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Properties

Label 350.2.a.b.1.1
Level $350$
Weight $2$
Character 350.1
Self dual yes
Analytic conductor $2.795$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,2,Mod(1,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.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: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 350.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +4.00000 q^{11} +6.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} -4.00000 q^{22} -6.00000 q^{26} +1.00000 q^{28} +6.00000 q^{29} +8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} -3.00000 q^{36} +10.0000 q^{37} +2.00000 q^{41} -4.00000 q^{43} +4.00000 q^{44} -8.00000 q^{47} +1.00000 q^{49} +6.00000 q^{52} +2.00000 q^{53} -1.00000 q^{56} -6.00000 q^{58} -8.00000 q^{59} -14.0000 q^{61} -8.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +12.0000 q^{67} -2.00000 q^{68} -16.0000 q^{71} +3.00000 q^{72} -2.00000 q^{73} -10.0000 q^{74} +4.00000 q^{77} -8.00000 q^{79} +9.00000 q^{81} -2.00000 q^{82} -8.00000 q^{83} +4.00000 q^{86} -4.00000 q^{88} +10.0000 q^{89} +6.00000 q^{91} +8.00000 q^{94} -2.00000 q^{97} -1.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 3.00000 0.707107
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 6.00000 0.832050
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −8.00000 −1.01600
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 − 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 3.00000 0.353553
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −2.00000 −0.220863
\(83\) −8.00000 −0.878114 −0.439057 − 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) −12.0000 −1.20605
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 − 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) −18.0000 −1.66410
\(118\) 8.00000 0.736460
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 14.0000 1.26750
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 − 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 − 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) 24.0000 2.00698
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −9.00000 −0.707107
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 22.0000 1.67263 0.836315 − 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 − 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 − 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −14.0000 −0.997459 −0.498729 − 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 12.0000 0.852803
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) −6.00000 −0.406371
\(219\) 0 0
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 18.0000 1.17670
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) 16.0000 1.03495 0.517477 − 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −3.00000 −0.188982
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 − 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) 16.0000 0.988483
\(263\) −8.00000 −0.493301 −0.246651 − 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 18.0000 1.08152 0.540758 − 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) −16.0000 −0.959616
\(279\) −24.0000 −1.43684
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 − 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) −32.0000 −1.90220 −0.951101 − 0.308879i \(-0.900046\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 2.00000 0.118056
\(288\) 3.00000 0.176777
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) −10.0000 −0.584206 −0.292103 − 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 8.00000 0.456584 0.228292 − 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 − 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −22.0000 −1.23564 −0.617822 − 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −8.00000 −0.439057
\(333\) −30.0000 −1.64399
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −22.0000 −1.18273
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 − 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 8.00000 0.422224 0.211112 − 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 14.0000 0.735824
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 − 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 − 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 36.0000 1.85409
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 − 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −24.0000 −1.22795
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 12.0000 0.609994
\(388\) −2.00000 −0.101535
\(389\) 14.0000 0.709828 0.354914 − 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 14.0000 0.705310
\(395\) 0 0
\(396\) −12.0000 −0.603023
\(397\) −10.0000 −0.501886 −0.250943 − 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 − 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 0 0
\(403\) 48.0000 2.39105
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 40.0000 1.98273
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 − 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) 0 0
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 − 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −4.00000 −0.194717
\(423\) 24.0000 1.16692
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) 0 0
\(427\) −14.0000 −0.677507
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 0 0
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 12.0000 0.570782
\(443\) 20.0000 0.950229 0.475114 − 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) 0 0
\(461\) 10.0000 0.465746 0.232873 − 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 40.0000 1.85098 0.925490 − 0.378773i \(-0.123654\pi\)
0.925490 + 0.378773i \(0.123654\pi\)
