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Properties

Label 6040.2.a.g.1.1
Level $6040$
Weight $2$
Character 6040.1
Self dual yes
Analytic conductor $48.230$
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] = [6040,2,Mod(1,6040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6040, 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("6040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6040.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: \(48.2296428209\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 6040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} -5.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} +2.00000 q^{17} +1.00000 q^{19} -1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +1.00000 q^{29} +3.00000 q^{31} -5.00000 q^{33} -1.00000 q^{39} -2.00000 q^{41} +6.00000 q^{43} +2.00000 q^{45} +6.00000 q^{47} -7.00000 q^{49} +2.00000 q^{51} -6.00000 q^{53} +5.00000 q^{55} +1.00000 q^{57} +13.0000 q^{59} +8.00000 q^{61} +1.00000 q^{65} -7.00000 q^{67} -1.00000 q^{69} +7.00000 q^{73} +1.00000 q^{75} -6.00000 q^{79} +1.00000 q^{81} +5.00000 q^{83} -2.00000 q^{85} +1.00000 q^{87} +10.0000 q^{89} +3.00000 q^{93} -1.00000 q^{95} -2.00000 q^{97} +10.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 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
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 − 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) −5.00000 −0.870388
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 13.0000 1.69246 0.846228 − 0.532821i \(-0.178868\pi\)
0.846228 + 0.532821i \(0.178868\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 − 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 − 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.00000 0.548821 0.274411 − 0.961613i \(-0.411517\pi\)
0.274411 + 0.961613i \(0.411517\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.00000 0.311086
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 10.0000 1.00504
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.0000 1.25676 0.628379 − 0.777908i \(-0.283719\pi\)
0.628379 + 0.777908i \(0.283719\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 − 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 − 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 − 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −2.00000 −0.174741 −0.0873704 − 0.996176i \(-0.527846\pi\)
−0.0873704 + 0.996176i \(0.527846\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 5.00000 0.430331
\(136\) 0 0
\(137\) 4.00000 0.341743 0.170872 − 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 − 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 5.00000 0.418121
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) −7.00000 −0.577350
\(148\) 0 0
\(149\) 24.0000 1.96616 0.983078 − 0.183186i \(-0.0586410\pi\)
0.983078 + 0.183186i \(0.0586410\pi\)
\(150\) 0 0
\(151\) 1.00000 0.0813788
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) −3.00000 −0.240966
\(156\) 0 0
\(157\) 13.0000 1.03751 0.518756 − 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.00000 0.0783260 0.0391630 − 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 0 0
\(165\) 5.00000 0.389249
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 8.00000 0.608229 0.304114 − 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 13.0000 0.977140
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 − 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 − 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −10.0000 −0.731272
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 − 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 − 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 − 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 6.00000 0.425329 0.212664 − 0.977125i \(-0.431786\pi\)
0.212664 + 0.977125i \(0.431786\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) 22.0000 1.47323 0.736614 − 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) −2.00000 −0.132745 −0.0663723 − 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 0 0
\(229\) −5.00000 −0.330409 −0.165205 − 0.986259i \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −21.0000 −1.37576 −0.687878 − 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 0 0
\(237\) −6.00000 −0.389742
\(238\) 0 0
\(239\) 15.0000 0.970269 0.485135 − 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 0 0
\(241\) −3.00000 −0.193247 −0.0966235 − 0.995321i \(-0.530804\pi\)
