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Properties

Label 1155.2.a.g.1.1
Level $1155$
Weight $2$
Character 1155.1
Self dual yes
Analytic conductor $9.223$
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] = [1155,2,Mod(1,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1155.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.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: \(9.22272143346\)
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 \(\chi\) \(=\) 1155.1

$q$-expansion

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −1.00000 −0.577350
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 2.00000 0.577350
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 4.00000 1.00000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 − 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 2.00000 0.447214
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 − 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) −7.00000 −1.29987 −0.649934 − 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) −2.00000 −0.333333
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 − 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) −2.00000 −0.301511
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −4.00000 −0.577350
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 − 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 3.00000 0.397360
\(58\) 0 0
\(59\) −5.00000 −0.650945 −0.325472 − 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) −2.00000 −0.258199
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 6.00000 0.733017 0.366508 − 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) −6.00000 −0.727607
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 − 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) 12.0000 1.40449 0.702247 − 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 6.00000 0.688247
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.00000 −0.329293 −0.164646 − 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 2.00000 0.218218
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) 7.00000 0.750479
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 3.00000 0.307794
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 − 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −2.00000 −0.200000
\(101\) −14.0000 −1.39305 −0.696526 − 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) −5.00000 −0.492665 −0.246332 − 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 2.00000 0.192450
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 4.00000 0.377964
\(113\) −19.0000 −1.78737 −0.893685 − 0.448695i \(-0.851889\pi\)
−0.893685 + 0.448695i \(0.851889\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 14.0000 1.29987
\(117\) 0 0
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 2.00000 0.180334
\(124\) −12.0000 −1.07763
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 19.0000 1.68598 0.842989 − 0.537931i \(-0.180794\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(128\) 0 0
\(129\) 5.00000 0.440225
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 2.00000 0.174078
\(133\) −3.00000 −0.260133
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 − 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 2.00000 0.169031
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 4.00000 0.333333
\(145\) 7.00000 0.581318
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 16.0000 1.31519
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 − 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) −1.00000 −0.0798087 −0.0399043 − 0.999204i \(-0.512705\pi\)
−0.0399043 + 0.999204i \(0.512705\pi\)
\(158\) 0 0
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 4.00000 0.312348
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) 10.0000 0.762493
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 4.00000 0.301511
\(177\) 5.00000 0.375823
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 − 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 2.00000 0.149071
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 7.00000 0.517455
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) 3.00000 0.219382
\(188\) 12.0000 0.875190
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 − 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 8.00000 0.577350
\(193\) 6.00000 0.431889 0.215945 − 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 − 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) 0 0
