Properties

Label 4029.2.a.a.1.1
Level $4029$
Weight $2$
Character 4029.1
Self dual yes
Analytic conductor $32.172$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4029,2,Mod(1,4029)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4029.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4029.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: \(32.1717269744\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4029.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} -3.00000 q^{15} +4.00000 q^{16} +1.00000 q^{17} +5.00000 q^{19} +6.00000 q^{20} +1.00000 q^{21} -7.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} -2.00000 q^{28} +10.0000 q^{29} +4.00000 q^{31} -4.00000 q^{33} -3.00000 q^{35} -2.00000 q^{36} -5.00000 q^{37} +2.00000 q^{39} -6.00000 q^{41} +6.00000 q^{43} +8.00000 q^{44} -3.00000 q^{45} -3.00000 q^{47} +4.00000 q^{48} -6.00000 q^{49} +1.00000 q^{51} -4.00000 q^{52} +3.00000 q^{53} +12.0000 q^{55} +5.00000 q^{57} -9.00000 q^{59} +6.00000 q^{60} +9.00000 q^{61} +1.00000 q^{63} -8.00000 q^{64} -6.00000 q^{65} +12.0000 q^{67} -2.00000 q^{68} -7.00000 q^{69} -6.00000 q^{71} -4.00000 q^{73} +4.00000 q^{75} -10.0000 q^{76} -4.00000 q^{77} -1.00000 q^{79} -12.0000 q^{80} +1.00000 q^{81} -16.0000 q^{83} -2.00000 q^{84} -3.00000 q^{85} +10.0000 q^{87} +12.0000 q^{89} +2.00000 q^{91} +14.0000 q^{92} +4.00000 q^{93} -15.0000 q^{95} +2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.00000 0.577350
\(4\) −2.00000 −1.00000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −2.00000 −0.577350
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 4.00000 1.00000
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 6.00000 1.34164
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −7.00000 −1.45960 −0.729800 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) −2.00000 −0.333333
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 8.00000 1.20605
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 4.00000 0.577350
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) −4.00000 −0.554700
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 5.00000 0.662266
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 6.00000 0.774597
\(61\) 9.00000 1.15233 0.576166 0.817333i \(-0.304548\pi\)
0.576166 + 0.817333i \(0.304548\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) −8.00000 −1.00000
\(65\) −6.00000 −0.744208
\(66\) 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
\(69\) −7.00000 −0.842701
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) −10.0000 −1.14708
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −1.00000 −0.112509
\(80\) −12.0000 −1.34164
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) −2.00000 −0.218218
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 14.0000 1.45960
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) −15.0000 −1.53897
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) −8.00000 −0.800000
\(101\) −16.0000 −1.59206 −0.796030 0.605257i \(-0.793070\pi\)
−0.796030 + 0.605257i \(0.793070\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) −2.00000 −0.192450
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −5.00000 −0.474579
\(112\) 4.00000 0.377964
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 0 0
\(115\) 21.0000 1.95826
\(116\) −20.0000 −1.85695
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) −8.00000 −0.718421
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −22.0000 −1.95218 −0.976092 0.217357i \(-0.930256\pi\)
−0.976092 + 0.217357i \(0.930256\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −3.00000 −0.262111 −0.131056 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) 8.00000 0.696311
\(133\) 5.00000 0.433555
\(134\) 0 0
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) 15.0000 1.28154 0.640768 0.767734i \(-0.278616\pi\)
