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Properties

Label 3920.2.a.c.1.1
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $0$
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] = [3920,2,Mod(1,3920)] 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("3920.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3,0,-1,0,0,0,6,0,5,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.3013575923\)
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 \(\chi\) \(=\) 3920.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -1.00000 q^{5} +6.00000 q^{9} +5.00000 q^{11} +5.00000 q^{13} +3.00000 q^{15} +7.00000 q^{17} -2.00000 q^{19} +2.00000 q^{23} +1.00000 q^{25} -9.00000 q^{27} +7.00000 q^{29} +4.00000 q^{31} -15.0000 q^{33} -6.00000 q^{37} -15.0000 q^{39} +12.0000 q^{41} +2.00000 q^{43} -6.00000 q^{45} +1.00000 q^{47} -21.0000 q^{51} -5.00000 q^{55} +6.00000 q^{57} -4.00000 q^{59} -4.00000 q^{61} -5.00000 q^{65} -8.00000 q^{67} -6.00000 q^{69} -6.00000 q^{73} -3.00000 q^{75} +3.00000 q^{79} +9.00000 q^{81} -4.00000 q^{83} -7.00000 q^{85} -21.0000 q^{87} -12.0000 q^{93} +2.00000 q^{95} -13.0000 q^{97} +30.0000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 − 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) 7.00000 1.69775 0.848875 − 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) 7.00000 1.29987 0.649934 − 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −15.0000 −2.61116
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −15.0000 −2.40192
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 − 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) −6.00000 −0.894427
\(46\) 0 0
\(47\) 1.00000 0.145865 0.0729325 − 0.997337i \(-0.476764\pi\)
0.0729325 + 0.997337i \(0.476764\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −21.0000 −2.94059
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 − 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.00000 −0.620174
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) −3.00000 −0.346410
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 3.00000 0.337526 0.168763 − 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −7.00000 −0.759257
\(86\) 0 0
\(87\) −21.0000 −2.25144
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −12.0000 −1.24434
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 − 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 0 0
\(99\) 30.0000 3.01511
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 − 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 13.0000 1.28093 0.640464 − 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −18.0000 −1.74013 −0.870063 − 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 − 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 18.0000 1.70848
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 0 0
\(117\) 30.0000 2.77350
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) −36.0000 −3.24601
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 − 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 9.00000 0.774597
\(136\) 0 0
\(137\) −8.00000 −0.683486 −0.341743 − 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 0 0
\(139\) 18.0000 1.52674 0.763370 − 0.645961i \(-0.223543\pi\)
0.763370 + 0.645961i \(0.223543\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) 25.0000 2.09061
\(144\) 0 0
\(145\) −7.00000 −0.581318
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −22.0000 −1.80231 −0.901155 − 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) 19.0000 1.54620 0.773099 − 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 0 0
\(153\) 42.0000 3.39550
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 14.0000 1.09656 0.548282 − 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 0 0
\(165\) 15.0000 1.16775
\(166\) 0 0
\(167\) −3.00000 −0.232147 −0.116073 − 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −12.0000 −0.917663
\(172\) 0 0
\(173\) 7.00000 0.532200 0.266100 − 0.963945i \(-0.414265\pi\)
