Properties

Label 4016.2.a.h.1.5
Level $4016$
Weight $2$
Character 4016.1
Self dual yes
Analytic conductor $32.068$
Analytic rank $1$
Dimension $9$
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] = [4016,2,Mod(1,4016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4016, 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("4016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4016.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 11x^{7} + 7x^{6} + 40x^{5} - 11x^{4} - 53x^{3} - 2x^{2} + 13x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2008)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.0778757\) of defining polynomial
Character \(\chi\) \(=\) 4016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.0778757 q^{3} +1.70938 q^{5} +3.05552 q^{7} -2.99394 q^{9} +O(q^{10})\) \(q-0.0778757 q^{3} +1.70938 q^{5} +3.05552 q^{7} -2.99394 q^{9} -2.72480 q^{11} -0.571803 q^{13} -0.133119 q^{15} -2.03094 q^{17} -1.65693 q^{19} -0.237951 q^{21} +1.92751 q^{23} -2.07801 q^{25} +0.466782 q^{27} -0.834774 q^{29} -6.56383 q^{31} +0.212195 q^{33} +5.22305 q^{35} -8.02102 q^{37} +0.0445296 q^{39} -2.26496 q^{41} +7.86429 q^{43} -5.11778 q^{45} -1.38606 q^{47} +2.33618 q^{49} +0.158161 q^{51} -10.7409 q^{53} -4.65772 q^{55} +0.129034 q^{57} +3.74126 q^{59} -11.0554 q^{61} -9.14802 q^{63} -0.977430 q^{65} +4.23468 q^{67} -0.150106 q^{69} -15.1893 q^{71} -3.47586 q^{73} +0.161827 q^{75} -8.32566 q^{77} +8.59705 q^{79} +8.94546 q^{81} +7.83230 q^{83} -3.47165 q^{85} +0.0650086 q^{87} -0.239990 q^{89} -1.74715 q^{91} +0.511163 q^{93} -2.83232 q^{95} -15.5490 q^{97} +8.15786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) \) 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
\(3\) −0.0778757 −0.0449616 −0.0224808 0.999747i \(-0.507156\pi\)
−0.0224808 + 0.999747i \(0.507156\pi\)
\(4\) 0 0
\(5\) 1.70938 0.764459 0.382229 0.924067i \(-0.375156\pi\)
0.382229 + 0.924067i \(0.375156\pi\)
\(6\) 0 0
\(7\) 3.05552 1.15488 0.577438 0.816434i \(-0.304052\pi\)
0.577438 + 0.816434i \(0.304052\pi\)
\(8\) 0 0
\(9\) −2.99394 −0.997978
\(10\) 0 0
\(11\) −2.72480 −0.821557 −0.410778 0.911735i \(-0.634743\pi\)
−0.410778 + 0.911735i \(0.634743\pi\)
\(12\) 0 0
\(13\) −0.571803 −0.158590 −0.0792949 0.996851i \(-0.525267\pi\)
−0.0792949 + 0.996851i \(0.525267\pi\)
\(14\) 0 0
\(15\) −0.133119 −0.0343713
\(16\) 0 0
\(17\) −2.03094 −0.492575 −0.246287 0.969197i \(-0.579211\pi\)
−0.246287 + 0.969197i \(0.579211\pi\)
\(18\) 0 0
\(19\) −1.65693 −0.380125 −0.190062 0.981772i \(-0.560869\pi\)
−0.190062 + 0.981772i \(0.560869\pi\)
\(20\) 0 0
\(21\) −0.237951 −0.0519251
\(22\) 0 0
\(23\) 1.92751 0.401913 0.200956 0.979600i \(-0.435595\pi\)
0.200956 + 0.979600i \(0.435595\pi\)
\(24\) 0 0
\(25\) −2.07801 −0.415603
\(26\) 0 0
\(27\) 0.466782 0.0898323
\(28\) 0 0
\(29\) −0.834774 −0.155014 −0.0775068 0.996992i \(-0.524696\pi\)
−0.0775068 + 0.996992i \(0.524696\pi\)
\(30\) 0 0
\(31\) −6.56383 −1.17890 −0.589449 0.807805i \(-0.700655\pi\)
−0.589449 + 0.807805i \(0.700655\pi\)
\(32\) 0 0
\(33\) 0.212195 0.0369385
\(34\) 0 0
\(35\) 5.22305 0.882856
\(36\) 0 0
\(37\) −8.02102 −1.31865 −0.659324 0.751859i \(-0.729157\pi\)
−0.659324 + 0.751859i \(0.729157\pi\)
\(38\) 0 0
\(39\) 0.0445296 0.00713044
\(40\) 0 0
\(41\) −2.26496 −0.353727 −0.176864 0.984235i \(-0.556595\pi\)
−0.176864 + 0.984235i \(0.556595\pi\)
\(42\) 0 0
\(43\) 7.86429 1.19929 0.599646 0.800265i \(-0.295308\pi\)
0.599646 + 0.800265i \(0.295308\pi\)
\(44\) 0 0
\(45\) −5.11778 −0.762914
\(46\) 0 0
\(47\) −1.38606 −0.202177 −0.101089 0.994877i \(-0.532233\pi\)
−0.101089 + 0.994877i \(0.532233\pi\)
\(48\) 0 0
\(49\) 2.33618 0.333740
\(50\) 0 0
\(51\) 0.158161 0.0221469
\(52\) 0 0
\(53\) −10.7409 −1.47538 −0.737690 0.675140i \(-0.764083\pi\)
−0.737690 + 0.675140i \(0.764083\pi\)
\(54\) 0 0
\(55\) −4.65772 −0.628046
\(56\) 0 0
\(57\) 0.129034 0.0170910
\(58\) 0 0
\(59\) 3.74126 0.487070 0.243535 0.969892i \(-0.421693\pi\)
0.243535 + 0.969892i \(0.421693\pi\)
\(60\) 0 0
\(61\) −11.0554 −1.41550 −0.707751 0.706462i \(-0.750290\pi\)
−0.707751 + 0.706462i \(0.750290\pi\)
\(62\) 0 0
\(63\) −9.14802 −1.15254
\(64\) 0 0
\(65\) −0.977430 −0.121235
\(66\) 0 0
