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Properties

Label 9680.2.a.f.1.1
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9680,2,Mod(1,9680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-1,0,-1,0,-1,0,-2,0,0,0,-2,0,1,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(77.2951891566\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2420)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9680.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} -2.00000 q^{13} +1.00000 q^{15} +2.00000 q^{19} +1.00000 q^{21} +1.00000 q^{25} +5.00000 q^{27} +6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{35} -4.00000 q^{37} +2.00000 q^{39} -9.00000 q^{41} -1.00000 q^{43} +2.00000 q^{45} +3.00000 q^{47} -6.00000 q^{49} -6.00000 q^{53} -2.00000 q^{57} +1.00000 q^{61} +2.00000 q^{63} +2.00000 q^{65} +13.0000 q^{67} +12.0000 q^{71} +16.0000 q^{73} -1.00000 q^{75} -10.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -6.00000 q^{87} -3.00000 q^{89} +2.00000 q^{91} -4.00000 q^{93} -2.00000 q^{95} -10.0000 q^{97} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 − 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 13.0000 1.58820 0.794101 − 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 − 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.00000 0.298511 0.149256 − 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 3.00000 0.290021 0.145010 − 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 − 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 − 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 9.00000 0.811503
\(124\) 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
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 − 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) 0 0
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) 6.00000 0.494872
\(148\) 0 0
\(149\) −9.00000 −0.737309 −0.368654 − 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 − 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −11.0000 −0.861586 −0.430793 − 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −21.0000 −1.62503 −0.812514 − 0.582941i \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 − 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 − 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 − 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) −2.00000 −0.143223
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 − 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) −13.0000 −0.916949
\(202\) 0 0
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 9.00000 0.628587
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 − 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) −16.0000 −1.08118
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −17.0000 −1.13840 −0.569202 − 0.822198i \(-0.692748\pi\)
−0.569202 + 0.822198i \(0.692748\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) 9.00000 0.597351 0.298675 − 0.954355i \(-0.403455\pi\)
0.298675 + 0.954355i \(0.403455\pi\)
\(228\) 0 0
\(229\) −13.0000 −0.859064 −0.429532 − 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 − 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) 0 0
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) 18.0000 1.16432 0.582162 − 0.813073i \(-0.302207\pi\)
0.582162 + 0.813073i \(0.302207\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 − 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −12.0000 −0.742781
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 3.00000 0.183597
\(268\) 0 0
\(269\) −3.00000 −0.182913 −0.0914566 − 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) −31.0000 −1.84276 −0.921379 − 0.388664i \(-0.872937\pi\)
−0.921379 + 0.388664i \(0.872937\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 9.00000 0.531253
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 0 0
\(293\) 24.0000 1.40209 0.701047 − 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) −3.00000 −0.172345
\(304\) 0 0
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 − 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 5.00000 0.276501
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 − 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 0 0
\(333\) 8.00000 0.438397
\(334\) 0 0
\(335\) −13.0000 −0.710266
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 − 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.00000 −0.161048 −0.0805242 − 0.996753i \(-0.525659\pi\)
−0.0805242 + 0.996753i \(0.525659\pi\)
\(348\) 0 0
\(349\) 34.0000 1.81998 0.909989 − 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) −10.0000 −0.533761
\(352\) 0 0
\(353\) 24.0000 1.27739 0.638696 − 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.00000 0.316668 0.158334 − 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −16.0000 −0.837478
\(366\) 0 0
\(367\) −5.00000 −0.260998 −0.130499 − 0.991448i \(-0.541658\pi\)
−0.130499 + 0.991448i \(0.541658\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) −32.0000 −1.65690 −0.828449 − 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −26.0000 −1.33553 −0.667765 − 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 0 0
\(381\) −17.0000 −0.870936
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.00000 0.101666
\(388\) 0 0
\(389\) −15.0000 −0.760530 −0.380265 − 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 18.0000 0.907980
\(394\) 0 0
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) −4.00000 −0.200754 −0.100377 − 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) 0 0
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) 3.00000 0.149813 0.0749064 − 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −11.0000 −0.543915 −0.271957 − 0.962309i \(-0.587671\pi\)
−0.271957 + 0.962309i \(0.587671\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) −18.0000 −0.879358 −0.439679 − 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) −37.0000 −1.80327 −0.901635 − 0.432498i \(-0.857632\pi\)
