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Properties

Label 1960.2.a.m.1.1
Level $1960$
Weight $2$
Character 1960.1
Self dual yes
Analytic conductor $15.651$
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] = [1960,2,Mod(1,1960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.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: \(15.6506787962\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} -1.00000 q^{11} +3.00000 q^{13} +2.00000 q^{15} +2.00000 q^{17} +5.00000 q^{19} +7.00000 q^{23} +1.00000 q^{25} -4.00000 q^{27} -6.00000 q^{29} -4.00000 q^{31} -2.00000 q^{33} -5.00000 q^{37} +6.00000 q^{39} +5.00000 q^{41} +6.00000 q^{43} +1.00000 q^{45} +9.00000 q^{47} +4.00000 q^{51} +11.0000 q^{53} -1.00000 q^{55} +10.0000 q^{57} -8.00000 q^{59} +12.0000 q^{61} +3.00000 q^{65} -4.00000 q^{67} +14.0000 q^{69} -4.00000 q^{71} -12.0000 q^{73} +2.00000 q^{75} +14.0000 q^{79} -11.0000 q^{81} +4.00000 q^{83} +2.00000 q^{85} -12.0000 q^{87} -6.00000 q^{89} -8.00000 q^{93} +5.00000 q^{95} -6.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 − 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 − 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.00000 1.45960 0.729800 − 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 − 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 − 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 11.0000 1.51097 0.755483 − 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 10.0000 1.32453
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 − 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 14.0000 1.68540
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −12.0000 −1.40449 −0.702247 − 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 0 0
\(75\) 2.00000 0.230940
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 − 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 5.00000 0.512989
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 − 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) −20.0000 −1.97066 −0.985329 − 0.170664i \(-0.945409\pi\)
−0.985329 + 0.170664i \(0.945409\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 − 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 20.0000 1.88144 0.940721 − 0.339182i \(-0.110150\pi\)
0.940721 + 0.339182i \(0.110150\pi\)
\(114\) 0 0
\(115\) 7.00000 0.652753
\(116\) 0 0
\(117\) 3.00000 0.277350
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 17.0000 1.50851 0.754253 − 0.656584i \(-0.227999\pi\)
0.754253 + 0.656584i \(0.227999\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) −7.00000 −0.611593 −0.305796 − 0.952097i \(-0.598923\pi\)
−0.305796 + 0.952097i \(0.598923\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 18.0000 1.51587
\(142\) 0 0
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.0000 0.819232 0.409616 − 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 − 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −5.00000 −0.399043 −0.199522 − 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(158\) 0 0
\(159\) 22.0000 1.74471
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) −2.00000 −0.155700
\(166\) 0 0
\(167\) −5.00000 −0.386912 −0.193456 − 0.981109i \(-0.561970\pi\)
−0.193456 + 0.981109i \(0.561970\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 5.00000 0.382360
\(172\) 0 0
\(173\) −19.0000 −1.44454 −0.722272 − 0.691609i \(-0.756902\pi\)
−0.722272 + 0.691609i \(0.756902\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −16.0000 −1.20263
\(178\) 0 0
\(179\) 9.00000 0.672692 0.336346 − 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 24.0000 1.77413
\(184\) 0 0
\(185\) −5.00000 −0.367607
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 − 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −20.0000 −1.43963 −0.719816 − 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) 0 0
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) −27.0000 −1.92367 −0.961835 − 0.273629i \(-0.911776\pi\)
−0.961835 + 0.273629i \(0.911776\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 − 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 5.00000 0.349215
\(206\) 0 0
\(207\) 7.00000 0.486534
\(208\) 0 0
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 − 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 0 0
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −24.0000 −1.62177
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 8.00000 0.530979 0.265489 − 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) 28.0000 1.85029 0.925146 − 0.379611i \(-0.123942\pi\)
0.925146 + 0.379611i \(0.123942\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.00000 0.131024 0.0655122 − 0.997852i \(-0.479132\pi\)
