) −48.0000 −2.05798
\(545\) −18.0000 −0.771035
\(546\) 12.0000 0.513553
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 12.0000 0.512615
\(549\) −2.00000 −0.0853579
\(550\) 40.0000 1.70561
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 45.0000 1.91359
\(554\) −20.0000 −0.849719
\(555\) −3.00000 −0.127343
\(556\) −20.0000 −0.848189
\(557\) 1.00000 0.0423714 0.0211857 0.999776i \(-0.493256\pi\)
0.0211857 + 0.999776i \(0.493256\pi\)
\(558\) 4.00000 0.169334
\(559\) 4.00000 0.169182
\(560\) −36.0000 −1.52128
\(561\) 30.0000 1.26660
\(562\) 30.0000 1.26547
\(563\) 8.00000 0.337160 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(564\) 2.00000 0.0842152
\(565\) 42.0000 1.76695
\(566\) 10.0000 0.420331
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) 21.0000 0.880366 0.440183 0.897908i \(-0.354914\pi\)
0.440183 + 0.897908i \(0.354914\pi\)
\(570\) −36.0000 −1.50787
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −20.0000 −0.836242
\(573\) 26.0000 1.08617
\(574\) 36.0000 1.50261
\(575\) 36.0000 1.50130
\(576\) −8.00000 −0.333333
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −38.0000 −1.58059
\(579\) 16.0000 0.664937
\(580\) −6.00000 −0.249136
\(581\) 12.0000 0.497844
\(582\) −2.00000 −0.0829027
\(583\) 0 0
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) −18.0000 −0.743573
\(587\) 20.0000 0.825488 0.412744 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) 4.00000 0.164957
\(589\) 12.0000 0.494451
\(590\) −72.0000 −2.96419
\(591\) −24.0000 −0.987228
\(592\) −4.00000 −0.164399
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 10.0000 0.410305
\(595\) −54.0000 −2.21378
\(596\) 0 0
\(597\) 10.0000 0.409273
\(598\) −36.0000 −1.47215
\(599\) 19.0000 0.776319 0.388159 0.921592i \(-0.373111\pi\)
0.388159 + 0.921592i \(0.373111\pi\)
\(600\) 0 0
\(601\) −25.0000 −1.01977 −0.509886 0.860242i \(-0.670312\pi\)
−0.509886 + 0.860242i \(0.670312\pi\)
\(602\) 12.0000 0.489083
\(603\) 2.00000 0.0814463
\(604\) −20.0000 −0.813788
\(605\) −42.0000 −1.70754
\(606\) 8.00000 0.324978
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) −48.0000 −1.94666
\(609\) −3.00000 −0.121566
\(610\) −12.0000 −0.485866
\(611\) 2.00000 0.0809113
\(612\) −12.0000 −0.485071
\(613\) 3.00000 0.121169 0.0605844 0.998163i \(-0.480704\pi\)
0.0605844 + 0.998163i \(0.480704\pi\)
\(614\) 34.0000 1.37213
\(615\) −18.0000 −0.725830
\(616\) 0 0
\(617\) −16.0000 −0.644136 −0.322068 0.946717i \(-0.604378\pi\)
−0.322068 + 0.946717i \(0.604378\pi\)
\(618\) 26.0000 1.04587
\(619\) −35.0000 −1.40677 −0.703384 0.710810i \(-0.748329\pi\)
−0.703384 + 0.710810i \(0.748329\pi\)
\(620\) 12.0000 0.481932
\(621\) 9.00000 0.361158
\(622\) −6.00000 −0.240578
\(623\) −30.0000 −1.20192
\(624\) −8.00000 −0.320256
\(625\) −29.0000 −1.16000
\(626\) 60.0000 2.39808
\(627\) 30.0000 1.19808
\(628\) 26.0000 1.03751
\(629\) −6.00000 −0.239236
\(630\) −18.0000 −0.717137
\(631\) 34.0000 1.35352 0.676759 0.736204i \(-0.263384\pi\)
0.676759 + 0.736204i \(0.263384\pi\)