−0.901635 + 0.432498i \(0.857632\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.00000 −0.0483934
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 − 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 − 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) 0 0
\(443\) −39.0000 −1.85295 −0.926473 − 0.376361i \(-0.877175\pi\)
−0.926473 + 0.376361i \(0.877175\pi\)
\(444\) 0 0
\(445\) 3.00000 0.142214
\(446\) 0 0
\(447\) 9.00000 0.425685
\(448\) 0 0
\(449\) −9.00000 −0.424736 −0.212368 − 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −20.0000 −0.939682
\(454\) 0 0
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) 28.0000 1.30978 0.654892 − 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −21.0000 −0.978068 −0.489034 − 0.872265i \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) 0 0
\(463\) −5.00000 −0.232370 −0.116185 − 0.993228i \(-0.537067\pi\)
−0.116185 + 0.993228i \(0.537067\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) −27.0000 −1.24941 −0.624705 − 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 0 0
\(469\) −13.0000 −0.600284
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) −18.0000 −0.822441 −0.411220 − 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10.0000 0.454077
\(486\) 0 0
\(487\) 28.0000 1.26880 0.634401 − 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 0 0
\(489\) 11.0000 0.497437
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) −26.0000 −1.16392 −0.581960 − 0.813217i \(-0.697714\pi\)
−0.581960 + 0.813217i \(0.697714\pi\)
\(500\) 0 0
\(501\) 21.0000 0.938211
\(502\) 0 0
\(503\) −21.0000 −0.936344 −0.468172 − 0.883637i \(-0.655087\pi\)
−0.468172 + 0.883637i \(0.655087\pi\)
\(504\) 0 0
\(505\) −3.00000 −0.133498
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) −21.0000 −0.930809 −0.465404 − 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 0 0
\(513\) 10.0000 0.441511
\(514\) 0 0
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 15.0000 0.657162 0.328581 − 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 18.0000 0.779667
\(534\) 0 0
\(535\) −3.00000 −0.129701
\(536\) 0 0
\(537\) 6.00000 0.258919
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 31.0000 1.33279 0.666397 − 0.745597i \(-0.267836\pi\)
0.666397 + 0.745597i \(0.267836\pi\)
\(542\) 0 0
\(543\) 7.00000 0.300399
\(544\) 0 0
\(545\) 5.00000 0.214176
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) 0 0
\(555\) −4.00000 −0.169791
\(556\) 0 0
\(557\) −36.0000 −1.52537 −0.762684 − 0.646771i \(-0.776119\pi\)
−0.762684 + 0.646771i \(0.776119\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 27.0000 1.13791 0.568957 − 0.822367i \(-0.307347\pi\)
0.568957 + 0.822367i \(0.307347\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −15.0000 −0.628833 −0.314416 − 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) 14.0000 0.585882 0.292941 − 0.956131i \(-0.405366\pi\)
0.292941 + 0.956131i \(0.405366\pi\)
\(572\) 0 0
\(573\) 18.0000 0.751961
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −16.0000 −0.666089 −0.333044 − 0.942911i \(-0.608076\pi\)
−0.333044 + 0.942911i \(0.608076\pi\)
\(578\) 0 0
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) 3.00000 0.123823 0.0619116 − 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −10.0000 −0.409273
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 − 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 − 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) −26.0000 −1.05880
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 − 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 − 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 0 0
\(615\) −9.00000 −0.362915
\(616\) 0 0
\(617\) 48.0000 1.93241 0.966204 − 0.257780i \(-0.0829910\pi\)
0.966204 + 0.257780i \(0.0829910\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.00000 0.120192
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 − 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 16.0000 0.635943
\(634\) 0 0
\(635\) −17.0000 −0.674624
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) −24.0000 −0.949425
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −47.0000 −1.85350 −0.926750 − 0.375680i \(-0.877409\pi\)
−0.926750 + 0.375680i \(0.877409\pi\)
\(644\) 0 0
\(645\) −1.00000 −0.0393750
\(646\) 0 0
\(647\) −27.0000 −1.06148 −0.530740 − 0.847535i \(-0.678086\pi\)
−0.530740 + 0.847535i \(0.678086\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 0 0
\(653\) 18.0000 0.704394 0.352197 − 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) 0 0
\(657\) −32.0000 −1.24844
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 − 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −49.0000 −1.90588 −0.952940 − 0.303160i \(-0.901958\pi\)
−0.952940 + 0.303160i \(0.901958\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.00000 0.0775567
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 17.0000 0.657258
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.00000 −0.308377 −0.154189 − 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) 48.0000 1.84479 0.922395 − 0.386248i \(-0.126229\pi\)
0.922395 + 0.386248i \(0.126229\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −9.00000 −0.344881
\(682\) 0 0
\(683\) −27.0000 −1.03313 −0.516563 − 0.856249i \(-0.672789\pi\)
−0.516563 + 0.856249i \(0.672789\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 0 0
\(687\) 13.0000 0.495981
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 − 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 − 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 3.00000 0.112987
\(706\) 0 0
\(707\) −3.00000 −0.112827
\(708\) 0 0
\(709\) 11.0000 0.413114 0.206557 − 0.978435i \(-0.433774\pi\)
0.206557 + 0.978435i \(0.433774\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −18.0000 −0.672222
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 0 0