0.266100 + 0.963945i \(0.414265\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 − 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 − 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 0 0
\(183\) 12.0000 0.887066
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 35.0000 2.55945
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.0000 0.940647 0.470323 − 0.882494i \(-0.344137\pi\)
0.470323 + 0.882494i \(0.344137\pi\)
\(192\) 0 0
\(193\) −8.00000 −0.575853 −0.287926 − 0.957653i \(-0.592966\pi\)
−0.287926 + 0.957653i \(0.592966\pi\)
\(194\) 0 0
\(195\) 15.0000 1.07417
\(196\) 0 0
\(197\) −8.00000 −0.569976 −0.284988 − 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 24.0000 1.69283
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) 12.0000 0.834058
\(208\) 0 0
\(209\) −10.0000 −0.691714
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 − 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 18.0000 1.21633
\(220\) 0 0
\(221\) 35.0000 2.35435
\(222\) 0 0
\(223\) −19.0000 −1.27233 −0.636167 − 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) 0 0
\(225\) 6.00000 0.400000
\(226\) 0 0
\(227\) 9.00000 0.597351 0.298675 − 0.954355i \(-0.403455\pi\)
0.298675 + 0.954355i \(0.403455\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 − 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 − 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) −1.00000 −0.0652328
\(236\) 0 0
\(237\) −9.00000 −0.584613
\(238\) 0 0
\(239\) −9.00000 −0.582162 −0.291081 − 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 − 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −10.0000 −0.636285
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 − 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 0 0
\(253\) 10.0000 0.628695
\(254\) 0 0
\(255\) 21.0000 1.31507
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 42.0000 2.59973
\(262\) 0 0
\(263\) 30.0000 1.84988 0.924940 − 0.380114i \(-0.124115\pi\)
0.924940 + 0.380114i \(0.124115\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −26.0000 −1.58525 −0.792624 − 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 − 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.00000 0.301511
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 − 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) 19.0000 1.13344 0.566722 − 0.823909i \(-0.308211\pi\)
0.566722 + 0.823909i \(0.308211\pi\)
\(282\) 0 0
\(283\) −25.0000 −1.48610 −0.743048 − 0.669238i \(-0.766621\pi\)
−0.743048 + 0.669238i \(0.766621\pi\)
\(284\) 0 0
\(285\) −6.00000 −0.355409
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) 39.0000 2.28622
\(292\) 0 0
\(293\) −13.0000 −0.759468 −0.379734 − 0.925096i \(-0.623985\pi\)
−0.379734 + 0.925096i \(0.623985\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) −45.0000 −2.61116
\(298\) 0 0
\(299\) 10.0000 0.578315
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −54.0000 −3.10222
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) −17.0000 −0.970241 −0.485121 − 0.874447i \(-0.661224\pi\)
−0.485121 + 0.874447i \(0.661224\pi\)
\(308\) 0 0
\(309\) −39.0000 −2.21863
\(310\) 0 0
\(311\) −34.0000 −1.92796 −0.963982 − 0.265969i \(-0.914308\pi\)
−0.963982 + 0.265969i \(0.914308\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 − 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 35.0000 1.95962
\(320\) 0 0
\(321\) 54.0000 3.01399
\(322\) 0 0
\(323\) −14.0000 −0.778981
\(324\) 0 0
\(325\) 5.00000 0.277350
\(326\) 0 0
\(327\) −15.0000 −0.829502
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 − 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) −36.0000 −1.97279
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 34.0000 1.85210 0.926049 − 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 0 0
\(347\) −26.0000 −1.39575 −0.697877 − 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 − 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) −45.0000 −2.40192
\(352\) 0 0
\(353\) −5.00000 −0.266123 −0.133062 − 0.991108i \(-0.542481\pi\)
−0.133062 + 0.991108i \(0.542481\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −42.0000 −2.20443