\(67\) 4.23468 0.517349 0.258674 0.965965i \(-0.416714\pi\)
0.258674 + 0.965965i \(0.416714\pi\)
\(68\) 0 0
\(69\) −0.150106 −0.0180706
\(70\) 0 0
\(71\) −15.1893 −1.80264 −0.901320 0.433154i \(-0.857401\pi\)
−0.901320 + 0.433154i \(0.857401\pi\)
\(72\) 0 0
\(73\) −3.47586 −0.406818 −0.203409 0.979094i \(-0.565202\pi\)
−0.203409 + 0.979094i \(0.565202\pi\)
\(74\) 0 0
\(75\) 0.161827 0.0186861
\(76\) 0 0
\(77\) −8.32566 −0.948797
\(78\) 0 0
\(79\) 8.59705 0.967244 0.483622 0.875277i \(-0.339321\pi\)
0.483622 + 0.875277i \(0.339321\pi\)
\(80\) 0 0
\(81\) 8.94546 0.993939
\(82\) 0 0
\(83\) 7.83230 0.859706 0.429853 0.902899i \(-0.358565\pi\)
0.429853 + 0.902899i \(0.358565\pi\)
\(84\) 0 0
\(85\) −3.47165 −0.376553
\(86\) 0 0
\(87\) 0.0650086 0.00696966
\(88\) 0 0
\(89\) −0.239990 −0.0254389 −0.0127195 0.999919i \(-0.504049\pi\)
−0.0127195 + 0.999919i \(0.504049\pi\)
\(90\) 0 0
\(91\) −1.74715 −0.183152
\(92\) 0 0
\(93\) 0.511163 0.0530051
\(94\) 0 0
\(95\) −2.83232 −0.290590
\(96\) 0 0
\(97\) −15.5490 −1.57876 −0.789379 0.613907i \(-0.789597\pi\)
−0.789379 + 0.613907i \(0.789597\pi\)
\(98\) 0 0
\(99\) 8.15786 0.819896
\(100\) 0 0
\(101\) 8.15007 0.810962 0.405481 0.914104i \(-0.367104\pi\)
0.405481 + 0.914104i \(0.367104\pi\)
\(102\) 0 0
\(103\) 9.00797 0.887582 0.443791 0.896130i \(-0.353633\pi\)
0.443791 + 0.896130i \(0.353633\pi\)
\(104\) 0 0
\(105\) −0.406749 −0.0396946
\(106\) 0 0
\(107\) −9.76227 −0.943754 −0.471877 0.881664i \(-0.656423\pi\)
−0.471877 + 0.881664i \(0.656423\pi\)
\(108\) 0 0
\(109\) 3.65061 0.349665 0.174832 0.984598i \(-0.444062\pi\)
0.174832 + 0.984598i \(0.444062\pi\)
\(110\) 0 0
\(111\) 0.624643 0.0592885
\(112\) 0 0
\(113\) 8.78722 0.826632 0.413316 0.910588i \(-0.364370\pi\)
0.413316 + 0.910588i \(0.364370\pi\)
\(114\) 0 0
\(115\) 3.29484 0.307246
\(116\) 0 0
\(117\) 1.71194 0.158269
\(118\) 0 0
\(119\) −6.20556 −0.568863
\(120\) 0 0
\(121\) −3.57549 −0.325045
\(122\) 0 0
\(123\) 0.176385 0.0159041
\(124\) 0 0
\(125\) −12.0990 −1.08217
\(126\) 0 0
\(127\) −17.0108 −1.50946 −0.754732 0.656034i \(-0.772233\pi\)
−0.754732 + 0.656034i \(0.772233\pi\)
\(128\) 0 0
\(129\) −0.612437 −0.0539221
\(130\) 0 0
\(131\) −3.00757 −0.262772 −0.131386 0.991331i \(-0.541943\pi\)
−0.131386 + 0.991331i \(0.541943\pi\)
\(132\) 0 0
\(133\) −5.06276 −0.438997
\(134\) 0 0
\(135\) 0.797909 0.0686731
\(136\) 0 0
\(137\) −1.58678 −0.135568 −0.0677838 0.997700i \(-0.521593\pi\)
−0.0677838 + 0.997700i \(0.521593\pi\)
\(138\) 0 0
\(139\) 5.70602 0.483979 0.241989 0.970279i \(-0.422200\pi\)
0.241989 + 0.970279i \(0.422200\pi\)
\(140\) 0 0
\(141\) 0.107940 0.00909021
\(142\) 0 0
\(143\) 1.55805 0.130290
\(144\) 0 0
\(145\) −1.42695 −0.118502
\(146\) 0 0
\(147\) −0.181932 −0.0150055
\(148\) 0 0
\(149\) 16.4824 1.35029 0.675145 0.737685i \(-0.264081\pi\)
0.675145 + 0.737685i \(0.264081\pi\)
\(150\) 0 0
\(151\) 4.92705 0.400958 0.200479 0.979698i \(-0.435750\pi\)
0.200479 + 0.979698i \(0.435750\pi\)
\(152\) 0 0
\(153\) 6.08049 0.491579
\(154\) 0 0
\(155\) −11.2201 −0.901220
\(156\) 0 0
\(157\) 12.3793 0.987976 0.493988 0.869469i \(-0.335539\pi\)
0.493988 + 0.869469i \(0.335539\pi\)
\(158\) 0 0
\(159\) 0.836457 0.0663354
\(160\) 0 0
\(161\) 5.88952 0.464160
\(162\) 0 0
\(163\) −12.4867 −0.978033 −0.489017 0.872274i \(-0.662644\pi\)
−0.489017 + 0.872274i \(0.662644\pi\)
\(164\) 0 0
\(165\) 0.362723 0.0282380
\(166\) 0 0
\(167\) 18.6685 1.44461 0.722306 0.691573i \(-0.243082\pi\)
0.722306 + 0.691573i \(0.243082\pi\)
\(168\) 0 0
\(169\) −12.6730 −0.974849
\(170\) 0 0
\(171\) 4.96073 0.379356
\(172\) 0 0
\(173\) −9.12101 −0.693458 −0.346729 0.937965i \(-0.612708\pi\)
−0.346729 + 0.937965i \(0.612708\pi\)
\(174\) 0 0
\(175\) −6.34940 −0.479970
\(176\) 0 0
\(177\) −0.291353 −0.0218994
\(178\) 0 0
\(179\) 8.71249 0.651202 0.325601 0.945507i \(-0.394433\pi\)
0.325601 + 0.945507i \(0.394433\pi\)
\(180\) 0 0
\(181\) 13.8288 1.02789 0.513945 0.857823i \(-0.328184\pi\)
0.513945 + 0.857823i \(0.328184\pi\)
\(182\) 0 0
\(183\) 0.860949 0.0636432
\(184\) 0 0
\(185\) −13.7110 −1.00805
\(186\) 0 0
\(187\) 5.53389 0.404678
\(188\) 0 0
\(189\) 1.42626 0.103745
\(190\) 0 0
\(191\) 16.0496 1.16131 0.580655 0.814150i \(-0.302796\pi\)
