Properties

Label 4840.2.a.y.1.2
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.09529\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.29496 q^{3} +1.00000 q^{5} -0.227777 q^{7} -1.32307 q^{9} +O(q^{10})\) \(q-1.29496 q^{3} +1.00000 q^{5} -0.227777 q^{7} -1.32307 q^{9} -3.34478 q^{13} -1.29496 q^{15} +6.04981 q^{17} -2.22137 q^{19} +0.294963 q^{21} -3.47726 q^{23} +1.00000 q^{25} +5.59822 q^{27} +0.0307867 q^{29} +2.21625 q^{31} -0.227777 q^{35} +8.74728 q^{37} +4.33136 q^{39} +10.2404 q^{41} -3.74411 q^{43} -1.32307 q^{45} -5.31399 q^{47} -6.94812 q^{49} -7.83428 q^{51} -10.2028 q^{53} +2.87660 q^{57} +8.34399 q^{59} -2.24553 q^{61} +0.301365 q^{63} -3.34478 q^{65} +3.47330 q^{67} +4.50292 q^{69} -15.0637 q^{71} -3.66389 q^{73} -1.29496 q^{75} -11.6212 q^{79} -3.28027 q^{81} +1.52274 q^{83} +6.04981 q^{85} -0.0398677 q^{87} -10.0058 q^{89} +0.761864 q^{91} -2.86996 q^{93} -2.22137 q^{95} -8.33570 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9} + 3 q^{13} - 2 q^{15} + 11 q^{17} + 2 q^{19} - 2 q^{21} - 11 q^{23} + 4 q^{25} + q^{27} + 4 q^{29} - 17 q^{31} - 7 q^{35} - 3 q^{37} + q^{39} + 13 q^{41} - 7 q^{43} - 4 q^{45} - q^{47} - 5 q^{49} - q^{51} - 15 q^{53} + 17 q^{57} - 17 q^{59} + 4 q^{61} + 15 q^{63} + 3 q^{65} + 7 q^{67} + 4 q^{69} - 15 q^{71} + 7 q^{73} - 2 q^{75} - 12 q^{79} - 8 q^{81} + 9 q^{83} + 11 q^{85} - 23 q^{87} - 12 q^{89} - 24 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.29496 −0.747647 −0.373824 0.927500i \(-0.621953\pi\)
−0.373824 + 0.927500i \(0.621953\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.227777 −0.0860917 −0.0430458 0.999073i \(-0.513706\pi\)
−0.0430458 + 0.999073i \(0.513706\pi\)
\(8\) 0 0
\(9\) −1.32307 −0.441024
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −3.34478 −0.927674 −0.463837 0.885921i \(-0.653528\pi\)
−0.463837 + 0.885921i \(0.653528\pi\)
\(14\) 0 0
\(15\) −1.29496 −0.334358
\(16\) 0 0
\(17\) 6.04981 1.46730 0.733648 0.679530i \(-0.237816\pi\)
0.733648 + 0.679530i \(0.237816\pi\)
\(18\) 0 0
\(19\) −2.22137 −0.509618 −0.254809 0.966991i \(-0.582013\pi\)
−0.254809 + 0.966991i \(0.582013\pi\)
\(20\) 0 0
\(21\) 0.294963 0.0643662
\(22\) 0 0
\(23\) −3.47726 −0.725059 −0.362529 0.931972i \(-0.618087\pi\)
−0.362529 + 0.931972i \(0.618087\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.59822 1.07738
\(28\) 0 0
\(29\) 0.0307867 0.00571695 0.00285848 0.999996i \(-0.499090\pi\)
0.00285848 + 0.999996i \(0.499090\pi\)
\(30\) 0 0
\(31\) 2.21625 0.398050 0.199025 0.979994i \(-0.436222\pi\)
0.199025 + 0.979994i \(0.436222\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.227777 −0.0385014
\(36\) 0 0
\(37\) 8.74728 1.43804 0.719022 0.694987i \(-0.244590\pi\)
0.719022 + 0.694987i \(0.244590\pi\)
\(38\) 0 0
\(39\) 4.33136 0.693573
\(40\) 0 0
\(41\) 10.2404 1.59928 0.799641 0.600478i \(-0.205023\pi\)
0.799641 + 0.600478i \(0.205023\pi\)
\(42\) 0 0
\(43\) −3.74411 −0.570972 −0.285486 0.958383i \(-0.592155\pi\)
−0.285486 + 0.958383i \(0.592155\pi\)
\(44\) 0 0
\(45\) −1.32307 −0.197232
\(46\) 0 0
\(47\) −5.31399 −0.775125 −0.387563 0.921843i \(-0.626683\pi\)
−0.387563 + 0.921843i \(0.626683\pi\)
\(48\) 0 0
\(49\) −6.94812 −0.992588
\(50\) 0 0
\(51\) −7.83428 −1.09702
\(52\) 0 0
\(53\) −10.2028 −1.40147 −0.700734 0.713423i \(-0.747144\pi\)
−0.700734 + 0.713423i \(0.747144\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.87660 0.381015
\(58\) 0 0
\(59\) 8.34399 1.08629 0.543147 0.839637i \(-0.317233\pi\)
0.543147 + 0.839637i \(0.317233\pi\)
\(60\) 0 0
\(61\) −2.24553 −0.287510 −0.143755 0.989613i \(-0.545918\pi\)
−0.143755 + 0.989613i \(0.545918\pi\)
\(62\) 0 0
\(63\) 0.301365 0.0379685
\(64\) 0 0
\(65\) −3.34478 −0.414869
\(66\) 0 0
\(67\) 3.47330 0.424332 0.212166 0.977234i \(-0.431948\pi\)
0.212166 + 0.977234i \(0.431948\pi\)
\(68\) 0 0
\(69\) 4.50292 0.542088
\(70\) 0 0
\(71\) −15.0637 −1.78773 −0.893867 0.448332i \(-0.852018\pi\)
−0.893867 + 0.448332i \(0.852018\pi\)
\(72\) 0 0
\(73\) −3.66389 −0.428826 −0.214413 0.976743i \(-0.568784\pi\)
−0.214413 + 0.976743i \(0.568784\pi\)
\(74\) 0 0
\(75\) −1.29496 −0.149529