0.640768 + 0.767734i \(0.278616\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 6.00000 0.507093
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 4.00000 0.333333
\(145\) −30.0000 −2.49136
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 10.0000 0.821995
\(149\) 11.0000 0.901155 0.450578 0.892737i \(-0.351218\pi\)
0.450578 + 0.892737i \(0.351218\pi\)
\(150\) 0 0
\(151\) 13.0000 1.05792 0.528962 0.848645i \(-0.322581\pi\)
0.528962 + 0.848645i \(0.322581\pi\)
\(152\) 0 0
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) −12.0000 −0.963863
\(156\) −4.00000 −0.320256
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 0 0
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −7.00000 −0.551677
\(162\) 0 0
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) 12.0000 0.937043
\(165\) 12.0000 0.934199
\(166\) 0 0
\(167\) 13.0000 1.00597 0.502985 0.864295i \(-0.332235\pi\)
0.502985 + 0.864295i \(0.332235\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 5.00000 0.382360
\(172\) −12.0000 −0.914991
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) −16.0000 −1.20605
\(177\) −9.00000 −0.676481
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 6.00000 0.447214
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 9.00000 0.665299
\(184\) 0 0
\(185\) 15.0000 1.10282
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 6.00000 0.437595
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 15.0000 1.08536 0.542681 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) −8.00000 −0.577350
\(193\) −15.0000 −1.07972 −0.539862 0.841754i \(-0.681524\pi\)
−0.539862 + 0.841754i \(0.681524\pi\)
\(194\) 0 0
\(195\) −6.00000 −0.429669
\(196\) 12.0000 0.857143
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 0 0
\(199\) −21.0000 −1.48865 −0.744325 0.667817i \(-0.767229\pi\)
−0.744325 + 0.667817i \(0.767229\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 10.0000 0.701862
\(204\) −2.00000 −0.140028
\(205\) 18.0000 1.25717
\(206\) 0 0
\(207\) −7.00000 −0.486534
\(208\) 8.00000 0.554700
\(209\) −20.0000 −1.38343
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) −6.00000 −0.412082
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) −18.0000 −1.22759
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) −24.0000 −1.61808
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) −7.00000 −0.468755 −0.234377 0.972146i \(-0.575305\pi\)
−0.234377 + 0.972146i \(0.575305\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) −10.0000 −0.662266
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 18.0000 1.17170
\(237\) −1.00000 −0.0649570
\(238\) 0 0
\(239\) −14.0000 −0.905585 −0.452792 0.891616i \(-0.649572\pi\)
−0.452792 + 0.891616i \(0.649572\pi\)
\(240\) −12.0000 −0.774597
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −18.0000 −1.15233
\(245\) 18.0000 1.14998
\(246\) 0 0
\(247\) 10.0000 0.636285
\(248\) 0 0
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) −2.00000 −0.125988
\(253\) 28.0000 1.76034
\(254\) 0 0
\(255\) −3.00000 −0.187867
\(256\) 16.0000 1.00000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) −5.00000 −0.310685
\(260\) 12.0000 0.744208
\(261\) 10.0000 0.618984
\(262\) 0 0
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) −24.0000 −1.46603
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 4.00000 0.242536
\(273\) 2.00000 0.121046
\(274\) 0 0
\(275\) −16.0000 −0.964836
\(276\) 14.0000 0.842701
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 12.0000 0.712069
\(285\) −15.0000 −0.888523
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 8.00000 0.468165
\(293\) 17.0000 0.993151 0.496575 0.867994i \(-0.334591\pi\)
0.496575 + 0.867994i \(0.334591\pi\)
\(294\) 0 0
\(295\) 27.0000 1.57200
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) −14.0000 −0.809641