0.580655 + 0.814150i \(0.302796\pi\)
\(192\) 0 0
\(193\) −17.5061 −1.26011 −0.630057 0.776549i \(-0.716968\pi\)
−0.630057 + 0.776549i \(0.716968\pi\)
\(194\) 0 0
\(195\) 0.0761181 0.00545093
\(196\) 0 0
\(197\) 14.8034 1.05470 0.527351 0.849648i \(-0.323185\pi\)
0.527351 + 0.849648i \(0.323185\pi\)
\(198\) 0 0
\(199\) −2.58294 −0.183100 −0.0915500 0.995800i \(-0.529182\pi\)
−0.0915500 + 0.995800i \(0.529182\pi\)
\(200\) 0 0
\(201\) −0.329779 −0.0232608
\(202\) 0 0
\(203\) −2.55066 −0.179022
\(204\) 0 0
\(205\) −3.87168 −0.270410
\(206\) 0 0
\(207\) −5.77083 −0.401100
\(208\) 0 0
\(209\) 4.51478 0.312294
\(210\) 0 0
\(211\) 3.28104 0.225876 0.112938 0.993602i \(-0.463974\pi\)
0.112938 + 0.993602i \(0.463974\pi\)
\(212\) 0 0
\(213\) 1.18288 0.0810495
\(214\) 0 0
\(215\) 13.4431 0.916810
\(216\) 0 0
\(217\) −20.0559 −1.36148
\(218\) 0 0
\(219\) 0.270685 0.0182912
\(220\) 0 0
\(221\) 1.16130 0.0781173
\(222\) 0 0
\(223\) 2.34057 0.156736 0.0783681 0.996924i \(-0.475029\pi\)
0.0783681 + 0.996924i \(0.475029\pi\)
\(224\) 0 0
\(225\) 6.22144 0.414762
\(226\) 0 0
\(227\) 0.175423 0.0116432 0.00582162 0.999983i \(-0.498147\pi\)
0.00582162 + 0.999983i \(0.498147\pi\)
\(228\) 0 0
\(229\) −27.3240 −1.80562 −0.902809 0.430041i \(-0.858499\pi\)
−0.902809 + 0.430041i \(0.858499\pi\)
\(230\) 0 0
\(231\) 0.648367 0.0426594
\(232\) 0 0
\(233\) −4.48043 −0.293523 −0.146761 0.989172i \(-0.546885\pi\)
−0.146761 + 0.989172i \(0.546885\pi\)
\(234\) 0 0
\(235\) −2.36930 −0.154556
\(236\) 0 0
\(237\) −0.669502 −0.0434888
\(238\) 0 0
\(239\) 1.79576 0.116158 0.0580791 0.998312i \(-0.481502\pi\)
0.0580791 + 0.998312i \(0.481502\pi\)
\(240\) 0 0
\(241\) 28.3951 1.82909 0.914544 0.404485i \(-0.132549\pi\)
0.914544 + 0.404485i \(0.132549\pi\)
\(242\) 0 0
\(243\) −2.09698 −0.134521
\(244\) 0 0
\(245\) 3.99343 0.255131
\(246\) 0 0
\(247\) 0.947435 0.0602839
\(248\) 0 0
\(249\) −0.609946 −0.0386538
\(250\) 0 0
\(251\) −1.00000 −0.0631194
\(252\) 0 0
\(253\) −5.25206 −0.330194
\(254\) 0 0
\(255\) 0.270357 0.0169304
\(256\) 0 0
\(257\) −7.17811 −0.447758 −0.223879 0.974617i \(-0.571872\pi\)
−0.223879 + 0.974617i \(0.571872\pi\)
\(258\) 0 0
\(259\) −24.5084 −1.52287
\(260\) 0 0
\(261\) 2.49926 0.154700
\(262\) 0 0
\(263\) −25.4933 −1.57198 −0.785991 0.618238i \(-0.787847\pi\)
−0.785991 + 0.618238i \(0.787847\pi\)
\(264\) 0 0
\(265\) −18.3603 −1.12787
\(266\) 0 0
\(267\) 0.0186894 0.00114377
\(268\) 0 0
\(269\) −25.6360 −1.56305 −0.781526 0.623872i \(-0.785559\pi\)
−0.781526 + 0.623872i \(0.785559\pi\)
\(270\) 0 0
\(271\) −24.8859 −1.51171 −0.755857 0.654737i \(-0.772779\pi\)
−0.755857 + 0.654737i \(0.772779\pi\)
\(272\) 0 0
\(273\) 0.136061 0.00823478
\(274\) 0 0
\(275\) 5.66216 0.341441
\(276\) 0 0
\(277\) 2.85404 0.171483 0.0857414 0.996317i \(-0.472674\pi\)
0.0857414 + 0.996317i \(0.472674\pi\)
\(278\) 0 0
\(279\) 19.6517 1.17652
\(280\) 0 0
\(281\) 16.0827 0.959414 0.479707 0.877429i \(-0.340743\pi\)
0.479707 + 0.877429i \(0.340743\pi\)
\(282\) 0 0
\(283\) 0.215076 0.0127849 0.00639246 0.999980i \(-0.497965\pi\)
0.00639246 + 0.999980i \(0.497965\pi\)
\(284\) 0 0
\(285\) 0.220569 0.0130654
\(286\) 0 0
\(287\) −6.92062 −0.408511
\(288\) 0 0
\(289\) −12.8753 −0.757370
\(290\) 0 0
\(291\) 1.21089 0.0709834
\(292\) 0 0
\(293\) −8.37322 −0.489169 −0.244584 0.969628i \(-0.578652\pi\)
−0.244584 + 0.969628i \(0.578652\pi\)
\(294\) 0 0
\(295\) 6.39524 0.372345
\(296\) 0 0
\(297\) −1.27189 −0.0738023
\(298\) 0 0
\(299\) −1.10215 −0.0637392
\(300\) 0 0
\(301\) 24.0295 1.38503
\(302\) 0 0
\(303\) −0.634692 −0.0364621
\(304\) 0 0
\(305\) −18.8979 −1.08209
\(306\) 0 0
\(307\) −8.45372 −0.482479 −0.241240 0.970466i \(-0.577554\pi\)
−0.241240 + 0.970466i \(0.577554\pi\)
\(308\) 0 0
\(309\) −0.701503 −0.0399071
\(310\) 0 0
\(311\) 7.13296 0.404473 0.202237 0.979337i \(-0.435179\pi\)
0.202237 + 0.979337i \(0.435179\pi\)
\(312\) 0 0
\(313\) −24.4935 −1.38445 −0.692227 0.721680i \(-0.743370\pi\)
−0.692227 + 0.721680i \(0.743370\pi\)
\(314\) 0 0
\(315\) −15.6375 −0.881071
\(316\) 0 0
\(317\) −2.34412 −0.131659 −0.0658295 0.997831i \(-0.520969\pi\)
−0.0658295 + 0.997831i \(0.520969\pi\)
\(318\) 0 0
\(319\) 2.27459 0.127352