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −11.6212 −1.30749 −0.653744 0.756716i \(-0.726803\pi\)
−0.653744 + 0.756716i \(0.726803\pi\)
\(80\) 0 0
\(81\) −3.28027 −0.364474
\(82\) 0 0
\(83\) 1.52274 0.167142 0.0835712 0.996502i \(-0.473367\pi\)
0.0835712 + 0.996502i \(0.473367\pi\)
\(84\) 0 0
\(85\) 6.04981 0.656194
\(86\) 0 0
\(87\) −0.0398677 −0.00427426
\(88\) 0 0
\(89\) −10.0058 −1.06062 −0.530309 0.847805i \(-0.677924\pi\)
−0.530309 + 0.847805i \(0.677924\pi\)
\(90\) 0 0
\(91\) 0.761864 0.0798650
\(92\) 0 0
\(93\) −2.86996 −0.297601
\(94\) 0 0
\(95\) −2.22137 −0.227908
\(96\) 0 0
\(97\) −8.33570 −0.846362 −0.423181 0.906045i \(-0.639087\pi\)
−0.423181 + 0.906045i \(0.639087\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.72279 0.171424 0.0857118 0.996320i \(-0.472684\pi\)
0.0857118 + 0.996320i \(0.472684\pi\)
\(102\) 0 0
\(103\) −1.95452 −0.192585 −0.0962923 0.995353i \(-0.530698\pi\)
−0.0962923 + 0.995353i \(0.530698\pi\)
\(104\) 0 0
\(105\) 0.294963 0.0287854
\(106\) 0 0
\(107\) 13.4004 1.29547 0.647735 0.761866i \(-0.275717\pi\)
0.647735 + 0.761866i \(0.275717\pi\)
\(108\) 0 0
\(109\) 7.46306 0.714831 0.357416 0.933945i \(-0.383658\pi\)
0.357416 + 0.933945i \(0.383658\pi\)
\(110\) 0 0
\(111\) −11.3274 −1.07515
\(112\) 0 0
\(113\) 5.29252 0.497878 0.248939 0.968519i \(-0.419918\pi\)
0.248939 + 0.968519i \(0.419918\pi\)
\(114\) 0 0
\(115\) −3.47726 −0.324256
\(116\) 0 0
\(117\) 4.42538 0.409126
\(118\) 0 0
\(119\) −1.37801 −0.126322
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −13.2609 −1.19570
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −12.1933 −1.08198 −0.540989 0.841030i \(-0.681950\pi\)
−0.540989 + 0.841030i \(0.681950\pi\)
\(128\) 0 0
\(129\) 4.84849 0.426886
\(130\) 0 0
\(131\) −2.06530 −0.180446 −0.0902229 0.995922i \(-0.528758\pi\)
−0.0902229 + 0.995922i \(0.528758\pi\)
\(132\) 0 0
\(133\) 0.505978 0.0438739
\(134\) 0 0
\(135\) 5.59822 0.481818
\(136\) 0 0
\(137\) 14.4436 1.23400 0.617001 0.786963i \(-0.288348\pi\)
0.617001 + 0.786963i \(0.288348\pi\)
\(138\) 0 0
\(139\) −16.3421 −1.38612 −0.693059 0.720881i \(-0.743738\pi\)
−0.693059 + 0.720881i \(0.743738\pi\)
\(140\) 0 0
\(141\) 6.88142 0.579520
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0.0307867 0.00255670
\(146\) 0 0
\(147\) 8.99755 0.742106
\(148\) 0 0
\(149\) 14.9817 1.22735 0.613674 0.789559i \(-0.289691\pi\)
0.613674 + 0.789559i \(0.289691\pi\)
\(150\) 0 0
\(151\) 12.3281 1.00325 0.501624 0.865086i \(-0.332736\pi\)
0.501624 + 0.865086i \(0.332736\pi\)
\(152\) 0 0
\(153\) −8.00433 −0.647112
\(154\) 0 0
\(155\) 2.21625 0.178014
\(156\) 0 0
\(157\) −20.8442 −1.66355 −0.831775 0.555112i \(-0.812675\pi\)
−0.831775 + 0.555112i \(0.812675\pi\)
\(158\) 0 0
\(159\) 13.2123 1.04780
\(160\) 0 0
\(161\) 0.792040 0.0624215
\(162\) 0 0
\(163\) 0.564984 0.0442530 0.0221265 0.999755i \(-0.492956\pi\)
0.0221265 + 0.999755i \(0.492956\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −22.4354 −1.73611 −0.868053 0.496471i \(-0.834629\pi\)
−0.868053 + 0.496471i \(0.834629\pi\)
\(168\) 0 0
\(169\) −1.81247 −0.139421
\(170\) 0 0
\(171\) 2.93904 0.224754
\(172\) 0 0
\(173\) 2.16572 0.164656 0.0823281 0.996605i \(-0.473764\pi\)
0.0823281 + 0.996605i \(0.473764\pi\)
\(174\) 0 0
\(175\) −0.227777 −0.0172183
\(176\) 0 0
\(177\) −10.8052 −0.812165
\(178\) 0 0
\(179\) −22.1568 −1.65608 −0.828038 0.560671i \(-0.810543\pi\)
−0.828038 + 0.560671i \(0.810543\pi\)
\(180\) 0 0
\(181\) −8.93677 −0.664265 −0.332132 0.943233i \(-0.607768\pi\)
−0.332132 + 0.943233i \(0.607768\pi\)
\(182\) 0 0
\(183\) 2.90787 0.214956
\(184\) 0 0
\(185\) 8.74728 0.643113
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.27515 −0.0927532
\(190\) 0 0
\(191\) −11.5080 −0.832693 −0.416347 0.909206i \(-0.636690\pi\)
−0.416347 + 0.909206i \(0.636690\pi\)
\(192\) 0 0
\(193\) 8.31192 0.598305 0.299153 0.954205i \(-0.403296\pi\)
0.299153 + 0.954205i \(0.403296\pi\)
\(194\) 0 0
\(195\) 4.33136 0.310175
\(196\) 0 0
\(197\) 3.64765 0.259885 0.129942 0.991522i \(-0.458521\pi\)
0.129942 + 0.991522i \(0.458521\pi\)
\(198\) 0 0
\(199\) −6.34757 −0.449967 −0.224984 0.974363i \(-0.572233\pi\)
