Properties

Label 3380.2.a.n.1.1
Level $3380$
Weight $2$
Character 3380.1
Self dual yes
Analytic conductor $26.989$
Analytic rank $0$
Dimension $3$
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] = [3380,2,Mod(1,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3380.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.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: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.756.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 2 \) 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 260)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.26180\) of defining polynomial
Character \(\chi\) \(=\) 3380.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.26180 q^{3} +1.00000 q^{5} +1.11575 q^{7} +2.11575 q^{9} +5.37755 q^{11} -2.26180 q^{15} -7.90116 q^{19} -2.52360 q^{21} +6.49330 q^{23} +1.00000 q^{25} +2.00000 q^{27} +3.63935 q^{29} +3.14605 q^{31} -12.1630 q^{33} +1.11575 q^{35} +7.40786 q^{37} +4.75510 q^{41} +4.78541 q^{43} +2.11575 q^{45} -6.16296 q^{47} -5.75510 q^{49} +0.292106 q^{53} +5.37755 q^{55} +17.8709 q^{57} -11.3776 q^{59} -5.34725 q^{61} +2.36065 q^{63} -11.8709 q^{67} -14.6866 q^{69} -3.43816 q^{71} +4.59214 q^{73} -2.26180 q^{75} +6.00000 q^{77} -10.8157 q^{79} -10.8709 q^{81} +7.40786 q^{83} -8.23150 q^{87} +14.8157 q^{89} -7.11575 q^{93} -7.90116 q^{95} +3.76850 q^{97} +11.3776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} + 3 q^{9} + 6 q^{11} + 6 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} - 6 q^{29} + 6 q^{31} - 6 q^{33} + 12 q^{37} - 6 q^{41} - 6 q^{43} + 3 q^{45} + 12 q^{47} + 3 q^{49} - 6 q^{53} + 6 q^{55}+ \cdots + 24 q^{99}+O(q^{100}) \) 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
\(3\) −2.26180 −1.30585 −0.652926 0.757422i \(-0.726459\pi\)
−0.652926 + 0.757422i \(0.726459\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.11575 0.421714 0.210857 0.977517i \(-0.432375\pi\)
0.210857 + 0.977517i \(0.432375\pi\)
\(8\) 0 0
\(9\) 2.11575 0.705250
\(10\) 0 0
\(11\) 5.37755 1.62139 0.810696 0.585467i \(-0.199089\pi\)
0.810696 + 0.585467i \(0.199089\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −2.26180 −0.583995
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −7.90116 −1.81265 −0.906325 0.422582i \(-0.861124\pi\)
−0.906325 + 0.422582i \(0.861124\pi\)
\(20\) 0 0
\(21\) −2.52360 −0.550696
\(22\) 0 0
\(23\) 6.49330 1.35395 0.676973 0.736007i \(-0.263291\pi\)
0.676973 + 0.736007i \(0.263291\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 2.00000 0.384900
\(28\) 0 0
\(29\) 3.63935 0.675811 0.337906 0.941180i \(-0.390282\pi\)
0.337906 + 0.941180i \(0.390282\pi\)
\(30\) 0 0
\(31\) 3.14605 0.565048 0.282524 0.959260i \(-0.408828\pi\)
0.282524 + 0.959260i \(0.408828\pi\)
\(32\) 0 0
\(33\) −12.1630 −2.11730
\(34\) 0 0
\(35\) 1.11575 0.188596
\(36\) 0 0
\(37\) 7.40786 1.21784 0.608922 0.793230i \(-0.291602\pi\)
0.608922 + 0.793230i \(0.291602\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.75510 0.742622 0.371311 0.928508i \(-0.378908\pi\)
0.371311 + 0.928508i \(0.378908\pi\)
\(42\) 0 0
\(43\) 4.78541 0.729768 0.364884 0.931053i \(-0.381109\pi\)
0.364884 + 0.931053i \(0.381109\pi\)
\(44\) 0 0
\(45\) 2.11575 0.315397
\(46\) 0 0
\(47\) −6.16296 −0.898960 −0.449480 0.893290i \(-0.648391\pi\)
−0.449480 + 0.893290i \(0.648391\pi\)
\(48\) 0 0
\(49\) −5.75510 −0.822158
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.292106 0.0401238 0.0200619 0.999799i \(-0.493614\pi\)
0.0200619 + 0.999799i \(0.493614\pi\)
\(54\) 0 0
\(55\) 5.37755 0.725109
\(56\) 0 0
\(57\) 17.8709 2.36705
\(58\) 0 0
\(59\) −11.3776 −1.48123 −0.740616 0.671929i \(-0.765466\pi\)
−0.740616 + 0.671929i \(0.765466\pi\)
\(60\) 0 0
\(61\) −5.34725 −0.684645 −0.342322 0.939583i \(-0.611214\pi\)
−0.342322 + 0.939583i \(0.611214\pi\)
\(62\) 0 0
\(63\) 2.36065 0.297413
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −11.8709 −1.45026 −0.725128 0.688614i \(-0.758219\pi\)
−0.725128 + 0.688614i \(0.758219\pi\)
\(68\) 0 0
\(69\) −14.6866 −1.76805
\(70\) 0 0
\(71\) −3.43816 −0.408034 −0.204017 0.978967i \(-0.565400\pi\)
−0.204017 + 0.978967i \(0.565400\pi\)