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) −7.00000 −0.365397 −0.182699 − 0.983169i \(-0.558483\pi\)
−0.182699 + 0.983169i \(0.558483\pi\)
\(368\) 0 0
\(369\) 72.0000 3.74817
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 16.0000 0.828449 0.414224 − 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) 35.0000 1.80259
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −36.0000 −1.84434
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 − 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 12.0000 0.609994
\(388\) 0 0
\(389\) −15.0000 −0.760530 −0.380265 − 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) 14.0000 0.708010
\(392\) 0 0
\(393\) 18.0000 0.907980
\(394\) 0 0
\(395\) −3.00000 −0.150946
\(396\) 0 0
\(397\) 23.0000 1.15434 0.577168 − 0.816625i \(-0.304158\pi\)
0.577168 + 0.816625i \(0.304158\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −15.0000 −0.749064 −0.374532 − 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) 20.0000 0.996271
\(404\) 0 0
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 24.0000 1.18383
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) −54.0000 −2.64439
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 − 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −3.00000 −0.146211 −0.0731055 − 0.997324i \(-0.523291\pi\)
−0.0731055 + 0.997324i \(0.523291\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) 7.00000 0.339550
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −75.0000 −3.62103
\(430\) 0 0
\(431\) 25.0000 1.20421 0.602104 − 0.798418i \(-0.294329\pi\)
0.602104 + 0.798418i \(0.294329\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 − 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 21.0000 1.00687
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 − 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6.00000 −0.285069 −0.142534 − 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 66.0000 3.12169
\(448\) 0 0
\(449\) −9.00000 −0.424736 −0.212368 − 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 60.0000 2.82529
\(452\) 0 0
\(453\) −57.0000 −2.67809
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 − 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 0 0
\(459\) −63.0000 −2.94059
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 − 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 0 0
\(465\) 12.0000 0.556487
\(466\) 0 0
\(467\) 11.0000 0.509019 0.254510 − 0.967070i \(-0.418086\pi\)
0.254510 + 0.967070i \(0.418086\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 30.0000 1.38233
\(472\) 0 0
\(473\) 10.0000 0.459800
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 34.0000 1.55350 0.776750 − 0.629809i \(-0.216867\pi\)
0.776750 + 0.629809i \(0.216867\pi\)
\(480\) 0 0
\(481\) −30.0000 −1.36788
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13.0000 0.590300
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 − 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) 0 0
\(489\) −42.0000 −1.89931
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 − 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) 49.0000 2.20685
\(494\) 0 0
\(495\) −30.0000 −1.34840
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.00000 0.0447661 0.0223831 − 0.999749i \(-0.492875\pi\)
0.0223831 + 0.999749i \(0.492875\pi\)
\(500\) 0 0
\(501\) 9.00000 0.402090
\(502\) 0 0
\(503\) 3.00000 0.133763 0.0668817 − 0.997761i \(-0.478695\pi\)
0.0668817 + 0.997761i \(0.478695\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) −36.0000 −1.59882
\(508\) 0 0
\(509\) 14.0000 0.620539 0.310270 − 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 18.0000 0.794719
\(514\) 0 0
\(515\) −13.0000 −0.572848
\(516\) 0 0
\(517\) 5.00000 0.219900
\(518\) 0 0
\(519\) −21.0000 −0.921798
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 − 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) −44.0000 −1.92399 −0.961993 − 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 28.0000 1.21970
