Properties

Label 4004.2.a.h.1.6
Level $4004$
Weight $2$
Character 4004.1
Self dual yes
Analytic conductor $31.972$
Analytic rank $0$
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] = [4004,2,Mod(1,4004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4004, 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("4004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.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: \(31.9721009693\)
Analytic rank: \(0\)
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: \( x^{9} - 3x^{8} - 19x^{7} + 51x^{6} + 116x^{5} - 247x^{4} - 249x^{3} + 288x^{2} + 189x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.32285\) of defining polynomial
Character \(\chi\) \(=\) 4004.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.32285 q^{3} -0.265081 q^{5} +1.00000 q^{7} -1.25008 q^{9} +O(q^{10})\) \(q+1.32285 q^{3} -0.265081 q^{5} +1.00000 q^{7} -1.25008 q^{9} +1.00000 q^{11} -1.00000 q^{13} -0.350661 q^{15} +6.26714 q^{17} -0.906059 q^{19} +1.32285 q^{21} +2.91706 q^{23} -4.92973 q^{25} -5.62220 q^{27} +9.05224 q^{29} -1.90381 q^{31} +1.32285 q^{33} -0.265081 q^{35} -4.42996 q^{37} -1.32285 q^{39} +10.9830 q^{41} +11.7925 q^{43} +0.331373 q^{45} -7.57057 q^{47} +1.00000 q^{49} +8.29046 q^{51} -7.67340 q^{53} -0.265081 q^{55} -1.19858 q^{57} +10.9826 q^{59} +10.5005 q^{61} -1.25008 q^{63} +0.265081 q^{65} -5.49875 q^{67} +3.85882 q^{69} +6.99897 q^{71} -14.5064 q^{73} -6.52127 q^{75} +1.00000 q^{77} +5.79013 q^{79} -3.68706 q^{81} -10.5193 q^{83} -1.66130 q^{85} +11.9747 q^{87} +17.0323 q^{89} -1.00000 q^{91} -2.51845 q^{93} +0.240179 q^{95} +7.68630 q^{97} -1.25008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + 5 q^{17} + 10 q^{19} + 3 q^{21} + 8 q^{23} + 3 q^{25} + 27 q^{27} + 14 q^{29} + 11 q^{31} + 3 q^{33} - 3 q^{39} + 14 q^{41} + 8 q^{43} + 4 q^{45} + 10 q^{47} + 9 q^{49} - 15 q^{51} + 21 q^{53} - 8 q^{57} + 23 q^{59} + 34 q^{61} + 20 q^{63} + 10 q^{67} - 16 q^{69} + 4 q^{71} + 9 q^{73} + 30 q^{75} + 9 q^{77} - 34 q^{79} + 69 q^{81} + 15 q^{83} + 5 q^{85} + 39 q^{87} - 9 q^{91} + 3 q^{93} - 64 q^{95} + 15 q^{97} + 20 q^{99}+O(q^{100}) \) 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
\(3\) 1.32285 0.763745 0.381873 0.924215i \(-0.375279\pi\)
0.381873 + 0.924215i \(0.375279\pi\)
\(4\) 0 0
\(5\) −0.265081 −0.118548 −0.0592739 0.998242i \(-0.518879\pi\)
−0.0592739 + 0.998242i \(0.518879\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −1.25008 −0.416693
