Properties

Label 2960.2.a.b.1.1
Level $2960$
Weight $2$
Character 2960.1
Self dual yes
Analytic conductor $23.636$
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] = [2960,2,Mod(1,2960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2960, 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("2960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.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: \(23.6357189983\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1480)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -2.00000 q^{15} +5.00000 q^{17} +2.00000 q^{19} +6.00000 q^{21} +6.00000 q^{23} +1.00000 q^{25} +4.00000 q^{27} +1.00000 q^{29} +1.00000 q^{31} +6.00000 q^{33} -3.00000 q^{35} +1.00000 q^{37} -5.00000 q^{41} -1.00000 q^{43} +1.00000 q^{45} -4.00000 q^{47} +2.00000 q^{49} -10.0000 q^{51} -9.00000 q^{53} -3.00000 q^{55} -4.00000 q^{57} -8.00000 q^{59} +5.00000 q^{61} -3.00000 q^{63} -12.0000 q^{69} +2.00000 q^{71} -4.00000 q^{73} -2.00000 q^{75} +9.00000 q^{77} -8.00000 q^{79} -11.0000 q^{81} +4.00000 q^{83} +5.00000 q^{85} -2.00000 q^{87} +10.0000 q^{89} -2.00000 q^{93} +2.00000 q^{95} -9.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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 \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −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
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −10.0000 −1.40028
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) −2.00000 −0.230940
\(76\) 0 0
\(77\) 9.00000 1.02565
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 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
\(85\) 5.00000 0.542326
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −9.00000 −0.913812 −0.456906 0.889515i \(-0.651042\pi\)
−0.456906 + 0.889515i \(0.651042\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 6.00000 0.585540
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 1.00000 0.0957826 0.0478913 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −15.0000 −1.37505
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 0 0
\(139\) −19.0000 −1.61156 −0.805779 0.592216i \(-0.798253\pi\)
−0.805779 + 0.592216i \(0.798253\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) −4.00000 −0.329914
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 5.00000 0.404226
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 0 0
\(159\) 18.0000 1.42749
\(160\) 0 0
\(161\) −18.0000 −1.41860
\(162\) 0 0
\(163\) 19.0000 1.48819 0.744097 0.668071i \(-0.232880\pi\)
0.744097 + 0.668071i \(0.232880\pi\)
\(164\) 0 0
\(165\) 6.00000 0.467099
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) −13.0000 −0.988372 −0.494186 0.869356i \(-0.664534\pi\)
−0.494186 + 0.869356i \(0.664534\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 0 0
\(177\) 16.0000 1.20263
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) −15.0000 −1.09691
\(188\) 0 0
\(189\) −12.0000 −0.872872
\(190\) 0 0
\(191\) −1.00000 −0.0723575 −0.0361787 0.999345i \(-0.511519\pi\)
−0.0361787 + 0.999345i \(0.511519\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) −5.00000 −0.349215
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 0 0
\(213\) −4.00000 −0.274075
\(214\) 0 0
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 0 0
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −5.00000 −0.334825 −0.167412 0.985887i \(-0.553541\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −18.0000 −1.18431
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) −21.0000 −1.35838 −0.679189 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 0 0
\(253\) −18.0000 −1.13165
\(254\) 0 0
\(255\) −10.0000 −0.626224
\(256\) 0 0
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 19.0000 1.17159 0.585795 0.810459i \(-0.300782\pi\)
0.585795 + 0.810459i \(0.300782\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 0 0
\(267\) −20.0000 −1.22398
\(268\) 0 0