\(632\) 0 0
\(633\) −16.0000 −0.635943
\(634\) 34.0000 1.35031
\(635\) −60.0000 −2.38103
\(636\) 0 0
\(637\) 4.00000 0.158486
\(638\) 10.0000 0.395904
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) −45.0000 −1.77739 −0.888697 0.458496i \(-0.848388\pi\)
−0.888697 + 0.458496i \(0.848388\pi\)
\(642\) 34.0000 1.34187
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) −54.0000 −2.12790
\(645\) −6.00000 −0.236250
\(646\) −72.0000 −2.83280
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) 60.0000 2.35521
\(650\) −16.0000 −0.627572
\(651\) 6.00000 0.235159
\(652\) 36.0000 1.40987
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −12.0000 −0.469237
\(655\) 66.0000 2.57883
\(656\) −24.0000 −0.937043
\(657\) −2.00000 −0.0780274
\(658\) 6.00000 0.233904
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 30.0000 1.16775
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) −40.0000 −1.55464
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) −54.0000 −2.09403
\(666\) −2.00000 −0.0774984
\(667\) 9.00000 0.348481
\(668\) −2.00000 −0.0773823
\(669\) −12.0000 −0.463947
\(670\) 12.0000 0.463600
\(671\) 10.0000 0.386046
\(672\) −24.0000 −0.925820
\(673\) 6.00000 0.231283 0.115642 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) 26.0000 1.00148
\(675\) 4.00000 0.153960
\(676\) −18.0000 −0.692308
\(677\) 46.0000 1.76792 0.883962 0.467559i \(-0.154866\pi\)
0.883962 + 0.467559i \(0.154866\pi\)
\(678\) 28.0000 1.07533
\(679\) −3.00000 −0.115129
\(680\) 0 0
\(681\) −3.00000 −0.114960
\(682\) −20.0000 −0.765840
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) −12.0000 −0.458831
\(685\) −18.0000 −0.687745
\(686\) −30.0000 −1.14541
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) 54.0000 2.05574
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 0 0
\(693\) 15.0000 0.569803
\(694\) 56.0000 2.12573
\(695\) 30.0000 1.13796
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 56.0000 2.11963
\(699\) 21.0000 0.794293
\(700\) −24.0000 −0.907115
\(701\) −43.0000 −1.62409 −0.812044 0.583597i \(-0.801645\pi\)
−0.812044 + 0.583597i \(0.801645\pi\)
\(702\) −4.00000 −0.150970
\(703\) −6.00000 −0.226294
\(704\) 40.0000 1.50756
\(705\) −3.00000 −0.112987
\(706\) −72.0000 −2.70976
\(707\) 12.0000 0.451306
\(708\) −24.0000 −0.901975
\(709\) −43.0000 −1.61490 −0.807449 0.589937i \(-0.799153\pi\)
−0.807449 + 0.589937i \(0.799153\pi\)
\(710\) −12.0000 −0.450352
\(711\) −15.0000 −0.562544
\(712\) 0 0
\(713\) −18.0000 −0.674105
\(714\) −36.0000 −1.34727
\(715\) 30.0000 1.12194
\(716\) −18.0000 −0.672692
\(717\) −4.00000 −0.149383
\(718\) −6.00000 −0.223918
\(719\) −26.0000 −0.969636 −0.484818 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) 12.0000 0.447214
\(721\) 39.0000 1.45244
\(722\) −34.0000 −1.26535
\(723\) −5.00000 −0.185952
\(724\) 24.0000 0.891953
\(725\) 4.00000 0.148556
\(726\) −28.0000 −1.03918
\(727\) 46.0000 1.70605 0.853023 0.521874i \(-0.174767\pi\)