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −0.350661 −0.0905404
\(16\) 0 0
\(17\) 6.26714 1.52001 0.760003 0.649920i \(-0.225198\pi\)
0.760003 + 0.649920i \(0.225198\pi\)
\(18\) 0 0
\(19\) −0.906059 −0.207864 −0.103932 0.994584i \(-0.533142\pi\)
−0.103932 + 0.994584i \(0.533142\pi\)
\(20\) 0 0
\(21\) 1.32285 0.288669
\(22\) 0 0
\(23\) 2.91706 0.608249 0.304125 0.952632i \(-0.401636\pi\)
0.304125 + 0.952632i \(0.401636\pi\)
\(24\) 0 0
\(25\) −4.92973 −0.985946
\(26\) 0 0
\(27\) −5.62220 −1.08199
\(28\) 0 0
\(29\) 9.05224 1.68096 0.840480 0.541843i \(-0.182273\pi\)
0.840480 + 0.541843i \(0.182273\pi\)
\(30\) 0 0
\(31\) −1.90381 −0.341935 −0.170967 0.985277i \(-0.554689\pi\)
−0.170967 + 0.985277i \(0.554689\pi\)
\(32\) 0 0
\(33\) 1.32285 0.230278
\(34\) 0 0
\(35\) −0.265081 −0.0448069
\(36\) 0 0
\(37\) −4.42996 −0.728280 −0.364140 0.931344i \(-0.618637\pi\)
−0.364140 + 0.931344i \(0.618637\pi\)
\(38\) 0 0
\(39\) −1.32285 −0.211825
\(40\) 0 0
\(41\) 10.9830 1.71526 0.857632 0.514265i \(-0.171935\pi\)
0.857632 + 0.514265i \(0.171935\pi\)
\(42\) 0 0
\(43\) 11.7925 1.79834 0.899169 0.437602i \(-0.144172\pi\)
0.899169 + 0.437602i \(0.144172\pi\)
\(44\) 0 0
\(45\) 0.331373 0.0493981
\(46\) 0 0
\(47\) −7.57057 −1.10428 −0.552140 0.833751i \(-0.686189\pi\)
−0.552140 + 0.833751i \(0.686189\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 8.29046 1.16090
\(52\) 0 0
\(53\) −7.67340 −1.05402 −0.527011 0.849858i \(-0.676687\pi\)
−0.527011 + 0.849858i \(0.676687\pi\)
\(54\) 0 0
\(55\) −0.265081 −0.0357435
\(56\) 0 0
\(57\) −1.19858 −0.158755
\(58\) 0 0
\(59\) 10.9826 1.42981 0.714904 0.699223i \(-0.246471\pi\)
0.714904 + 0.699223i \(0.246471\pi\)
\(60\) 0 0
\(61\) 10.5005 1.34446 0.672228 0.740344i \(-0.265338\pi\)
0.672228 + 0.740344i \(0.265338\pi\)
\(62\) 0 0
\(63\) −1.25008 −0.157495
\(64\) 0 0
\(65\) 0.265081 0.0328793
\(66\) 0 0
\(67\) −5.49875 −0.671780 −0.335890 0.941901i \(-0.609037\pi\)
−0.335890 + 0.941901i \(0.609037\pi\)
\(68\) 0 0
\(69\) 3.85882 0.464547
\(70\) 0 0
\(71\) 6.99897 0.830625 0.415313 0.909679i \(-0.363672\pi\)
0.415313 + 0.909679i \(0.363672\pi\)
\(72\) 0 0
\(73\) −14.5064 −1.69785 −0.848924 0.528515i \(-0.822749\pi\)
−0.848924 + 0.528515i \(0.822749\pi\)
\(74\) 0 0
\(75\) −6.52127 −0.753012
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 5.79013 0.651440 0.325720 0.945466i \(-0.394393\pi\)
0.325720 + 0.945466i \(0.394393\pi\)
\(80\) 0 0
\(81\) −3.68706 −0.409673
\(82\) 0 0
\(83\) −10.5193 −1.15464 −0.577320 0.816518i \(-0.695901\pi\)
−0.577320 + 0.816518i \(0.695901\pi\)