\(320\) 0 0
\(321\) 0.760244 0.0424327
\(322\) 0 0
\(323\) 3.36511 0.187240
\(324\) 0 0
\(325\) 1.18821 0.0659103
\(326\) 0 0
\(327\) −0.284294 −0.0157215
\(328\) 0 0
\(329\) −4.23512 −0.233490
\(330\) 0 0
\(331\) −21.1288 −1.16135 −0.580673 0.814137i \(-0.697210\pi\)
−0.580673 + 0.814137i \(0.697210\pi\)
\(332\) 0 0
\(333\) 24.0144 1.31598
\(334\) 0 0
\(335\) 7.23869 0.395492
\(336\) 0 0
\(337\) 10.2670 0.559277 0.279638 0.960105i \(-0.409785\pi\)
0.279638 + 0.960105i \(0.409785\pi\)
\(338\) 0 0
\(339\) −0.684311 −0.0371667
\(340\) 0 0
\(341\) 17.8851 0.968532
\(342\) 0 0
\(343\) −14.2504 −0.769448
\(344\) 0 0
\(345\) −0.256588 −0.0138143
\(346\) 0 0
\(347\) 18.3461 0.984869 0.492434 0.870350i \(-0.336107\pi\)
0.492434 + 0.870350i \(0.336107\pi\)
\(348\) 0 0
\(349\) 25.6875 1.37502 0.687511 0.726174i \(-0.258703\pi\)
0.687511 + 0.726174i \(0.258703\pi\)
\(350\) 0 0
\(351\) −0.266908 −0.0142465
\(352\) 0 0
\(353\) 9.79528 0.521350 0.260675 0.965427i \(-0.416055\pi\)
0.260675 + 0.965427i \(0.416055\pi\)
\(354\) 0 0
\(355\) −25.9643 −1.37804
\(356\) 0 0
\(357\) 0.483263 0.0255770
\(358\) 0 0
\(359\) −4.10793 −0.216808 −0.108404 0.994107i \(-0.534574\pi\)
−0.108404 + 0.994107i \(0.534574\pi\)
\(360\) 0 0
\(361\) −16.2546 −0.855505
\(362\) 0 0
\(363\) 0.278444 0.0146145
\(364\) 0 0
\(365\) −5.94157 −0.310996
\(366\) 0 0
\(367\) −25.2139 −1.31616 −0.658078 0.752950i \(-0.728630\pi\)
−0.658078 + 0.752950i \(0.728630\pi\)
\(368\) 0 0
\(369\) 6.78114 0.353012
\(370\) 0 0
\(371\) −32.8191 −1.70388
\(372\) 0 0
\(373\) −7.13770 −0.369576 −0.184788 0.982778i \(-0.559160\pi\)
−0.184788 + 0.982778i \(0.559160\pi\)
\(374\) 0 0
\(375\) 0.942221 0.0486561
\(376\) 0 0
\(377\) 0.477326 0.0245836
\(378\) 0 0
\(379\) −36.6756 −1.88390 −0.941949 0.335755i \(-0.891008\pi\)
−0.941949 + 0.335755i \(0.891008\pi\)
\(380\) 0 0
\(381\) 1.32473 0.0678678
\(382\) 0 0
\(383\) −32.5767 −1.66459 −0.832295 0.554332i \(-0.812974\pi\)
−0.832295 + 0.554332i \(0.812974\pi\)
\(384\) 0 0
\(385\) −14.2317 −0.725316
\(386\) 0 0
\(387\) −23.5452 −1.19687
\(388\) 0 0
\(389\) −10.8625 −0.550752 −0.275376 0.961337i \(-0.588802\pi\)
−0.275376 + 0.961337i \(0.588802\pi\)
\(390\) 0 0
\(391\) −3.91464 −0.197972
\(392\) 0 0
\(393\) 0.234217 0.0118147
\(394\) 0 0
\(395\) 14.6957 0.739419
\(396\) 0 0
\(397\) 25.0798 1.25872 0.629359 0.777115i \(-0.283317\pi\)
0.629359 + 0.777115i \(0.283317\pi\)
\(398\) 0 0
\(399\) 0.394266 0.0197380
\(400\) 0 0
\(401\) 7.05363 0.352241 0.176121 0.984369i \(-0.443645\pi\)
0.176121 + 0.984369i \(0.443645\pi\)
\(402\) 0 0
\(403\) 3.75322 0.186961
\(404\) 0 0
\(405\) 15.2912 0.759826
\(406\) 0 0
\(407\) 21.8556 1.08334
\(408\) 0 0
\(409\) 22.4077 1.10799 0.553996 0.832519i \(-0.313102\pi\)
0.553996 + 0.832519i \(0.313102\pi\)
\(410\) 0 0
\(411\) 0.123572 0.00609534
\(412\) 0 0
\(413\) 11.4315 0.562506
\(414\) 0 0
\(415\) 13.3884 0.657210
\(416\) 0 0
\(417\) −0.444361 −0.0217604
\(418\) 0 0
\(419\) 21.9728 1.07344 0.536721 0.843760i \(-0.319663\pi\)
0.536721 + 0.843760i \(0.319663\pi\)
\(420\) 0 0
\(421\) 19.3286 0.942019 0.471009 0.882128i \(-0.343890\pi\)
0.471009 + 0.882128i \(0.343890\pi\)
\(422\) 0 0
\(423\) 4.14977 0.201768
\(424\) 0 0
\(425\) 4.22031 0.204715
\(426\) 0 0
\(427\) −33.7800 −1.63473
\(428\) 0 0
\(429\) −0.121334 −0.00585806
\(430\) 0 0
\(431\) −16.3393 −0.787037 −0.393518 0.919317i \(-0.628742\pi\)
−0.393518 + 0.919317i \(0.628742\pi\)
\(432\) 0 0
\(433\) −6.05445 −0.290958 −0.145479 0.989361i \(-0.546472\pi\)
−0.145479 + 0.989361i \(0.546472\pi\)
\(434\) 0 0
\(435\) 0.111125 0.00532802
\(436\) 0 0
\(437\) −3.19373 −0.152777
\(438\) 0 0
\(439\) 0.447066 0.0213373 0.0106686 0.999943i \(-0.496604\pi\)
0.0106686 + 0.999943i \(0.496604\pi\)
\(440\) 0 0
\(441\) −6.99438 −0.333066
\(442\) 0 0
\(443\) −2.85194 −0.135500 −0.0677498 0.997702i \(-0.521582\pi\)
−0.0677498 + 0.997702i \(0.521582\pi\)
\(444\) 0 0
\(445\) −0.410235 −0.0194470
\(446\) 0 0
\(447\) −1.28358 −0.0607112
\(448\) 0 0
\(449\) 5.54935 0.261890 0.130945 0.991390i \(-0.458199\pi\)
0.130945 + 0.991390i \(0.458199\pi\)
\(450\) 0 0
\(451\) 6.17155 0.290607
\(452\) 0 0
\(453\) −0.383698 −0.0180277
\(454\) 0 0
\(455\) −2.98655 −0.140012