\(72\) 0 0
\(73\) 4.59214 0.537470 0.268735 0.963214i \(-0.413394\pi\)
0.268735 + 0.963214i \(0.413394\pi\)
\(74\) 0 0
\(75\) −2.26180 −0.261170
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) −10.8157 −1.21686 −0.608431 0.793607i \(-0.708201\pi\)
−0.608431 + 0.793607i \(0.708201\pi\)
\(80\) 0 0
\(81\) −10.8709 −1.20787
\(82\) 0 0
\(83\) 7.40786 0.813118 0.406559 0.913625i \(-0.366729\pi\)
0.406559 + 0.913625i \(0.366729\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.23150 −0.882509
\(88\) 0 0
\(89\) 14.8157 1.57046 0.785231 0.619203i \(-0.212544\pi\)
0.785231 + 0.619203i \(0.212544\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −7.11575 −0.737869
\(94\) 0 0
\(95\) −7.90116 −0.810642
\(96\) 0 0
\(97\) 3.76850 0.382633 0.191317 0.981528i \(-0.438724\pi\)
0.191317 + 0.981528i \(0.438724\pi\)
\(98\) 0 0
\(99\) 11.3776 1.14349
\(100\) 0 0
\(101\) 0.292106 0.0290656 0.0145328 0.999894i \(-0.495374\pi\)
0.0145328 + 0.999894i \(0.495374\pi\)
\(102\) 0 0
\(103\) −4.03030 −0.397118 −0.198559 0.980089i \(-0.563626\pi\)
−0.198559 + 0.980089i \(0.563626\pi\)
\(104\) 0 0
\(105\) −2.52360 −0.246279
\(106\) 0 0
\(107\) 5.50670 0.532353 0.266176 0.963924i \(-0.414240\pi\)
0.266176 + 0.963924i \(0.414240\pi\)
\(108\) 0 0
\(109\) −1.27871 −0.122478 −0.0612390 0.998123i \(-0.519505\pi\)
−0.0612390 + 0.998123i \(0.519505\pi\)
\(110\) 0 0
\(111\) −16.7551 −1.59032
\(112\) 0 0
\(113\) 19.2787 1.81359 0.906794 0.421574i \(-0.138522\pi\)
0.906794 + 0.421574i \(0.138522\pi\)
\(114\) 0 0
\(115\) 6.49330 0.605503
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 17.9181 1.62891
\(122\) 0 0
\(123\) −10.7551 −0.969755
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −7.96970 −0.707196 −0.353598 0.935397i \(-0.615042\pi\)
−0.353598 + 0.935397i \(0.615042\pi\)
\(128\) 0 0
\(129\) −10.8236 −0.952969
\(130\) 0 0
\(131\) 7.27871 0.635944 0.317972 0.948100i \(-0.396998\pi\)
0.317972 + 0.948100i \(0.396998\pi\)
\(132\) 0 0
\(133\) −8.81571 −0.764419
\(134\) 0 0
\(135\) 2.00000 0.172133
\(136\) 0 0
\(137\) 4.75510 0.406256 0.203128 0.979152i \(-0.434889\pi\)
0.203128 + 0.979152i \(0.434889\pi\)
\(138\) 0 0
\(139\) −16.3259 −1.38475 −0.692373 0.721540i \(-0.743435\pi\)
−0.692373 + 0.721540i \(0.743435\pi\)
\(140\) 0 0
\(141\) 13.9394 1.17391
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 3.63935 0.302232
\(146\) 0 0
\(147\) 13.0169 1.07362
\(148\) 0 0
\(149\) 8.81571 0.722211 0.361106 0.932525i \(-0.382399\pi\)
0.361106 + 0.932525i \(0.382399\pi\)
\(150\) 0 0
\(151\) 19.9012 1.61953 0.809767 0.586752i \(-0.199594\pi\)
0.809767 + 0.586752i \(0.199594\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.14605 0.252697
\(156\) 0 0
\(157\) 18.7551 1.49682 0.748410 0.663236i \(-0.230818\pi\)
0.748410 + 0.663236i \(0.230818\pi\)
\(158\) 0 0
\(159\) −0.660685 −0.0523958
\(160\) 0 0
\(161\) 7.24490 0.570978
\(162\) 0 0
\(163\) 13.4417 1.05283 0.526416 0.850227i \(-0.323535\pi\)
0.526416 + 0.850227i \(0.323535\pi\)
\(164\) 0 0
\(165\) −12.1630 −0.946885
\(166\) 0 0
\(167\) −4.91806 −0.380571 −0.190286 0.981729i \(-0.560941\pi\)
−0.190286 + 0.981729i \(0.560941\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −16.7169 −1.27837
\(172\) 0 0
\(173\) 19.5708 1.48794 0.743971 0.668212i \(-0.232940\pi\)
0.743971 + 0.668212i \(0.232940\pi\)
\(174\) 0 0
\(175\) 1.11575 0.0843427
\(176\) 0 0
\(177\) 25.7338 1.93427
\(178\) 0 0
\(179\) 12.9866 0.970664 0.485332 0.874330i \(-0.338699\pi\)
0.485332 + 0.874330i \(0.338699\pi\)
\(180\) 0 0
\(181\) −9.40786 −0.699280 −0.349640 0.936884i \(-0.613696\pi\)
−0.349640 + 0.936884i \(0.613696\pi\)
\(182\) 0 0
\(183\) 12.0944 0.894045
\(184\) 0 0
\(185\) 7.40786 0.544636
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.23150 0.162318
\(190\) 0 0
\(191\) 26.5574 1.92163 0.960814 0.277195i \(-0.0894049\pi\)
0.960814 + 0.277195i \(0.0894049\pi\)
\(192\) 0 0
\(193\) 21.8023 1.56936 0.784682 0.619898i \(-0.212826\pi\)
0.784682 + 0.619898i \(0.212826\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.57081 0.539398 0.269699 0.962945i \(-0.413076\pi\)
0.269699 + 0.962945i \(0.413076\pi\)
\(198\) 0 0