−0.224984 + 0.974363i \(0.572233\pi\)
\(200\) 0 0
\(201\) −4.49780 −0.317250
\(202\) 0 0
\(203\) −0.00701251 −0.000492182 0
\(204\) 0 0
\(205\) 10.2404 0.715221
\(206\) 0 0
\(207\) 4.60066 0.319768
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −19.2543 −1.32552 −0.662761 0.748831i \(-0.730615\pi\)
−0.662761 + 0.748831i \(0.730615\pi\)
\(212\) 0 0
\(213\) 19.5070 1.33659
\(214\) 0 0
\(215\) −3.74411 −0.255347
\(216\) 0 0
\(217\) −0.504811 −0.0342688
\(218\) 0 0
\(219\) 4.74460 0.320611
\(220\) 0 0
\(221\) −20.2353 −1.36117
\(222\) 0 0
\(223\) −21.7301 −1.45516 −0.727579 0.686024i \(-0.759355\pi\)
−0.727579 + 0.686024i \(0.759355\pi\)
\(224\) 0 0
\(225\) −1.32307 −0.0882047
\(226\) 0 0
\(227\) −6.00475 −0.398549 −0.199275 0.979944i \(-0.563859\pi\)
−0.199275 + 0.979944i \(0.563859\pi\)
\(228\) 0 0
\(229\) −23.2079 −1.53362 −0.766810 0.641874i \(-0.778157\pi\)
−0.766810 + 0.641874i \(0.778157\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.2183 1.58659 0.793296 0.608836i \(-0.208363\pi\)
0.793296 + 0.608836i \(0.208363\pi\)
\(234\) 0 0
\(235\) −5.31399 −0.346646
\(236\) 0 0
\(237\) 15.0490 0.977539
\(238\) 0 0
\(239\) −18.3873 −1.18937 −0.594686 0.803958i \(-0.702724\pi\)
−0.594686 + 0.803958i \(0.702724\pi\)
\(240\) 0 0
\(241\) −12.3990 −0.798692 −0.399346 0.916800i \(-0.630763\pi\)
−0.399346 + 0.916800i \(0.630763\pi\)
\(242\) 0 0
\(243\) −12.5468 −0.804879
\(244\) 0 0
\(245\) −6.94812 −0.443899
\(246\) 0 0
\(247\) 7.43000 0.472760
\(248\) 0 0
\(249\) −1.97189 −0.124964
\(250\) 0 0
\(251\) −21.7939 −1.37562 −0.687810 0.725890i \(-0.741428\pi\)
−0.687810 + 0.725890i \(0.741428\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −7.83428 −0.490602
\(256\) 0 0
\(257\) 14.7089 0.917518 0.458759 0.888561i \(-0.348294\pi\)
0.458759 + 0.888561i \(0.348294\pi\)
\(258\) 0 0
\(259\) −1.99243 −0.123804
\(260\) 0 0
\(261\) −0.0407330 −0.00252131
\(262\) 0 0
\(263\) 27.4694 1.69384 0.846918 0.531724i \(-0.178456\pi\)
0.846918 + 0.531724i \(0.178456\pi\)
\(264\) 0 0
\(265\) −10.2028 −0.626755
\(266\) 0 0
\(267\) 12.9572 0.792968
\(268\) 0 0
\(269\) −1.41916 −0.0865274 −0.0432637 0.999064i \(-0.513776\pi\)
−0.0432637 + 0.999064i \(0.513776\pi\)
\(270\) 0 0
\(271\) 10.3498 0.628708 0.314354 0.949306i \(-0.398212\pi\)
0.314354 + 0.949306i \(0.398212\pi\)
\(272\) 0 0
\(273\) −0.986585 −0.0597108
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 17.8058 1.06985 0.534923 0.844901i \(-0.320340\pi\)
0.534923 + 0.844901i \(0.320340\pi\)
\(278\) 0 0
\(279\) −2.93226 −0.175550
\(280\) 0 0
\(281\) 17.1393 1.02245 0.511223 0.859448i \(-0.329193\pi\)
0.511223 + 0.859448i \(0.329193\pi\)
\(282\) 0 0
\(283\) 9.92524 0.589995 0.294997 0.955498i \(-0.404681\pi\)
0.294997 + 0.955498i \(0.404681\pi\)
\(284\) 0 0
\(285\) 2.87660 0.170395
\(286\) 0 0
\(287\) −2.33253 −0.137685
\(288\) 0 0
\(289\) 19.6002 1.15296
\(290\) 0 0
\(291\) 10.7944 0.632780
\(292\) 0 0
\(293\) −18.1035 −1.05762 −0.528809 0.848741i \(-0.677361\pi\)
−0.528809 + 0.848741i \(0.677361\pi\)
\(294\) 0 0
\(295\) 8.34399 0.485806
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 11.6307 0.672618
\(300\) 0 0
\(301\) 0.852824 0.0491559
\(302\) 0 0
\(303\) −2.23094 −0.128164
\(304\) 0 0
\(305\) −2.24553 −0.128578
\(306\) 0 0
\(307\) −6.00433 −0.342685 −0.171343 0.985211i \(-0.554811\pi\)
−0.171343 + 0.985211i \(0.554811\pi\)
\(308\) 0 0
\(309\) 2.53103 0.143985
\(310\) 0 0
\(311\) −14.8766 −0.843574 −0.421787 0.906695i \(-0.638597\pi\)
−0.421787 + 0.906695i \(0.638597\pi\)
\(312\) 0 0
\(313\) −2.41939 −0.136752 −0.0683760 0.997660i \(-0.521782\pi\)
−0.0683760 + 0.997660i \(0.521782\pi\)
\(314\) 0 0
\(315\) 0.301365 0.0169800
\(316\) 0 0
\(317\) 19.2375 1.08049 0.540244 0.841508i \(-0.318332\pi\)
0.540244 + 0.841508i \(0.318332\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −17.3531 −0.968554
\(322\) 0 0
\(323\) −13.4389 −0.747761
\(324\) 0 0
\(325\) −3.34478 −0.185535
\(326\) 0 0
\(327\) −9.66438 −0.534441
\(328\) 0 0
\(329\) 1.21041 0.0667318
\(330\) 0 0
\(331\) −13.8876 −0.763330 −0.381665 0.924301i \(-0.624649\pi\)
−0.381665 + 0.924301i \(0.624649\pi\)
\(332\) 0 0
\(333\) −11.5733 −0.634212