\(300\) −8.00000 −0.461880
\(301\) 6.00000 0.345834
\(302\) 0 0
\(303\) −16.0000 −0.919176
\(304\) 20.0000 1.14708
\(305\) −27.0000 −1.54602
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 8.00000 0.455842
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −26.0000 −1.47432 −0.737162 0.675716i \(-0.763835\pi\)
−0.737162 + 0.675716i \(0.763835\pi\)
\(312\) 0 0
\(313\) 32.0000 1.80875 0.904373 0.426742i \(-0.140339\pi\)
0.904373 + 0.426742i \(0.140339\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 2.00000 0.112509
\(317\) −9.00000 −0.505490 −0.252745 0.967533i \(-0.581333\pi\)
−0.252745 + 0.967533i \(0.581333\pi\)
\(318\) 0 0
\(319\) −40.0000 −2.23957
\(320\) 24.0000 1.34164
\(321\) −2.00000 −0.111629
\(322\) 0 0
\(323\) 5.00000 0.278207
\(324\) −2.00000 −0.111111
\(325\) 8.00000 0.443760
\(326\) 0 0
\(327\) −14.0000 −0.774202
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 32.0000 1.75623
\(333\) −5.00000 −0.273998
\(334\) 0 0
\(335\) −36.0000 −1.96689
\(336\) 4.00000 0.218218
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 0 0
\(339\) −8.00000 −0.434500
\(340\) 6.00000 0.325396
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 21.0000 1.13060
\(346\) 0 0
\(347\) −11.0000 −0.590511 −0.295255 0.955418i \(-0.595405\pi\)
−0.295255 + 0.955418i \(0.595405\pi\)
\(348\) −20.0000 −1.07211
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 15.0000 0.798369 0.399185 0.916871i \(-0.369293\pi\)
0.399185 + 0.916871i \(0.369293\pi\)
\(354\) 0 0
\(355\) 18.0000 0.955341
\(356\) −24.0000 −1.27200
\(357\) 1.00000 0.0529256
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) −4.00000 −0.209657
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) 10.0000 0.521996 0.260998 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) −28.0000 −1.45960
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) −8.00000 −0.414781
\(373\) −20.0000 −1.03556 −0.517780 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) 20.0000 1.03005
\(378\) 0 0
\(379\) −15.0000 −0.770498 −0.385249 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 30.0000 1.53897
\(381\) −22.0000 −1.12709
\(382\) 0 0
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) 0 0
\(387\) 6.00000 0.304997
\(388\) −4.00000 −0.203069
\(389\) 20.0000 1.01404 0.507020 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) 0 0
\(391\) −7.00000 −0.354005
\(392\) 0 0
\(393\) −3.00000 −0.151330
\(394\) 0 0
\(395\) 3.00000 0.150946
\(396\) 8.00000 0.402015
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 5.00000 0.250313
\(400\) 16.0000 0.800000
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 32.0000 1.59206
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) 38.0000 1.87898 0.939490 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 0 0
\(411\) 15.0000 0.739895
\(412\) 32.0000 1.57653
\(413\) −9.00000 −0.442861
\(414\) 0 0
\(415\) 48.0000 2.35623
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −40.0000 −1.95413 −0.977064 0.212946i \(-0.931694\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) 6.00000 0.292770
\(421\) −23.0000 −1.12095 −0.560476 0.828171i \(-0.689382\pi\)
−0.560476 + 0.828171i \(0.689382\pi\)
\(422\) 0 0
\(423\) −3.00000 −0.145865
\(424\) 0 0
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) 9.00000 0.435541
\(428\) 4.00000 0.193347
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 33.0000 1.58955 0.794777 0.606902i \(-0.207588\pi\)
0.794777 + 0.606902i \(0.207588\pi\)
\(432\) 4.00000 0.192450
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) −30.0000 −1.43839
\(436\) 28.0000 1.34096
\(437\) −35.0000 −1.67428
\(438\) 0 0
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) −29.0000 −1.37783 −0.688916 0.724841i \(-0.741913\pi\)
−0.688916 + 0.724841i \(0.741913\pi\)
\(444\) 10.0000 0.474579
\(445\) −36.0000 −1.70656
\(446\) 0 0