\(199\) −17.5708 −1.24556 −0.622781 0.782396i \(-0.713997\pi\)
−0.622781 + 0.782396i \(0.713997\pi\)
\(200\) 0 0
\(201\) 26.8495 1.89382
\(202\) 0 0
\(203\) 4.06061 0.284999
\(204\) 0 0
\(205\) 4.75510 0.332111
\(206\) 0 0
\(207\) 13.7382 0.954871
\(208\) 0 0
\(209\) −42.4889 −2.93902
\(210\) 0 0
\(211\) −0.0606069 −0.00417235 −0.00208618 0.999998i \(-0.500664\pi\)
−0.00208618 + 0.999998i \(0.500664\pi\)
\(212\) 0 0
\(213\) 7.77643 0.532833
\(214\) 0 0
\(215\) 4.78541 0.326362
\(216\) 0 0
\(217\) 3.51021 0.238288
\(218\) 0 0
\(219\) −10.3865 −0.701856
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 22.6260 1.51515 0.757573 0.652750i \(-0.226385\pi\)
0.757573 + 0.652750i \(0.226385\pi\)
\(224\) 0 0
\(225\) 2.11575 0.141050
\(226\) 0 0
\(227\) −4.59214 −0.304791 −0.152396 0.988320i \(-0.548699\pi\)
−0.152396 + 0.988320i \(0.548699\pi\)
\(228\) 0 0
\(229\) −0.986602 −0.0651965 −0.0325983 0.999469i \(-0.510378\pi\)
−0.0325983 + 0.999469i \(0.510378\pi\)
\(230\) 0 0
\(231\) −13.5708 −0.892894
\(232\) 0 0
\(233\) −13.2787 −0.869917 −0.434959 0.900450i \(-0.643237\pi\)
−0.434959 + 0.900450i \(0.643237\pi\)
\(234\) 0 0
\(235\) −6.16296 −0.402027
\(236\) 0 0
\(237\) 24.4630 1.58904
\(238\) 0 0
\(239\) 25.6429 1.65870 0.829349 0.558730i \(-0.188711\pi\)
0.829349 + 0.558730i \(0.188711\pi\)
\(240\) 0 0
\(241\) 14.5236 0.935548 0.467774 0.883848i \(-0.345056\pi\)
0.467774 + 0.883848i \(0.345056\pi\)
\(242\) 0 0
\(243\) 18.5877 1.19240
\(244\) 0 0
\(245\) −5.75510 −0.367680
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −16.7551 −1.06181
\(250\) 0 0
\(251\) −20.2653 −1.27914 −0.639568 0.768735i \(-0.720887\pi\)
−0.639568 + 0.768735i \(0.720887\pi\)
\(252\) 0 0
\(253\) 34.9181 2.19528
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.58421 −0.410712 −0.205356 0.978687i \(-0.565835\pi\)
−0.205356 + 0.978687i \(0.565835\pi\)
\(258\) 0 0
\(259\) 8.26531 0.513581
\(260\) 0 0
\(261\) 7.69996 0.476616
\(262\) 0 0
\(263\) −18.7854 −1.15836 −0.579179 0.815200i \(-0.696627\pi\)
−0.579179 + 0.815200i \(0.696627\pi\)
\(264\) 0 0
\(265\) 0.292106 0.0179439
\(266\) 0 0
\(267\) −33.5102 −2.05079
\(268\) 0 0
\(269\) −14.8495 −0.905391 −0.452696 0.891665i \(-0.649538\pi\)
−0.452696 + 0.891665i \(0.649538\pi\)
\(270\) 0 0
\(271\) −20.8878 −1.26884 −0.634420 0.772988i \(-0.718761\pi\)
−0.634420 + 0.772988i \(0.718761\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.37755 0.324279
\(276\) 0 0
\(277\) −22.2653 −1.33779 −0.668896 0.743356i \(-0.733233\pi\)
−0.668896 + 0.743356i \(0.733233\pi\)
\(278\) 0 0
\(279\) 6.65626 0.398500
\(280\) 0 0
\(281\) 7.57081 0.451637 0.225818 0.974169i \(-0.427494\pi\)
0.225818 + 0.974169i \(0.427494\pi\)
\(282\) 0 0
\(283\) −0.724800 −0.0430849 −0.0215424 0.999768i \(-0.506858\pi\)
−0.0215424 + 0.999768i \(0.506858\pi\)
\(284\) 0 0
\(285\) 17.8709 1.05858
\(286\) 0 0
\(287\) 5.30550 0.313174
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −8.52360 −0.499663
\(292\) 0 0
\(293\) 10.2236 0.597267 0.298634 0.954368i \(-0.403469\pi\)
0.298634 + 0.954368i \(0.403469\pi\)
\(294\) 0 0
\(295\) −11.3776 −0.662427
\(296\) 0 0
\(297\) 10.7551 0.624074
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 5.33931 0.307753
\(302\) 0 0
\(303\) −0.660685 −0.0379554
\(304\) 0 0
\(305\) −5.34725 −0.306183
\(306\) 0 0
\(307\) 21.6394 1.23502 0.617512 0.786562i \(-0.288141\pi\)
0.617512 + 0.786562i \(0.288141\pi\)
\(308\) 0 0
\(309\) 9.11575 0.518577
\(310\) 0 0
\(311\) −0.986602 −0.0559451 −0.0279725 0.999609i \(-0.508905\pi\)
−0.0279725 + 0.999609i \(0.508905\pi\)
\(312\) 0 0
\(313\) 25.8361 1.46034 0.730172 0.683263i \(-0.239440\pi\)
0.730172 + 0.683263i \(0.239440\pi\)
\(314\) 0 0
\(315\) 2.36065 0.133007
\(316\) 0 0
\(317\) −26.4283 −1.48436 −0.742180 0.670201i \(-0.766208\pi\)
−0.742180 + 0.670201i \(0.766208\pi\)
\(318\) 0 0
\(319\) 19.5708 1.09576
\(320\) 0 0
\(321\) −12.4551 −0.695174
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.89218 0.159938
\(328\) 0 0
\(329\) −6.87632 −0.379104
\(330\) 0 0
\(331\) −14.8878 −0.818305 −0.409153 0.912466i \(-0.634176\pi\)
−0.409153 + 0.912466i \(0.634176\pi\)
\(332\) 0 0
\(333\) 15.6732 0.858884
\(334\) 0 0