\(334\) 0 0
\(335\) 3.47330 0.189767
\(336\) 0 0
\(337\) −3.08479 −0.168039 −0.0840196 0.996464i \(-0.526776\pi\)
−0.0840196 + 0.996464i \(0.526776\pi\)
\(338\) 0 0
\(339\) −6.85361 −0.372237
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 3.17706 0.171545
\(344\) 0 0
\(345\) 4.50292 0.242429
\(346\) 0 0
\(347\) −28.6219 −1.53650 −0.768252 0.640148i \(-0.778873\pi\)
−0.768252 + 0.640148i \(0.778873\pi\)
\(348\) 0 0
\(349\) 16.8202 0.900365 0.450182 0.892937i \(-0.351359\pi\)
0.450182 + 0.892937i \(0.351359\pi\)
\(350\) 0 0
\(351\) −18.7248 −0.999455
\(352\) 0 0
\(353\) −14.6656 −0.780573 −0.390287 0.920693i \(-0.627624\pi\)
−0.390287 + 0.920693i \(0.627624\pi\)
\(354\) 0 0
\(355\) −15.0637 −0.799499
\(356\) 0 0
\(357\) 1.78447 0.0944442
\(358\) 0 0
\(359\) −30.8757 −1.62956 −0.814780 0.579771i \(-0.803142\pi\)
−0.814780 + 0.579771i \(0.803142\pi\)
\(360\) 0 0
\(361\) −14.0655 −0.740289
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.66389 −0.191777
\(366\) 0 0
\(367\) −3.63145 −0.189560 −0.0947800 0.995498i \(-0.530215\pi\)
−0.0947800 + 0.995498i \(0.530215\pi\)
\(368\) 0 0
\(369\) −13.5488 −0.705321
\(370\) 0 0
\(371\) 2.32397 0.120655
\(372\) 0 0
\(373\) 29.2244 1.51318 0.756590 0.653890i \(-0.226864\pi\)
0.756590 + 0.653890i \(0.226864\pi\)
\(374\) 0 0
\(375\) −1.29496 −0.0668716
\(376\) 0 0
\(377\) −0.102975 −0.00530347
\(378\) 0 0
\(379\) 3.65327 0.187656 0.0938278 0.995588i \(-0.470090\pi\)
0.0938278 + 0.995588i \(0.470090\pi\)
\(380\) 0 0
\(381\) 15.7898 0.808937
\(382\) 0 0
\(383\) −14.2462 −0.727949 −0.363975 0.931409i \(-0.618581\pi\)
−0.363975 + 0.931409i \(0.618581\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.95373 0.251812
\(388\) 0 0
\(389\) −8.24632 −0.418105 −0.209052 0.977904i \(-0.567038\pi\)
−0.209052 + 0.977904i \(0.567038\pi\)
\(390\) 0 0
\(391\) −21.0368 −1.06388
\(392\) 0 0
\(393\) 2.67448 0.134910
\(394\) 0 0
\(395\) −11.6212 −0.584726
\(396\) 0 0
\(397\) −2.57097 −0.129033 −0.0645167 0.997917i \(-0.520551\pi\)
−0.0645167 + 0.997917i \(0.520551\pi\)
\(398\) 0 0
\(399\) −0.655223 −0.0328022
\(400\) 0 0
\(401\) −38.9710 −1.94612 −0.973060 0.230551i \(-0.925947\pi\)
−0.973060 + 0.230551i \(0.925947\pi\)
\(402\) 0 0
\(403\) −7.41286 −0.369261
\(404\) 0 0
\(405\) −3.28027 −0.162998
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −3.42256 −0.169234 −0.0846172 0.996414i \(-0.526967\pi\)
−0.0846172 + 0.996414i \(0.526967\pi\)
\(410\) 0 0
\(411\) −18.7039 −0.922598
\(412\) 0 0
\(413\) −1.90057 −0.0935209
\(414\) 0 0
\(415\) 1.52274 0.0747484
\(416\) 0 0
\(417\) 21.1624 1.03633
\(418\) 0 0
\(419\) 10.5914 0.517426 0.258713 0.965954i \(-0.416702\pi\)
0.258713 + 0.965954i \(0.416702\pi\)
\(420\) 0 0
\(421\) 2.37055 0.115534 0.0577668 0.998330i \(-0.481602\pi\)
0.0577668 + 0.998330i \(0.481602\pi\)
\(422\) 0 0
\(423\) 7.03079 0.341849
\(424\) 0 0
\(425\) 6.04981 0.293459
\(426\) 0 0
\(427\) 0.511479 0.0247522
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −26.2360 −1.26374 −0.631872 0.775073i \(-0.717713\pi\)
−0.631872 + 0.775073i \(0.717713\pi\)
\(432\) 0 0
\(433\) −20.0338 −0.962762 −0.481381 0.876512i \(-0.659865\pi\)
−0.481381 + 0.876512i \(0.659865\pi\)
\(434\) 0 0
\(435\) −0.0398677 −0.00191151
\(436\) 0 0
\(437\) 7.72430 0.369503
\(438\) 0 0
\(439\) 21.9546 1.04783 0.523917 0.851769i \(-0.324470\pi\)
0.523917 + 0.851769i \(0.324470\pi\)
\(440\) 0 0
\(441\) 9.19285 0.437755
\(442\) 0 0
\(443\) 16.3419 0.776428 0.388214 0.921569i \(-0.373092\pi\)
0.388214 + 0.921569i \(0.373092\pi\)
\(444\) 0 0
\(445\) −10.0058 −0.474323
\(446\) 0 0
\(447\) −19.4007 −0.917623
\(448\) 0 0
\(449\) −38.0476 −1.79558 −0.897788 0.440429i \(-0.854826\pi\)
−0.897788 + 0.440429i \(0.854826\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −15.9645 −0.750076
\(454\) 0 0
\(455\) 0.761864 0.0357167
\(456\) 0 0
\(457\) −29.3016 −1.37067 −0.685336 0.728227i \(-0.740344\pi\)
−0.685336 + 0.728227i \(0.740344\pi\)
\(458\) 0 0
\(459\) 33.8682 1.58083
\(460\) 0 0
\(461\) −12.1136 −0.564188 −0.282094 0.959387i \(-0.591029\pi\)
−0.282094 + 0.959387i \(0.591029\pi\)
\(462\) 0 0
\(463\) −3.92452 −0.182388 −0.0911940 0.995833i \(-0.529068\pi\)