\(203\) −7.00000 −0.491304
\(204\) 6.00000 0.420084
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −18.0000 −1.23917 −0.619586 − 0.784929i \(-0.712699\pi\)
−0.619586 + 0.784929i \(0.712699\pi\)
\(212\) 6.00000 0.412082
\(213\) 10.0000 0.685189
\(214\) 0 0
\(215\) 5.00000 0.340997
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) −12.0000 −0.810885
\(220\) 2.00000 0.134840
\(221\) 0 0
\(222\) 0 0
\(223\) 1.00000 0.0669650 0.0334825 − 0.999439i \(-0.489340\pi\)
0.0334825 + 0.999439i \(0.489340\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −3.00000 −0.199117 −0.0995585 − 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) −6.00000 −0.397360
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) −16.0000 −1.04819 −0.524097 − 0.851658i \(-0.675597\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 10.0000 0.650945
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) −21.0000 −1.35838 −0.679189 − 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 4.00000 0.258199
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 14.0000 0.896258
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 3.00000 0.190117
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −2.00000 −0.125988
\(253\) −1.00000 −0.0628695
\(254\) 0 0
\(255\) 3.00000 0.187867
\(256\) 16.0000 1.00000
\(257\) 20.0000 1.24757 0.623783 − 0.781598i \(-0.285595\pi\)
0.623783 + 0.781598i \(0.285595\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −7.00000 −0.433289
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 3.00000 0.184289
\(266\) 0 0
\(267\) −3.00000 −0.183597
\(268\) −12.0000 −0.733017
\(269\) −1.00000 −0.0609711 −0.0304855 − 0.999535i \(-0.509705\pi\)
−0.0304855 + 0.999535i \(0.509705\pi\)
\(270\) 0 0
\(271\) −1.00000 −0.0607457 −0.0303728 − 0.999539i \(-0.509669\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 12.0000 0.727607
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) −2.00000 −0.120386
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 − 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) −8.00000 −0.475551 −0.237775 − 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 20.0000 1.18678
\(285\) −3.00000 −0.177705
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 1.00000 0.0586210
\(292\) −24.0000 −1.40449
\(293\) 19.0000 1.10999 0.554996 − 0.831853i \(-0.312720\pi\)
0.554996 + 0.831853i \(0.312720\pi\)
\(294\) 0 0
\(295\) 5.00000 0.291111
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 0 0
\(300\) 2.00000 0.115470
\(301\) −5.00000 −0.288195
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) −12.0000 −0.688247
\(305\) 7.00000 0.400819
\(306\) 0 0
\(307\) 26.0000 1.48390 0.741949 − 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) −2.00000 −0.113961
\(309\) 5.00000 0.284440
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 − 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) 0 0
\(313\) 21.0000 1.18699 0.593495 − 0.804838i \(-0.297748\pi\)
0.593495 + 0.804838i \(0.297748\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 32.0000 1.80014
\(317\) 2.00000 0.112331 0.0561656 − 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) −7.00000 −0.391925
\(320\) 8.00000 0.447214
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) −9.00000 −0.500773
\(324\) −2.00000 −0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 − 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 6.00000 0.329293
\(333\) −8.00000 −0.438397
\(334\) 0 0
\(335\) −6.00000 −0.327815
\(336\) −4.00000 −0.218218
\(337\) −1.00000 −0.0544735 −0.0272367 − 0.999629i \(-0.508671\pi\)
−0.0272367 + 0.999629i \(0.508671\pi\)
\(338\) 0 0
\(339\) 19.0000 1.03194
\(340\) 6.00000 0.325396
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −8.00000 −0.429463 −0.214731 − 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) −14.0000 −0.750479
\(349\) −19.0000 −1.01705 −0.508523 − 0.861048i \(-0.669808\pi\)
−0.508523 + 0.861048i \(0.669808\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 − 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 10.0000 0.530745
\(356\) −6.00000 −0.317999
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) −21.0000 −1.10834 −0.554169 − 0.832404i \(-0.686964\pi\)
−0.554169 + 0.832404i \(0.686964\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −12.0000 −0.628109
\(366\) 0 0
\(367\) 7.00000 0.365397 0.182699 − 0.983169i \(-0.441517\pi\)
0.182699 + 0.983169i \(0.441517\pi\)