0.0655122 + 0.997852i \(0.479132\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 0 0
\(237\) 28.0000 1.81880
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 − 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) −23.0000 −1.48156 −0.740780 − 0.671748i \(-0.765544\pi\)
−0.740780 + 0.671748i \(0.765544\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 15.0000 0.954427
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) −29.0000 −1.83046 −0.915232 − 0.402928i \(-0.867993\pi\)
−0.915232 + 0.402928i \(0.867993\pi\)
\(252\) 0 0
\(253\) −7.00000 −0.440086
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) 0 0
\(257\) −12.0000 −0.748539 −0.374270 − 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 − 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) 11.0000 0.675725
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 0 0
\(269\) 12.0000 0.731653 0.365826 − 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 − 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −3.00000 −0.178965 −0.0894825 − 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) 0 0
\(283\) 22.0000 1.30776 0.653882 − 0.756596i \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) 0 0
\(285\) 10.0000 0.592349
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) 0 0
\(293\) −21.0000 −1.22683 −0.613417 − 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) 21.0000 1.21446
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −24.0000 −1.37876
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 6.00000 0.342438 0.171219 − 0.985233i \(-0.445229\pi\)
0.171219 + 0.985233i \(0.445229\pi\)
\(308\) 0 0
\(309\) −40.0000 −2.27552
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 − 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −16.0000 −0.904373 −0.452187 − 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 − 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) 10.0000 0.556415
\(324\) 0 0
\(325\) 3.00000 0.166410
\(326\) 0 0
\(327\) −8.00000 −0.442401
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −27.0000 −1.48405 −0.742027 − 0.670370i \(-0.766135\pi\)
−0.742027 + 0.670370i \(0.766135\pi\)
\(332\) 0 0
\(333\) −5.00000 −0.273998
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 − 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 0 0
\(339\) 40.0000 2.17250
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 14.0000 0.753735
\(346\) 0 0
\(347\) 8.00000 0.429463 0.214731 − 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −12.0000 −0.640513
\(352\) 0 0
\(353\) −20.0000 −1.06449 −0.532246 − 0.846590i \(-0.678652\pi\)
−0.532246 + 0.846590i \(0.678652\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 30.0000 1.58334 0.791670 − 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) −20.0000 −1.04973
\(364\) 0 0
\(365\) −12.0000 −0.628109
\(366\) 0 0
\(367\) −19.0000 −0.991792 −0.495896 − 0.868382i \(-0.665160\pi\)
−0.495896 + 0.868382i \(0.665160\pi\)
\(368\) 0 0
\(369\) 5.00000 0.260290
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 2.00000 0.103280
\(376\) 0 0
\(377\) −18.0000 −0.927047
\(378\) 0 0
\(379\) −21.0000 −1.07870 −0.539349 − 0.842082i \(-0.681330\pi\)
−0.539349 + 0.842082i \(0.681330\pi\)
\(380\) 0 0
\(381\) 34.0000 1.74187
\(382\) 0 0
\(383\) 21.0000 1.07305 0.536525 − 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.00000 0.304997
\(388\) 0 0
\(389\) −16.0000 −0.811232 −0.405616 − 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) 14.0000 0.708010
\(392\) 0 0
\(393\) −14.0000 −0.706207
\(394\) 0 0
\(395\) 14.0000 0.704416
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 − 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.0000 0.649189 0.324595 − 0.945853i \(-0.394772\pi\)
0.324595 + 0.945853i \(0.394772\pi\)
\(402\) 0 0
\(403\) −12.0000 −0.597763
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) 5.00000 0.247841
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 − 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −24.0000 −1.18383
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) 5.00000 0.244266 0.122133 − 0.992514i \(-0.461027\pi\)
0.122133 + 0.992514i \(0.461027\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 − 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) 9.00000 0.437595
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 − 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) −24.0000 −1.15337 −0.576683 − 0.816968i \(-0.695653\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 0 0
\(435\) −12.0000 −0.575356
\(436\) 0 0
\(437\) 35.0000 1.67428
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 − 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −32.0000 −1.52037 −0.760183 − 0.649709i \(-0.774891\pi\)