\(456\) 0 0
\(457\) 3.68251 0.172261 0.0861303 0.996284i \(-0.472550\pi\)
0.0861303 + 0.996284i \(0.472550\pi\)
\(458\) 0 0
\(459\) −0.948005 −0.0442491
\(460\) 0 0
\(461\) 3.87267 0.180368 0.0901842 0.995925i \(-0.471254\pi\)
0.0901842 + 0.995925i \(0.471254\pi\)
\(462\) 0 0
\(463\) −29.6014 −1.37569 −0.687847 0.725855i \(-0.741444\pi\)
−0.687847 + 0.725855i \(0.741444\pi\)
\(464\) 0 0
\(465\) 0.873773 0.0405203
\(466\) 0 0
\(467\) −7.13850 −0.330331 −0.165165 0.986266i \(-0.552816\pi\)
−0.165165 + 0.986266i \(0.552816\pi\)
\(468\) 0 0
\(469\) 12.9391 0.597474
\(470\) 0 0
\(471\) −0.964048 −0.0444210
\(472\) 0 0
\(473\) −21.4286 −0.985287
\(474\) 0 0
\(475\) 3.44311 0.157981
\(476\) 0 0
\(477\) 32.1576 1.47240
\(478\) 0 0
\(479\) −17.5680 −0.802702 −0.401351 0.915924i \(-0.631459\pi\)
−0.401351 + 0.915924i \(0.631459\pi\)
\(480\) 0 0
\(481\) 4.58644 0.209124
\(482\) 0 0
\(483\) −0.458651 −0.0208693
\(484\) 0 0
\(485\) −26.5791 −1.20689
\(486\) 0 0
\(487\) 14.7763 0.669578 0.334789 0.942293i \(-0.391335\pi\)
0.334789 + 0.942293i \(0.391335\pi\)
\(488\) 0 0
\(489\) 0.972410 0.0439739
\(490\) 0 0
\(491\) 36.2125 1.63425 0.817124 0.576462i \(-0.195567\pi\)
0.817124 + 0.576462i \(0.195567\pi\)
\(492\) 0 0
\(493\) 1.69537 0.0763557
\(494\) 0 0
\(495\) 13.9449 0.626777
\(496\) 0 0
\(497\) −46.4112 −2.08183
\(498\) 0 0
\(499\) 20.9435 0.937561 0.468781 0.883315i \(-0.344693\pi\)
0.468781 + 0.883315i \(0.344693\pi\)
\(500\) 0 0
\(501\) −1.45382 −0.0649521
\(502\) 0 0
\(503\) 12.0839 0.538793 0.269396 0.963029i \(-0.413176\pi\)
0.269396 + 0.963029i \(0.413176\pi\)
\(504\) 0 0
\(505\) 13.9316 0.619947
\(506\) 0 0
\(507\) 0.986922 0.0438308
\(508\) 0 0
\(509\) −15.0509 −0.667119 −0.333560 0.942729i \(-0.608250\pi\)
−0.333560 + 0.942729i \(0.608250\pi\)
\(510\) 0 0
\(511\) −10.6205 −0.469825
\(512\) 0 0
\(513\) −0.773423 −0.0341475
\(514\) 0 0
\(515\) 15.3981 0.678520
\(516\) 0 0
\(517\) 3.77672 0.166100
\(518\) 0 0
\(519\) 0.710306 0.0311790
\(520\) 0 0
\(521\) −9.29342 −0.407152 −0.203576 0.979059i \(-0.565256\pi\)
−0.203576 + 0.979059i \(0.565256\pi\)
\(522\) 0 0
\(523\) −11.5177 −0.503633 −0.251817 0.967775i \(-0.581028\pi\)
−0.251817 + 0.967775i \(0.581028\pi\)
\(524\) 0 0
\(525\) 0.494464 0.0215802
\(526\) 0 0
\(527\) 13.3307 0.580695
\(528\) 0 0
\(529\) −19.2847 −0.838466
\(530\) 0 0
\(531\) −11.2011 −0.486086
\(532\) 0 0
\(533\) 1.29511 0.0560975
\(534\) 0 0
\(535\) −16.6874 −0.721461
\(536\) 0 0
\(537\) −0.678492 −0.0292791
\(538\) 0 0
\(539\) −6.36562 −0.274187
\(540\) 0 0
\(541\) 6.17430 0.265454 0.132727 0.991153i \(-0.457627\pi\)
0.132727 + 0.991153i \(0.457627\pi\)
\(542\) 0 0
\(543\) −1.07693 −0.0462156
\(544\) 0 0
\(545\) 6.24028 0.267304
\(546\) 0 0
\(547\) 18.0123 0.770152 0.385076 0.922885i \(-0.374175\pi\)
0.385076 + 0.922885i \(0.374175\pi\)
\(548\) 0 0
\(549\) 33.0992 1.41264
\(550\) 0 0
\(551\) 1.38316 0.0589245
\(552\) 0 0
\(553\) 26.2684 1.11705
\(554\) 0 0
\(555\) 1.06775 0.0453236
\(556\) 0 0
\(557\) 22.8043 0.966248 0.483124 0.875552i \(-0.339502\pi\)
0.483124 + 0.875552i \(0.339502\pi\)
\(558\) 0 0
\(559\) −4.49682 −0.190195
\(560\) 0 0
\(561\) −0.430956 −0.0181950
\(562\) 0 0
\(563\) 10.0959 0.425493 0.212746 0.977107i \(-0.431759\pi\)
0.212746 + 0.977107i \(0.431759\pi\)
\(564\) 0 0
\(565\) 15.0207 0.631926
\(566\) 0 0
\(567\) 27.3330 1.14788
\(568\) 0 0
\(569\) −12.4147 −0.520451 −0.260225 0.965548i \(-0.583797\pi\)
−0.260225 + 0.965548i \(0.583797\pi\)
\(570\) 0 0
\(571\) 40.6169 1.69976 0.849882 0.526973i \(-0.176673\pi\)
0.849882 + 0.526973i \(0.176673\pi\)
\(572\) 0 0
\(573\) −1.24988 −0.0522144
\(574\) 0 0
\(575\) −4.00538 −0.167036
\(576\) 0 0
\(577\) 27.2850 1.13589 0.567945 0.823067i \(-0.307739\pi\)
0.567945 + 0.823067i \(0.307739\pi\)
\(578\) 0 0
\(579\) 1.36330 0.0566567
\(580\) 0 0
\(581\) 23.9317 0.992855
\(582\) 0 0
\(583\) 29.2668 1.21211
\(584\) 0 0
\(585\) 2.92636 0.120990
\(586\) 0 0
\(587\) 2.37884 0.0981852 0.0490926 0.998794i \(-0.484367\pi\)
0.0490926 + 0.998794i \(0.484367\pi\)
\(588\) 0 0
\(589\) 10.8758 0.448128
\(590\) 0 0
\(591\) −1.15283 −0.0474210
\(592\) 0 0
\(593\) −26.9090 −1.10502 −0.552510 0.833506i \(-0.686330\pi\)