\(447\) 11.0000 0.520282
\(448\) −8.00000 −0.377964
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 16.0000 0.752577
\(453\) 13.0000 0.610793
\(454\) 0 0
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) 5.00000 0.233890 0.116945 0.993138i \(-0.462690\pi\)
0.116945 + 0.993138i \(0.462690\pi\)
\(458\) 0 0
\(459\) 1.00000 0.0466760
\(460\) −42.0000 −1.95826
\(461\) 42.0000 1.95614 0.978068 0.208288i \(-0.0667892\pi\)
0.978068 + 0.208288i \(0.0667892\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 40.0000 1.85695
\(465\) −12.0000 −0.556487
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) −4.00000 −0.184900
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) −24.0000 −1.10352
\(474\) 0 0
\(475\) 20.0000 0.917663
\(476\) −2.00000 −0.0916698
\(477\) 3.00000 0.137361
\(478\) 0 0
\(479\) −17.0000 −0.776750 −0.388375 0.921501i \(-0.626963\pi\)
−0.388375 + 0.921501i \(0.626963\pi\)
\(480\) 0 0
\(481\) −10.0000 −0.455961
\(482\) 0 0
\(483\) −7.00000 −0.318511
\(484\) −10.0000 −0.454545
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) −14.0000 −0.634401 −0.317200 0.948359i \(-0.602743\pi\)
−0.317200 + 0.948359i \(0.602743\pi\)
\(488\) 0 0
\(489\) 6.00000 0.271329
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 12.0000 0.541002
\(493\) 10.0000 0.450377
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) 16.0000 0.718421
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 30.0000 1.34298 0.671492 0.741012i \(-0.265654\pi\)
0.671492 + 0.741012i \(0.265654\pi\)
\(500\) −6.00000 −0.268328
\(501\) 13.0000 0.580797
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 48.0000 2.13597
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 44.0000 1.95218
\(509\) −3.00000 −0.132973 −0.0664863 0.997787i \(-0.521179\pi\)
−0.0664863 + 0.997787i \(0.521179\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) 0 0
\(513\) 5.00000 0.220755
\(514\) 0 0
\(515\) 48.0000 2.11513
\(516\) −12.0000 −0.528271
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) −40.0000 −1.75243 −0.876216 0.481919i \(-0.839940\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(522\) 0 0
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) 6.00000 0.262111
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) 4.00000 0.174243
\(528\) −16.0000 −0.696311
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) −9.00000 −0.390567
\(532\) −10.0000 −0.433555
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) −10.0000 −0.431532
\(538\) 0 0
\(539\) 24.0000 1.03375
\(540\) 6.00000 0.258199
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 42.0000 1.79908
\(546\) 0 0
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) −30.0000 −1.28154
\(549\) 9.00000 0.384111
\(550\) 0 0
\(551\) 50.0000 2.13007
\(552\) 0 0
\(553\) −1.00000 −0.0425243
\(554\) 0 0
\(555\) 15.0000 0.636715
\(556\) 24.0000 1.01783
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) −12.0000 −0.507093
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 6.00000 0.252646
\(565\) 24.0000 1.00969
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −4.00000 −0.167689 −0.0838444 0.996479i \(-0.526720\pi\)
−0.0838444 + 0.996479i \(0.526720\pi\)
\(570\) 0 0
\(571\) 10.0000 0.418487 0.209243 0.977864i \(-0.432900\pi\)
0.209243 + 0.977864i \(0.432900\pi\)
\(572\) 16.0000 0.668994
\(573\) 15.0000 0.626634
\(574\) 0 0
\(575\) −28.0000 −1.16768
\(576\) −8.00000 −0.333333
\(577\) −8.00000 −0.333044 −0.166522 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) 0 0
\(579\) −15.0000 −0.623379
\(580\) 60.0000 2.49136
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) −39.0000 −1.60970 −0.804851 0.593477i \(-0.797755\pi\)
−0.804851 + 0.593477i \(0.797755\pi\)
\(588\) 12.0000 0.494872
\(589\) 20.0000 0.824086
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) −20.0000 −0.821995