\(723\) 5.00000 0.185952
\(724\) 0 0
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 37.0000 1.37225 0.686127 − 0.727482i \(-0.259309\pi\)
0.686127 + 0.727482i \(0.259309\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 − 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −10.0000 −0.367856 −0.183928 − 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 3.00000 0.110059 0.0550297 − 0.998485i \(-0.482475\pi\)
0.0550297 + 0.998485i \(0.482475\pi\)
\(744\) 0 0
\(745\) 9.00000 0.329734
\(746\) 0 0
\(747\) −24.0000 −0.878114
\(748\) 0 0
\(749\) −3.00000 −0.109618
\(750\) 0 0
\(751\) 22.0000 0.802791 0.401396 − 0.915905i \(-0.368525\pi\)
0.401396 + 0.915905i \(0.368525\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) −20.0000 −0.727875
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 − 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 5.00000 0.181012
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 − 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 0 0
\(773\) 36.0000 1.29483 0.647415 − 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) −18.0000 −0.644917
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 30.0000 1.07211
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) 35.0000 1.24762 0.623808 − 0.781578i \(-0.285585\pi\)
0.623808 + 0.781578i \(0.285585\pi\)
\(788\) 0 0
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) −6.00000 −0.212798
\(796\) 0 0
\(797\) 12.0000 0.425062 0.212531 − 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.00000 0.105605
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 − 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 0 0
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) 11.0000 0.385313
\(816\) 0 0
\(817\) −2.00000 −0.0699711
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 21.0000 0.732905 0.366453 − 0.930437i \(-0.380572\pi\)
0.366453 + 0.930437i \(0.380572\pi\)
\(822\) 0 0
\(823\) −11.0000 −0.383436 −0.191718 − 0.981450i \(-0.561406\pi\)
−0.191718 + 0.981450i \(0.561406\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.0000 0.938882 0.469441 − 0.882964i \(-0.344455\pi\)
0.469441 + 0.882964i \(0.344455\pi\)
\(828\) 0 0
\(829\) −31.0000 −1.07667 −0.538337 − 0.842729i \(-0.680947\pi\)
−0.538337 + 0.842729i \(0.680947\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 21.0000 0.726735
\(836\) 0 0
\(837\) 20.0000 0.691301
\(838\) 0 0
\(839\) 6.00000 0.207143 0.103572 − 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −30.0000 −1.03325
\(844\) 0 0
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 31.0000 1.06392
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −44.0000 −1.50653 −0.753266 − 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −26.0000 −0.887109 −0.443554 − 0.896248i \(-0.646283\pi\)
−0.443554 + 0.896248i \(0.646283\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) 0 0
\(863\) 33.0000 1.12333 0.561667 − 0.827364i \(-0.310160\pi\)
0.561667 + 0.827364i \(0.310160\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −26.0000 −0.880976
\(872\) 0 0
\(873\) 20.0000 0.676897
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −32.0000 −1.08056 −0.540282 − 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 0 0
\(879\) −24.0000 −0.809500
\(880\) 0 0
\(881\) −27.0000 −0.909653 −0.454827 − 0.890580i \(-0.650299\pi\)
−0.454827 + 0.890580i \(0.650299\pi\)
\(882\) 0 0
\(883\) −56.0000 −1.88455 −0.942275 − 0.334840i \(-0.891318\pi\)
−0.942275 + 0.334840i \(0.891318\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 51.0000 1.71241 0.856206 − 0.516634i \(-0.172815\pi\)
0.856206 + 0.516634i \(0.172815\pi\)
\(888\) 0 0
\(889\) −17.0000 −0.570162
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −1.00000 −0.0332779
\(904\) 0 0
\(905\) 7.00000 0.232688
\(906\) 0 0
\(907\) 7.00000 0.232431 0.116216 − 0.993224i \(-0.462924\pi\)
0.116216 + 0.993224i \(0.462924\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 − 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 1.00000 0.0330590
\(916\) 0 0
\(917\) 18.0000 0.594412
\(918\) 0 0
\(919\) −22.0000 −0.725713 −0.362857 − 0.931845i \(-0.618198\pi\)
−0.362857 + 0.931845i \(0.618198\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −42.0000 −1.37798 −0.688988 − 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 0 0
\(933\) −18.0000 −0.589294
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 − 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) −33.0000 −1.07577 −0.537885 − 0.843018i \(-0.680776\pi\)
−0.537885 + 0.843018i \(0.680776\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 5.00000 0.162650
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −32.0000 −1.03876
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 − 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 0 0
\(955\) 18.0000 0.582466
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 0 0
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 − 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) 0 0
\(975\) 2.00000 0.0640513
\(976\) 0 0
\(977\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) 33.0000 1.05254 0.526268 − 0.850319i \(-0.323591\pi\)
0.526268 + 0.850319i \(0.323591\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 3.00000 0.0954911
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 58.0000 1.84243 0.921215 − 0.389053i \(-0.127198\pi\)
0.921215 + 0.389053i \(0.127198\pi\)
\(992\) 0 0
\(993\) −10.0000 −0.317340
\(994\) 0 0
\(995\) −10.0000 −0.317021
\(996\) 0 0
\(997\) 22.0000 0.696747 0.348373 − 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) 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 9680.2.a.f.1.1 1
4.3 odd 2 2420.2.a.f.1.1 yes 1
11.10 odd 2 9680.2.a.h.1.1 1
44.43 even 2 2420.2.a.e.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2420.2.a.e.1.1 ✓ 1 44.43 even 2
2420.2.a.f.1.1 yes 1 4.3 odd 2
9680.2.a.f.1.1 1 1.1 even 1 trivial
9680.2.a.h.1.1 1 11.10 odd 2