−0.552510 + 0.833506i \(0.686330\pi\)
\(594\) 0 0
\(595\) −10.6077 −0.434872
\(596\) 0 0
\(597\) 0.201149 0.00823247
\(598\) 0 0
\(599\) 31.8560 1.30160 0.650801 0.759248i \(-0.274433\pi\)
0.650801 + 0.759248i \(0.274433\pi\)
\(600\) 0 0
\(601\) −7.23787 −0.295239 −0.147620 0.989044i \(-0.547161\pi\)
−0.147620 + 0.989044i \(0.547161\pi\)
\(602\) 0 0
\(603\) −12.6784 −0.516303
\(604\) 0 0
\(605\) −6.11188 −0.248483
\(606\) 0 0
\(607\) 9.34152 0.379161 0.189580 0.981865i \(-0.439287\pi\)
0.189580 + 0.981865i \(0.439287\pi\)
\(608\) 0 0
\(609\) 0.198635 0.00804909
\(610\) 0 0
\(611\) 0.792552 0.0320632
\(612\) 0 0
\(613\) 31.9349 1.28984 0.644918 0.764252i \(-0.276891\pi\)
0.644918 + 0.764252i \(0.276891\pi\)
\(614\) 0 0
\(615\) 0.301510 0.0121581
\(616\) 0 0
\(617\) 8.11548 0.326717 0.163358 0.986567i \(-0.447767\pi\)
0.163358 + 0.986567i \(0.447767\pi\)
\(618\) 0 0
\(619\) −11.0527 −0.444246 −0.222123 0.975019i \(-0.571299\pi\)
−0.222123 + 0.975019i \(0.571299\pi\)
\(620\) 0 0
\(621\) 0.899725 0.0361047
\(622\) 0 0
\(623\) −0.733295 −0.0293788
\(624\) 0 0
\(625\) −10.2918 −0.411672
\(626\) 0 0
\(627\) −0.351592 −0.0140412
\(628\) 0 0
\(629\) 16.2902 0.649532
\(630\) 0 0
\(631\) 13.2530 0.527594 0.263797 0.964578i \(-0.415025\pi\)
0.263797 + 0.964578i \(0.415025\pi\)
\(632\) 0 0
\(633\) −0.255513 −0.0101557
\(634\) 0 0
\(635\) −29.0779 −1.15392
\(636\) 0 0
\(637\) −1.33584 −0.0529278
\(638\) 0 0
\(639\) 45.4758 1.79900
\(640\) 0 0
\(641\) 0.287590 0.0113591 0.00567956 0.999984i \(-0.498192\pi\)
0.00567956 + 0.999984i \(0.498192\pi\)
\(642\) 0 0
\(643\) −9.46321 −0.373193 −0.186596 0.982437i \(-0.559746\pi\)
−0.186596 + 0.982437i \(0.559746\pi\)
\(644\) 0 0
\(645\) −1.04689 −0.0412212
\(646\) 0 0
\(647\) 19.4672 0.765335 0.382668 0.923886i \(-0.375005\pi\)
0.382668 + 0.923886i \(0.375005\pi\)
\(648\) 0 0
\(649\) −10.1942 −0.400156
\(650\) 0 0
\(651\) 1.56187 0.0612144
\(652\) 0 0
\(653\) 9.50880 0.372108 0.186054 0.982540i \(-0.440430\pi\)
0.186054 + 0.982540i \(0.440430\pi\)
\(654\) 0 0
\(655\) −5.14108 −0.200879
\(656\) 0 0
\(657\) 10.4065 0.405996
\(658\) 0 0
\(659\) −17.9280 −0.698375 −0.349187 0.937053i \(-0.613542\pi\)
−0.349187 + 0.937053i \(0.613542\pi\)
\(660\) 0 0
\(661\) 15.5342 0.604210 0.302105 0.953275i \(-0.402311\pi\)
0.302105 + 0.953275i \(0.402311\pi\)
\(662\) 0 0
\(663\) −0.0904368 −0.00351228
\(664\) 0 0
\(665\) −8.65420 −0.335595
\(666\) 0 0
\(667\) −1.60903 −0.0623019
\(668\) 0 0
\(669\) −0.182274 −0.00704711
\(670\) 0 0
\(671\) 30.1237 1.16291
\(672\) 0 0
\(673\) −17.7362 −0.683682 −0.341841 0.939758i \(-0.611050\pi\)
−0.341841 + 0.939758i \(0.611050\pi\)
\(674\) 0 0
\(675\) −0.969979 −0.0373345
\(676\) 0 0
\(677\) −14.1083 −0.542224 −0.271112 0.962548i \(-0.587391\pi\)
−0.271112 + 0.962548i \(0.587391\pi\)
\(678\) 0 0
\(679\) −47.5101 −1.82327
\(680\) 0 0
\(681\) −0.0136612 −0.000523498 0
\(682\) 0 0
\(683\) 29.6593 1.13488 0.567441 0.823414i \(-0.307934\pi\)
0.567441 + 0.823414i \(0.307934\pi\)
\(684\) 0 0
\(685\) −2.71241 −0.103636
\(686\) 0 0
\(687\) 2.12787 0.0811835
\(688\) 0 0
\(689\) 6.14170 0.233980
\(690\) 0 0
\(691\) 41.5743 1.58156 0.790780 0.612100i \(-0.209675\pi\)
0.790780 + 0.612100i \(0.209675\pi\)
\(692\) 0 0
\(693\) 24.9265 0.946879
\(694\) 0 0
\(695\) 9.75378 0.369982
\(696\) 0 0
\(697\) 4.59999 0.174237
\(698\) 0 0
\(699\) 0.348917 0.0131972
\(700\) 0 0
\(701\) 23.2450 0.877951 0.438975 0.898499i \(-0.355342\pi\)
0.438975 + 0.898499i \(0.355342\pi\)
\(702\) 0 0
\(703\) 13.2902 0.501250
\(704\) 0 0
\(705\) 0.184511 0.00694909
\(706\) 0 0
\(707\) 24.9027 0.936561
\(708\) 0 0
\(709\) 19.5771 0.735234 0.367617 0.929977i \(-0.380174\pi\)
0.367617 + 0.929977i \(0.380174\pi\)
\(710\) 0 0
\(711\) −25.7390 −0.965289
\(712\) 0 0
\(713\) −12.6518 −0.473814
\(714\) 0 0
\(715\) 2.66330 0.0996017
\(716\) 0 0
\(717\) −0.139846 −0.00522266
\(718\) 0 0
\(719\) −32.1957 −1.20070 −0.600349 0.799738i \(-0.704972\pi\)
−0.600349 + 0.799738i \(0.704972\pi\)
\(720\) 0 0
\(721\) 27.5240 1.02505
\(722\) 0 0
\(723\) −2.21129 −0.0822387
\(724\) 0 0
\(725\) 1.73467 0.0644240
\(726\) 0 0
\(727\) 26.6658 0.988979 0.494490 0.869184i \(-0.335355\pi\)
0.494490 + 0.869184i \(0.335355\pi\)
\(728\) 0 0