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) −24.0000 −1.04151
\(532\) 0 0
\(533\) 60.0000 2.59889
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.00000 0.300954 0.150477 − 0.988614i \(-0.451919\pi\)
0.150477 + 0.988614i \(0.451919\pi\)
\(542\) 0 0
\(543\) 24.0000 1.02994
\(544\) 0 0
\(545\) −5.00000 −0.214176
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) −24.0000 −1.02430
\(550\) 0 0
\(551\) −14.0000 −0.596420
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −18.0000 −0.764057
\(556\) 0 0
\(557\) −16.0000 −0.677942 −0.338971 − 0.940797i \(-0.610079\pi\)
−0.338971 + 0.940797i \(0.610079\pi\)
\(558\) 0 0
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) −105.000 −4.43310
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 − 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 − 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) −39.0000 −1.62925
\(574\) 0 0
\(575\) 2.00000 0.0834058
\(576\) 0 0
\(577\) 37.0000 1.54033 0.770165 − 0.637845i \(-0.220174\pi\)
0.770165 + 0.637845i \(0.220174\pi\)
\(578\) 0 0
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −30.0000 −1.24035
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) 0 0
\(593\) −27.0000 −1.10876 −0.554379 − 0.832265i \(-0.687044\pi\)
−0.554379 + 0.832265i \(0.687044\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 12.0000 0.491127
\(598\) 0 0
\(599\) −15.0000 −0.612883 −0.306442 − 0.951889i \(-0.599138\pi\)
−0.306442 + 0.951889i \(0.599138\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) −48.0000 −1.95471
\(604\) 0 0
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) 3.00000 0.121766 0.0608831 − 0.998145i \(-0.480608\pi\)
0.0608831 + 0.998145i \(0.480608\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5.00000 0.202278
\(612\) 0 0
\(613\) −30.0000 −1.21169 −0.605844 − 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 0 0
\(615\) 36.0000 1.45166
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 − 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) −18.0000 −0.722315
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 30.0000 1.19808
\(628\) 0 0
\(629\) −42.0000 −1.67465
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 − 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 0 0
\(633\) 15.0000 0.596196
\(634\) 0 0
\(635\) −12.0000 −0.476205
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 21.0000 0.828159 0.414080 − 0.910241i \(-0.364104\pi\)
0.414080 + 0.910241i \(0.364104\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 − 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 38.0000 1.48705 0.743527 − 0.668705i \(-0.233151\pi\)
0.743527 + 0.668705i \(0.233151\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 0 0
\(657\) −36.0000 −1.40449
\(658\) 0 0
\(659\) −15.0000 −0.584317 −0.292159 − 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) 0 0
\(661\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) −105.000 −4.07786
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 14.0000 0.542082
\(668\) 0 0
\(669\) 57.0000 2.20375
\(670\) 0 0
\(671\) −20.0000 −0.772091
\(672\) 0 0
\(673\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) −9.00000 −0.346410
\(676\) 0 0
\(677\) 29.0000 1.11456 0.557280 − 0.830324i \(-0.311845\pi\)
0.557280 + 0.830324i \(0.311845\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −27.0000 −1.03464
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 8.00000 0.305664
\(686\) 0 0
\(687\) 12.0000 0.457829
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 − 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −18.0000 −0.682779
\(696\) 0 0
\(697\) 84.0000 3.18173
\(698\) 0 0
\(699\) 72.0000 2.72329
\(700\) 0 0
\(701\) 35.0000 1.32193 0.660966 − 0.750416i \(-0.270147\pi\)
0.660966 + 0.750416i \(0.270147\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) 0 0
\(705\) 3.00000 0.112987
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 − 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 0 0
\(711\) 18.0000 0.675053
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −25.0000 −0.934947