\(269\) −22.0000 −1.34136 −0.670682 0.741745i \(-0.733998\pi\)
−0.670682 + 0.741745i \(0.733998\pi\)
\(270\) 0 0
\(271\) 18.0000 1.09342 0.546711 0.837321i \(-0.315880\pi\)
0.546711 + 0.837321i \(0.315880\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) −12.0000 −0.721010 −0.360505 0.932757i \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 20.0000 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) 15.0000 0.885422
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) 0 0
\(293\) 25.0000 1.46052 0.730258 0.683172i \(-0.239400\pi\)
0.730258 + 0.683172i \(0.239400\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) −12.0000 −0.696311
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 3.00000 0.172917
\(302\) 0 0
\(303\) 36.0000 2.06815
\(304\) 0 0
\(305\) 5.00000 0.286299
\(306\) 0 0
\(307\) −26.0000 −1.48390 −0.741949 0.670456i \(-0.766098\pi\)
−0.741949 + 0.670456i \(0.766098\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 3.00000 0.170114 0.0850572 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) −9.00000 −0.505490 −0.252745 0.967533i \(-0.581333\pi\)
−0.252745 + 0.967533i \(0.581333\pi\)
\(318\) 0 0
\(319\) −3.00000 −0.167968
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 10.0000 0.556415
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.00000 −0.110600
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 0 0
\(333\) 1.00000 0.0547997
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) 0 0
\(339\) −30.0000 −1.62938
\(340\) 0 0
\(341\) −3.00000 −0.162459
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) −12.0000 −0.646058
\(346\) 0 0
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3.00000 −0.159674 −0.0798369 0.996808i \(-0.525440\pi\)
−0.0798369 + 0.996808i \(0.525440\pi\)
\(354\) 0 0
\(355\) 2.00000 0.106149
\(356\) 0 0
\(357\) 30.0000 1.58777
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 4.00000 0.209946
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) −7.00000 −0.365397 −0.182699 0.983169i \(-0.558483\pi\)
−0.182699 + 0.983169i \(0.558483\pi\)
\(368\) 0 0
\(369\) −5.00000 −0.260290
\(370\) 0 0
\(371\) 27.0000 1.40177
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 0 0
\(385\) 9.00000 0.458682
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) 0 0
\(389\) −37.0000 −1.87597 −0.937987 0.346670i \(-0.887312\pi\)
−0.937987 + 0.346670i \(0.887312\pi\)
\(390\) 0 0
\(391\) 30.0000 1.51717
\(392\) 0 0
\(393\) 32.0000 1.61419
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 0 0
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) −3.00000 −0.148704
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 0 0
\(411\) −8.00000 −0.394611
\(412\) 0 0
\(413\) 24.0000 1.18096
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 38.0000 1.86087
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) −4.00000 −0.194487
\(424\) 0 0
\(425\) 5.00000 0.242536
\(426\) 0 0
\(427\) −15.0000 −0.725901
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −21.0000 −1.01153 −0.505767 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) 0 0
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 0 0
\(437\) 12.0000 0.574038
\(438\) 0 0
\(439\) 7.00000 0.334092 0.167046 0.985949i \(-0.446577\pi\)
0.167046 + 0.985949i \(0.446577\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) −34.0000 −1.61539 −0.807694 0.589601i \(-0.799285\pi\)
−0.807694 + 0.589601i \(0.799285\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) 36.0000 1.70274
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 15.0000 0.706322
\(452\) 0 0
\(453\) −24.0000 −1.12762
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17.0000 −0.795226 −0.397613 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 0 0
\(459\) 20.0000 0.933520
\(460\) 0 0
\(461\) −5.00000 −0.232873 −0.116437 0.993198i \(-0.537147\pi\)
−0.116437 + 0.993198i \(0.537147\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 0 0