−0.0911940 + 0.995833i \(0.529068\pi\)
\(464\) 0 0
\(465\) −2.86996 −0.133091
\(466\) 0 0
\(467\) 35.1795 1.62792 0.813958 0.580924i \(-0.197309\pi\)
0.813958 + 0.580924i \(0.197309\pi\)
\(468\) 0 0
\(469\) −0.791139 −0.0365314
\(470\) 0 0
\(471\) 26.9925 1.24375
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −2.22137 −0.101924
\(476\) 0 0
\(477\) 13.4991 0.618080
\(478\) 0 0
\(479\) 34.9726 1.59794 0.798969 0.601372i \(-0.205379\pi\)
0.798969 + 0.601372i \(0.205379\pi\)
\(480\) 0 0
\(481\) −29.2577 −1.33404
\(482\) 0 0
\(483\) −1.02566 −0.0466693
\(484\) 0 0
\(485\) −8.33570 −0.378504
\(486\) 0 0
\(487\) 10.6381 0.482058 0.241029 0.970518i \(-0.422515\pi\)
0.241029 + 0.970518i \(0.422515\pi\)
\(488\) 0 0
\(489\) −0.731634 −0.0330856
\(490\) 0 0
\(491\) 29.0985 1.31320 0.656598 0.754241i \(-0.271995\pi\)
0.656598 + 0.754241i \(0.271995\pi\)
\(492\) 0 0
\(493\) 0.186254 0.00838846
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.43117 0.153909
\(498\) 0 0
\(499\) −23.1487 −1.03628 −0.518140 0.855296i \(-0.673375\pi\)
−0.518140 + 0.855296i \(0.673375\pi\)
\(500\) 0 0
\(501\) 29.0531 1.29799
\(502\) 0 0
\(503\) 12.6609 0.564521 0.282261 0.959338i \(-0.408916\pi\)
0.282261 + 0.959338i \(0.408916\pi\)
\(504\) 0 0
\(505\) 1.72279 0.0766630
\(506\) 0 0
\(507\) 2.34708 0.104237
\(508\) 0 0
\(509\) 9.86065 0.437066 0.218533 0.975830i \(-0.429873\pi\)
0.218533 + 0.975830i \(0.429873\pi\)
\(510\) 0 0
\(511\) 0.834550 0.0369183
\(512\) 0 0
\(513\) −12.4357 −0.549051
\(514\) 0 0
\(515\) −1.95452 −0.0861264
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −2.80452 −0.123105
\(520\) 0 0
\(521\) −2.52164 −0.110475 −0.0552376 0.998473i \(-0.517592\pi\)
−0.0552376 + 0.998473i \(0.517592\pi\)
\(522\) 0 0
\(523\) 20.7213 0.906079 0.453039 0.891490i \(-0.350340\pi\)
0.453039 + 0.891490i \(0.350340\pi\)
\(524\) 0 0
\(525\) 0.294963 0.0128732
\(526\) 0 0
\(527\) 13.4079 0.584057
\(528\) 0 0
\(529\) −10.9087 −0.474290
\(530\) 0 0
\(531\) −11.0397 −0.479082
\(532\) 0 0
\(533\) −34.2519 −1.48361
\(534\) 0 0
\(535\) 13.4004 0.579351
\(536\) 0 0
\(537\) 28.6922 1.23816
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 26.7777 1.15126 0.575631 0.817710i \(-0.304757\pi\)
0.575631 + 0.817710i \(0.304757\pi\)
\(542\) 0 0
\(543\) 11.5728 0.496636
\(544\) 0 0
\(545\) 7.46306 0.319682
\(546\) 0 0
\(547\) −4.57034 −0.195414 −0.0977068 0.995215i \(-0.531151\pi\)
−0.0977068 + 0.995215i \(0.531151\pi\)
\(548\) 0 0
\(549\) 2.97099 0.126799
\(550\) 0 0
\(551\) −0.0683889 −0.00291346
\(552\) 0 0
\(553\) 2.64704 0.112564
\(554\) 0 0
\(555\) −11.3274 −0.480822
\(556\) 0 0
\(557\) −31.7834 −1.34670 −0.673352 0.739322i \(-0.735146\pi\)
−0.673352 + 0.739322i \(0.735146\pi\)
\(558\) 0 0
\(559\) 12.5232 0.529676
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −9.29858 −0.391888 −0.195944 0.980615i \(-0.562777\pi\)
−0.195944 + 0.980615i \(0.562777\pi\)
\(564\) 0 0
\(565\) 5.29252 0.222658
\(566\) 0 0
\(567\) 0.747170 0.0313782
\(568\) 0 0
\(569\) −47.1667 −1.97733 −0.988665 0.150139i \(-0.952028\pi\)
−0.988665 + 0.150139i \(0.952028\pi\)
\(570\) 0 0
\(571\) −4.32635 −0.181052 −0.0905260 0.995894i \(-0.528855\pi\)
−0.0905260 + 0.995894i \(0.528855\pi\)
\(572\) 0 0
\(573\) 14.9025 0.622561
\(574\) 0 0
\(575\) −3.47726 −0.145012
\(576\) 0 0
\(577\) 8.77746 0.365410 0.182705 0.983168i \(-0.441515\pi\)
0.182705 + 0.983168i \(0.441515\pi\)
\(578\) 0 0
\(579\) −10.7636 −0.447321
\(580\) 0 0
\(581\) −0.346845 −0.0143896
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 4.42538 0.182967
\(586\) 0 0
\(587\) −20.6919 −0.854047 −0.427024 0.904240i \(-0.640438\pi\)
−0.427024 + 0.904240i \(0.640438\pi\)
\(588\) 0 0
\(589\) −4.92312 −0.202854
\(590\) 0 0
\(591\) −4.72358 −0.194302
\(592\) 0 0
\(593\) −46.8273 −1.92296 −0.961482 0.274866i \(-0.911366\pi\)
−0.961482 + 0.274866i \(0.911366\pi\)
\(594\) 0 0
\(595\) −1.37801 −0.0564929
\(596\) 0 0
\(597\) 8.21986 0.336417
\(598\) 0 0
\(599\) −31.0055 −1.26685 −0.633425 0.773804i \(-0.718351\pi\)
−0.633425 + 0.773804i \(0.718351\pi\)
\(600\) 0 0
\(601\) 42.0290 1.71440 0.857199 0.514986i \(-0.172203\pi\)