\(368\) −4.00000 −0.208514
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −3.00000 −0.155752
\(372\) 12.0000 0.622171
\(373\) 11.0000 0.569558 0.284779 − 0.958593i \(-0.408080\pi\)
0.284779 + 0.958593i \(0.408080\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 5.00000 0.256833 0.128416 − 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) −6.00000 −0.307794
\(381\) −19.0000 −0.973399
\(382\) 0 0
\(383\) 32.0000 1.63512 0.817562 − 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −5.00000 −0.254164
\(388\) 2.00000 0.101535
\(389\) 12.0000 0.608424 0.304212 − 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) 16.0000 0.805047
\(396\) −2.00000 −0.100504
\(397\) −10.0000 −0.501886 −0.250943 − 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 0 0
\(399\) 3.00000 0.150188
\(400\) 4.00000 0.200000
\(401\) −10.0000 −0.499376 −0.249688 − 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 28.0000 1.39305
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 − 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 10.0000 0.492665
\(413\) −5.00000 −0.246034
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 33.0000 1.61216 0.806078 − 0.591810i \(-0.201586\pi\)
0.806078 + 0.591810i \(0.201586\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −23.0000 −1.12095 −0.560476 − 0.828171i \(-0.689382\pi\)
−0.560476 + 0.828171i \(0.689382\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) −7.00000 −0.338754
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 − 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) −4.00000 −0.192450
\(433\) 22.0000 1.05725 0.528626 − 0.848855i \(-0.322707\pi\)
0.528626 + 0.848855i \(0.322707\pi\)
\(434\) 0 0
\(435\) −7.00000 −0.335624
\(436\) 4.00000 0.191565
\(437\) 3.00000 0.143509
\(438\) 0 0
\(439\) 1.00000 0.0477274 0.0238637 − 0.999715i \(-0.492403\pi\)
0.0238637 + 0.999715i \(0.492403\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 32.0000 1.52037 0.760183 − 0.649709i \(-0.225109\pi\)
0.760183 + 0.649709i \(0.225109\pi\)
\(444\) −16.0000 −0.759326
\(445\) −3.00000 −0.142214
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) −8.00000 −0.377964
\(449\) 26.0000 1.22702 0.613508 − 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 38.0000 1.78737
\(453\) −14.0000 −0.657777
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −37.0000 −1.73079 −0.865393 − 0.501093i \(-0.832931\pi\)
−0.865393 + 0.501093i \(0.832931\pi\)
\(458\) 0 0
\(459\) −3.00000 −0.140028
\(460\) −2.00000 −0.0932505
\(461\) 6.00000 0.279448 0.139724 − 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −18.0000 −0.836531 −0.418265 − 0.908325i \(-0.637362\pi\)
−0.418265 + 0.908325i \(0.637362\pi\)
\(464\) −28.0000 −1.29987
\(465\) 6.00000 0.278243
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 6.00000 0.277054
\(470\) 0 0
\(471\) 1.00000 0.0460776
\(472\) 0 0
\(473\) −5.00000 −0.229900
\(474\) 0 0
\(475\) −3.00000 −0.137649
\(476\) −6.00000 −0.275010
\(477\) −3.00000 −0.137361
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) −2.00000 −0.0909091
\(485\) 1.00000 0.0454077
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 − 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 − 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) −4.00000 −0.180334
\(493\) −21.0000 −0.945792
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 24.0000 1.07763
\(497\) −10.0000 −0.448561
\(498\) 0 0
\(499\) −21.0000 −0.940089 −0.470045 − 0.882643i \(-0.655762\pi\)
−0.470045 + 0.882643i \(0.655762\pi\)
\(500\) 2.00000 0.0894427
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) −13.0000 −0.579641 −0.289821 − 0.957081i \(-0.593596\pi\)
−0.289821 + 0.957081i \(0.593596\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) 13.0000 0.577350
\(508\) −38.0000 −1.68598
\(509\) −11.0000 −0.487566 −0.243783 − 0.969830i \(-0.578389\pi\)
−0.243783 + 0.969830i \(0.578389\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) 0 0
\(513\) 3.00000 0.132453
\(514\) 0 0
\(515\) 5.00000 0.220326
\(516\) −10.0000 −0.440225
\(517\) −6.00000 −0.263880
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −27.0000 −1.18289 −0.591446 − 0.806345i \(-0.701443\pi\)
−0.591446 + 0.806345i \(0.701443\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −8.00000 −0.349482
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 18.0000 0.784092
\(528\) −4.00000 −0.174078
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) −5.00000 −0.216982