0.853023 + 0.521874i \(0.174767\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −12.0000 −0.444140
\(731\) −12.0000 −0.443836
\(732\) −4.00000 −0.147844
\(733\) −7.00000 −0.258551 −0.129275 0.991609i \(-0.541265\pi\)
−0.129275 + 0.991609i \(0.541265\pi\)
\(734\) 48.0000 1.77171
\(735\) −6.00000 −0.221313
\(736\) 72.0000 2.65396
\(737\) −10.0000 −0.368355
\(738\) −12.0000 −0.441726
\(739\) 41.0000 1.50821 0.754105 0.656754i \(-0.228071\pi\)
0.754105 + 0.656754i \(0.228071\pi\)
\(740\) −6.00000 −0.220564
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −40.0000 −1.46450
\(747\) −4.00000 −0.146352
\(748\) 60.0000 2.19382
\(749\) 51.0000 1.86350
\(750\) −6.00000 −0.219089
\(751\) −2.00000 −0.0729810 −0.0364905 0.999334i \(-0.511618\pi\)
−0.0364905 + 0.999334i \(0.511618\pi\)
\(752\) −4.00000 −0.145865
\(753\) 14.0000 0.510188
\(754\) −4.00000 −0.145671
\(755\) 30.0000 1.09181
\(756\) −6.00000 −0.218218
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 10.0000 0.363216
\(759\) −45.0000 −1.63340
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −40.0000 −1.44905
\(763\) −18.0000 −0.651644
\(764\) 52.0000 1.88129
\(765\) 18.0000 0.650791
\(766\) 16.0000 0.578103
\(767\) −24.0000 −0.866590
\(768\) 16.0000 0.577350
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 90.0000 3.24337
\(771\) 11.0000 0.396155
\(772\) 32.0000 1.15171
\(773\) 12.0000 0.431610 0.215805 0.976436i \(-0.430762\pi\)
0.215805 + 0.976436i \(0.430762\pi\)
\(774\) −4.00000 −0.143777
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) −3.00000 −0.107624
\(778\) −4.00000 −0.143407
\(779\) −36.0000 −1.28983
\(780\) −12.0000 −0.429669
\(781\) 10.0000 0.357828
\(782\) 108.000 3.86207
\(783\) 1.00000 0.0357371
\(784\) −8.00000 −0.285714
\(785\) −39.0000 −1.39197
\(786\) 44.0000 1.56943
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) −48.0000 −1.70993
\(789\) −12.0000 −0.427211
\(790\) −90.0000 −3.20206
\(791\) 42.0000 1.49335
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 36.0000 1.27759
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −33.0000 −1.16892 −0.584460 0.811423i \(-0.698694\pi\)
−0.584460 + 0.811423i \(0.698694\pi\)
\(798\) −36.0000 −1.27439
\(799\) −6.00000 −0.212265
\(800\) 32.0000 1.13137
\(801\) 10.0000 0.353333
\(802\) −24.0000 −0.847469
\(803\) 10.0000 0.352892
\(804\) 4.00000 0.141069
\(805\) 81.0000 2.85487
\(806\) 8.00000 0.281788
\(807\) −4.00000 −0.140807
\(808\) 0 0
\(809\) 31.0000 1.08990 0.544951 0.838468i \(-0.316548\pi\)
0.544951 + 0.838468i \(0.316548\pi\)
\(810\) 6.00000 0.210819
\(811\) −37.0000 −1.29925 −0.649623 0.760257i \(-0.725073\pi\)
−0.649623 + 0.760257i \(0.725073\pi\)
\(812\) −6.00000 −0.210559
\(813\) 9.00000 0.315644
\(814\) 10.0000 0.350500
\(815\) −54.0000 −1.89154
\(816\) 24.0000 0.840168
\(817\) −12.0000 −0.419827
\(818\) −32.0000 −1.11885
\(819\) −6.00000 −0.209657
\(820\) −36.0000 −1.25717
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) −12.0000 −0.418548
\(823\) −39.0000 −1.35945 −0.679727 0.733465i \(-0.737902\pi\)