\(84\) 0 0
\(85\) −1.66130 −0.180193
\(86\) 0 0
\(87\) 11.9747 1.28382
\(88\) 0 0
\(89\) 17.0323 1.80542 0.902709 0.430251i \(-0.141575\pi\)
0.902709 + 0.430251i \(0.141575\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −2.51845 −0.261151
\(94\) 0 0
\(95\) 0.240179 0.0246419
\(96\) 0 0
\(97\) 7.68630 0.780426 0.390213 0.920725i \(-0.372401\pi\)
0.390213 + 0.920725i \(0.372401\pi\)
\(98\) 0 0
\(99\) −1.25008 −0.125638
\(100\) 0 0
\(101\) −5.82065 −0.579177 −0.289588 0.957151i \(-0.593518\pi\)
−0.289588 + 0.957151i \(0.593518\pi\)
\(102\) 0 0
\(103\) 17.8559 1.75939 0.879695 0.475539i \(-0.157747\pi\)
0.879695 + 0.475539i \(0.157747\pi\)
\(104\) 0 0
\(105\) −0.350661 −0.0342210
\(106\) 0 0
\(107\) 18.5806 1.79625 0.898126 0.439739i \(-0.144929\pi\)
0.898126 + 0.439739i \(0.144929\pi\)
\(108\) 0 0
\(109\) −2.75718 −0.264090 −0.132045 0.991244i \(-0.542154\pi\)
−0.132045 + 0.991244i \(0.542154\pi\)
\(110\) 0 0
\(111\) −5.86015 −0.556220
\(112\) 0 0
\(113\) −10.2154 −0.960981 −0.480490 0.877000i \(-0.659541\pi\)
−0.480490 + 0.877000i \(0.659541\pi\)
\(114\) 0 0
\(115\) −0.773258 −0.0721067
\(116\) 0 0
\(117\) 1.25008 0.115570
\(118\) 0 0
\(119\) 6.26714 0.574508
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 14.5289 1.31002
\(124\) 0 0
\(125\) 2.63218 0.235430
\(126\) 0 0
\(127\) −4.99069 −0.442852 −0.221426 0.975177i \(-0.571071\pi\)
−0.221426 + 0.975177i \(0.571071\pi\)
\(128\) 0 0
\(129\) 15.5996 1.37347
\(130\) 0 0
\(131\) 10.0962 0.882107 0.441054 0.897481i \(-0.354605\pi\)
0.441054 + 0.897481i \(0.354605\pi\)
\(132\) 0 0
\(133\) −0.906059 −0.0785653
\(134\) 0 0
\(135\) 1.49034 0.128268
\(136\) 0 0
\(137\) 6.99537 0.597655 0.298827 0.954307i \(-0.403405\pi\)
0.298827 + 0.954307i \(0.403405\pi\)
\(138\) 0 0
\(139\) −9.69846 −0.822613 −0.411306 0.911497i \(-0.634927\pi\)
−0.411306 + 0.911497i \(0.634927\pi\)
\(140\) 0 0
\(141\) −10.0147 −0.843389
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −2.39958 −0.199274
\(146\) 0 0
\(147\) 1.32285 0.109106
\(148\) 0 0
\(149\) 20.9921 1.71974 0.859870 0.510513i \(-0.170545\pi\)
0.859870 + 0.510513i \(0.170545\pi\)
\(150\) 0 0
\(151\) 14.8409 1.20773 0.603867 0.797085i \(-0.293626\pi\)
0.603867 + 0.797085i \(0.293626\pi\)
\(152\) 0 0
\(153\) −7.83443 −0.633376
\(154\) 0 0
\(155\) 0.504664 0.0405356
\(156\) 0 0
\(157\) −8.92728 −0.712475 −0.356237 0.934395i \(-0.615941\pi\)
−0.356237 + 0.934395i \(0.615941\pi\)
\(158\) 0 0
\(159\) −10.1507 −0.805004
\(160\) 0 0
\(161\) 2.91706 0.229897