\(335\) −11.8709 −0.648574
\(336\) 0 0
\(337\) 20.0000 1.08947 0.544735 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 0 0
\(339\) −43.6046 −2.36828
\(340\) 0 0
\(341\) 16.9181 0.916164
\(342\) 0 0
\(343\) −14.2315 −0.768429
\(344\) 0 0
\(345\) −14.6866 −0.790698
\(346\) 0 0
\(347\) −26.0641 −1.39919 −0.699597 0.714537i \(-0.746637\pi\)
−0.699597 + 0.714537i \(0.746637\pi\)
\(348\) 0 0
\(349\) −10.7551 −0.575707 −0.287854 0.957674i \(-0.592942\pi\)
−0.287854 + 0.957674i \(0.592942\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −35.6126 −1.89547 −0.947733 0.319066i \(-0.896631\pi\)
−0.947733 + 0.319066i \(0.896631\pi\)
\(354\) 0 0
\(355\) −3.43816 −0.182479
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.07205 0.320470 0.160235 0.987079i \(-0.448775\pi\)
0.160235 + 0.987079i \(0.448775\pi\)
\(360\) 0 0
\(361\) 43.4283 2.28570
\(362\) 0 0
\(363\) −40.5271 −2.12712
\(364\) 0 0
\(365\) 4.59214 0.240364
\(366\) 0 0
\(367\) 26.7854 1.39819 0.699093 0.715030i \(-0.253587\pi\)
0.699093 + 0.715030i \(0.253587\pi\)
\(368\) 0 0
\(369\) 10.0606 0.523734
\(370\) 0 0
\(371\) 0.325917 0.0169208
\(372\) 0 0
\(373\) 10.8763 0.563154 0.281577 0.959539i \(-0.409142\pi\)
0.281577 + 0.959539i \(0.409142\pi\)
\(374\) 0 0
\(375\) −2.26180 −0.116799
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 5.96176 0.306235 0.153118 0.988208i \(-0.451069\pi\)
0.153118 + 0.988208i \(0.451069\pi\)
\(380\) 0 0
\(381\) 18.0259 0.923494
\(382\) 0 0
\(383\) −19.4079 −0.991695 −0.495848 0.868410i \(-0.665143\pi\)
−0.495848 + 0.868410i \(0.665143\pi\)
\(384\) 0 0
\(385\) 6.00000 0.305788
\(386\) 0 0
\(387\) 10.1247 0.514669
\(388\) 0 0
\(389\) 8.55742 0.433878 0.216939 0.976185i \(-0.430393\pi\)
0.216939 + 0.976185i \(0.430393\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −16.4630 −0.830448
\(394\) 0 0
\(395\) −10.8157 −0.544197
\(396\) 0 0
\(397\) 14.9449 0.750061 0.375030 0.927012i \(-0.377632\pi\)
0.375030 + 0.927012i \(0.377632\pi\)
\(398\) 0 0
\(399\) 19.9394 0.998218
\(400\) 0 0
\(401\) 27.1416 1.35539 0.677694 0.735344i \(-0.262979\pi\)
0.677694 + 0.735344i \(0.262979\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −10.8709 −0.540177
\(406\) 0 0
\(407\) 39.8361 1.97460
\(408\) 0 0
\(409\) 33.2181 1.64253 0.821265 0.570547i \(-0.193269\pi\)
0.821265 + 0.570547i \(0.193269\pi\)
\(410\) 0 0
\(411\) −10.7551 −0.530510
\(412\) 0 0
\(413\) −12.6945 −0.624655
\(414\) 0 0
\(415\) 7.40786 0.363637
\(416\) 0 0
\(417\) 36.9260 1.80827
\(418\) 0 0
\(419\) −17.7079 −0.865087 −0.432544 0.901613i \(-0.642384\pi\)
−0.432544 + 0.901613i \(0.642384\pi\)
\(420\) 0 0
\(421\) 22.2047 1.08219 0.541096 0.840961i \(-0.318010\pi\)
0.541096 + 0.840961i \(0.318010\pi\)
\(422\) 0 0
\(423\) −13.0393 −0.633991
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −5.96619 −0.288724
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.6831 0.514585 0.257292 0.966334i \(-0.417170\pi\)
0.257292 + 0.966334i \(0.417170\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) −8.23150 −0.394670
\(436\) 0 0
\(437\) −51.3046 −2.45423
\(438\) 0 0
\(439\) 5.24490 0.250325 0.125163 0.992136i \(-0.460055\pi\)
0.125163 + 0.992136i \(0.460055\pi\)
\(440\) 0 0
\(441\) −12.1764 −0.579826
\(442\) 0 0
\(443\) 25.4799 1.21059 0.605293 0.796002i \(-0.293056\pi\)
0.605293 + 0.796002i \(0.293056\pi\)
\(444\) 0 0
\(445\) 14.8157 0.702332
\(446\) 0 0
\(447\) −19.9394 −0.943101
\(448\) 0 0
\(449\) 13.9394 0.657841 0.328920 0.944358i \(-0.393315\pi\)
0.328920 + 0.944358i \(0.393315\pi\)
\(450\) 0 0
\(451\) 25.5708 1.20408
\(452\) 0 0
\(453\) −45.0125 −2.11487
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 12.6945 0.593823 0.296912 0.954905i \(-0.404043\pi\)
0.296912 + 0.954905i \(0.404043\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −23.0810 −1.07499 −0.537495 0.843267i \(-0.680629\pi\)
−0.537495 + 0.843267i \(0.680629\pi\)
\(462\) 0 0
\(463\) −6.16296 −0.286417 −0.143208 0.989693i \(-0.545742\pi\)
−0.143208 + 0.989693i \(0.545742\pi\)
\(464\) 0 0
\(465\) −7.11575 −0.329985
\(466\) 0 0
\(467\) 23.5067 1.08776 0.543880 0.839163i \(-0.316955\pi\)
0.543880 + 0.839163i \(0.316955\pi\)
\(468\) 0 0