0.857199 + 0.514986i \(0.172203\pi\)
\(602\) 0 0
\(603\) −4.59543 −0.187140
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 10.8594 0.440769 0.220385 0.975413i \(-0.429269\pi\)
0.220385 + 0.975413i \(0.429269\pi\)
\(608\) 0 0
\(609\) 0.00908094 0.000367978 0
\(610\) 0 0
\(611\) 17.7741 0.719064
\(612\) 0 0
\(613\) −37.0203 −1.49524 −0.747618 0.664129i \(-0.768803\pi\)
−0.747618 + 0.664129i \(0.768803\pi\)
\(614\) 0 0
\(615\) −13.2609 −0.534733
\(616\) 0 0
\(617\) 37.3620 1.50414 0.752068 0.659085i \(-0.229056\pi\)
0.752068 + 0.659085i \(0.229056\pi\)
\(618\) 0 0
\(619\) −39.7254 −1.59670 −0.798350 0.602194i \(-0.794293\pi\)
−0.798350 + 0.602194i \(0.794293\pi\)
\(620\) 0 0
\(621\) −19.4665 −0.781162
\(622\) 0 0
\(623\) 2.27910 0.0913103
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 52.9194 2.11004
\(630\) 0 0
\(631\) 7.44019 0.296189 0.148095 0.988973i \(-0.452686\pi\)
0.148095 + 0.988973i \(0.452686\pi\)
\(632\) 0 0
\(633\) 24.9336 0.991022
\(634\) 0 0
\(635\) −12.1933 −0.483875
\(636\) 0 0
\(637\) 23.2399 0.920798
\(638\) 0 0
\(639\) 19.9304 0.788433
\(640\) 0 0
\(641\) −1.86589 −0.0736984 −0.0368492 0.999321i \(-0.511732\pi\)
−0.0368492 + 0.999321i \(0.511732\pi\)
\(642\) 0 0
\(643\) 45.7333 1.80354 0.901772 0.432212i \(-0.142267\pi\)
0.901772 + 0.432212i \(0.142267\pi\)
\(644\) 0 0
\(645\) 4.84849 0.190909
\(646\) 0 0
\(647\) −34.0395 −1.33823 −0.669115 0.743159i \(-0.733327\pi\)
−0.669115 + 0.743159i \(0.733327\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.653712 0.0256210
\(652\) 0 0
\(653\) 3.10780 0.121617 0.0608087 0.998149i \(-0.480632\pi\)
0.0608087 + 0.998149i \(0.480632\pi\)
\(654\) 0 0
\(655\) −2.06530 −0.0806978
\(656\) 0 0
\(657\) 4.84759 0.189122
\(658\) 0 0
\(659\) −10.1518 −0.395459 −0.197729 0.980257i \(-0.563357\pi\)
−0.197729 + 0.980257i \(0.563357\pi\)
\(660\) 0 0
\(661\) −44.7642 −1.74113 −0.870564 0.492056i \(-0.836246\pi\)
−0.870564 + 0.492056i \(0.836246\pi\)
\(662\) 0 0
\(663\) 26.2039 1.01768
\(664\) 0 0
\(665\) 0.505978 0.0196210
\(666\) 0 0
\(667\) −0.107053 −0.00414513
\(668\) 0 0
\(669\) 28.1397 1.08794
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −5.36472 −0.206795 −0.103397 0.994640i \(-0.532971\pi\)
−0.103397 + 0.994640i \(0.532971\pi\)
\(674\) 0 0
\(675\) 5.59822 0.215475
\(676\) 0 0
\(677\) −11.3147 −0.434860 −0.217430 0.976076i \(-0.569767\pi\)
−0.217430 + 0.976076i \(0.569767\pi\)
\(678\) 0 0
\(679\) 1.89868 0.0728647
\(680\) 0 0
\(681\) 7.77592 0.297974
\(682\) 0 0
\(683\) 31.0817 1.18931 0.594655 0.803981i \(-0.297289\pi\)
0.594655 + 0.803981i \(0.297289\pi\)
\(684\) 0 0
\(685\) 14.4436 0.551862
\(686\) 0 0
\(687\) 30.0534 1.14661
\(688\) 0 0
\(689\) 34.1262 1.30011
\(690\) 0 0
\(691\) 3.95742 0.150547 0.0752737 0.997163i \(-0.476017\pi\)
0.0752737 + 0.997163i \(0.476017\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.3421 −0.619891
\(696\) 0 0
\(697\) 61.9525 2.34662
\(698\) 0 0
\(699\) −31.3618 −1.18621
\(700\) 0 0
\(701\) 22.9309 0.866089 0.433045 0.901372i \(-0.357439\pi\)
0.433045 + 0.901372i \(0.357439\pi\)
\(702\) 0 0
\(703\) −19.4310 −0.732854
\(704\) 0 0
\(705\) 6.88142 0.259169
\(706\) 0 0
\(707\) −0.392411 −0.0147581
\(708\) 0 0
\(709\) 32.5067 1.22081 0.610407 0.792088i \(-0.291006\pi\)
0.610407 + 0.792088i \(0.291006\pi\)
\(710\) 0 0
\(711\) 15.3757 0.576633
\(712\) 0 0
\(713\) −7.70648 −0.288610
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 23.8108 0.889231
\(718\) 0 0
\(719\) 30.4360 1.13507 0.567536 0.823348i \(-0.307897\pi\)
0.567536 + 0.823348i \(0.307897\pi\)
\(720\) 0 0
\(721\) 0.445195 0.0165799
\(722\) 0 0
\(723\) 16.0563 0.597140
\(724\) 0 0
\(725\) 0.0307867 0.00114339
\(726\) 0 0
\(727\) 36.6338 1.35867 0.679337 0.733827i \(-0.262268\pi\)
0.679337 + 0.733827i \(0.262268\pi\)
\(728\) 0 0
\(729\) 26.0885 0.966240
\(730\) 0 0
\(731\) −22.6512 −0.837785
\(732\) 0 0
\(733\) 32.1710 1.18826 0.594132 0.804368i \(-0.297496\pi\)
0.594132 + 0.804368i \(0.297496\pi\)
\(734\) 0 0
\(735\) 8.99755 0.331880
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20.7794 0.764382 0.382191 0.924083i \(-0.375170\pi\)