−0.679727 + 0.733465i \(0.737902\pi\)
\(824\) 0 0
\(825\) −20.0000 −0.696311
\(826\) −72.0000 −2.50520
\(827\) −44.0000 −1.53003 −0.765015 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 18.0000 0.625543
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) −24.0000 −0.833052
\(831\) 10.0000 0.346896
\(832\) −16.0000 −0.554700
\(833\) −12.0000 −0.415775
\(834\) 20.0000 0.692543
\(835\) 3.00000 0.103819
\(836\) 60.0000 2.07514
\(837\) −2.00000 −0.0691301
\(838\) −6.00000 −0.207267
\(839\) −21.0000 −0.725001 −0.362500 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −40.0000 −1.37849
\(843\) −15.0000 −0.516627
\(844\) −32.0000 −1.10149
\(845\) 27.0000 0.928828
\(846\) −2.00000 −0.0687614
\(847\) −42.0000 −1.44314
\(848\) 0 0
\(849\) −5.00000 −0.171600
\(850\) 48.0000 1.64639
\(851\) 9.00000 0.308516
\(852\) −4.00000 −0.137038
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) −12.0000 −0.410632
\(855\) 18.0000 0.615587
\(856\) 0 0
\(857\) 41.0000 1.40053 0.700267 0.713881i \(-0.253064\pi\)
0.700267 + 0.713881i \(0.253064\pi\)
\(858\) 20.0000 0.682789
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) −12.0000 −0.409197
\(861\) −18.0000 −0.613438
\(862\) −36.0000 −1.22616
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 8.00000 0.272166
\(865\) 0 0
\(866\) 0 0
\(867\) 19.0000 0.645274
\(868\) 12.0000 0.407307
\(869\) 75.0000 2.54420
\(870\) 6.00000 0.203419
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) 1.00000 0.0338449
\(874\) 108.000 3.65315
\(875\) −9.00000 −0.304256
\(876\) −4.00000 −0.135147
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 30.0000 1.01245
\(879\) 9.00000 0.303562
\(880\) −60.0000 −2.02260
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) −4.00000 −0.134687
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) −24.0000 −0.807207
\(885\) 36.0000 1.21013
\(886\) 24.0000 0.806296
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) −60.0000 −2.01234
\(890\) 60.0000 2.01120
\(891\) −5.00000 −0.167506
\(892\) −24.0000 −0.803579
\(893\) −6.00000 −0.200782
\(894\) 0 0
\(895\) 27.0000 0.902510
\(896\) 0 0
\(897\) 18.0000 0.601003
\(898\) −10.0000 −0.333704
\(899\) −2.00000 −0.0667037
\(900\) 8.00000 0.266667
\(901\) 0 0
\(902\) 60.0000 1.99778
\(903\) −6.00000 −0.199667
\(904\) 0 0
\(905\) −36.0000 −1.19668
\(906\) 20.0000 0.664455
\(907\) 13.0000 0.431658 0.215829 0.976431i \(-0.430755\pi\)
0.215829 + 0.976431i \(0.430755\pi\)
\(908\) −6.00000 −0.199117
\(909\) −4.00000 −0.132672
\(910\) −36.0000 −1.19339
\(911\) 22.0000 0.728893 0.364446 0.931224i \(-0.381258\pi\)
0.364446 + 0.931224i \(0.381258\pi\)
\(912\) 24.0000 0.794719
\(913\) 20.0000 0.661903
\(914\) −22.0000 −0.727695
\(915\) 6.00000 0.198354
\(916\) 0 0
\(917\) 66.0000 2.17951
\(918\) 12.0000 0.396059
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) −17.0000 −0.560169
\(922\) 54.0000 1.77840
\(923\) −4.00000 −0.131662
\(924\) 30.0000 0.986928
\(925\) 4.00000 0.131519
\(926\) 16.0000 0.525793