\(465\) −2.00000 −0.0927478
\(466\) 0 0
\(467\) −21.0000 −0.971764 −0.485882 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 0 0
\(473\) 3.00000 0.137940
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) 0 0
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 36.0000 1.63806
\(484\) 0 0
\(485\) −9.00000 −0.408669
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) −38.0000 −1.71842
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) 5.00000 0.225189
\(494\) 0 0
\(495\) −3.00000 −0.134840
\(496\) 0 0
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 6.00000 0.268597 0.134298 0.990941i \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) 0 0
\(501\) 32.0000 1.42965
\(502\) 0 0
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 26.0000 1.15470
\(508\) 0 0
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) 0 0
\(513\) 8.00000 0.353209
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 26.0000 1.14127
\(520\) 0 0
\(521\) 1.00000 0.0438108 0.0219054 0.999760i \(-0.493027\pi\)
0.0219054 + 0.999760i \(0.493027\pi\)
\(522\) 0 0
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 0 0
\(525\) 6.00000 0.261861
\(526\) 0 0
\(527\) 5.00000 0.217803
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) 8.00000 0.345225
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) 20.0000 0.858282
\(544\) 0 0
\(545\) 1.00000 0.0428353
\(546\) 0 0
\(547\) −9.00000 −0.384812 −0.192406 0.981315i \(-0.561629\pi\)
−0.192406 + 0.981315i \(0.561629\pi\)
\(548\) 0 0
\(549\) 5.00000 0.213395
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) 24.0000 1.02058
\(554\) 0 0
\(555\) −2.00000 −0.0848953
\(556\) 0 0
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 30.0000 1.26660
\(562\) 0 0
\(563\) 21.0000 0.885044 0.442522 0.896758i \(-0.354084\pi\)
0.442522 + 0.896758i \(0.354084\pi\)
\(564\) 0 0
\(565\) 15.0000 0.631055
\(566\) 0 0
\(567\) 33.0000 1.38587
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 15.0000 0.627730 0.313865 0.949468i \(-0.398376\pi\)
0.313865 + 0.949468i \(0.398376\pi\)
\(572\) 0 0
\(573\) 2.00000 0.0835512
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 28.0000 1.16364
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 27.0000 1.11823
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −45.0000 −1.85735 −0.928674 0.370896i \(-0.879051\pi\)
−0.928674 + 0.370896i \(0.879051\pi\)
\(588\) 0 0
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) −4.00000 −0.164538
\(592\) 0 0
\(593\) 44.0000 1.80686 0.903432 0.428732i \(-0.141040\pi\)
0.903432 + 0.428732i \(0.141040\pi\)
\(594\) 0 0
\(595\) −15.0000 −0.614940
\(596\) 0 0
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −23.0000 −0.928961 −0.464481 0.885583i \(-0.653759\pi\)
−0.464481 + 0.885583i \(0.653759\pi\)
\(614\) 0 0
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) 16.0000 0.644136 0.322068 0.946717i \(-0.395622\pi\)
0.322068 + 0.946717i \(0.395622\pi\)
\(618\) 0 0
\(619\) −19.0000 −0.763674 −0.381837 0.924230i \(-0.624709\pi\)
−0.381837 + 0.924230i \(0.624709\pi\)
\(620\) 0 0
\(621\) 24.0000 0.963087
\(622\) 0 0
\(623\) −30.0000 −1.20192
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 12.0000 0.479234
\(628\) 0 0
\(629\) 5.00000 0.199363
\(630\) 0 0
\(631\) 13.0000 0.517522 0.258761 0.965941i \(-0.416686\pi\)
0.258761 + 0.965941i \(0.416686\pi\)
\(632\) 0 0
\(633\) 26.0000 1.03341
\(634\) 0 0
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) 39.0000 1.54041 0.770204 0.637798i \(-0.220155\pi\)
0.770204 + 0.637798i \(0.220155\pi\)
\(642\) 0 0
\(643\) −29.0000 −1.14365 −0.571824 0.820376i \(-0.693764\pi\)
−0.571824 + 0.820376i \(0.693764\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) 4.00000 0.157256 0.0786281 0.996904i \(-0.474946\pi\)
0.0786281 + 0.996904i \(0.474946\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) 0 0
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 0 0
\(655\) −16.0000 −0.625172