0.382191 + 0.924083i \(0.375170\pi\)
\(740\) 0 0
\(741\) −9.62158 −0.353457
\(742\) 0 0
\(743\) −37.8417 −1.38828 −0.694140 0.719840i \(-0.744215\pi\)
−0.694140 + 0.719840i \(0.744215\pi\)
\(744\) 0 0
\(745\) 14.9817 0.548887
\(746\) 0 0
\(747\) −2.01469 −0.0737138
\(748\) 0 0
\(749\) −3.05231 −0.111529
\(750\) 0 0
\(751\) −13.0447 −0.476007 −0.238004 0.971264i \(-0.576493\pi\)
−0.238004 + 0.971264i \(0.576493\pi\)
\(752\) 0 0
\(753\) 28.2223 1.02848
\(754\) 0 0
\(755\) 12.3281 0.448666
\(756\) 0 0
\(757\) 41.1704 1.49636 0.748181 0.663495i \(-0.230927\pi\)
0.748181 + 0.663495i \(0.230927\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −43.7344 −1.58537 −0.792686 0.609630i \(-0.791318\pi\)
−0.792686 + 0.609630i \(0.791318\pi\)
\(762\) 0 0
\(763\) −1.69991 −0.0615410
\(764\) 0 0
\(765\) −8.00433 −0.289397
\(766\) 0 0
\(767\) −27.9088 −1.00773
\(768\) 0 0
\(769\) −17.3914 −0.627151 −0.313575 0.949563i \(-0.601527\pi\)
−0.313575 + 0.949563i \(0.601527\pi\)
\(770\) 0 0
\(771\) −19.0475 −0.685979
\(772\) 0 0
\(773\) 24.8805 0.894890 0.447445 0.894311i \(-0.352334\pi\)
0.447445 + 0.894311i \(0.352334\pi\)
\(774\) 0 0
\(775\) 2.21625 0.0796101
\(776\) 0 0
\(777\) 2.58012 0.0925614
\(778\) 0 0
\(779\) −22.7478 −0.815023
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0.172351 0.00615931
\(784\) 0 0
\(785\) −20.8442 −0.743963
\(786\) 0 0
\(787\) 27.0641 0.964731 0.482365 0.875970i \(-0.339778\pi\)
0.482365 + 0.875970i \(0.339778\pi\)
\(788\) 0 0
\(789\) −35.5718 −1.26639
\(790\) 0 0
\(791\) −1.20551 −0.0428632
\(792\) 0 0
\(793\) 7.51078 0.266716
\(794\) 0 0
\(795\) 13.2123 0.468592
\(796\) 0 0
\(797\) −12.0392 −0.426449 −0.213224 0.977003i \(-0.568397\pi\)
−0.213224 + 0.977003i \(0.568397\pi\)
\(798\) 0 0
\(799\) −32.1487 −1.13734
\(800\) 0 0
\(801\) 13.2384 0.467757
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0.792040 0.0279157
\(806\) 0 0
\(807\) 1.83775 0.0646920
\(808\) 0 0
\(809\) 27.3914 0.963031 0.481516 0.876438i \(-0.340086\pi\)
0.481516 + 0.876438i \(0.340086\pi\)
\(810\) 0 0
\(811\) 54.3420 1.90820 0.954102 0.299481i \(-0.0968136\pi\)
0.954102 + 0.299481i \(0.0968136\pi\)
\(812\) 0 0
\(813\) −13.4026 −0.470051
\(814\) 0 0
\(815\) 0.564984 0.0197905
\(816\) 0 0
\(817\) 8.31708 0.290978
\(818\) 0 0
\(819\) −1.00800 −0.0352224
\(820\) 0 0
\(821\) −39.7997 −1.38902 −0.694509 0.719484i \(-0.744378\pi\)
−0.694509 + 0.719484i \(0.744378\pi\)
\(822\) 0 0
\(823\) 23.8835 0.832526 0.416263 0.909244i \(-0.363340\pi\)
0.416263 + 0.909244i \(0.363340\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.83591 0.272481 0.136241 0.990676i \(-0.456498\pi\)
0.136241 + 0.990676i \(0.456498\pi\)
\(828\) 0 0
\(829\) 36.1073 1.25406 0.627028 0.778996i \(-0.284271\pi\)
0.627028 + 0.778996i \(0.284271\pi\)
\(830\) 0 0
\(831\) −23.0578 −0.799868
\(832\) 0 0
\(833\) −42.0348 −1.45642
\(834\) 0 0
\(835\) −22.4354 −0.776410
\(836\) 0 0
\(837\) 12.4071 0.428850
\(838\) 0 0
\(839\) 25.3952 0.876741 0.438371 0.898794i \(-0.355556\pi\)
0.438371 + 0.898794i \(0.355556\pi\)
\(840\) 0 0
\(841\) −28.9991 −0.999967
\(842\) 0 0
\(843\) −22.1948 −0.764428
\(844\) 0 0
\(845\) −1.81247 −0.0623508
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −12.8528 −0.441108
\(850\) 0 0
\(851\) −30.4166 −1.04267
\(852\) 0 0
\(853\) −17.0660 −0.584329 −0.292165 0.956368i \(-0.594376\pi\)
−0.292165 + 0.956368i \(0.594376\pi\)
\(854\) 0 0
\(855\) 2.93904 0.100513
\(856\) 0 0
\(857\) 31.2827 1.06860 0.534298 0.845296i \(-0.320576\pi\)
0.534298 + 0.845296i \(0.320576\pi\)
\(858\) 0 0
\(859\) 50.6101 1.72679 0.863396 0.504526i \(-0.168333\pi\)
0.863396 + 0.504526i \(0.168333\pi\)
\(860\) 0 0
\(861\) 3.02054 0.102940
\(862\) 0 0
\(863\) −29.9414 −1.01922 −0.509609 0.860406i \(-0.670210\pi\)
−0.509609 + 0.860406i \(0.670210\pi\)
\(864\) 0 0
\(865\) 2.16572 0.0736365
\(866\) 0 0
\(867\) −25.3816 −0.862004
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −11.6174 −0.393641
\(872\) 0 0
\(873\) 11.0287 0.373266
\(874\) 0 0
\(875\) −0.227777 −0.00770027
\(876\) 0 0
\(877\) 23.7971 0.803571 0.401785 0.915734i \(-0.368390\pi\)
0.401785 + 0.915734i \(0.368390\pi\)