\(162\) 0 0
\(163\) 5.66814 0.443963 0.221982 0.975051i \(-0.428748\pi\)
0.221982 + 0.975051i \(0.428748\pi\)
\(164\) 0 0
\(165\) −0.350661 −0.0272990
\(166\) 0 0
\(167\) −3.52644 −0.272884 −0.136442 0.990648i \(-0.543567\pi\)
−0.136442 + 0.990648i \(0.543567\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 1.13265 0.0866156
\(172\) 0 0
\(173\) −23.5914 −1.79362 −0.896812 0.442411i \(-0.854123\pi\)
−0.896812 + 0.442411i \(0.854123\pi\)
\(174\) 0 0
\(175\) −4.92973 −0.372653
\(176\) 0 0
\(177\) 14.5282 1.09201
\(178\) 0 0
\(179\) −4.88830 −0.365369 −0.182684 0.983172i \(-0.558479\pi\)
−0.182684 + 0.983172i \(0.558479\pi\)
\(180\) 0 0
\(181\) 6.70984 0.498738 0.249369 0.968408i \(-0.419777\pi\)
0.249369 + 0.968408i \(0.419777\pi\)
\(182\) 0 0
\(183\) 13.8906 1.02682
\(184\) 0 0
\(185\) 1.17430 0.0863361
\(186\) 0 0
\(187\) 6.26714 0.458299
\(188\) 0 0
\(189\) −5.62220 −0.408955
\(190\) 0 0
\(191\) −8.75265 −0.633319 −0.316660 0.948539i \(-0.602561\pi\)
−0.316660 + 0.948539i \(0.602561\pi\)
\(192\) 0 0
\(193\) −9.95829 −0.716813 −0.358407 0.933566i \(-0.616680\pi\)
−0.358407 + 0.933566i \(0.616680\pi\)
\(194\) 0 0
\(195\) 0.350661 0.0251114
\(196\) 0 0
\(197\) 7.55983 0.538615 0.269308 0.963054i \(-0.413205\pi\)
0.269308 + 0.963054i \(0.413205\pi\)
\(198\) 0 0
\(199\) −2.50958 −0.177899 −0.0889497 0.996036i \(-0.528351\pi\)
−0.0889497 + 0.996036i \(0.528351\pi\)
\(200\) 0 0
\(201\) −7.27400 −0.513068
\(202\) 0 0
\(203\) 9.05224 0.635343
\(204\) 0 0
\(205\) −2.91140 −0.203341
\(206\) 0 0
\(207\) −3.64656 −0.253453
\(208\) 0 0
\(209\) −0.906059 −0.0626734
\(210\) 0 0
\(211\) 21.4873 1.47924 0.739622 0.673023i \(-0.235004\pi\)
0.739622 + 0.673023i \(0.235004\pi\)
\(212\) 0 0
\(213\) 9.25856 0.634386
\(214\) 0 0
\(215\) −3.12597 −0.213189
\(216\) 0 0
\(217\) −1.90381 −0.129239
\(218\) 0 0
\(219\) −19.1898 −1.29672
\(220\) 0 0
\(221\) −6.26714 −0.421574
\(222\) 0 0
\(223\) 16.2219 1.08630 0.543150 0.839635i \(-0.317231\pi\)
0.543150 + 0.839635i \(0.317231\pi\)
\(224\) 0 0
\(225\) 6.16256 0.410837
\(226\) 0 0
\(227\) −19.8036 −1.31441 −0.657207 0.753710i \(-0.728262\pi\)
−0.657207 + 0.753710i \(0.728262\pi\)
\(228\) 0 0
\(229\) 14.1527 0.935240 0.467620 0.883930i \(-0.345112\pi\)
0.467620 + 0.883930i \(0.345112\pi\)
\(230\) 0 0
\(231\) 1.32285 0.0870368
\(232\) 0 0
\(233\) −9.56928 −0.626904 −0.313452 0.949604i \(-0.601486\pi\)
−0.313452 + 0.949604i \(0.601486\pi\)
\(234\) 0 0
\(235\) 2.00681 0.130910
\(236\) 0 0
\(237\) 7.65944 0.497534
\(238\) 0 0