\(716\) 0 0
\(717\) 27.0000 1.00833
\(718\) 0 0
\(719\) −14.0000 −0.522112 −0.261056 − 0.965324i \(-0.584071\pi\)
−0.261056 + 0.965324i \(0.584071\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −66.0000 −2.45457
\(724\) 0 0
\(725\) 7.00000 0.259973
\(726\) 0 0
\(727\) −40.0000 −1.48352 −0.741759 − 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 14.0000 0.517809
\(732\) 0 0
\(733\) 33.0000 1.21888 0.609441 − 0.792831i \(-0.291394\pi\)
0.609441 + 0.792831i \(0.291394\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −40.0000 −1.47342
\(738\) 0 0
\(739\) −51.0000 −1.87607 −0.938033 − 0.346547i \(-0.887354\pi\)
−0.938033 + 0.346547i \(0.887354\pi\)
\(740\) 0 0
\(741\) 30.0000 1.10208
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 22.0000 0.806018
\(746\) 0 0
\(747\) −24.0000 −0.878114
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 3.00000 0.109472 0.0547358 − 0.998501i \(-0.482568\pi\)
0.0547358 + 0.998501i \(0.482568\pi\)
\(752\) 0 0
\(753\) −18.0000 −0.655956
\(754\) 0 0
\(755\) −19.0000 −0.691481
\(756\) 0 0
\(757\) −40.0000 −1.45382 −0.726912 − 0.686730i \(-0.759045\pi\)
−0.726912 + 0.686730i \(0.759045\pi\)
\(758\) 0 0
\(759\) −30.0000 −1.08893
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 − 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −42.0000 −1.51851
\(766\) 0 0
\(767\) −20.0000 −0.722158
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 54.0000 1.94476
\(772\) 0 0
\(773\) 21.0000 0.755318 0.377659 − 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −63.0000 −2.25144
\(784\) 0 0
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) 17.0000 0.605985 0.302992 − 0.952993i \(-0.402014\pi\)
0.302992 + 0.952993i \(0.402014\pi\)
\(788\) 0 0
\(789\) −90.0000 −3.20408
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 39.0000 1.38145 0.690725 − 0.723117i \(-0.257291\pi\)
0.690725 + 0.723117i \(0.257291\pi\)
\(798\) 0 0
\(799\) 7.00000 0.247642
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −30.0000 −1.05868
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 78.0000 2.74573
\(808\) 0 0
\(809\) −39.0000 −1.37117 −0.685583 − 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) −10.0000 −0.351147 −0.175574 − 0.984466i \(-0.556178\pi\)
−0.175574 + 0.984466i \(0.556178\pi\)
\(812\) 0 0
\(813\) −36.0000 −1.26258
\(814\) 0 0
\(815\) −14.0000 −0.490399
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.00000 0.0349002 0.0174501 − 0.999848i \(-0.494445\pi\)
0.0174501 + 0.999848i \(0.494445\pi\)
\(822\) 0 0
\(823\) −28.0000 −0.976019 −0.488009 − 0.872838i \(-0.662277\pi\)
−0.488009 + 0.872838i \(0.662277\pi\)
\(824\) 0 0
\(825\) −15.0000 −0.522233
\(826\) 0 0
\(827\) −6.00000 −0.208640 −0.104320 − 0.994544i \(-0.533267\pi\)
−0.104320 + 0.994544i \(0.533267\pi\)
\(828\) 0 0
\(829\) 20.0000 0.694629 0.347314 − 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 3.00000 0.103819
\(836\) 0 0
\(837\) −36.0000 −1.24434
\(838\) 0 0
\(839\) −34.0000 −1.17381 −0.586905 − 0.809656i \(-0.699654\pi\)
−0.586905 + 0.809656i \(0.699654\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 0 0
\(843\) −57.0000 −1.96318
\(844\) 0 0
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 75.0000 2.57399
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 − 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 12.0000 0.410391
\(856\) 0 0
\(857\) −6.00000 −0.204956 −0.102478 − 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −48.0000 −1.63394 −0.816970 − 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 0 0
\(865\) −7.00000 −0.238007
\(866\) 0 0
\(867\) −96.0000 −3.26033
\(868\) 0 0
\(869\) 15.0000 0.508840
\(870\) 0 0
\(871\) −40.0000 −1.35535
\(872\) 0 0
\(873\) −78.0000 −2.63990
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 0 0
\(879\) 39.0000 1.31544
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 40.0000 1.34611 0.673054 − 0.739594i \(-0.264982\pi\)
0.673054 + 0.739594i \(0.264982\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) 0 0