−0.0966235 + 0.995321i \(0.530804\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 7.00000 0.447214
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 0 0
\(249\) 5.00000 0.316862
\(250\) 0 0
\(251\) −9.00000 −0.568075 −0.284037 − 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 0 0
\(253\) 5.00000 0.314347
\(254\) 0 0
\(255\) −2.00000 −0.125245
\(256\) 0 0
\(257\) −27.0000 −1.68421 −0.842107 − 0.539311i \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) −9.00000 −0.554964 −0.277482 − 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 0 0
\(269\) 19.0000 1.15845 0.579225 − 0.815168i \(-0.303355\pi\)
0.579225 + 0.815168i \(0.303355\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 − 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 − 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 − 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) −13.0000 −0.772770 −0.386385 − 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) −13.0000 −0.756889
\(296\) 0 0
\(297\) 25.0000 1.45065
\(298\) 0 0
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 10.0000 0.574485
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) 10.0000 0.570730 0.285365 − 0.958419i \(-0.407885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 27.0000 1.53103 0.765515 − 0.643418i \(-0.222484\pi\)
0.765515 + 0.643418i \(0.222484\pi\)
\(312\) 0 0
\(313\) −20.0000 −1.13047 −0.565233 − 0.824931i \(-0.691214\pi\)
−0.565233 + 0.824931i \(0.691214\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.0000 1.29181 0.645904 − 0.763418i \(-0.276480\pi\)
0.645904 + 0.763418i \(0.276480\pi\)
\(318\) 0 0
\(319\) −5.00000 −0.279946
\(320\) 0 0
\(321\) 13.0000 0.725589
\(322\) 0 0
\(323\) 2.00000 0.111283
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) −16.0000 −0.884802
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 − 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.00000 0.382451
\(336\) 0 0
\(337\) 9.00000 0.490261 0.245131 − 0.969490i \(-0.421169\pi\)
0.245131 + 0.969490i \(0.421169\pi\)
\(338\) 0 0
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 27.0000 1.44528 0.722638 − 0.691226i \(-0.242929\pi\)
0.722638 + 0.691226i \(0.242929\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) −1.00000 −0.0532246 −0.0266123 − 0.999646i \(-0.508472\pi\)
−0.0266123 + 0.999646i \(0.508472\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.0000 0.527780 0.263890 − 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) −7.00000 −0.366397
\(366\) 0 0
\(367\) −20.0000 −1.04399 −0.521996 − 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 0 0
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −13.0000 −0.673114 −0.336557 − 0.941663i \(-0.609263\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −1.00000 −0.0515026
\(378\) 0 0
\(379\) −6.00000 −0.308199 −0.154100 − 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 18.0000 0.919757 0.459879 − 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −12.0000 −0.609994
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) 0 0
\(393\) −2.00000 −0.100887
\(394\) 0 0
\(395\) 6.00000 0.301893
\(396\) 0 0
\(397\) 12.0000 0.602263 0.301131 − 0.953583i \(-0.402636\pi\)
0.301131 + 0.953583i \(0.402636\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) −3.00000 −0.149441
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 − 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 4.00000 0.197305
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −5.00000 −0.245440
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) 14.0000 0.683945 0.341972 − 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) −28.0000 −1.36464 −0.682318 − 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 0 0
\(423\) −12.0000 −0.583460
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 5.00000 0.241402
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 − 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) 30.0000 1.44171 0.720854 − 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 0 0
\(435\) −1.00000 −0.0479463
\(436\) 0 0
\(437\) −1.00000 −0.0478365
\(438\) 0 0
\(439\) 33.0000 1.57500 0.787502 − 0.616312i \(-0.211374\pi\)
0.787502 + 0.616312i \(0.211374\pi\)
\(440\) 0 0
\(441\) 14.0000 0.666667
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 − 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) −10.0000 −0.474045
\(446\) 0 0
\(447\) 24.0000 1.13516