\(239\) −22.2425 −1.43875 −0.719374 0.694623i \(-0.755571\pi\)
−0.719374 + 0.694623i \(0.755571\pi\)
\(240\) 0 0
\(241\) −9.94928 −0.640889 −0.320445 0.947267i \(-0.603832\pi\)
−0.320445 + 0.947267i \(0.603832\pi\)
\(242\) 0 0
\(243\) 11.9892 0.769107
\(244\) 0 0
\(245\) −0.265081 −0.0169354
\(246\) 0 0
\(247\) 0.906059 0.0576511
\(248\) 0 0
\(249\) −13.9154 −0.881850
\(250\) 0 0
\(251\) −11.4389 −0.722017 −0.361008 0.932563i \(-0.617567\pi\)
−0.361008 + 0.932563i \(0.617567\pi\)
\(252\) 0 0
\(253\) 2.91706 0.183394
\(254\) 0 0
\(255\) −2.19765 −0.137622
\(256\) 0 0
\(257\) 16.8781 1.05282 0.526412 0.850229i \(-0.323537\pi\)
0.526412 + 0.850229i \(0.323537\pi\)
\(258\) 0 0
\(259\) −4.42996 −0.275264
\(260\) 0 0
\(261\) −11.3160 −0.700445
\(262\) 0 0
\(263\) −24.5628 −1.51461 −0.757305 0.653061i \(-0.773484\pi\)
−0.757305 + 0.653061i \(0.773484\pi\)
\(264\) 0 0
\(265\) 2.03407 0.124952
\(266\) 0 0
\(267\) 22.5311 1.37888
\(268\) 0 0
\(269\) 26.0062 1.58562 0.792812 0.609466i \(-0.208616\pi\)
0.792812 + 0.609466i \(0.208616\pi\)
\(270\) 0 0
\(271\) −6.47841 −0.393536 −0.196768 0.980450i \(-0.563045\pi\)
−0.196768 + 0.980450i \(0.563045\pi\)
\(272\) 0 0
\(273\) −1.32285 −0.0800622
\(274\) 0 0
\(275\) −4.92973 −0.297274
\(276\) 0 0
\(277\) 4.37897 0.263107 0.131553 0.991309i \(-0.458004\pi\)
0.131553 + 0.991309i \(0.458004\pi\)
\(278\) 0 0
\(279\) 2.37992 0.142482
\(280\) 0 0
\(281\) −21.2357 −1.26681 −0.633407 0.773819i \(-0.718344\pi\)
−0.633407 + 0.773819i \(0.718344\pi\)
\(282\) 0 0
\(283\) −4.25664 −0.253031 −0.126515 0.991965i \(-0.540379\pi\)
−0.126515 + 0.991965i \(0.540379\pi\)
\(284\) 0 0
\(285\) 0.317720 0.0188201
\(286\) 0 0
\(287\) 10.9830 0.648308
\(288\) 0 0
\(289\) 22.2771 1.31042
\(290\) 0 0
\(291\) 10.1678 0.596046
\(292\) 0 0
\(293\) −15.6572 −0.914706 −0.457353 0.889285i \(-0.651202\pi\)
−0.457353 + 0.889285i \(0.651202\pi\)
\(294\) 0 0
\(295\) −2.91127 −0.169501
\(296\) 0 0
\(297\) −5.62220 −0.326233
\(298\) 0 0
\(299\) −2.91706 −0.168698
\(300\) 0 0
\(301\) 11.7925 0.679708
\(302\) 0 0
\(303\) −7.69983 −0.442343
\(304\) 0 0
\(305\) −2.78349 −0.159382
\(306\) 0 0
\(307\) 25.9546 1.48131 0.740654 0.671886i \(-0.234516\pi\)
0.740654 + 0.671886i \(0.234516\pi\)
\(308\) 0 0
\(309\) 23.6205 1.34373
\(310\) 0 0
\(311\) −19.7324 −1.11892 −0.559461 0.828857i \(-0.688992\pi\)
−0.559461 + 0.828857i \(0.688992\pi\)
\(312\) 0 0
\(313\) 10.5661 0.597229 0.298614 0.954374i \(-0.403476\pi\)
0.298614 + 0.954374i \(0.403476\pi\)
\(314\) 0 0
\(315\) 0.331373 0.0186707
\(316\) 0 0