\(887\) 16.0000 0.537227 0.268614 − 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 45.0000 1.50756
\(892\) 0 0
\(893\) −2.00000 −0.0669274
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) −30.0000 −1.00167
\(898\) 0 0
\(899\) 28.0000 0.933852
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8.00000 0.265929
\(906\) 0 0
\(907\) 18.0000 0.597680 0.298840 − 0.954303i \(-0.403400\pi\)
0.298840 + 0.954303i \(0.403400\pi\)
\(908\) 0 0
\(909\) 108.000 3.58213
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 − 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) −20.0000 −0.661903
\(914\) 0 0
\(915\) −12.0000 −0.396708
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 55.0000 1.81428 0.907141 − 0.420826i \(-0.138260\pi\)
0.907141 + 0.420826i \(0.138260\pi\)
\(920\) 0 0
\(921\) 51.0000 1.68051
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 0 0
\(927\) 78.0000 2.56186
\(928\) 0 0
\(929\) 32.0000 1.04989 0.524943 − 0.851137i \(-0.324087\pi\)
0.524943 + 0.851137i \(0.324087\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 102.000 3.33933
\(934\) 0 0
\(935\) −35.0000 −1.14462
\(936\) 0 0
\(937\) −13.0000 −0.424691 −0.212346 − 0.977195i \(-0.568110\pi\)
−0.212346 + 0.977195i \(0.568110\pi\)
\(938\) 0 0
\(939\) −3.00000 −0.0979013
\(940\) 0 0
\(941\) −28.0000 −0.912774 −0.456387 − 0.889781i \(-0.650857\pi\)
−0.456387 + 0.889781i \(0.650857\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −40.0000 −1.29983 −0.649913 − 0.760009i \(-0.725195\pi\)
−0.649913 + 0.760009i \(0.725195\pi\)
\(948\) 0 0
\(949\) −30.0000 −0.973841
\(950\) 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
\(955\) −13.0000 −0.420670
\(956\) 0 0
\(957\) −105.000 −3.39417
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −108.000 −3.48025
\(964\) 0 0
\(965\) 8.00000 0.257529
\(966\) 0 0
\(967\) −6.00000 −0.192947 −0.0964735 − 0.995336i \(-0.530756\pi\)
−0.0964735 + 0.995336i \(0.530756\pi\)
\(968\) 0 0
\(969\) 42.0000 1.34923
\(970\) 0 0
\(971\) −16.0000 −0.513464 −0.256732 − 0.966483i \(-0.582646\pi\)
−0.256732 + 0.966483i \(0.582646\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −15.0000 −0.480384
\(976\) 0 0
\(977\) 26.0000 0.831814 0.415907 − 0.909407i \(-0.363464\pi\)
0.415907 + 0.909407i \(0.363464\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 30.0000 0.957826
\(982\) 0 0
\(983\) 3.00000 0.0956851 0.0478426 − 0.998855i \(-0.484765\pi\)
0.0478426 + 0.998855i \(0.484765\pi\)
\(984\) 0 0
\(985\) 8.00000 0.254901
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 − 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) −36.0000 −1.14243
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) 3.00000 0.0950110 0.0475055 − 0.998871i \(-0.484873\pi\)
0.0475055 + 0.998871i \(0.484873\pi\)
\(998\) 0 0
\(999\) 54.0000 1.70848
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 3920.2.a.c.1.1 1
4.3 odd 2 1960.2.a.o.1.1 1
7.6 odd 2 560.2.a.f.1.1 1
20.19 odd 2 9800.2.a.a.1.1 1
21.20 even 2 5040.2.a.a.1.1 1
28.3 even 6 1960.2.q.o.961.1 2
28.11 odd 6 1960.2.q.a.961.1 2
28.19 even 6 1960.2.q.o.361.1 2
28.23 odd 6 1960.2.q.a.361.1 2
28.27 even 2 280.2.a.a.1.1 ✓ 1
35.13 even 4 2800.2.g.b.449.2 2
35.27 even 4 2800.2.g.b.449.1 2
35.34 odd 2 2800.2.a.c.1.1 1
56.13 odd 2 2240.2.a.a.1.1 1
56.27 even 2 2240.2.a.z.1.1 1
84.83 odd 2 2520.2.a.i.1.1 1
140.27 odd 4 1400.2.g.a.449.2 2
140.83 odd 4 1400.2.g.a.449.1 2
140.139 even 2 1400.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.a.1.1 ✓ 1 28.27 even 2
560.2.a.f.1.1 1 7.6 odd 2
1400.2.a.n.1.1 1 140.139 even 2
1400.2.g.a.449.1 2 140.83 odd 4
1400.2.g.a.449.2 2 140.27 odd 4
1960.2.a.o.1.1 1 4.3 odd 2
1960.2.q.a.361.1 2 28.23 odd 6
1960.2.q.a.961.1 2 28.11 odd 6
1960.2.q.o.361.1 2 28.19 even 6
1960.2.q.o.961.1 2 28.3 even 6
2240.2.a.a.1.1 1 56.13 odd 2
2240.2.a.z.1.1 1 56.27 even 2
2520.2.a.i.1.1 1 84.83 odd 2
2800.2.a.c.1.1 1 35.34 odd 2
2800.2.g.b.449.1 2 35.27 even 4
2800.2.g.b.449.2 2 35.13 even 4
3920.2.a.c.1.1 1 1.1 even 1 trivial
5040.2.a.a.1.1 1 21.20 even 2
9800.2.a.a.1.1 1 20.19 odd 2