\(468\) −18.0000 −0.832050
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) 8.00000 0.368230
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) −6.00000 −0.274721
\(478\) −16.0000 −0.731823
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 60.0000 2.73576
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) −32.0000 −1.45006 −0.725029 − 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 14.0000 0.633750
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −12.0000 −0.537194 −0.268597 − 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 10.0000 0.443242 0.221621 − 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −22.0000 −0.970378
\(515\) 0 0
\(516\) 0 0
\(517\) −32.0000 −1.40736
\(518\) −10.0000 −0.439375
\(519\) 0 0
\(520\) 0 0
\(521\) 2.00000 0.0876216 0.0438108 − 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) 18.0000 0.787839
\(523\) −16.0000 −0.699631 −0.349816 − 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −16.0000 −0.698963
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 − 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 − 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 6.00000 0.256307
\(549\) 42.0000 1.79252
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) −18.0000 −0.764747
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −22.0000 −0.932170 −0.466085 − 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) 24.0000 1.01600
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) −26.0000 −1.09674
\(563\) 16.0000 0.674320 0.337160 − 0.941447i \(-0.390534\pi\)
0.337160 + 0.941447i \(0.390534\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 32.0000 1.34506
\(567\) 9.00000 0.377964
\(568\) 16.0000 0.671345
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 − 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 24.0000 1.00349
\(573\) 0 0
\(574\) −2.00000 −0.0834784
\(575\) 0 0
\(576\) −3.00000 −0.125000
\(577\) −10.0000 −0.416305 −0.208153 − 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 10.0000 0.413096
\(587\) −8.00000 −0.330195 −0.165098 − 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 10.0000 0.410997
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 − 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 4.00000 0.163028
\(603\) −36.0000 −1.46603
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 − 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 6.00000 0.242536
\(613\) −6.00000 −0.242338 −0.121169 − 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) 22.0000 0.885687 0.442843 − 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) 8.00000 0.321547 0.160774 − 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 − 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 22.0000 0.873732
\(635\) 0 0
\(636\) 0 0
\(637\) 6.00000 0.237729
\(638\) −24.0000 −0.950169
\(639\) 48.0000 1.89885
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 − 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 32.0000 1.26196 0.630978 − 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −16.0000 −0.629025 −0.314512 − 0.949253i \(-0.601841\pi\)
−0.314512 + 0.949253i \(0.601841\pi\)
\(648\) −9.00000 −0.353553
\(649\) −32.0000 −1.25611
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −14.0000 −0.547862 −0.273931 − 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 6.00000 0.234082
\(658\) 8.00000 0.311872
\(659\) 28.0000 1.09073 0.545363 − 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 − 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 8.00000 0.310460
\(665\) 0 0
\(666\) 30.0000 1.16248
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −56.0000 −2.16186
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 − 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) −32.0000 −1.22534
\(683\) 4.00000 0.153056 0.0765279 − 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 − 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 22.0000 0.836315
\(693\) −12.0000 −0.455842
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) −10.0000 −0.378506
\(699\) 0 0
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 − 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −2.00000 −0.0751116 −0.0375558 − 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) 0 0
\(711\) 24.0000 0.900070
\(712\) −10.0000 −0.374766
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) −14.0000 −0.520306
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) −6.00000 −0.222375
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 − 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 0 0
\(737\) 48.0000 1.76810
\(738\) 6.00000 0.220863
\(739\) 28.0000 1.03000 0.514998 − 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.00000 −0.0734223
\(743\) −48.0000 −1.76095 −0.880475 − 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) 24.0000 0.878114
\(748\) −8.00000 −0.292509
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 − 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 − 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 12.0000 0.435860
\(759\) 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
\(763\) 6.00000 0.217215
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −48.0000 −1.73318
\(768\) 0 0
\(769\) 10.0000 0.360609 0.180305 − 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) −42.0000 −1.51064 −0.755318 − 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −14.0000 −0.501924
\(779\) 0 0
\(780\) 0 0
\(781\) −64.0000 −2.29010
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 − 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) −14.0000 −0.498729
\(789\) 0 0
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 12.0000 0.426401
\(793\) −84.0000 −2.98293