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) −22.0000 −0.901155
\(597\) −21.0000 −0.859473
\(598\) 0 0
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) −25.0000 −1.01977 −0.509886 0.860242i \(-0.670312\pi\)
−0.509886 + 0.860242i \(0.670312\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) −26.0000 −1.05792
\(605\) −15.0000 −0.609837
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 0 0
\(609\) 10.0000 0.405220
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) −2.00000 −0.0808452
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 0 0
\(615\) 18.0000 0.725830
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 15.0000 0.602901 0.301450 0.953482i \(-0.402529\pi\)
0.301450 + 0.953482i \(0.402529\pi\)
\(620\) 24.0000 0.963863
\(621\) −7.00000 −0.280900
\(622\) 0 0
\(623\) 12.0000 0.480770
\(624\) 8.00000 0.320256
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) −20.0000 −0.798723
\(628\) −8.00000 −0.319235
\(629\) −5.00000 −0.199363
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 0 0
\(633\) −5.00000 −0.198732
\(634\) 0 0
\(635\) 66.0000 2.61913
\(636\) −6.00000 −0.237915
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 13.0000 0.513469 0.256735 0.966482i \(-0.417353\pi\)
0.256735 + 0.966482i \(0.417353\pi\)
\(642\) 0 0
\(643\) 38.0000 1.49857 0.749287 0.662246i \(-0.230396\pi\)
0.749287 + 0.662246i \(0.230396\pi\)
\(644\) 14.0000 0.551677
\(645\) −18.0000 −0.708749
\(646\) 0 0
\(647\) −35.0000 −1.37599 −0.687996 0.725714i \(-0.741509\pi\)
−0.687996 + 0.725714i \(0.741509\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) −12.0000 −0.469956
\(653\) −39.0000 −1.52619 −0.763094 0.646288i \(-0.776321\pi\)
−0.763094 + 0.646288i \(0.776321\pi\)
\(654\) 0 0
\(655\) 9.00000 0.351659
\(656\) −24.0000 −0.937043
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −1.00000 −0.0389545 −0.0194772 0.999810i \(-0.506200\pi\)
−0.0194772 + 0.999810i \(0.506200\pi\)
\(660\) −24.0000 −0.934199
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 0 0
\(663\) 2.00000 0.0776736
\(664\) 0 0
\(665\) −15.0000 −0.581675
\(666\) 0 0
\(667\) −70.0000 −2.71041
\(668\) −26.0000 −1.00597
\(669\) −7.00000 −0.270636
\(670\) 0 0
\(671\) −36.0000 −1.38976
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 18.0000 0.692308
\(677\) 39.0000 1.49889 0.749446 0.662066i \(-0.230320\pi\)
0.749446 + 0.662066i \(0.230320\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 7.00000 0.267848 0.133924 0.990992i \(-0.457242\pi\)
0.133924 + 0.990992i \(0.457242\pi\)
\(684\) −10.0000 −0.382360
\(685\) −45.0000 −1.71936
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 24.0000 0.914991
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) 43.0000 1.63580 0.817899 0.575362i \(-0.195139\pi\)
0.817899 + 0.575362i \(0.195139\pi\)
\(692\) −4.00000 −0.152057
\(693\) −4.00000 −0.151947
\(694\) 0 0
\(695\) 36.0000 1.36556
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) 0 0
\(699\) 26.0000 0.983410
\(700\) −8.00000 −0.302372
\(701\) −21.0000 −0.793159 −0.396580 0.918000i \(-0.629803\pi\)
−0.396580 + 0.918000i \(0.629803\pi\)
\(702\) 0 0
\(703\) −25.0000 −0.942893
\(704\) 32.0000 1.20605
\(705\) 9.00000 0.338960
\(706\) 0 0
\(707\) −16.0000 −0.601742
\(708\) 18.0000 0.676481
\(709\) −43.0000 −1.61490 −0.807449 0.589937i \(-0.799153\pi\)
−0.807449 + 0.589937i \(0.799153\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) 0 0
\(713\) −28.0000 −1.04861
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 20.0000 0.747435
\(717\) −14.0000 −0.522840
\(718\) 0 0
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) −12.0000 −0.447214
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) −22.0000 −0.818189
\(724\) 0 0
\(725\) 40.0000 1.48556
\(726\) 0 0
\(727\) 35.0000 1.29808 0.649039 0.760755i \(-0.275171\pi\)