\(532\) 6.00000 0.260133
\(533\) 0 0
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) 18.0000 0.776757
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) −2.00000 −0.0860663
\(541\) −14.0000 −0.601907 −0.300954 − 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 13.0000 0.555840 0.277920 − 0.960604i \(-0.410355\pi\)
0.277920 + 0.960604i \(0.410355\pi\)
\(548\) 4.00000 0.170872
\(549\) −7.00000 −0.298753
\(550\) 0 0
\(551\) 21.0000 0.894630
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) −8.00000 −0.339581
\(556\) −8.00000 −0.339276
\(557\) 6.00000 0.254228 0.127114 − 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −4.00000 −0.169031
\(561\) −3.00000 −0.126660
\(562\) 0 0
\(563\) −4.00000 −0.168580 −0.0842900 − 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) −12.0000 −0.505291
\(565\) 19.0000 0.799336
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 13.0000 0.544988 0.272494 − 0.962157i \(-0.412151\pi\)
0.272494 + 0.962157i \(0.412151\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 − 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) −8.00000 −0.333333
\(577\) 18.0000 0.749350 0.374675 − 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 0 0
\(579\) −6.00000 −0.249351
\(580\) −14.0000 −0.581318
\(581\) −3.00000 −0.124461
\(582\) 0 0
\(583\) −3.00000 −0.124247
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.0000 1.48588 0.742940 − 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 2.00000 0.0824786
\(589\) −18.0000 −0.741677
\(590\) 0 0
\(591\) 0 0
\(592\) −32.0000 −1.31519
\(593\) −34.0000 −1.39621 −0.698106 − 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 12.0000 0.491539
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 − 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) 15.0000 0.611863 0.305931 − 0.952054i \(-0.401032\pi\)
0.305931 + 0.952054i \(0.401032\pi\)
\(602\) 0 0
\(603\) 6.00000 0.244339
\(604\) −28.0000 −1.13930
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −38.0000 −1.54237 −0.771186 − 0.636610i \(-0.780336\pi\)
−0.771186 + 0.636610i \(0.780336\pi\)
\(608\) 0 0
\(609\) 7.00000 0.283654
\(610\) 0 0
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) −38.0000 −1.53481 −0.767403 − 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 0 0
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) 2.00000 0.0805170 0.0402585 − 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) 46.0000 1.84890 0.924448 − 0.381308i \(-0.124526\pi\)
0.924448 + 0.381308i \(0.124526\pi\)
\(620\) 12.0000 0.481932
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 3.00000 0.120192
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 3.00000 0.119808
\(628\) 2.00000 0.0798087
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 − 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 0 0
\(633\) 18.0000 0.715436
\(634\) 0 0
\(635\) −19.0000 −0.753992
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 − 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 31.0000 1.22252 0.611260 − 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(644\) 2.00000 0.0788110
\(645\) −5.00000 −0.196875
\(646\) 0 0
\(647\) −4.00000 −0.157256 −0.0786281 − 0.996904i \(-0.525054\pi\)
−0.0786281 + 0.996904i \(0.525054\pi\)
\(648\) 0 0
\(649\) −5.00000 −0.196267
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) −24.0000 −0.939913
\(653\) 39.0000 1.52619 0.763094 − 0.646288i \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) −8.00000 −0.312348
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) −15.0000 −0.584317 −0.292159 − 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) −2.00000 −0.0778499
\(661\) −18.0000 −0.700119 −0.350059 − 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.00000 0.116335
\(666\) 0 0
\(667\) 7.00000 0.271041
\(668\) −24.0000 −0.928588
\(669\) −1.00000 −0.0386622
\(670\) 0 0
\(671\) −7.00000 −0.270232
\(672\) 0 0
\(673\) −43.0000 −1.65753 −0.828764 − 0.559598i \(-0.810955\pi\)
−0.828764 + 0.559598i \(0.810955\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 26.0000 1.00000
\(677\) −25.0000 −0.960828 −0.480414 − 0.877042i \(-0.659514\pi\)
−0.480414 + 0.877042i \(0.659514\pi\)
\(678\) 0 0
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 − 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 6.00000 0.229416
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) −20.0000 −0.762493
\(689\) 0 0
\(690\) 0 0