\(794\) 10.0000 0.354887
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 6.00000 0.212531 0.106265 − 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 14.0000 0.494357
\(803\) −8.00000 −0.282314
\(804\) 0 0
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 − 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 6.00000 0.210559
\(813\) 0 0
\(814\) −40.0000 −1.40200
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 30.0000 1.04893
\(819\) −18.0000 −0.628971
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 − 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 − 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) −44.0000 −1.53003 −0.765015 − 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 0 0
\(829\) −46.0000 −1.59765 −0.798823 − 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 6.00000 0.208013
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −24.0000 −0.829066
\(839\) −16.0000 −0.552381 −0.276191 − 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) −24.0000 −0.825137
\(847\) 5.00000 0.171802
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 − 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 14.0000 0.479070
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 54.0000 1.84460 0.922302 − 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 0 0
\(859\) −48.0000 −1.63774 −0.818869 − 0.573980i \(-0.805399\pi\)
−0.818869 + 0.573980i \(0.805399\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 32.0000 1.08929 0.544646 − 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 72.0000 2.43963
\(872\) −6.00000 −0.203186
\(873\) 6.00000 0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −14.0000 −0.472746 −0.236373 − 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 − 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 3.00000 0.101015
\(883\) 12.0000 0.403832 0.201916 − 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) 16.0000 0.537227 0.268614 − 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 36.0000 1.20605
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) −8.00000 −0.266371
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 0 0
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 − 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) −8.00000 −0.265489
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −32.0000 −1.05905
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) −16.0000 −0.528367
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −10.0000 −0.329332
\(923\) −96.0000 −3.15988
\(924\) 0 0
\(925\) 0 0
\(926\) −16.0000 −0.525793
\(927\) 48.0000 1.57653
\(928\) −6.00000 −0.196960
\(929\) 58.0000 1.90292 0.951459 − 0.307775i \(-0.0995844\pi\)
0.951459 + 0.307775i \(0.0995844\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) −40.0000 −1.30884
\(935\) 0 0
\(936\) 18.0000 0.588348
\(937\) −50.0000 −1.63343 −0.816714 − 0.577042i \(-0.804207\pi\)
−0.816714 + 0.577042i \(0.804207\pi\)
\(938\) −12.0000 −0.391814
\(939\) 0 0
\(940\) 0 0
\(941\) 2.00000 0.0651981 0.0325991 − 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) −44.0000 −1.42981 −0.714904 − 0.699223i \(-0.753530\pi\)
−0.714904 + 0.699223i \(0.753530\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 2.00000 0.0648204
\(953\) 54.0000 1.74923 0.874616 − 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −60.0000 −1.93448
\(963\) 36.0000 1.16008
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 − 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 − 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 0 0
\(973\) 16.0000 0.512936
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 40.0000 1.27841
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) −12.0000 −0.382935
\(983\) 16.0000 0.510321 0.255160 − 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 − 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) −8.00000 −0.254000
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) 0 0
\(997\) 22.0000 0.696747 0.348373 − 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) 12.0000 0.379853
\(999\) 0 0
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 350.2.a.b.1.1 1
3.2 odd 2 3150.2.a.bj.1.1 1
4.3 odd 2 2800.2.a.m.1.1 1
5.2 odd 4 350.2.c.b.99.1 2
5.3 odd 4 350.2.c.b.99.2 2
5.4 even 2 70.2.a.a.1.1 ✓ 1
7.6 odd 2 2450.2.a.l.1.1 1
15.2 even 4 3150.2.g.c.2899.2 2
15.8 even 4 3150.2.g.c.2899.1 2
15.14 odd 2 630.2.a.d.1.1 1
20.3 even 4 2800.2.g.n.449.2 2
20.7 even 4 2800.2.g.n.449.1 2
20.19 odd 2 560.2.a.d.1.1 1
35.4 even 6 490.2.e.d.471.1 2
35.9 even 6 490.2.e.d.361.1 2
35.13 even 4 2450.2.c.k.99.2 2
35.19 odd 6 490.2.e.c.361.1 2
35.24 odd 6 490.2.e.c.471.1 2
35.27 even 4 2450.2.c.k.99.1 2
35.34 odd 2 490.2.a.h.1.1 1
40.19 odd 2 2240.2.a.q.1.1 1
40.29 even 2 2240.2.a.n.1.1 1
55.54 odd 2 8470.2.a.j.1.1 1
60.59 even 2 5040.2.a.bm.1.1 1
105.104 even 2 4410.2.a.b.1.1 1
140.139 even 2 3920.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.a.a.1.1 ✓ 1 5.4 even 2
350.2.a.b.1.1 1 1.1 even 1 trivial
350.2.c.b.99.1 2 5.2 odd 4
350.2.c.b.99.2 2 5.3 odd 4
490.2.a.h.1.1 1 35.34 odd 2
490.2.e.c.361.1 2 35.19 odd 6
490.2.e.c.471.1 2 35.24 odd 6
490.2.e.d.361.1 2 35.9 even 6
490.2.e.d.471.1 2 35.4 even 6
560.2.a.d.1.1 1 20.19 odd 2
630.2.a.d.1.1 1 15.14 odd 2
2240.2.a.n.1.1 1 40.29 even 2
2240.2.a.q.1.1 1 40.19 odd 2
2450.2.a.l.1.1 1 7.6 odd 2
2450.2.c.k.99.1 2 35.27 even 4
2450.2.c.k.99.2 2 35.13 even 4
2800.2.a.m.1.1 1 4.3 odd 2
2800.2.g.n.449.1 2 20.7 even 4
2800.2.g.n.449.2 2 20.3 even 4
3150.2.a.bj.1.1 1 3.2 odd 2
3150.2.g.c.2899.1 2 15.8 even 4
3150.2.g.c.2899.2 2 15.2 even 4
3920.2.a.t.1.1 1 140.139 even 2
4410.2.a.b.1.1 1 105.104 even 2
5040.2.a.bm.1.1 1 60.59 even 2
8470.2.a.j.1.1 1 55.54 odd 2