\(729\) −26.6731 −0.987891
\(730\) 0 0
\(731\) −15.9719 −0.590741
\(732\) 0 0
\(733\) −2.38186 −0.0879760 −0.0439880 0.999032i \(-0.514006\pi\)
−0.0439880 + 0.999032i \(0.514006\pi\)
\(734\) 0 0
\(735\) −0.310991 −0.0114711
\(736\) 0 0
\(737\) −11.5386 −0.425031
\(738\) 0 0
\(739\) 7.55884 0.278056 0.139028 0.990288i \(-0.455602\pi\)
0.139028 + 0.990288i \(0.455602\pi\)
\(740\) 0 0
\(741\) −0.0737822 −0.00271046
\(742\) 0 0
\(743\) 46.3474 1.70032 0.850160 0.526524i \(-0.176505\pi\)
0.850160 + 0.526524i \(0.176505\pi\)
\(744\) 0 0
\(745\) 28.1747 1.03224
\(746\) 0 0
\(747\) −23.4494 −0.857968
\(748\) 0 0
\(749\) −29.8288 −1.08992
\(750\) 0 0
\(751\) −22.0561 −0.804838 −0.402419 0.915456i \(-0.631830\pi\)
−0.402419 + 0.915456i \(0.631830\pi\)
\(752\) 0 0
\(753\) 0.0778757 0.00283795
\(754\) 0 0
\(755\) 8.42222 0.306516
\(756\) 0 0
\(757\) −9.93932 −0.361251 −0.180625 0.983552i \(-0.557812\pi\)
−0.180625 + 0.983552i \(0.557812\pi\)
\(758\) 0 0
\(759\) 0.409008 0.0148460
\(760\) 0 0
\(761\) −4.99238 −0.180973 −0.0904867 0.995898i \(-0.528842\pi\)
−0.0904867 + 0.995898i \(0.528842\pi\)
\(762\) 0 0
\(763\) 11.1545 0.403820
\(764\) 0 0
\(765\) 10.3939 0.375792
\(766\) 0 0
\(767\) −2.13926 −0.0772443
\(768\) 0 0
\(769\) −0.336906 −0.0121491 −0.00607456 0.999982i \(-0.501934\pi\)
−0.00607456 + 0.999982i \(0.501934\pi\)
\(770\) 0 0
\(771\) 0.559000 0.0201319
\(772\) 0 0
\(773\) 17.4451 0.627458 0.313729 0.949513i \(-0.398422\pi\)
0.313729 + 0.949513i \(0.398422\pi\)
\(774\) 0 0
\(775\) 13.6397 0.489953
\(776\) 0 0
\(777\) 1.90861 0.0684709
\(778\) 0 0
\(779\) 3.75287 0.134460
\(780\) 0 0
\(781\) 41.3878 1.48097
\(782\) 0 0
\(783\) −0.389657 −0.0139252
\(784\) 0 0
\(785\) 21.1610 0.755267
\(786\) 0 0
\(787\) −25.8946 −0.923044 −0.461522 0.887129i \(-0.652697\pi\)
−0.461522 + 0.887129i \(0.652697\pi\)
\(788\) 0 0
\(789\) 1.98531 0.0706788
\(790\) 0 0
\(791\) 26.8495 0.954658
\(792\) 0 0
\(793\) 6.32152 0.224484
\(794\) 0 0
\(795\) 1.42983 0.0507107
\(796\) 0 0
\(797\) −31.7022 −1.12295 −0.561475 0.827493i \(-0.689766\pi\)
−0.561475 + 0.827493i \(0.689766\pi\)
\(798\) 0 0
\(799\) 2.81499 0.0995873
\(800\) 0 0
\(801\) 0.718516 0.0253875
\(802\) 0 0
\(803\) 9.47100 0.334224
\(804\) 0 0
\(805\) 10.0674 0.354831
\(806\) 0 0
\(807\) 1.99642 0.0702773
\(808\) 0 0
\(809\) 26.5422 0.933174 0.466587 0.884475i \(-0.345484\pi\)
0.466587 + 0.884475i \(0.345484\pi\)
\(810\) 0 0
\(811\) 16.6485 0.584609 0.292305 0.956325i \(-0.405578\pi\)
0.292305 + 0.956325i \(0.405578\pi\)
\(812\) 0 0
\(813\) 1.93801 0.0679690
\(814\) 0 0
\(815\) −21.3445 −0.747666
\(816\) 0 0
\(817\) −13.0305 −0.455881
\(818\) 0 0
\(819\) 5.23087 0.182781
\(820\) 0 0
\(821\) 44.7418 1.56150 0.780749 0.624845i \(-0.214838\pi\)
0.780749 + 0.624845i \(0.214838\pi\)
\(822\) 0 0
\(823\) −51.9554 −1.81105 −0.905525 0.424293i \(-0.860523\pi\)
−0.905525 + 0.424293i \(0.860523\pi\)
\(824\) 0 0
\(825\) −0.440945 −0.0153517
\(826\) 0 0
\(827\) −9.64570 −0.335414 −0.167707 0.985837i \(-0.553636\pi\)
−0.167707 + 0.985837i \(0.553636\pi\)
\(828\) 0 0
\(829\) −35.5294 −1.23399 −0.616993 0.786968i \(-0.711650\pi\)
−0.616993 + 0.786968i \(0.711650\pi\)
\(830\) 0 0
\(831\) −0.222261 −0.00771013
\(832\) 0 0
\(833\) −4.74464 −0.164392
\(834\) 0 0
\(835\) 31.9116 1.10435
\(836\) 0 0
\(837\) −3.06388 −0.105903
\(838\) 0 0
\(839\) −19.1025 −0.659490 −0.329745 0.944070i \(-0.606963\pi\)
−0.329745 + 0.944070i \(0.606963\pi\)
\(840\) 0 0
\(841\) −28.3032 −0.975971
\(842\) 0 0
\(843\) −1.25245 −0.0431368
\(844\) 0 0
\(845\) −21.6631 −0.745232
\(846\) 0 0
\(847\) −10.9250 −0.375386
\(848\) 0 0
\(849\) −0.0167492 −0.000574830 0
\(850\) 0 0
\(851\) −15.4606 −0.529981
\(852\) 0 0
\(853\) −25.4326 −0.870795 −0.435397 0.900238i \(-0.643392\pi\)
−0.435397 + 0.900238i \(0.643392\pi\)
\(854\) 0 0
\(855\) 8.47978 0.290002
\(856\) 0 0
\(857\) −42.2512 −1.44327 −0.721637 0.692271i \(-0.756610\pi\)
−0.721637 + 0.692271i \(0.756610\pi\)
\(858\) 0 0
\(859\) −4.64578 −0.158512 −0.0792560 0.996854i \(-0.525254\pi\)
−0.0792560 + 0.996854i \(0.525254\pi\)
\(860\) 0 0
\(861\) 0.538948 0.0183673
\(862\) 0 0
\(863\) 20.7931 0.707804 0.353902 0.935282i \(-0.384855\pi\)