\(469\) −13.2449 −0.611593
\(470\) 0 0
\(471\) −42.4203 −1.95463
\(472\) 0 0
\(473\) 25.7338 1.18324
\(474\) 0 0
\(475\) −7.90116 −0.362530
\(476\) 0 0
\(477\) 0.618022 0.0282973
\(478\) 0 0
\(479\) 33.2137 1.51757 0.758786 0.651340i \(-0.225793\pi\)
0.758786 + 0.651340i \(0.225793\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −16.3865 −0.745613
\(484\) 0 0
\(485\) 3.76850 0.171119
\(486\) 0 0
\(487\) −18.7472 −0.849515 −0.424758 0.905307i \(-0.639641\pi\)
−0.424758 + 0.905307i \(0.639641\pi\)
\(488\) 0 0
\(489\) −30.4024 −1.37484
\(490\) 0 0
\(491\) −26.5574 −1.19852 −0.599260 0.800555i \(-0.704538\pi\)
−0.599260 + 0.800555i \(0.704538\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 11.3776 0.511383
\(496\) 0 0
\(497\) −3.83612 −0.172074
\(498\) 0 0
\(499\) −26.8539 −1.20215 −0.601074 0.799193i \(-0.705260\pi\)
−0.601074 + 0.799193i \(0.705260\pi\)
\(500\) 0 0
\(501\) 11.1237 0.496970
\(502\) 0 0
\(503\) −20.4665 −0.912556 −0.456278 0.889837i \(-0.650818\pi\)
−0.456278 + 0.889837i \(0.650818\pi\)
\(504\) 0 0
\(505\) 0.292106 0.0129985
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 26.9598 1.19497 0.597486 0.801879i \(-0.296166\pi\)
0.597486 + 0.801879i \(0.296166\pi\)
\(510\) 0 0
\(511\) 5.12368 0.226658
\(512\) 0 0
\(513\) −15.8023 −0.697689
\(514\) 0 0
\(515\) −4.03030 −0.177596
\(516\) 0 0
\(517\) −33.1416 −1.45757
\(518\) 0 0
\(519\) −44.2653 −1.94303
\(520\) 0 0
\(521\) −8.36065 −0.366287 −0.183143 0.983086i \(-0.558627\pi\)
−0.183143 + 0.983086i \(0.558627\pi\)
\(522\) 0 0
\(523\) −18.7248 −0.818778 −0.409389 0.912360i \(-0.634258\pi\)
−0.409389 + 0.912360i \(0.634258\pi\)
\(524\) 0 0
\(525\) −2.52360 −0.110139
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 19.1630 0.833172
\(530\) 0 0
\(531\) −24.0720 −1.04464
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 5.50670 0.238075
\(536\) 0 0
\(537\) −29.3731 −1.26754
\(538\) 0 0
\(539\) −30.9484 −1.33304
\(540\) 0 0
\(541\) −42.9439 −1.84630 −0.923152 0.384435i \(-0.874396\pi\)
−0.923152 + 0.384435i \(0.874396\pi\)
\(542\) 0 0
\(543\) 21.2787 0.913157
\(544\) 0 0
\(545\) −1.27871 −0.0547738
\(546\) 0 0
\(547\) 14.0909 0.602484 0.301242 0.953548i \(-0.402599\pi\)
0.301242 + 0.953548i \(0.402599\pi\)
\(548\) 0 0
\(549\) −11.3134 −0.482846
\(550\) 0 0
\(551\) −28.7551 −1.22501
\(552\) 0 0
\(553\) −12.0676 −0.513167
\(554\) 0 0
\(555\) −16.7551 −0.711215
\(556\) 0 0
\(557\) −0.713359 −0.0302260 −0.0151130 0.999886i \(-0.504811\pi\)
−0.0151130 + 0.999886i \(0.504811\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.2012 0.514219 0.257110 0.966382i \(-0.417230\pi\)
0.257110 + 0.966382i \(0.417230\pi\)
\(564\) 0 0
\(565\) 19.2787 0.811061
\(566\) 0 0
\(567\) −12.1291 −0.509376
\(568\) 0 0
\(569\) −16.6260 −0.696996 −0.348498 0.937309i \(-0.613308\pi\)
−0.348498 + 0.937309i \(0.613308\pi\)
\(570\) 0 0
\(571\) 20.3259 0.850613 0.425307 0.905049i \(-0.360166\pi\)
0.425307 + 0.905049i \(0.360166\pi\)
\(572\) 0 0
\(573\) −60.0676 −2.50936
\(574\) 0 0
\(575\) 6.49330 0.270789
\(576\) 0 0
\(577\) 19.1496 0.797207 0.398603 0.917123i \(-0.369495\pi\)
0.398603 + 0.917123i \(0.369495\pi\)
\(578\) 0 0
\(579\) −49.3125 −2.04936
\(580\) 0 0
\(581\) 8.26531 0.342903
\(582\) 0 0
\(583\) 1.57081 0.0650564
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 30.4889 1.25841 0.629205 0.777239i \(-0.283380\pi\)
0.629205 + 0.777239i \(0.283380\pi\)
\(588\) 0 0
\(589\) −24.8575 −1.02423
\(590\) 0 0
\(591\) −17.1237 −0.704374
\(592\) 0 0
\(593\) −42.3259 −1.73812 −0.869059 0.494709i \(-0.835275\pi\)
−0.869059 + 0.494709i \(0.835275\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 39.7417 1.62652
\(598\) 0 0
\(599\) 15.5440 0.635111 0.317556 0.948240i \(-0.397138\pi\)
0.317556 + 0.948240i \(0.397138\pi\)
\(600\) 0 0
\(601\) 25.1416 1.02555 0.512774 0.858524i \(-0.328618\pi\)
0.512774 + 0.858524i \(0.328618\pi\)
\(602\) 0 0
\(603\) −25.1157 −1.02279
\(604\) 0 0
\(605\) 17.9181 0.728473
\(606\) 0 0
\(607\) −10.0909 −0.409577 −0.204789 0.978806i \(-0.565651\pi\)
−0.204789 + 0.978806i \(0.565651\pi\)
\(608\) 0 0
\(609\) −9.18429 −0.372166
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 37.0204 1.49524 0.747620 0.664127i \(-0.231196\pi\)