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 39.0000 1.51692 0.758462 0.651717i \(-0.225951\pi\)
0.758462 + 0.651717i \(0.225951\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.00000 −0.232670
\(666\) 0 0
\(667\) 6.00000 0.232321
\(668\) 0 0
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) −15.0000 −0.579069
\(672\) 0 0
\(673\) −50.0000 −1.92736 −0.963679 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) 27.0000 1.03616
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) 0 0
\(683\) −39.0000 −1.49229 −0.746147 0.665782i \(-0.768098\pi\)
−0.746147 + 0.665782i \(0.768098\pi\)
\(684\) 0 0
\(685\) 4.00000 0.152832
\(686\) 0 0
\(687\) 20.0000 0.763048
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 19.0000 0.722794 0.361397 0.932412i \(-0.382300\pi\)
0.361397 + 0.932412i \(0.382300\pi\)
\(692\) 0 0
\(693\) 9.00000 0.341882
\(694\) 0 0
\(695\) −19.0000 −0.720711
\(696\) 0 0
\(697\) −25.0000 −0.946943
\(698\) 0 0
\(699\) −48.0000 −1.81553
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) 8.00000 0.301297
\(706\) 0 0
\(707\) 54.0000 2.03088
\(708\) 0 0
\(709\) −15.0000 −0.563337 −0.281668 0.959512i \(-0.590888\pi\)
−0.281668 + 0.959512i \(0.590888\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 42.0000 1.56852
\(718\) 0 0
\(719\) −34.0000 −1.26799 −0.633993 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 0 0
\(723\) 24.0000 0.892570
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −5.00000 −0.184932
\(732\) 0 0
\(733\) 37.0000 1.36663 0.683313 0.730125i \(-0.260538\pi\)
0.683313 + 0.730125i \(0.260538\pi\)
\(734\) 0 0
\(735\) −4.00000 −0.147542
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −27.0000 −0.993211 −0.496606 0.867976i \(-0.665420\pi\)
−0.496606 + 0.867976i \(0.665420\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.00000 −0.0366864 −0.0183432 0.999832i \(-0.505839\pi\)
−0.0183432 + 0.999832i \(0.505839\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) −30.0000 −1.09472 −0.547358 0.836899i \(-0.684366\pi\)
−0.547358 + 0.836899i \(0.684366\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) 36.0000 1.30672
\(760\) 0 0
\(761\) 11.0000 0.398750 0.199375 0.979923i \(-0.436109\pi\)
0.199375 + 0.979923i \(0.436109\pi\)
\(762\) 0 0
\(763\) −3.00000 −0.108607
\(764\) 0 0
\(765\) 5.00000 0.180775
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 4.00000 0.144056
\(772\) 0 0
\(773\) 45.0000 1.61854 0.809269 0.587439i \(-0.199864\pi\)
0.809269 + 0.587439i \(0.199864\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 0 0
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 0 0
\(783\) 4.00000 0.142948
\(784\) 0 0
\(785\) 7.00000 0.249841
\(786\) 0 0
\(787\) −18.0000 −0.641631 −0.320815 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(788\) 0 0
\(789\) −38.0000 −1.35284
\(790\) 0 0
\(791\) −45.0000 −1.60002
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 18.0000 0.638394
\(796\) 0 0
\(797\) 46.0000 1.62940 0.814702 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) 0 0
\(799\) −20.0000 −0.707549
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) −18.0000 −0.634417
\(806\) 0 0
\(807\) 44.0000 1.54887
\(808\) 0 0
\(809\) 14.0000 0.492214 0.246107 0.969243i \(-0.420849\pi\)
0.246107 + 0.969243i \(0.420849\pi\)
\(810\) 0 0
\(811\) 48.0000 1.68551 0.842754 0.538299i \(-0.180933\pi\)
0.842754 + 0.538299i \(0.180933\pi\)
\(812\) 0 0
\(813\) −36.0000 −1.26258
\(814\) 0 0
\(815\) 19.0000 0.665541
\(816\) 0 0
\(817\) −2.00000 −0.0699711
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 0 0
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) −19.0000 −0.660695 −0.330347 0.943859i \(-0.607166\pi\)
−0.330347 + 0.943859i \(0.607166\pi\)
\(828\) 0 0
\(829\) 13.0000 0.451509 0.225754 0.974184i \(-0.427515\pi\)
0.225754 + 0.974184i \(0.427515\pi\)
\(830\) 0 0
\(831\) 24.0000 0.832551
\(832\) 0 0
\(833\) 10.0000 0.346479