\(317\) 24.9493 1.40129 0.700646 0.713509i \(-0.252895\pi\)
0.700646 + 0.713509i \(0.252895\pi\)
\(318\) 0 0
\(319\) 9.05224 0.506828
\(320\) 0 0
\(321\) 24.5792 1.37188
\(322\) 0 0
\(323\) −5.67840 −0.315955
\(324\) 0 0
\(325\) 4.92973 0.273452
\(326\) 0 0
\(327\) −3.64733 −0.201698
\(328\) 0 0
\(329\) −7.57057 −0.417379
\(330\) 0 0
\(331\) −19.6362 −1.07931 −0.539653 0.841888i \(-0.681444\pi\)
−0.539653 + 0.841888i \(0.681444\pi\)
\(332\) 0 0
\(333\) 5.53780 0.303470
\(334\) 0 0
\(335\) 1.45762 0.0796381
\(336\) 0 0
\(337\) 17.6678 0.962425 0.481212 0.876604i \(-0.340197\pi\)
0.481212 + 0.876604i \(0.340197\pi\)
\(338\) 0 0
\(339\) −13.5134 −0.733945
\(340\) 0 0
\(341\) −1.90381 −0.103097
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −1.02290 −0.0550711
\(346\) 0 0
\(347\) 7.63288 0.409754 0.204877 0.978788i \(-0.434321\pi\)
0.204877 + 0.978788i \(0.434321\pi\)
\(348\) 0 0
\(349\) −13.6761 −0.732066 −0.366033 0.930602i \(-0.619284\pi\)
−0.366033 + 0.930602i \(0.619284\pi\)
\(350\) 0 0
\(351\) 5.62220 0.300091
\(352\) 0 0
\(353\) 14.1251 0.751805 0.375902 0.926659i \(-0.377333\pi\)
0.375902 + 0.926659i \(0.377333\pi\)
\(354\) 0 0
\(355\) −1.85530 −0.0984689
\(356\) 0 0
\(357\) 8.29046 0.438778
\(358\) 0 0
\(359\) −18.0695 −0.953670 −0.476835 0.878993i \(-0.658216\pi\)
−0.476835 + 0.878993i \(0.658216\pi\)
\(360\) 0 0
\(361\) −18.1791 −0.956793
\(362\) 0 0
\(363\) 1.32285 0.0694314
\(364\) 0 0
\(365\) 3.84538 0.201276
\(366\) 0 0
\(367\) 13.7815 0.719389 0.359694 0.933070i \(-0.382881\pi\)
0.359694 + 0.933070i \(0.382881\pi\)
\(368\) 0 0
\(369\) −13.7297 −0.714739
\(370\) 0 0
\(371\) −7.67340 −0.398383
\(372\) 0 0
\(373\) 31.1620 1.61350 0.806752 0.590890i \(-0.201223\pi\)
0.806752 + 0.590890i \(0.201223\pi\)
\(374\) 0 0
\(375\) 3.48197 0.179808
\(376\) 0 0
\(377\) −9.05224 −0.466214
\(378\) 0 0
\(379\) −18.7474 −0.962990 −0.481495 0.876449i \(-0.659906\pi\)
−0.481495 + 0.876449i \(0.659906\pi\)
\(380\) 0 0
\(381\) −6.60191 −0.338226
\(382\) 0 0
\(383\) −27.1478 −1.38719 −0.693593 0.720367i \(-0.743973\pi\)
−0.693593 + 0.720367i \(0.743973\pi\)
\(384\) 0 0
\(385\) −0.265081 −0.0135098
\(386\) 0 0
\(387\) −14.7416 −0.749355
\(388\) 0 0
\(389\) −30.3823 −1.54045 −0.770223 0.637775i \(-0.779855\pi\)
−0.770223 + 0.637775i \(0.779855\pi\)
\(390\) 0 0
\(391\) 18.2816 0.924542
\(392\) 0 0
\(393\) 13.3557 0.673705
\(394\) 0 0
\(395\) −1.53485 −0.0772269
\(396\) 0 0
\(397\) −28.3928 −1.42499 −0.712497 0.701675i \(-0.752436\pi\)
−0.712497 + 0.701675i \(0.752436\pi\)