−0.760183 + 0.649709i \(0.774891\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) 20.0000 0.945968
\(448\) 0 0
\(449\) 13.0000 0.613508 0.306754 − 0.951789i \(-0.400757\pi\)
0.306754 + 0.951789i \(0.400757\pi\)
\(450\) 0 0
\(451\) −5.00000 −0.235441
\(452\) 0 0
\(453\) −20.0000 −0.939682
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 − 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 9.00000 0.418265 0.209133 − 0.977887i \(-0.432936\pi\)
0.209133 + 0.977887i \(0.432936\pi\)
\(464\) 0 0
\(465\) −8.00000 −0.370991
\(466\) 0 0
\(467\) 10.0000 0.462745 0.231372 − 0.972865i \(-0.425678\pi\)
0.231372 + 0.972865i \(0.425678\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 5.00000 0.229416
\(476\) 0 0
\(477\) 11.0000 0.503655
\(478\) 0 0
\(479\) 16.0000 0.731059 0.365529 − 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −15.0000 −0.683941
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 − 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 − 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) −10.0000 −0.446767
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) −8.00000 −0.355292
\(508\) 0 0
\(509\) 18.0000 0.797836 0.398918 − 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −20.0000 −0.883022
\(514\) 0 0
\(515\) −20.0000 −0.881305
\(516\) 0 0
\(517\) −9.00000 −0.395820
\(518\) 0 0
\(519\) −38.0000 −1.66801
\(520\) 0 0
\(521\) 15.0000 0.657162 0.328581 − 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 − 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 15.0000 0.649722
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 18.0000 0.776757
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −40.0000 −1.71973 −0.859867 − 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) 0 0
\(543\) −4.00000 −0.171656
\(544\) 0 0
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 − 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) −30.0000 −1.27804
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −10.0000 −0.424476
\(556\) 0 0
\(557\) 21.0000 0.889799 0.444899 − 0.895581i \(-0.353239\pi\)
0.444899 + 0.895581i \(0.353239\pi\)
\(558\) 0 0
\(559\) 18.0000 0.761319
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) −6.00000 −0.252870 −0.126435 − 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 0 0
\(565\) 20.0000 0.841406
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −11.0000 −0.461144 −0.230572 − 0.973055i \(-0.574060\pi\)
−0.230572 + 0.973055i \(0.574060\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 − 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 24.0000 1.00261
\(574\) 0 0
\(575\) 7.00000 0.291920
\(576\) 0 0
\(577\) 4.00000 0.166522 0.0832611 − 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) 0 0
\(579\) −40.0000 −1.66234
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −11.0000 −0.455573
\(584\) 0 0
\(585\) 3.00000 0.124035
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) −54.0000 −2.22126
\(592\) 0 0
\(593\) 12.0000 0.492781 0.246390 − 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) 34.0000 1.38920 0.694601 − 0.719395i \(-0.255581\pi\)
0.694601 + 0.719395i \(0.255581\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 − 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) −33.0000 −1.33943 −0.669714 − 0.742619i \(-0.733583\pi\)
−0.669714 + 0.742619i \(0.733583\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 27.0000 1.09230
\(612\) 0 0
\(613\) −41.0000 −1.65597 −0.827987 − 0.560747i \(-0.810514\pi\)
−0.827987 + 0.560747i \(0.810514\pi\)
\(614\) 0 0
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 − 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) −35.0000 −1.40677 −0.703384 − 0.710810i \(-0.748329\pi\)
−0.703384 + 0.710810i \(0.748329\pi\)
\(620\) 0 0
\(621\) −28.0000 −1.12360
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −10.0000 −0.399362
\(628\) 0 0
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 − 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) −26.0000 −1.03341
\(634\) 0 0
\(635\) 17.0000 0.674624
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) 35.0000 1.38242 0.691208 − 0.722655i \(-0.257079\pi\)
0.691208 + 0.722655i \(0.257079\pi\)
\(642\) 0 0
\(643\) 34.0000 1.34083 0.670415 − 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 0 0
\(645\) 12.0000 0.472500
\(646\) 0 0
\(647\) −23.0000 −0.904223 −0.452112 − 0.891961i \(-0.649329\pi\)
−0.452112 + 0.891961i \(0.649329\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 45.0000 1.76099 0.880493 − 0.474059i \(-0.157212\pi\)
0.880493 + 0.474059i \(0.157212\pi\)
\(654\) 0 0
\(655\) −7.00000 −0.273513
\(656\) 0 0
\(657\) −12.0000 −0.468165
\(658\) 0 0