\(927\) −13.0000 −0.426976
\(928\) 8.00000 0.262613
\(929\) −50.0000 −1.64045 −0.820223 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) −12.0000 −0.393496
\(931\) −12.0000 −0.393284
\(932\) 42.0000 1.37576
\(933\) 3.00000 0.0982156
\(934\) 46.0000 1.50517
\(935\) −90.0000 −2.94331
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 12.0000 0.391814
\(939\) −30.0000 −0.979013
\(940\) −6.00000 −0.195698
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) −26.0000 −0.847126
\(943\) 54.0000 1.75848
\(944\) 48.0000 1.56227
\(945\) 9.00000 0.292770
\(946\) 20.0000 0.650256
\(947\) 58.0000 1.88475 0.942373 0.334563i \(-0.108589\pi\)
0.942373 + 0.334563i \(0.108589\pi\)
\(948\) −30.0000 −0.974355
\(949\) −4.00000 −0.129845
\(950\) 48.0000 1.55733
\(951\) −17.0000 −0.551263
\(952\) 0 0
\(953\) −11.0000 −0.356325 −0.178162 0.984001i \(-0.557015\pi\)
−0.178162 + 0.984001i \(0.557015\pi\)
\(954\) 0 0
\(955\) −78.0000 −2.52402
\(956\) −8.00000 −0.258738
\(957\) −5.00000 −0.161627
\(958\) −48.0000 −1.55081
\(959\) −18.0000 −0.581250
\(960\) 24.0000 0.774597
\(961\) −27.0000 −0.870968
\(962\) −4.00000 −0.128965
\(963\) −17.0000 −0.547817
\(964\) −10.0000 −0.322078
\(965\) −48.0000 −1.54517
\(966\) 54.0000 1.73742
\(967\) 43.0000 1.38279 0.691393 0.722478i \(-0.256997\pi\)
0.691393 + 0.722478i \(0.256997\pi\)
\(968\) 0 0
\(969\) 36.0000 1.15649
\(970\) 6.00000 0.192648
\(971\) −61.0000 −1.95758 −0.978792 0.204859i \(-0.934327\pi\)
−0.978792 + 0.204859i \(0.934327\pi\)
\(972\) 2.00000 0.0641500
\(973\) 30.0000 0.961756
\(974\) 16.0000 0.512673
\(975\) 8.00000 0.256205
\(976\) 8.00000 0.256074
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) −36.0000 −1.15115
\(979\) −50.0000 −1.59801
\(980\) −12.0000 −0.383326
\(981\) 6.00000 0.191565
\(982\) −56.0000 −1.78703
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) 72.0000 2.29411
\(986\) 12.0000 0.382158
\(987\) −3.00000 −0.0954911
\(988\) −24.0000 −0.763542
\(989\) 18.0000 0.572367
\(990\) −30.0000 −0.953463
\(991\) −33.0000 −1.04828 −0.524140 0.851632i \(-0.675613\pi\)
−0.524140 + 0.851632i \(0.675613\pi\)
\(992\) −16.0000 −0.508001
\(993\) 20.0000 0.634681
\(994\) −12.0000 −0.380617
\(995\) −30.0000 −0.951064
\(996\) −8.00000 −0.253490
\(997\) 20.0000 0.633406 0.316703 0.948525i \(-0.397424\pi\)
0.316703 + 0.948525i \(0.397424\pi\)
\(998\) −4.00000 −0.126618
\(999\) 1.00000 0.0316386
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 141.2.a.a.1.1 1
3.2 odd 2 423.2.a.f.1.1 1
4.3 odd 2 2256.2.a.c.1.1 1
5.4 even 2 3525.2.a.m.1.1 1
7.6 odd 2 6909.2.a.a.1.1 1
8.3 odd 2 9024.2.a.bv.1.1 1
8.5 even 2 9024.2.a.t.1.1 1
12.11 even 2 6768.2.a.t.1.1 1
47.46 odd 2 6627.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
141.2.a.a.1.1 1 1.1 even 1 trivial
423.2.a.f.1.1 1 3.2 odd 2
2256.2.a.c.1.1 1 4.3 odd 2
3525.2.a.m.1.1 1 5.4 even 2
6627.2.a.a.1.1 1 47.46 odd 2
6768.2.a.t.1.1 1 12.11 even 2
6909.2.a.a.1.1 1 7.6 odd 2
9024.2.a.t.1.1 1 8.5 even 2
9024.2.a.bv.1.1 1 8.3 odd 2