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 − 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 0 0
\(453\) 1.00000 0.0469841
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 − 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) −10.0000 −0.466760
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 − 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 0 0
\(465\) −3.00000 −0.139122
\(466\) 0 0
\(467\) 20.0000 0.925490 0.462745 − 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 13.0000 0.599008
\(472\) 0 0
\(473\) −30.0000 −1.37940
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) 36.0000 1.64488 0.822441 − 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) 34.0000 1.54069 0.770344 − 0.637629i \(-0.220085\pi\)
0.770344 + 0.637629i \(0.220085\pi\)
\(488\) 0 0
\(489\) 1.00000 0.0452216
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 − 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) 2.00000 0.0900755
\(494\) 0 0
\(495\) −10.0000 −0.449467
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 22.0000 0.984855 0.492428 − 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 0 0
\(509\) 4.00000 0.177297 0.0886484 − 0.996063i \(-0.471745\pi\)
0.0886484 + 0.996063i \(0.471745\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −5.00000 −0.220755
\(514\) 0 0
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) −30.0000 −1.31940
\(518\) 0 0
\(519\) 8.00000 0.351161
\(520\) 0 0
\(521\) 17.0000 0.744784 0.372392 − 0.928076i \(-0.378538\pi\)
0.372392 + 0.928076i \(0.378538\pi\)
\(522\) 0 0
\(523\) 11.0000 0.480996 0.240498 − 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000 0.261364
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) −26.0000 −1.12830
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) −13.0000 −0.562039
\(536\) 0 0
\(537\) 20.0000 0.863064
\(538\) 0 0
\(539\) 35.0000 1.50756
\(540\) 0 0
\(541\) −7.00000 −0.300954 −0.150477 − 0.988614i \(-0.548081\pi\)
−0.150477 + 0.988614i \(0.548081\pi\)
\(542\) 0 0
\(543\) 14.0000 0.600798
\(544\) 0 0
\(545\) 16.0000 0.685365
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) −16.0000 −0.682863
\(550\) 0 0
\(551\) 1.00000 0.0426014
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.00000 −0.381342 −0.190671 − 0.981654i \(-0.561066\pi\)
−0.190671 + 0.981654i \(0.561066\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) −10.0000 −0.422200
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 − 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −15.0000 −0.627730 −0.313865 − 0.949468i \(-0.601624\pi\)
−0.313865 + 0.949468i \(0.601624\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 16.0000 0.666089 0.333044 − 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) 0 0
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 30.0000 1.24247
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 3.00000 0.123613
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 0 0
\(593\) 21.0000 0.862367 0.431183 − 0.902264i \(-0.358096\pi\)
0.431183 + 0.902264i \(0.358096\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.00000 0.245564
\(598\) 0 0
\(599\) 14.0000 0.572024 0.286012 − 0.958226i \(-0.407670\pi\)
0.286012 + 0.958226i \(0.407670\pi\)
\(600\) 0 0
\(601\) −31.0000 −1.26452 −0.632258 − 0.774758i \(-0.717872\pi\)
−0.632258 + 0.774758i \(0.717872\pi\)
\(602\) 0 0
\(603\) 14.0000 0.570124
\(604\) 0 0
\(605\) −14.0000 −0.569181
\(606\) 0 0
\(607\) −20.0000 −0.811775 −0.405887 − 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 28.0000 1.13091 0.565455 − 0.824779i \(-0.308701\pi\)
0.565455 + 0.824779i \(0.308701\pi\)
\(614\) 0 0
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) −21.0000 −0.845428 −0.422714 − 0.906263i \(-0.638923\pi\)
−0.422714 + 0.906263i \(0.638923\pi\)
\(618\) 0 0
\(619\) −24.0000 −0.964641 −0.482321 − 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) 5.00000 0.200643
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −5.00000 −0.199681
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 34.0000 1.35352 0.676759 − 0.736204i \(-0.263384\pi\)
0.676759 + 0.736204i \(0.263384\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 7.00000 0.277350
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −15.0000 −0.592464 −0.296232 − 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 − 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) −6.00000 −0.236250
\(646\) 0 0