0.747620 + 0.664127i \(0.231196\pi\)
\(614\) 0 0
\(615\) −10.7551 −0.433688
\(616\) 0 0
\(617\) 8.63389 0.347587 0.173794 0.984782i \(-0.444397\pi\)
0.173794 + 0.984782i \(0.444397\pi\)
\(618\) 0 0
\(619\) −9.47197 −0.380711 −0.190355 0.981715i \(-0.560964\pi\)
−0.190355 + 0.981715i \(0.560964\pi\)
\(620\) 0 0
\(621\) 12.9866 0.521134
\(622\) 0 0
\(623\) 16.5306 0.662285
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 96.1014 3.83792
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 18.0382 0.718091 0.359045 0.933320i \(-0.383102\pi\)
0.359045 + 0.933320i \(0.383102\pi\)
\(632\) 0 0
\(633\) 0.137081 0.00544847
\(634\) 0 0
\(635\) −7.96970 −0.316268
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −7.27428 −0.287766
\(640\) 0 0
\(641\) −17.4158 −0.687882 −0.343941 0.938991i \(-0.611762\pi\)
−0.343941 + 0.938991i \(0.611762\pi\)
\(642\) 0 0
\(643\) 15.3472 0.605236 0.302618 0.953112i \(-0.402139\pi\)
0.302618 + 0.953112i \(0.402139\pi\)
\(644\) 0 0
\(645\) −10.8236 −0.426181
\(646\) 0 0
\(647\) 3.05072 0.119936 0.0599680 0.998200i \(-0.480900\pi\)
0.0599680 + 0.998200i \(0.480900\pi\)
\(648\) 0 0
\(649\) −61.1834 −2.40166
\(650\) 0 0
\(651\) −7.93939 −0.311169
\(652\) 0 0
\(653\) −25.5708 −1.00066 −0.500332 0.865834i \(-0.666789\pi\)
−0.500332 + 0.865834i \(0.666789\pi\)
\(654\) 0 0
\(655\) 7.27871 0.284403
\(656\) 0 0
\(657\) 9.71583 0.379051
\(658\) 0 0
\(659\) 6.29211 0.245106 0.122553 0.992462i \(-0.460892\pi\)
0.122553 + 0.992462i \(0.460892\pi\)
\(660\) 0 0
\(661\) −39.5102 −1.53677 −0.768384 0.639989i \(-0.778939\pi\)
−0.768384 + 0.639989i \(0.778939\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.81571 −0.341859
\(666\) 0 0
\(667\) 23.6314 0.915012
\(668\) 0 0
\(669\) −51.1754 −1.97856
\(670\) 0 0
\(671\) −28.7551 −1.11008
\(672\) 0 0
\(673\) −1.30550 −0.0503235 −0.0251617 0.999683i \(-0.508010\pi\)
−0.0251617 + 0.999683i \(0.508010\pi\)
\(674\) 0 0
\(675\) 2.00000 0.0769800
\(676\) 0 0
\(677\) 5.01340 0.192681 0.0963403 0.995348i \(-0.469286\pi\)
0.0963403 + 0.995348i \(0.469286\pi\)
\(678\) 0 0
\(679\) 4.20470 0.161362
\(680\) 0 0
\(681\) 10.3865 0.398012
\(682\) 0 0
\(683\) −19.4079 −0.742621 −0.371310 0.928509i \(-0.621091\pi\)
−0.371310 + 0.928509i \(0.621091\pi\)
\(684\) 0 0
\(685\) 4.75510 0.181683
\(686\) 0 0
\(687\) 2.23150 0.0851370
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −2.56184 −0.0974570 −0.0487285 0.998812i \(-0.515517\pi\)
−0.0487285 + 0.998812i \(0.515517\pi\)
\(692\) 0 0
\(693\) 12.6945 0.482224
\(694\) 0 0
\(695\) −16.3259 −0.619277
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 30.0338 1.13598
\(700\) 0 0
\(701\) −38.8495 −1.46733 −0.733663 0.679513i \(-0.762191\pi\)
−0.733663 + 0.679513i \(0.762191\pi\)
\(702\) 0 0
\(703\) −58.5306 −2.20752
\(704\) 0 0
\(705\) 13.9394 0.524988
\(706\) 0 0
\(707\) 0.325917 0.0122574
\(708\) 0 0
\(709\) 35.3393 1.32720 0.663598 0.748089i \(-0.269029\pi\)
0.663598 + 0.748089i \(0.269029\pi\)
\(710\) 0 0
\(711\) −22.8833 −0.858192
\(712\) 0 0
\(713\) 20.4283 0.765045
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −57.9991 −2.16602
\(718\) 0 0
\(719\) −6.11028 −0.227875 −0.113938 0.993488i \(-0.536346\pi\)
−0.113938 + 0.993488i \(0.536346\pi\)
\(720\) 0 0
\(721\) −4.49681 −0.167470
\(722\) 0 0
\(723\) −32.8495 −1.22169
\(724\) 0 0
\(725\) 3.63935 0.135162
\(726\) 0 0
\(727\) −1.05072 −0.0389689 −0.0194845 0.999810i \(-0.506202\pi\)
−0.0194845 + 0.999810i \(0.506202\pi\)
\(728\) 0 0
\(729\) −9.42919 −0.349229
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −31.5708 −1.16609 −0.583047 0.812438i \(-0.698140\pi\)
−0.583047 + 0.812438i \(0.698140\pi\)
\(734\) 0 0
\(735\) 13.0169 0.480136
\(736\) 0 0
\(737\) −63.8361 −2.35143
\(738\) 0 0
\(739\) −31.8673 −1.17226 −0.586130 0.810217i \(-0.699349\pi\)
−0.586130 + 0.810217i \(0.699349\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 39.8550 1.46214 0.731069 0.682304i \(-0.239022\pi\)
0.731069 + 0.682304i \(0.239022\pi\)
\(744\) 0 0
\(745\) 8.81571 0.322983
\(746\) 0 0
\(747\) 15.6732 0.573451
\(748\) 0 0
\(749\) 6.14410 0.224500
\(750\) 0 0