\(691\) 2.00000 0.0760836 0.0380418 − 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) 28.0000 1.06440
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) 0 0
\(699\) 16.0000 0.605176
\(700\) −2.00000 −0.0755929
\(701\) 11.0000 0.415464 0.207732 − 0.978186i \(-0.433392\pi\)
0.207732 + 0.978186i \(0.433392\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) −8.00000 −0.301511
\(705\) −6.00000 −0.225973
\(706\) 0 0
\(707\) −14.0000 −0.526524
\(708\) −10.0000 −0.375823
\(709\) −15.0000 −0.563337 −0.281668 − 0.959512i \(-0.590888\pi\)
−0.281668 + 0.959512i \(0.590888\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 0 0
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 36.0000 1.34538
\(717\) 21.0000 0.784259
\(718\) 0 0
\(719\) 15.0000 0.559406 0.279703 − 0.960087i \(-0.409764\pi\)
0.279703 + 0.960087i \(0.409764\pi\)
\(720\) −4.00000 −0.149071
\(721\) −5.00000 −0.186210
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) −20.0000 −0.743294
\(725\) −7.00000 −0.259973
\(726\) 0 0
\(727\) −7.00000 −0.259616 −0.129808 − 0.991539i \(-0.541436\pi\)
−0.129808 + 0.991539i \(0.541436\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −15.0000 −0.554795
\(732\) −14.0000 −0.517455
\(733\) −6.00000 −0.221615 −0.110808 − 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) 6.00000 0.221013
\(738\) 0 0
\(739\) 22.0000 0.809283 0.404642 − 0.914475i \(-0.367396\pi\)
0.404642 + 0.914475i \(0.367396\pi\)
\(740\) −16.0000 −0.588172
\(741\) 0 0
\(742\) 0 0
\(743\) 18.0000 0.660356 0.330178 − 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) −3.00000 −0.109764
\(748\) −6.00000 −0.219382
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 53.0000 1.93400 0.966999 − 0.254781i \(-0.0820034\pi\)
0.966999 + 0.254781i \(0.0820034\pi\)
\(752\) −24.0000 −0.875190
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) −14.0000 −0.509512
\(756\) 2.00000 0.0727393
\(757\) −6.00000 −0.218074 −0.109037 − 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 0 0
\(759\) 1.00000 0.0362977
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) 32.0000 1.15772
\(765\) −3.00000 −0.108465
\(766\) 0 0
\(767\) 0 0
\(768\) −16.0000 −0.577350
\(769\) 9.00000 0.324548 0.162274 − 0.986746i \(-0.448117\pi\)
0.162274 + 0.986746i \(0.448117\pi\)
\(770\) 0 0
\(771\) −20.0000 −0.720282
\(772\) −12.0000 −0.431889
\(773\) 26.0000 0.935155 0.467578 − 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) 0 0
\(777\) 8.00000 0.286998
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) −10.0000 −0.357828
\(782\) 0 0
\(783\) 7.00000 0.250160
\(784\) 4.00000 0.142857
\(785\) 1.00000 0.0356915
\(786\) 0 0
\(787\) −40.0000 −1.42585 −0.712923 − 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) 0 0
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) −19.0000 −0.675562
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −3.00000 −0.106399
\(796\) −48.0000 −1.70131
\(797\) −30.0000 −1.06265 −0.531327 − 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 0 0
\(801\) 3.00000 0.106000
\(802\) 0 0
\(803\) 12.0000 0.423471
\(804\) 12.0000 0.423207
\(805\) 1.00000 0.0352454
\(806\) 0 0
\(807\) 1.00000 0.0352017
\(808\) 0 0
\(809\) 10.0000 0.351581 0.175791 − 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 − 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 14.0000 0.491304
\(813\) 1.00000 0.0350715
\(814\) 0 0
\(815\) −12.0000 −0.420342
\(816\) −12.0000 −0.420084
\(817\) 15.0000 0.524784
\(818\) 0 0
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 23.0000 0.802706 0.401353 − 0.915924i \(-0.368540\pi\)
0.401353 + 0.915924i \(0.368540\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 − 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 − 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 2.00000 0.0695048
\(829\) 38.0000 1.31979 0.659897 − 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) 6.00000 0.207514
\(837\) −6.00000 −0.207390
\(838\) 0 0
\(839\) −17.0000 −0.586905 −0.293453 − 0.955974i \(-0.594804\pi\)
−0.293453 + 0.955974i \(0.594804\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 0 0
\(843\) 22.0000 0.757720
\(844\) 36.0000 1.23917
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) −12.0000 −0.412082
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) −20.0000 −0.685189
\(853\) −10.0000 −0.342393 −0.171197 − 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 3.00000 0.102598
\(856\) 0 0
\(857\) 50.0000 1.70797 0.853984 − 0.520300i \(-0.174180\pi\)