\(647\) 28.0000 1.10079 0.550397 − 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) −65.0000 −2.55147
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26.0000 1.01746 0.508729 − 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) 2.00000 0.0781465
\(656\) 0 0
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −1.00000 −0.0389545 −0.0194772 − 0.999810i \(-0.506200\pi\)
−0.0194772 + 0.999810i \(0.506200\pi\)
\(660\) 0 0
\(661\) −44.0000 −1.71140 −0.855701 − 0.517471i \(-0.826874\pi\)
−0.855701 + 0.517471i \(0.826874\pi\)
\(662\) 0 0
\(663\) −2.00000 −0.0776736
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) 22.0000 0.850569
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) 42.0000 1.61419 0.807096 − 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −2.00000 −0.0766402
\(682\) 0 0
\(683\) 15.0000 0.573959 0.286980 − 0.957937i \(-0.407349\pi\)
0.286980 + 0.957937i \(0.407349\pi\)
\(684\) 0 0
\(685\) −4.00000 −0.152832
\(686\) 0 0
\(687\) −5.00000 −0.190762
\(688\) 0 0
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 − 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.0000 −0.606915
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0 0
\(699\) −21.0000 −0.794293
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 − 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −6.00000 −0.225973
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 29.0000 1.08912 0.544559 − 0.838723i \(-0.316697\pi\)
0.544559 + 0.838723i \(0.316697\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 0 0
\(713\) −3.00000 −0.112351
\(714\) 0 0
\(715\) −5.00000 −0.186989
\(716\) 0 0
\(717\) 15.0000 0.560185
\(718\) 0 0
\(719\) 2.00000 0.0745874 0.0372937 − 0.999304i \(-0.488126\pi\)
0.0372937 + 0.999304i \(0.488126\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −3.00000 −0.111571
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 − 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 0 0
\(733\) −13.0000 −0.480166 −0.240083 − 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(734\) 0 0
\(735\) 7.00000 0.258199
\(736\) 0 0
\(737\) 35.0000 1.28924
\(738\) 0 0
\(739\) 34.0000 1.25071 0.625355 − 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) 0 0
\(741\) −1.00000 −0.0367359
\(742\) 0 0
\(743\) 10.0000 0.366864 0.183432 − 0.983032i \(-0.441279\pi\)
0.183432 + 0.983032i \(0.441279\pi\)
\(744\) 0 0
\(745\) −24.0000 −0.879292
\(746\) 0 0
\(747\) −10.0000 −0.365881
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 10.0000 0.364905 0.182453 − 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(752\) 0 0
\(753\) −9.00000 −0.327978
\(754\) 0 0
\(755\) −1.00000 −0.0363937
\(756\) 0 0
\(757\) −20.0000 −0.726912 −0.363456 − 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 0 0
\(759\) 5.00000 0.181489
\(760\) 0 0
\(761\) 2.00000 0.0724999 0.0362500 − 0.999343i \(-0.488459\pi\)
0.0362500 + 0.999343i \(0.488459\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 4.00000 0.144620
\(766\) 0 0
\(767\) −13.0000 −0.469403
\(768\) 0 0
\(769\) 16.0000 0.576975 0.288487 − 0.957484i \(-0.406848\pi\)
0.288487 + 0.957484i \(0.406848\pi\)
\(770\) 0 0
\(771\) −27.0000 −0.972381
\(772\) 0 0
\(773\) −28.0000 −1.00709 −0.503545 − 0.863969i \(-0.667971\pi\)
−0.503545 + 0.863969i \(0.667971\pi\)
\(774\) 0 0
\(775\) 3.00000 0.107763
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −5.00000 −0.178685
\(784\) 0 0
\(785\) −13.0000 −0.463990
\(786\) 0 0
\(787\) 52.0000 1.85360 0.926800 − 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 0 0
\(789\) −9.00000 −0.320408
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) 0 0
\(795\) 6.00000 0.212798
\(796\) 0 0
\(797\) 4.00000 0.141687 0.0708436 − 0.997487i \(-0.477431\pi\)
0.0708436 + 0.997487i \(0.477431\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) −20.0000 −0.706665
\(802\) 0 0
\(803\) −35.0000 −1.23512
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 19.0000 0.668832
\(808\) 0 0
\(809\) −48.0000 −1.68759 −0.843795 − 0.536666i \(-0.819684\pi\)
−0.843795 + 0.536666i \(0.819684\pi\)
\(810\) 0 0
\(811\) −26.0000 −0.912983 −0.456492 − 0.889728i \(-0.650894\pi\)
−0.456492 + 0.889728i \(0.650894\pi\)
\(812\) 0 0
\(813\) 12.0000 0.420858
\(814\) 0 0
\(815\) −1.00000 −0.0350285
\(816\) 0 0
\(817\) 6.00000 0.209913
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −38.0000 −1.32621 −0.663105 − 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 0 0
\(823\) 14.0000 0.488009 0.244005 − 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) 0 0