0.649039 + 0.760755i \(0.275171\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) −18.0000 −0.665299
\(733\) 19.0000 0.701781 0.350891 0.936416i \(-0.385879\pi\)
0.350891 + 0.936416i \(0.385879\pi\)
\(734\) 0 0
\(735\) 18.0000 0.663940
\(736\) 0 0
\(737\) −48.0000 −1.76810
\(738\) 0 0
\(739\) 14.0000 0.514998 0.257499 0.966279i \(-0.417102\pi\)
0.257499 + 0.966279i \(0.417102\pi\)
\(740\) −30.0000 −1.10282
\(741\) 10.0000 0.367359
\(742\) 0 0
\(743\) 43.0000 1.57752 0.788759 0.614703i \(-0.210724\pi\)
0.788759 + 0.614703i \(0.210724\pi\)
\(744\) 0 0
\(745\) −33.0000 −1.20903
\(746\) 0 0
\(747\) −16.0000 −0.585409
\(748\) 8.00000 0.292509
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) −48.0000 −1.75154 −0.875772 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) −12.0000 −0.437595
\(753\) −21.0000 −0.765283
\(754\) 0 0
\(755\) −39.0000 −1.41936
\(756\) −2.00000 −0.0727393
\(757\) 23.0000 0.835949 0.417975 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) 0 0
\(759\) 28.0000 1.01634
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) −14.0000 −0.506834
\(764\) −30.0000 −1.08536
\(765\) −3.00000 −0.108465
\(766\) 0 0
\(767\) −18.0000 −0.649942
\(768\) 16.0000 0.577350
\(769\) 4.00000 0.144244 0.0721218 0.997396i \(-0.477023\pi\)
0.0721218 + 0.997396i \(0.477023\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 30.0000 1.07972
\(773\) −40.0000 −1.43870 −0.719350 0.694648i \(-0.755560\pi\)
−0.719350 + 0.694648i \(0.755560\pi\)
\(774\) 0 0
\(775\) 16.0000 0.574737
\(776\) 0 0
\(777\) −5.00000 −0.179374
\(778\) 0 0
\(779\) −30.0000 −1.07486
\(780\) 12.0000 0.429669
\(781\) 24.0000 0.858788
\(782\) 0 0
\(783\) 10.0000 0.357371
\(784\) −24.0000 −0.857143
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) 2.00000 0.0712923 0.0356462 0.999364i \(-0.488651\pi\)
0.0356462 + 0.999364i \(0.488651\pi\)
\(788\) 48.0000 1.70993
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) −8.00000 −0.284447
\(792\) 0 0
\(793\) 18.0000 0.639199
\(794\) 0 0
\(795\) −9.00000 −0.319197
\(796\) 42.0000 1.48865
\(797\) −43.0000 −1.52314 −0.761569 0.648084i \(-0.775571\pi\)
−0.761569 + 0.648084i \(0.775571\pi\)
\(798\) 0 0
\(799\) −3.00000 −0.106132
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) 0 0
\(803\) 16.0000 0.564628
\(804\) −24.0000 −0.846415
\(805\) 21.0000 0.740153
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) 32.0000 1.12367 0.561836 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) −20.0000 −0.701862
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) −18.0000 −0.630512
\(816\) 4.00000 0.140028
\(817\) 30.0000 1.04957
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) −36.0000 −1.25717
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) 0 0
\(823\) −25.0000 −0.871445 −0.435723 0.900081i \(-0.643507\pi\)
−0.435723 + 0.900081i \(0.643507\pi\)
\(824\) 0 0
\(825\) −16.0000 −0.557048
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 14.0000 0.486534
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) −4.00000 −0.138758
\(832\) −16.0000 −0.554700
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −39.0000 −1.34965
\(836\) 40.0000 1.38343
\(837\) 4.00000 0.138260
\(838\) 0 0
\(839\) 5.00000 0.172619 0.0863096 0.996268i \(-0.472493\pi\)
0.0863096 + 0.996268i \(0.472493\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) −20.0000 −0.688837
\(844\) 10.0000 0.344214
\(845\) 27.0000 0.928828
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 12.0000 0.412082
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 35.0000 1.19978
\(852\) 12.0000 0.411113
\(853\) −5.00000 −0.171197 −0.0855984 0.996330i \(-0.527280\pi\)
−0.0855984 + 0.996330i \(0.527280\pi\)
\(854\) 0 0
\(855\) −15.0000 −0.512989
\(856\) 0 0
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 0 0
\(859\) 24.0000 0.818869 0.409435 0.912339i \(-0.365726\pi\)