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) 0 0
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) −40.0000 −1.37767
\(844\) 0 0
\(845\) −13.0000 −0.447214
\(846\) 0 0
\(847\) 6.00000 0.206162
\(848\) 0 0
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) 0 0
\(853\) −38.0000 −1.30110 −0.650548 0.759465i \(-0.725461\pi\)
−0.650548 + 0.759465i \(0.725461\pi\)
\(854\) 0 0
\(855\) 2.00000 0.0683986
\(856\) 0 0
\(857\) −33.0000 −1.12726 −0.563629 0.826028i \(-0.690595\pi\)
−0.563629 + 0.826028i \(0.690595\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) −30.0000 −1.02240
\(862\) 0 0
\(863\) 21.0000 0.714848 0.357424 0.933942i \(-0.383655\pi\)
0.357424 + 0.933942i \(0.383655\pi\)
\(864\) 0 0
\(865\) −13.0000 −0.442013
\(866\) 0 0
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −9.00000 −0.304604
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) 23.0000 0.776655 0.388327 0.921521i \(-0.373053\pi\)
0.388327 + 0.921521i \(0.373053\pi\)
\(878\) 0 0
\(879\) −50.0000 −1.68646
\(880\) 0 0
\(881\) −33.0000 −1.11180 −0.555899 0.831250i \(-0.687626\pi\)
−0.555899 + 0.831250i \(0.687626\pi\)
\(882\) 0 0
\(883\) −35.0000 −1.17784 −0.588922 0.808190i \(-0.700447\pi\)
−0.588922 + 0.808190i \(0.700447\pi\)
\(884\) 0 0
\(885\) 16.0000 0.537834
\(886\) 0 0
\(887\) −21.0000 −0.705111 −0.352555 0.935791i \(-0.614687\pi\)
−0.352555 + 0.935791i \(0.614687\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) 33.0000 1.10554
\(892\) 0 0
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.00000 0.0333519
\(900\) 0 0
\(901\) −45.0000 −1.49917
\(902\) 0 0
\(903\) −6.00000 −0.199667
\(904\) 0 0
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) −56.0000 −1.85536 −0.927681 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) −10.0000 −0.330590
\(916\) 0 0
\(917\) 48.0000 1.58510
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 52.0000 1.71346
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 5.00000 0.164045 0.0820223 0.996630i \(-0.473862\pi\)
0.0820223 + 0.996630i \(0.473862\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 0 0
\(933\) −6.00000 −0.196431
\(934\) 0 0
\(935\) −15.0000 −0.490552
\(936\) 0 0
\(937\) −46.0000 −1.50275 −0.751377 0.659873i \(-0.770610\pi\)
−0.751377 + 0.659873i \(0.770610\pi\)
\(938\) 0 0
\(939\) −12.0000 −0.391605
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 0 0
\(943\) −30.0000 −0.976934
\(944\) 0 0
\(945\) −12.0000 −0.390360
\(946\) 0 0
\(947\) −55.0000 −1.78726 −0.893630 0.448805i \(-0.851850\pi\)
−0.893630 + 0.448805i \(0.851850\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) −1.00000 −0.0323592
\(956\) 0 0
\(957\) 6.00000 0.193952
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 0 0
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 52.0000 1.67221 0.836104 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) 0 0
\(969\) −20.0000 −0.642493
\(970\) 0 0
\(971\) 23.0000 0.738105 0.369053 0.929409i \(-0.379682\pi\)
0.369053 + 0.929409i \(0.379682\pi\)
\(972\) 0 0
\(973\) 57.0000 1.82734
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 13.0000 0.415907 0.207953 0.978139i \(-0.433320\pi\)
0.207953 + 0.978139i \(0.433320\pi\)
\(978\) 0 0
\(979\) −30.0000 −0.958804
\(980\) 0 0
\(981\) 1.00000 0.0319275
\(982\) 0 0
\(983\) −39.0000 −1.24391 −0.621953 0.783054i \(-0.713661\pi\)
−0.621953 + 0.783054i \(0.713661\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) 19.0000 0.603555 0.301777 0.953378i \(-0.402420\pi\)
0.301777 + 0.953378i \(0.402420\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) 60.0000 1.90022 0.950110 0.311916i \(-0.100971\pi\)
0.950110 + 0.311916i \(0.100971\pi\)
\(998\) 0 0
\(999\) 4.00000 0.126554
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 2960.2.a.b.1.1 1
4.3 odd 2 1480.2.a.d.1.1 1
20.19 odd 2 7400.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1480.2.a.d.1.1 1 4.3 odd 2
2960.2.a.b.1.1 1 1.1 even 1 trivial
7400.2.a.a.1.1 1 20.19 odd 2