0.853984 + 0.520300i \(0.174180\pi\)
\(858\) 0 0
\(859\) 46.0000 1.56950 0.784750 − 0.619813i \(-0.212791\pi\)
0.784750 + 0.619813i \(0.212791\pi\)
\(860\) −10.0000 −0.340997
\(861\) 2.00000 0.0681598
\(862\) 0 0
\(863\) 39.0000 1.32758 0.663788 − 0.747921i \(-0.268948\pi\)
0.663788 + 0.747921i \(0.268948\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) −12.0000 −0.407307
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −1.00000 −0.0338449
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 24.0000 0.810885
\(877\) 45.0000 1.51954 0.759771 − 0.650191i \(-0.225311\pi\)
0.759771 + 0.650191i \(0.225311\pi\)
\(878\) 0 0
\(879\) −19.0000 −0.640854
\(880\) −4.00000 −0.134840
\(881\) 15.0000 0.505363 0.252681 − 0.967550i \(-0.418688\pi\)
0.252681 + 0.967550i \(0.418688\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) −5.00000 −0.168073
\(886\) 0 0
\(887\) 41.0000 1.37665 0.688323 − 0.725405i \(-0.258347\pi\)
0.688323 + 0.725405i \(0.258347\pi\)
\(888\) 0 0
\(889\) 19.0000 0.637240
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) −2.00000 −0.0669650
\(893\) 18.0000 0.602347
\(894\) 0 0
\(895\) 18.0000 0.601674
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −42.0000 −1.40078
\(900\) −2.00000 −0.0666667
\(901\) −9.00000 −0.299833
\(902\) 0 0
\(903\) 5.00000 0.166390
\(904\) 0 0
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 − 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 6.00000 0.199117
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) −4.00000 −0.132526 −0.0662630 − 0.997802i \(-0.521108\pi\)
−0.0662630 + 0.997802i \(0.521108\pi\)
\(912\) 12.0000 0.397360
\(913\) −3.00000 −0.0992855
\(914\) 0 0
\(915\) −7.00000 −0.231413
\(916\) 28.0000 0.925146
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) −14.0000 −0.461817 −0.230909 − 0.972975i \(-0.574170\pi\)
−0.230909 + 0.972975i \(0.574170\pi\)
\(920\) 0 0
\(921\) −26.0000 −0.856729
\(922\) 0 0
\(923\) 0 0
\(924\) 2.00000 0.0657952
\(925\) −8.00000 −0.263038
\(926\) 0 0
\(927\) −5.00000 −0.164222
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 − 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 32.0000 1.04819
\(933\) 32.0000 1.04763
\(934\) 0 0
\(935\) −3.00000 −0.0981105
\(936\) 0 0
\(937\) −38.0000 −1.24141 −0.620703 − 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) −21.0000 −0.685309
\(940\) −12.0000 −0.391397
\(941\) 40.0000 1.30396 0.651981 − 0.758235i \(-0.273938\pi\)
0.651981 + 0.758235i \(0.273938\pi\)
\(942\) 0 0
\(943\) 2.00000 0.0651290
\(944\) −20.0000 −0.650945
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −45.0000 −1.46230 −0.731152 − 0.682215i \(-0.761017\pi\)
−0.731152 + 0.682215i \(0.761017\pi\)
\(948\) −32.0000 −1.03931
\(949\) 0 0
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) 30.0000 0.971795 0.485898 − 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) 42.0000 1.35838
\(957\) 7.00000 0.226278
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) −8.00000 −0.258199
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) −20.0000 −0.644157
\(965\) −6.00000 −0.193147
\(966\) 0 0
\(967\) −17.0000 −0.546683 −0.273342 − 0.961917i \(-0.588129\pi\)
−0.273342 + 0.961917i \(0.588129\pi\)
\(968\) 0 0
\(969\) 9.00000 0.289122
\(970\) 0 0
\(971\) 47.0000 1.50830 0.754151 − 0.656701i \(-0.228049\pi\)
0.754151 + 0.656701i \(0.228049\pi\)
\(972\) 2.00000 0.0641500
\(973\) 4.00000 0.128234
\(974\) 0 0
\(975\) 0 0
\(976\) −28.0000 −0.896258
\(977\) 49.0000 1.56765 0.783824 − 0.620982i \(-0.213266\pi\)
0.783824 + 0.620982i \(0.213266\pi\)
\(978\) 0 0
\(979\) 3.00000 0.0958804
\(980\) 2.00000 0.0638877
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) −6.00000 −0.191370 −0.0956851 − 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) 5.00000 0.158991
\(990\) 0 0
\(991\) 31.0000 0.984747 0.492374 − 0.870384i \(-0.336129\pi\)
0.492374 + 0.870384i \(0.336129\pi\)
\(992\) 0 0
\(993\) −17.0000 −0.539479
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) −6.00000 −0.190117
\(997\) −42.0000 −1.33015 −0.665077 − 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 0 0
\(999\) 8.00000 0.253109
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 1155.2.a.g.1.1 ✓ 1
3.2 odd 2 3465.2.a.i.1.1 1
5.4 even 2 5775.2.a.o.1.1 1
7.6 odd 2 8085.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.g.1.1 ✓ 1 1.1 even 1 trivial
3465.2.a.i.1.1 1 3.2 odd 2
5775.2.a.o.1.1 1 5.4 even 2
8085.2.a.q.1.1 1 7.6 odd 2