0.409435 + 0.912339i \(0.365726\pi\)
\(860\) 36.0000 1.22759
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) −8.00000 −0.271538
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 8.00000 0.270295
\(877\) −8.00000 −0.270141 −0.135070 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(878\) 0 0
\(879\) 17.0000 0.573396
\(880\) 48.0000 1.61808
\(881\) 28.0000 0.943344 0.471672 0.881774i \(-0.343651\pi\)
0.471672 + 0.881774i \(0.343651\pi\)
\(882\) 0 0
\(883\) −2.00000 −0.0673054 −0.0336527 0.999434i \(-0.510714\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) −4.00000 −0.134535
\(885\) 27.0000 0.907595
\(886\) 0 0
\(887\) −25.0000 −0.839418 −0.419709 0.907659i \(-0.637868\pi\)
−0.419709 + 0.907659i \(0.637868\pi\)
\(888\) 0 0
\(889\) −22.0000 −0.737856
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 14.0000 0.468755
\(893\) −15.0000 −0.501956
\(894\) 0 0
\(895\) 30.0000 1.00279
\(896\) 0 0
\(897\) −14.0000 −0.467446
\(898\) 0 0
\(899\) 40.0000 1.33407
\(900\) −8.00000 −0.266667
\(901\) 3.00000 0.0999445
\(902\) 0 0
\(903\) 6.00000 0.199667
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −38.0000 −1.26177 −0.630885 0.775877i \(-0.717308\pi\)
−0.630885 + 0.775877i \(0.717308\pi\)
\(908\) 40.0000 1.32745
\(909\) −16.0000 −0.530687
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 20.0000 0.662266
\(913\) 64.0000 2.11809
\(914\) 0 0
\(915\) −27.0000 −0.892592
\(916\) 20.0000 0.660819
\(917\) −3.00000 −0.0990687
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 8.00000 0.263181
\(925\) −20.0000 −0.657596
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) 12.0000 0.393707 0.196854 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 0 0
\(931\) −30.0000 −0.983210
\(932\) −52.0000 −1.70332
\(933\) −26.0000 −0.851202
\(934\) 0 0
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 0 0
\(939\) 32.0000 1.04428
\(940\) −18.0000 −0.587095
\(941\) −19.0000 −0.619382 −0.309691 0.950837i \(-0.600226\pi\)
−0.309691 + 0.950837i \(0.600226\pi\)
\(942\) 0 0
\(943\) 42.0000 1.36771
\(944\) −36.0000 −1.17170
\(945\) −3.00000 −0.0975900
\(946\) 0 0
\(947\) −30.0000 −0.974869 −0.487435 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) 2.00000 0.0649570
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) −9.00000 −0.291845
\(952\) 0 0
\(953\) −54.0000 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 0 0
\(955\) −45.0000 −1.45617
\(956\) 28.0000 0.905585
\(957\) −40.0000 −1.29302
\(958\) 0 0
\(959\) 15.0000 0.484375
\(960\) 24.0000 0.774597
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −2.00000 −0.0644491
\(964\) 44.0000 1.41714
\(965\) 45.0000 1.44860
\(966\) 0 0
\(967\) 7.00000 0.225105 0.112552 0.993646i \(-0.464097\pi\)
0.112552 + 0.993646i \(0.464097\pi\)
\(968\) 0 0
\(969\) 5.00000 0.160623
\(970\) 0 0
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) −2.00000 −0.0641500
\(973\) −12.0000 −0.384702
\(974\) 0 0
\(975\) 8.00000 0.256205
\(976\) 36.0000 1.15233
\(977\) −53.0000 −1.69562 −0.847810 0.530300i \(-0.822079\pi\)
−0.847810 + 0.530300i \(0.822079\pi\)
\(978\) 0 0
\(979\) −48.0000 −1.53409
\(980\) −36.0000 −1.14998
\(981\) −14.0000 −0.446986
\(982\) 0 0
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) 72.0000 2.29411
\(986\) 0 0
\(987\) −3.00000 −0.0954911
\(988\) −20.0000 −0.636285
\(989\) −42.0000 −1.33552
\(990\) 0 0
\(991\) −39.0000 −1.23888 −0.619438 0.785046i \(-0.712639\pi\)
−0.619438 + 0.785046i \(0.712639\pi\)
\(992\) 0 0
\(993\) −32.0000 −1.01549
\(994\) 0 0
\(995\) 63.0000 1.99723
\(996\) 32.0000 1.01396
\(997\) −58.0000 −1.83688 −0.918439 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) 0 0
\(999\) −5.00000 −0.158193
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 4029.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4029.2.a.a.1.1 1 1.1 even 1 trivial