\(751\) −9.24490 −0.337351 −0.168676 0.985672i \(-0.553949\pi\)
−0.168676 + 0.985672i \(0.553949\pi\)
\(752\) 0 0
\(753\) 45.8361 1.67036
\(754\) 0 0
\(755\) 19.9012 0.724277
\(756\) 0 0
\(757\) −22.5912 −0.821092 −0.410546 0.911840i \(-0.634662\pi\)
−0.410546 + 0.911840i \(0.634662\pi\)
\(758\) 0 0
\(759\) −78.9778 −2.86671
\(760\) 0 0
\(761\) −16.2047 −0.587420 −0.293710 0.955895i \(-0.594890\pi\)
−0.293710 + 0.955895i \(0.594890\pi\)
\(762\) 0 0
\(763\) −1.42672 −0.0516506
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −16.7889 −0.605424 −0.302712 0.953082i \(-0.597892\pi\)
−0.302712 + 0.953082i \(0.597892\pi\)
\(770\) 0 0
\(771\) 14.8922 0.536329
\(772\) 0 0
\(773\) −16.5921 −0.596778 −0.298389 0.954444i \(-0.596449\pi\)
−0.298389 + 0.954444i \(0.596449\pi\)
\(774\) 0 0
\(775\) 3.14605 0.113010
\(776\) 0 0
\(777\) −18.6945 −0.670661
\(778\) 0 0
\(779\) −37.5708 −1.34611
\(780\) 0 0
\(781\) −18.4889 −0.661584
\(782\) 0 0
\(783\) 7.27871 0.260120
\(784\) 0 0
\(785\) 18.7551 0.669398
\(786\) 0 0
\(787\) 31.1496 1.11036 0.555181 0.831730i \(-0.312649\pi\)
0.555181 + 0.831730i \(0.312649\pi\)
\(788\) 0 0
\(789\) 42.4889 1.51264
\(790\) 0 0
\(791\) 21.5102 0.764815
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −0.660685 −0.0234321
\(796\) 0 0
\(797\) 0.402391 0.0142534 0.00712670 0.999975i \(-0.497731\pi\)
0.00712670 + 0.999975i \(0.497731\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 31.3463 1.10757
\(802\) 0 0
\(803\) 24.6945 0.871450
\(804\) 0 0
\(805\) 7.24490 0.255349
\(806\) 0 0
\(807\) 33.5867 1.18231
\(808\) 0 0
\(809\) −55.3652 −1.94654 −0.973268 0.229671i \(-0.926235\pi\)
−0.973268 + 0.229671i \(0.926235\pi\)
\(810\) 0 0
\(811\) −27.1461 −0.953227 −0.476613 0.879113i \(-0.658136\pi\)
−0.476613 + 0.879113i \(0.658136\pi\)
\(812\) 0 0
\(813\) 47.2440 1.65692
\(814\) 0 0
\(815\) 13.4417 0.470841
\(816\) 0 0
\(817\) −37.8102 −1.32281
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −27.8361 −0.971487 −0.485744 0.874101i \(-0.661451\pi\)
−0.485744 + 0.874101i \(0.661451\pi\)
\(822\) 0 0
\(823\) −25.2146 −0.878925 −0.439463 0.898261i \(-0.644831\pi\)
−0.439463 + 0.898261i \(0.644831\pi\)
\(824\) 0 0
\(825\) −12.1630 −0.423460
\(826\) 0 0
\(827\) 1.03928 0.0361391 0.0180696 0.999837i \(-0.494248\pi\)
0.0180696 + 0.999837i \(0.494248\pi\)
\(828\) 0 0
\(829\) −6.97867 −0.242379 −0.121190 0.992629i \(-0.538671\pi\)
−0.121190 + 0.992629i \(0.538671\pi\)
\(830\) 0 0
\(831\) 50.3597 1.74696
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −4.91806 −0.170197
\(836\) 0 0
\(837\) 6.29211 0.217487
\(838\) 0 0
\(839\) −13.3169 −0.459752 −0.229876 0.973220i \(-0.573832\pi\)
−0.229876 + 0.973220i \(0.573832\pi\)
\(840\) 0 0
\(841\) −15.7551 −0.543279
\(842\) 0 0
\(843\) −17.1237 −0.589771
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 19.9921 0.686936
\(848\) 0 0
\(849\) 1.63935 0.0562625
\(850\) 0 0
\(851\) 48.1014 1.64890
\(852\) 0 0
\(853\) 30.5653 1.04654 0.523269 0.852168i \(-0.324712\pi\)
0.523269 + 0.852168i \(0.324712\pi\)
\(854\) 0 0
\(855\) −16.7169 −0.571705
\(856\) 0 0
\(857\) 52.4203 1.79064 0.895322 0.445419i \(-0.146945\pi\)
0.895322 + 0.445419i \(0.146945\pi\)
\(858\) 0 0
\(859\) 19.1416 0.653104 0.326552 0.945179i \(-0.394113\pi\)
0.326552 + 0.945179i \(0.394113\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 0 0
\(863\) −36.8575 −1.25464 −0.627321 0.778761i \(-0.715849\pi\)
−0.627321 + 0.778761i \(0.715849\pi\)
\(864\) 0 0
\(865\) 19.5708 0.665428
\(866\) 0 0
\(867\) 38.4506 1.30585
\(868\) 0 0
\(869\) −58.1620 −1.97301
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 7.97320 0.269852
\(874\) 0 0
\(875\) 1.11575 0.0377192
\(876\) 0 0
\(877\) 37.5370 1.26753 0.633767 0.773524i \(-0.281508\pi\)
0.633767 + 0.773524i \(0.281508\pi\)
\(878\) 0 0
\(879\) −23.1237 −0.779942
\(880\) 0 0
\(881\) 15.8212 0.533029 0.266514 0.963831i \(-0.414128\pi\)
0.266514 + 0.963831i \(0.414128\pi\)
\(882\) 0 0
\(883\) −49.0507 −1.65069 −0.825344 0.564630i \(-0.809019\pi\)
−0.825344 + 0.564630i \(0.809019\pi\)
\(884\) 0 0
\(885\) 25.7338 0.865031
\(886\) 0 0
\(887\) 12.0909 0.405973 0.202987 0.979182i \(-0.434935\pi\)