0.353902 + 0.935282i \(0.384855\pi\)
\(864\) 0 0
\(865\) −15.5913 −0.530120
\(866\) 0 0
\(867\) 1.00267 0.0340526
\(868\) 0 0
\(869\) −23.4252 −0.794646
\(870\) 0 0
\(871\) −2.42140 −0.0820462
\(872\) 0 0
\(873\) 46.5526 1.57557
\(874\) 0 0
\(875\) −36.9688 −1.24977
\(876\) 0 0
\(877\) 4.97099 0.167858 0.0839291 0.996472i \(-0.473253\pi\)
0.0839291 + 0.996472i \(0.473253\pi\)
\(878\) 0 0
\(879\) 0.652071 0.0219938
\(880\) 0 0
\(881\) 30.3935 1.02398 0.511992 0.858990i \(-0.328908\pi\)
0.511992 + 0.858990i \(0.328908\pi\)
\(882\) 0 0
\(883\) 27.4333 0.923204 0.461602 0.887087i \(-0.347275\pi\)
0.461602 + 0.887087i \(0.347275\pi\)
\(884\) 0 0
\(885\) −0.498034 −0.0167412
\(886\) 0 0
\(887\) 19.8209 0.665522 0.332761 0.943011i \(-0.392020\pi\)
0.332761 + 0.943011i \(0.392020\pi\)
\(888\) 0 0
\(889\) −51.9767 −1.74324
\(890\) 0 0
\(891\) −24.3745 −0.816578
\(892\) 0 0
\(893\) 2.29659 0.0768526
\(894\) 0 0
\(895\) 14.8930 0.497817
\(896\) 0 0
\(897\) 0.0858311 0.00286582
\(898\) 0 0
\(899\) 5.47931 0.182745
\(900\) 0 0
\(901\) 21.8141 0.726734
\(902\) 0 0
\(903\) −1.87131 −0.0622733
\(904\) 0 0
\(905\) 23.6388 0.785780
\(906\) 0 0
\(907\) 14.3531 0.476585 0.238293 0.971193i \(-0.423412\pi\)
0.238293 + 0.971193i \(0.423412\pi\)
\(908\) 0 0
\(909\) −24.4008 −0.809322
\(910\) 0 0
\(911\) 18.0811 0.599054 0.299527 0.954088i \(-0.403171\pi\)
0.299527 + 0.954088i \(0.403171\pi\)
\(912\) 0 0
\(913\) −21.3414 −0.706297
\(914\) 0 0
\(915\) 1.47169 0.0486526
\(916\) 0 0
\(917\) −9.18967 −0.303470
\(918\) 0 0
\(919\) 1.07012 0.0353001 0.0176500 0.999844i \(-0.494382\pi\)
0.0176500 + 0.999844i \(0.494382\pi\)
\(920\) 0 0
\(921\) 0.658340 0.0216930
\(922\) 0 0
\(923\) 8.68530 0.285880
\(924\) 0 0
\(925\) 16.6678 0.548033
\(926\) 0 0
\(927\) −26.9693 −0.885788
\(928\) 0 0
\(929\) −32.0153 −1.05039 −0.525193 0.850983i \(-0.676007\pi\)
−0.525193 + 0.850983i \(0.676007\pi\)
\(930\) 0 0
\(931\) −3.87088 −0.126863
\(932\) 0 0
\(933\) −0.555484 −0.0181857
\(934\) 0 0
\(935\) 9.45953 0.309360
\(936\) 0 0
\(937\) −45.6597 −1.49164 −0.745818 0.666149i \(-0.767941\pi\)
−0.745818 + 0.666149i \(0.767941\pi\)
\(938\) 0 0
\(939\) 1.90745 0.0622472
\(940\) 0 0
\(941\) 30.9241 1.00810 0.504048 0.863676i \(-0.331843\pi\)
0.504048 + 0.863676i \(0.331843\pi\)
\(942\) 0 0
\(943\) −4.36572 −0.142167
\(944\) 0 0
\(945\) 2.43802 0.0793089
\(946\) 0 0
\(947\) 51.4681 1.67249 0.836244 0.548358i \(-0.184747\pi\)
0.836244 + 0.548358i \(0.184747\pi\)
\(948\) 0 0
\(949\) 1.98751 0.0645172
\(950\) 0 0
\(951\) 0.182550 0.00591960
\(952\) 0 0
\(953\) −41.2982 −1.33778 −0.668891 0.743361i \(-0.733231\pi\)
−0.668891 + 0.743361i \(0.733231\pi\)
\(954\) 0 0
\(955\) 27.4350 0.887774
\(956\) 0 0
\(957\) −0.177135 −0.00572597
\(958\) 0 0
\(959\) −4.84843 −0.156564
\(960\) 0 0
\(961\) 12.0839 0.389802
\(962\) 0 0
\(963\) 29.2276 0.941846
\(964\) 0 0
\(965\) −29.9245 −0.963305
\(966\) 0 0
\(967\) −44.8806 −1.44326 −0.721632 0.692277i \(-0.756608\pi\)
−0.721632 + 0.692277i \(0.756608\pi\)
\(968\) 0 0
\(969\) −0.262060 −0.00841860
\(970\) 0 0
\(971\) 8.77648 0.281651 0.140825 0.990034i \(-0.455024\pi\)
0.140825 + 0.990034i \(0.455024\pi\)
\(972\) 0 0
\(973\) 17.4349 0.558936
\(974\) 0 0
\(975\) −0.0925331 −0.00296343
\(976\) 0 0
\(977\) −11.2264 −0.359165 −0.179582 0.983743i \(-0.557475\pi\)
−0.179582 + 0.983743i \(0.557475\pi\)
\(978\) 0 0
\(979\) 0.653925 0.0208995
\(980\) 0 0
\(981\) −10.9297 −0.348958
\(982\) 0 0
\(983\) 32.8297 1.04710 0.523552 0.851993i \(-0.324606\pi\)
0.523552 + 0.851993i \(0.324606\pi\)
\(984\) 0 0
\(985\) 25.3047 0.806276
\(986\) 0 0
\(987\) 0.329813 0.0104981
\(988\) 0 0
\(989\) 15.1585 0.482011
\(990\) 0 0
\(991\) −28.4687 −0.904337 −0.452169 0.891933i \(-0.649349\pi\)
−0.452169 + 0.891933i \(0.649349\pi\)
\(992\) 0 0
\(993\) 1.64542 0.0522159
\(994\) 0 0
\(995\) −4.41524 −0.139972
\(996\) 0 0
\(997\) 50.2836 1.59250 0.796249 0.604969i \(-0.206814\pi\)
0.796249 + 0.604969i \(0.206814\pi\)
\(998\) 0 0
\(999\) −3.74407 −0.118457
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 4016.2.a.h.1.5 9
4.3 odd 2 2008.2.a.a.1.5 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2008.2.a.a.1.5 9 4.3 odd 2
4016.2.a.h.1.5 9 1.1 even 1 trivial