\(659\) −16.0000 −0.623272 −0.311636 − 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) 36.0000 1.40024 0.700119 − 0.714026i \(-0.253130\pi\)
0.700119 + 0.714026i \(0.253130\pi\)
\(662\) 0 0
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −42.0000 −1.62625
\(668\) 0 0
\(669\) 32.0000 1.23719
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −42.0000 −1.61898 −0.809491 − 0.587133i \(-0.800257\pi\)
−0.809491 + 0.587133i \(0.800257\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −21.0000 −0.807096 −0.403548 − 0.914959i \(-0.632223\pi\)
−0.403548 + 0.914959i \(0.632223\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 16.0000 0.613121
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 − 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 56.0000 2.13653
\(688\) 0 0
\(689\) 33.0000 1.25720
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 − 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 10.0000 0.378777
\(698\) 0 0
\(699\) 4.00000 0.151294
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 − 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) −25.0000 −0.942893
\(704\) 0 0
\(705\) 18.0000 0.677919
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −18.0000 −0.676004 −0.338002 − 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) 0 0
\(713\) −28.0000 −1.04861
\(714\) 0 0
\(715\) −3.00000 −0.112194
\(716\) 0 0
\(717\) 12.0000 0.448148
\(718\) 0 0
\(719\) 36.0000 1.34257 0.671287 − 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −46.0000 −1.71076
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −41.0000 −1.52061 −0.760303 − 0.649569i \(-0.774949\pi\)
−0.760303 + 0.649569i \(0.774949\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 0 0
\(733\) −31.0000 −1.14501 −0.572506 − 0.819901i \(-0.694029\pi\)
−0.572506 + 0.819901i \(0.694029\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −41.0000 −1.50821 −0.754105 − 0.656754i \(-0.771929\pi\)
−0.754105 + 0.656754i \(0.771929\pi\)
\(740\) 0 0
\(741\) 30.0000 1.10208
\(742\) 0 0
\(743\) −3.00000 −0.110059 −0.0550297 − 0.998485i \(-0.517525\pi\)
−0.0550297 + 0.998485i \(0.517525\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 18.0000 0.656829 0.328415 − 0.944534i \(-0.393486\pi\)
0.328415 + 0.944534i \(0.393486\pi\)
\(752\) 0 0
\(753\) −58.0000 −2.11364
\(754\) 0 0
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 − 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 0 0
\(759\) −14.0000 −0.508168
\(760\) 0 0
\(761\) 3.00000 0.108750 0.0543750 − 0.998521i \(-0.482683\pi\)
0.0543750 + 0.998521i \(0.482683\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) 41.0000 1.47850 0.739249 − 0.673432i \(-0.235181\pi\)
0.739249 + 0.673432i \(0.235181\pi\)
\(770\) 0 0
\(771\) −24.0000 −0.864339
\(772\) 0 0
\(773\) −15.0000 −0.539513 −0.269756 − 0.962929i \(-0.586943\pi\)
−0.269756 + 0.962929i \(0.586943\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 25.0000 0.895718
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 0 0
\(785\) −5.00000 −0.178458
\(786\) 0 0
\(787\) 18.0000 0.641631 0.320815 − 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) 0 0
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 36.0000 1.27840
\(794\) 0 0
\(795\) 22.0000 0.780260
\(796\) 0 0
\(797\) −2.00000 −0.0708436 −0.0354218 − 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 24.0000 0.844840
\(808\) 0 0
\(809\) 55.0000 1.93370 0.966849 − 0.255351i \(-0.0821909\pi\)
0.966849 + 0.255351i \(0.0821909\pi\)
\(810\) 0 0
\(811\) 5.00000 0.175574 0.0877869 − 0.996139i \(-0.472021\pi\)
0.0877869 + 0.996139i \(0.472021\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 30.0000 1.04957
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 − 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 − 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 0 0
\(825\) −2.00000 −0.0696311
\(826\) 0 0
\(827\) −14.0000 −0.486828 −0.243414 − 0.969923i \(-0.578267\pi\)
−0.243414 + 0.969923i \(0.578267\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 4.00000 0.138758
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −5.00000 −0.173032
\(836\) 0 0
\(837\) 16.0000 0.553041
\(838\) 0 0
\(839\) 14.0000 0.483334 0.241667 − 0.970359i \(-0.422306\pi\)
0.241667 + 0.970359i \(0.422306\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −6.00000 −0.206651
\(844\) 0 0
\(845\) −4.00000 −0.137604
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 44.0000 1.51008
\(850\) 0 0
\(851\) −35.0000 −1.19978
\(852\) 0 0
\(853\) −23.0000 −0.787505 −0.393753 − 0.919216i \(-0.628823\pi\)
−0.393753 + 0.919216i \(0.628823\pi\)
\(854\) 0 0
\(855\) 5.00000 0.170996
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 − 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −13.0000 −0.442525 −0.221263 − 0.975214i \(-0.571018\pi\)