\(825\) −5.00000 −0.174078
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 − 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −35.0000 −1.21560 −0.607800 − 0.794090i \(-0.707948\pi\)
−0.607800 + 0.794090i \(0.707948\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −15.0000 −0.518476
\(838\) 0 0
\(839\) 8.00000 0.276191 0.138095 − 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) −30.0000 −1.03325
\(844\) 0 0
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −13.0000 −0.446159
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 4.00000 0.136957 0.0684787 − 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) 0 0
\(855\) 2.00000 0.0683986
\(856\) 0 0
\(857\) 33.0000 1.12726 0.563629 − 0.826028i \(-0.309405\pi\)
0.563629 + 0.826028i \(0.309405\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 − 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −52.0000 −1.77010 −0.885050 − 0.465495i \(-0.845876\pi\)
−0.885050 + 0.465495i \(0.845876\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.272008
\(866\) 0 0
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 30.0000 1.01768
\(870\) 0 0
\(871\) 7.00000 0.237186
\(872\) 0 0
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.00000 0.236373 0.118187 − 0.992991i \(-0.462292\pi\)
0.118187 + 0.992991i \(0.462292\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 36.0000 1.21287 0.606435 − 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) 0 0
\(883\) 52.0000 1.74994 0.874970 − 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 0 0
\(885\) −13.0000 −0.436990
\(886\) 0 0
\(887\) −29.0000 −0.973725 −0.486862 − 0.873479i \(-0.661859\pi\)
−0.486862 + 0.873479i \(0.661859\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 0 0
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) 0 0
\(897\) 1.00000 0.0333890
\(898\) 0 0
\(899\) 3.00000 0.100056
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) 22.0000 0.730498 0.365249 − 0.930910i \(-0.380984\pi\)
0.365249 + 0.930910i \(0.380984\pi\)
\(908\) 0 0
\(909\) −20.0000 −0.663358
\(910\) 0 0
\(911\) 11.0000 0.364446 0.182223 − 0.983257i \(-0.441671\pi\)
0.182223 + 0.983257i \(0.441671\pi\)
\(912\) 0 0
\(913\) −25.0000 −0.827379
\(914\) 0 0
\(915\) −8.00000 −0.264472
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 12.0000 0.395843 0.197922 − 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) 0 0
\(921\) 10.0000 0.329511
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 32.0000 1.05102
\(928\) 0 0
\(929\) 26.0000 0.853032 0.426516 − 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 0 0
\(931\) −7.00000 −0.229416
\(932\) 0 0
\(933\) 27.0000 0.883940
\(934\) 0 0
\(935\) 10.0000 0.327035
\(936\) 0 0
\(937\) 4.00000 0.130674 0.0653372 − 0.997863i \(-0.479188\pi\)
0.0653372 + 0.997863i \(0.479188\pi\)
\(938\) 0 0
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) 12.0000 0.391189 0.195594 − 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) 0 0
\(943\) 2.00000 0.0651290
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15.0000 0.487435 0.243717 − 0.969846i \(-0.421633\pi\)
0.243717 + 0.969846i \(0.421633\pi\)
\(948\) 0 0
\(949\) −7.00000 −0.227230
\(950\) 0 0
\(951\) 23.0000 0.745826
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 − 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) 0 0
\(957\) −5.00000 −0.161627
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) −26.0000 −0.837838
\(964\) 0 0
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) 13.0000 0.418052 0.209026 − 0.977910i \(-0.432971\pi\)
0.209026 + 0.977910i \(0.432971\pi\)
\(968\) 0 0
\(969\) 2.00000 0.0642493
\(970\) 0 0
\(971\) 24.0000 0.770197 0.385098 − 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −1.00000 −0.0320256
\(976\) 0 0
\(977\) 2.00000 0.0639857 0.0319928 − 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 0 0
\(979\) −50.0000 −1.59801
\(980\) 0 0
\(981\) 32.0000 1.02168
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 − 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) −17.0000 −0.540023 −0.270011 − 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) −6.00000 −0.190213
\(996\) 0 0
\(997\) −46.0000 −1.45683 −0.728417 − 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) 0 0
\(999\) 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 6040.2.a.g.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6040.2.a.g.1.1 ✓ 1 1.1 even 1 trivial