\(878\) 0 0
\(879\) 23.4434 0.790726
\(880\) 0 0
\(881\) 34.7141 1.16955 0.584773 0.811197i \(-0.301184\pi\)
0.584773 + 0.811197i \(0.301184\pi\)
\(882\) 0 0
\(883\) −7.25118 −0.244022 −0.122011 0.992529i \(-0.538934\pi\)
−0.122011 + 0.992529i \(0.538934\pi\)
\(884\) 0 0
\(885\) −10.8052 −0.363211
\(886\) 0 0
\(887\) −14.3106 −0.480503 −0.240251 0.970711i \(-0.577230\pi\)
−0.240251 + 0.970711i \(0.577230\pi\)
\(888\) 0 0
\(889\) 2.77735 0.0931492
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11.8044 0.395018
\(894\) 0 0
\(895\) −22.1568 −0.740620
\(896\) 0 0
\(897\) −15.0613 −0.502881
\(898\) 0 0
\(899\) 0.0682311 0.00227564
\(900\) 0 0
\(901\) −61.7253 −2.05637
\(902\) 0 0
\(903\) −1.10437 −0.0367513
\(904\) 0 0
\(905\) −8.93677 −0.297068
\(906\) 0 0
\(907\) 55.7899 1.85247 0.926237 0.376941i \(-0.123024\pi\)
0.926237 + 0.376941i \(0.123024\pi\)
\(908\) 0 0
\(909\) −2.27937 −0.0756019
\(910\) 0 0
\(911\) 38.8903 1.28849 0.644246 0.764819i \(-0.277171\pi\)
0.644246 + 0.764819i \(0.277171\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 2.90787 0.0961313
\(916\) 0 0
\(917\) 0.470427 0.0155349
\(918\) 0 0
\(919\) −37.3365 −1.23162 −0.615808 0.787896i \(-0.711170\pi\)
−0.615808 + 0.787896i \(0.711170\pi\)
\(920\) 0 0
\(921\) 7.77539 0.256208
\(922\) 0 0
\(923\) 50.3848 1.65844
\(924\) 0 0
\(925\) 8.74728 0.287609
\(926\) 0 0
\(927\) 2.58597 0.0849344
\(928\) 0 0
\(929\) 46.8843 1.53822 0.769112 0.639114i \(-0.220699\pi\)
0.769112 + 0.639114i \(0.220699\pi\)
\(930\) 0 0
\(931\) 15.4344 0.505841
\(932\) 0 0
\(933\) 19.2646 0.630696
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 53.2268 1.73884 0.869422 0.494070i \(-0.164491\pi\)
0.869422 + 0.494070i \(0.164491\pi\)
\(938\) 0 0
\(939\) 3.13302 0.102242
\(940\) 0 0
\(941\) −26.1871 −0.853676 −0.426838 0.904328i \(-0.640372\pi\)
−0.426838 + 0.904328i \(0.640372\pi\)
\(942\) 0 0
\(943\) −35.6085 −1.15957
\(944\) 0 0
\(945\) −1.27515 −0.0414805
\(946\) 0 0
\(947\) 27.9451 0.908095 0.454047 0.890977i \(-0.349980\pi\)
0.454047 + 0.890977i \(0.349980\pi\)
\(948\) 0 0
\(949\) 12.2549 0.397811
\(950\) 0 0
\(951\) −24.9119 −0.807824
\(952\) 0 0
\(953\) −7.31788 −0.237049 −0.118525 0.992951i \(-0.537816\pi\)
−0.118525 + 0.992951i \(0.537816\pi\)
\(954\) 0 0
\(955\) −11.5080 −0.372392
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.28992 −0.106237
\(960\) 0 0
\(961\) −26.0882 −0.841556
\(962\) 0 0
\(963\) −17.7297 −0.571333
\(964\) 0 0
\(965\) 8.31192 0.267570
\(966\) 0 0
\(967\) −39.7018 −1.27673 −0.638363 0.769736i \(-0.720388\pi\)
−0.638363 + 0.769736i \(0.720388\pi\)
\(968\) 0 0
\(969\) 17.4029 0.559061
\(970\) 0 0
\(971\) 59.0922 1.89636 0.948179 0.317737i \(-0.102923\pi\)
0.948179 + 0.317737i \(0.102923\pi\)
\(972\) 0 0
\(973\) 3.72236 0.119333
\(974\) 0 0
\(975\) 4.33136 0.138715
\(976\) 0 0
\(977\) −51.0338 −1.63272 −0.816359 0.577545i \(-0.804011\pi\)
−0.816359 + 0.577545i \(0.804011\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −9.87415 −0.315257
\(982\) 0 0
\(983\) −21.5904 −0.688626 −0.344313 0.938855i \(-0.611888\pi\)
−0.344313 + 0.938855i \(0.611888\pi\)
\(984\) 0 0
\(985\) 3.64765 0.116224
\(986\) 0 0
\(987\) −1.56743 −0.0498918
\(988\) 0 0
\(989\) 13.0193 0.413988
\(990\) 0 0
\(991\) 33.7197 1.07114 0.535570 0.844491i \(-0.320097\pi\)
0.535570 + 0.844491i \(0.320097\pi\)
\(992\) 0 0
\(993\) 17.9839 0.570701
\(994\) 0 0
\(995\) −6.34757 −0.201231
\(996\) 0 0
\(997\) 14.5044 0.459360 0.229680 0.973266i \(-0.426232\pi\)
0.229680 + 0.973266i \(0.426232\pi\)
\(998\) 0 0
\(999\) 48.9692 1.54932
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 4840.2.a.y.1.2 4
4.3 odd 2 9680.2.a.cu.1.3 4
11.5 even 5 440.2.y.a.201.1 yes 8
11.9 even 5 440.2.y.a.81.1 8
11.10 odd 2 4840.2.a.z.1.2 4
44.27 odd 10 880.2.bo.d.641.2 8
44.31 odd 10 880.2.bo.d.81.2 8
44.43 even 2 9680.2.a.ct.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.a.81.1 8 11.9 even 5
440.2.y.a.201.1 yes 8 11.5 even 5
880.2.bo.d.81.2 8 44.31 odd 10
880.2.bo.d.641.2 8 44.27 odd 10
4840.2.a.y.1.2 4 1.1 even 1 trivial
4840.2.a.z.1.2 4 11.10 odd 2
9680.2.a.ct.1.3 4 44.43 even 2
9680.2.a.cu.1.3 4 4.3 odd 2