0.202987 + 0.979182i \(0.434935\pi\)
\(888\) 0 0
\(889\) −8.89218 −0.298234
\(890\) 0 0
\(891\) −58.4586 −1.95844
\(892\) 0 0
\(893\) 48.6945 1.62950
\(894\) 0 0
\(895\) 12.9866 0.434094
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 11.4496 0.381866
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −12.0765 −0.401880
\(904\) 0 0
\(905\) −9.40786 −0.312728
\(906\) 0 0
\(907\) −9.54051 −0.316787 −0.158394 0.987376i \(-0.550632\pi\)
−0.158394 + 0.987376i \(0.550632\pi\)
\(908\) 0 0
\(909\) 0.618022 0.0204985
\(910\) 0 0
\(911\) −26.7392 −0.885910 −0.442955 0.896544i \(-0.646070\pi\)
−0.442955 + 0.896544i \(0.646070\pi\)
\(912\) 0 0
\(913\) 39.8361 1.31838
\(914\) 0 0
\(915\) 12.0944 0.399829
\(916\) 0 0
\(917\) 8.12121 0.268186
\(918\) 0 0
\(919\) 51.0810 1.68501 0.842504 0.538691i \(-0.181081\pi\)
0.842504 + 0.538691i \(0.181081\pi\)
\(920\) 0 0
\(921\) −48.9439 −1.61276
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 7.40786 0.243569
\(926\) 0 0
\(927\) −8.52711 −0.280067
\(928\) 0 0
\(929\) −11.1237 −0.364956 −0.182478 0.983210i \(-0.558412\pi\)
−0.182478 + 0.983210i \(0.558412\pi\)
\(930\) 0 0
\(931\) 45.4720 1.49028
\(932\) 0 0
\(933\) 2.23150 0.0730560
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 15.3055 0.500009 0.250005 0.968245i \(-0.419568\pi\)
0.250005 + 0.968245i \(0.419568\pi\)
\(938\) 0 0
\(939\) −58.4362 −1.90699
\(940\) 0 0
\(941\) −28.7124 −0.935999 −0.467999 0.883729i \(-0.655025\pi\)
−0.467999 + 0.883729i \(0.655025\pi\)
\(942\) 0 0
\(943\) 30.8763 1.00547
\(944\) 0 0
\(945\) 2.23150 0.0725907
\(946\) 0 0
\(947\) −14.1024 −0.458265 −0.229132 0.973395i \(-0.573589\pi\)
−0.229132 + 0.973395i \(0.573589\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 59.7755 1.93835
\(952\) 0 0
\(953\) −15.8361 −0.512982 −0.256491 0.966547i \(-0.582566\pi\)
−0.256491 + 0.966547i \(0.582566\pi\)
\(954\) 0 0
\(955\) 26.5574 0.859378
\(956\) 0 0
\(957\) −44.2653 −1.43089
\(958\) 0 0
\(959\) 5.30550 0.171324
\(960\) 0 0
\(961\) −21.1024 −0.680721
\(962\) 0 0
\(963\) 11.6508 0.375442
\(964\) 0 0
\(965\) 21.8023 0.701841
\(966\) 0 0
\(967\) −0.713359 −0.0229401 −0.0114700 0.999934i \(-0.503651\pi\)
−0.0114700 + 0.999934i \(0.503651\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −11.8182 −0.379263 −0.189632 0.981855i \(-0.560729\pi\)
−0.189632 + 0.981855i \(0.560729\pi\)
\(972\) 0 0
\(973\) −18.2156 −0.583966
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −16.9181 −0.541257 −0.270628 0.962684i \(-0.587231\pi\)
−0.270628 + 0.962684i \(0.587231\pi\)
\(978\) 0 0
\(979\) 79.6722 2.54634
\(980\) 0 0
\(981\) −2.70543 −0.0863776
\(982\) 0 0
\(983\) −12.5315 −0.399694 −0.199847 0.979827i \(-0.564045\pi\)
−0.199847 + 0.979827i \(0.564045\pi\)
\(984\) 0 0
\(985\) 7.57081 0.241226
\(986\) 0 0
\(987\) 15.5529 0.495053
\(988\) 0 0
\(989\) 31.0731 0.988067
\(990\) 0 0
\(991\) −49.6543 −1.57732 −0.788660 0.614829i \(-0.789225\pi\)
−0.788660 + 0.614829i \(0.789225\pi\)
\(992\) 0 0
\(993\) 33.6732 1.06859
\(994\) 0 0
\(995\) −17.5708 −0.557032
\(996\) 0 0
\(997\) −54.5912 −1.72892 −0.864461 0.502700i \(-0.832340\pi\)
−0.864461 + 0.502700i \(0.832340\pi\)
\(998\) 0 0
\(999\) 14.8157 0.468748
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 3380.2.a.n.1.1 3
13.5 odd 4 260.2.f.a.181.1 6
13.8 odd 4 260.2.f.a.181.2 yes 6
13.12 even 2 3380.2.a.m.1.1 3
39.5 even 4 2340.2.c.d.181.5 6
39.8 even 4 2340.2.c.d.181.2 6
52.31 even 4 1040.2.k.c.961.5 6
52.47 even 4 1040.2.k.c.961.6 6
65.8 even 4 1300.2.d.c.649.2 6
65.18 even 4 1300.2.d.d.649.2 6
65.34 odd 4 1300.2.f.e.701.6 6
65.44 odd 4 1300.2.f.e.701.5 6
65.47 even 4 1300.2.d.d.649.5 6
65.57 even 4 1300.2.d.c.649.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.f.a.181.1 6 13.5 odd 4
260.2.f.a.181.2 yes 6 13.8 odd 4
1040.2.k.c.961.5 6 52.31 even 4
1040.2.k.c.961.6 6 52.47 even 4
1300.2.d.c.649.2 6 65.8 even 4
1300.2.d.c.649.5 6 65.57 even 4
1300.2.d.d.649.2 6 65.18 even 4
1300.2.d.d.649.5 6 65.47 even 4
1300.2.f.e.701.5 6 65.44 odd 4
1300.2.f.e.701.6 6 65.34 odd 4
2340.2.c.d.181.2 6 39.8 even 4
2340.2.c.d.181.5 6 39.5 even 4
3380.2.a.m.1.1 3 13.12 even 2
3380.2.a.n.1.1 3 1.1 even 1 trivial