−0.221263 + 0.975214i \(0.571018\pi\)
\(864\) 0 0
\(865\) −19.0000 −0.646019
\(866\) 0 0
\(867\) −26.0000 −0.883006
\(868\) 0 0
\(869\) −14.0000 −0.474917
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 9.00000 0.303908 0.151954 − 0.988388i \(-0.451443\pi\)
0.151954 + 0.988388i \(0.451443\pi\)
\(878\) 0 0
\(879\) −42.0000 −1.41662
\(880\) 0 0
\(881\) 7.00000 0.235836 0.117918 − 0.993023i \(-0.462378\pi\)
0.117918 + 0.993023i \(0.462378\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 − 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) −16.0000 −0.537834
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) 0 0
\(893\) 45.0000 1.50587
\(894\) 0 0
\(895\) 9.00000 0.300837
\(896\) 0 0
\(897\) 42.0000 1.40234
\(898\) 0 0
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 22.0000 0.732926
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.00000 −0.0664822
\(906\) 0 0
\(907\) 22.0000 0.730498 0.365249 − 0.930910i \(-0.380984\pi\)
0.365249 + 0.930910i \(0.380984\pi\)
\(908\) 0 0
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) 42.0000 1.39152 0.695761 − 0.718273i \(-0.255067\pi\)
0.695761 + 0.718273i \(0.255067\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) 0 0
\(915\) 24.0000 0.793416
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −18.0000 −0.593765 −0.296883 − 0.954914i \(-0.595947\pi\)
−0.296883 + 0.954914i \(0.595947\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −5.00000 −0.164399
\(926\) 0 0
\(927\) −20.0000 −0.656886
\(928\) 0 0
\(929\) 31.0000 1.01708 0.508539 − 0.861039i \(-0.330186\pi\)
0.508539 + 0.861039i \(0.330186\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 − 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) −32.0000 −1.04428
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 − 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 0 0
\(943\) 35.0000 1.13976
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14.0000 0.454939 0.227469 − 0.973785i \(-0.426955\pi\)
0.227469 + 0.973785i \(0.426955\pi\)
\(948\) 0 0
\(949\) −36.0000 −1.16861
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) −20.0000 −0.647864 −0.323932 − 0.946080i \(-0.605005\pi\)
−0.323932 + 0.946080i \(0.605005\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) 0 0
\(957\) 12.0000 0.387905
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) −20.0000 −0.643823
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 − 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 20.0000 0.642493
\(970\) 0 0
\(971\) −7.00000 −0.224641 −0.112320 − 0.993672i \(-0.535828\pi\)
−0.112320 + 0.993672i \(0.535828\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 6.00000 0.192154
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 − 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) 3.00000 0.0956851 0.0478426 − 0.998855i \(-0.484765\pi\)
0.0478426 + 0.998855i \(0.484765\pi\)
\(984\) 0 0
\(985\) −27.0000 −0.860292
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 42.0000 1.33552
\(990\) 0 0
\(991\) 44.0000 1.39771 0.698853 − 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) 0 0
\(993\) −54.0000 −1.71364
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) −46.0000 −1.45683 −0.728417 − 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) 0 0
\(999\) 20.0000 0.632772
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 1960.2.a.m.1.1 1
4.3 odd 2 3920.2.a.i.1.1 1
5.4 even 2 9800.2.a.g.1.1 1
7.2 even 3 1960.2.q.c.361.1 2
7.3 odd 6 280.2.q.c.121.1 yes 2
7.4 even 3 1960.2.q.c.961.1 2
7.5 odd 6 280.2.q.c.81.1 ✓ 2
7.6 odd 2 1960.2.a.a.1.1 1
21.5 even 6 2520.2.bi.e.361.1 2
21.17 even 6 2520.2.bi.e.1801.1 2
28.3 even 6 560.2.q.c.401.1 2
28.19 even 6 560.2.q.c.81.1 2
28.27 even 2 3920.2.a.bf.1.1 1
35.3 even 12 1400.2.bh.e.849.1 4
35.12 even 12 1400.2.bh.e.249.1 4
35.17 even 12 1400.2.bh.e.849.2 4
35.19 odd 6 1400.2.q.a.1201.1 2
35.24 odd 6 1400.2.q.a.401.1 2
35.33 even 12 1400.2.bh.e.249.2 4
35.34 odd 2 9800.2.a.bi.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.c.81.1 ✓ 2 7.5 odd 6
280.2.q.c.121.1 yes 2 7.3 odd 6
560.2.q.c.81.1 2 28.19 even 6
560.2.q.c.401.1 2 28.3 even 6
1400.2.q.a.401.1 2 35.24 odd 6
1400.2.q.a.1201.1 2 35.19 odd 6
1400.2.bh.e.249.1 4 35.12 even 12
1400.2.bh.e.249.2 4 35.33 even 12
1400.2.bh.e.849.1 4 35.3 even 12
1400.2.bh.e.849.2 4 35.17 even 12
1960.2.a.a.1.1 1 7.6 odd 2
1960.2.a.m.1.1 1 1.1 even 1 trivial
1960.2.q.c.361.1 2 7.2 even 3
1960.2.q.c.961.1 2 7.4 even 3
2520.2.bi.e.361.1 2 21.5 even 6
2520.2.bi.e.1801.1 2 21.17 even 6
3920.2.a.i.1.1 1 4.3 odd 2
3920.2.a.bf.1.1 1 28.27 even 2
9800.2.a.g.1.1 1 5.4 even 2
9800.2.a.bi.1.1 1 35.34 odd 2