\(398\) 0 0
\(399\) −1.19858 −0.0600038
\(400\) 0 0
\(401\) 10.3809 0.518396 0.259198 0.965824i \(-0.416542\pi\)
0.259198 + 0.965824i \(0.416542\pi\)
\(402\) 0 0
\(403\) 1.90381 0.0948356
\(404\) 0 0
\(405\) 0.977370 0.0485659
\(406\) 0 0
\(407\) −4.42996 −0.219585
\(408\) 0 0
\(409\) −7.29337 −0.360634 −0.180317 0.983609i \(-0.557712\pi\)
−0.180317 + 0.983609i \(0.557712\pi\)
\(410\) 0 0
\(411\) 9.25379 0.456456
\(412\) 0 0
\(413\) 10.9826 0.540416
\(414\) 0 0
\(415\) 2.78846 0.136880
\(416\) 0 0
\(417\) −12.8296 −0.628266
\(418\) 0 0
\(419\) 9.70466 0.474104 0.237052 0.971497i \(-0.423819\pi\)
0.237052 + 0.971497i \(0.423819\pi\)
\(420\) 0 0
\(421\) 12.4760 0.608042 0.304021 0.952665i \(-0.401671\pi\)
0.304021 + 0.952665i \(0.401671\pi\)
\(422\) 0 0
\(423\) 9.46382 0.460146
\(424\) 0 0
\(425\) −30.8953 −1.49864
\(426\) 0 0
\(427\) 10.5005 0.508156
\(428\) 0 0
\(429\) −1.32285 −0.0638676
\(430\) 0 0
\(431\) −4.80807 −0.231597 −0.115798 0.993273i \(-0.536943\pi\)
−0.115798 + 0.993273i \(0.536943\pi\)
\(432\) 0 0
\(433\) 18.5011 0.889105 0.444552 0.895753i \(-0.353363\pi\)
0.444552 + 0.895753i \(0.353363\pi\)
\(434\) 0 0
\(435\) −3.17427 −0.152195
\(436\) 0 0
\(437\) −2.64303 −0.126433
\(438\) 0 0
\(439\) 6.91267 0.329924 0.164962 0.986300i \(-0.447250\pi\)
0.164962 + 0.986300i \(0.447250\pi\)
\(440\) 0 0
\(441\) −1.25008 −0.0595276
\(442\) 0 0
\(443\) −12.8863 −0.612247 −0.306123 0.951992i \(-0.599032\pi\)
−0.306123 + 0.951992i \(0.599032\pi\)
\(444\) 0 0
\(445\) −4.51494 −0.214029
\(446\) 0 0
\(447\) 27.7693 1.31344
\(448\) 0 0
\(449\) 2.73958 0.129289 0.0646445 0.997908i \(-0.479409\pi\)
0.0646445 + 0.997908i \(0.479409\pi\)
\(450\) 0 0
\(451\) 10.9830 0.517171
\(452\) 0 0
\(453\) 19.6322 0.922401
\(454\) 0 0
\(455\) 0.265081 0.0124272
\(456\) 0 0
\(457\) −25.7361 −1.20388 −0.601941 0.798540i \(-0.705606\pi\)
−0.601941 + 0.798540i \(0.705606\pi\)
\(458\) 0 0
\(459\) −35.2351 −1.64463
\(460\) 0 0
\(461\) 1.15202 0.0536551 0.0268275 0.999640i \(-0.491460\pi\)
0.0268275 + 0.999640i \(0.491460\pi\)
\(462\) 0 0
\(463\) 7.60927 0.353633 0.176816 0.984244i \(-0.443420\pi\)
0.176816 + 0.984244i \(0.443420\pi\)
\(464\) 0 0
\(465\) 0.667593 0.0309589
\(466\) 0 0
\(467\) −22.9299 −1.06107 −0.530536 0.847663i \(-0.678009\pi\)
−0.530536 + 0.847663i \(0.678009\pi\)
\(468\) 0 0
\(469\) −5.49875 −0.253909
\(470\) 0 0
\(471\) −11.8094 −0.544149
\(472\) 0 0
\(473\) 11.7925 0.542219
\(474\) 0 0
\(475\) 4.46663 0.204943
\(476\) 0 0
\(477\) 9.59236 0.439204
\(478\) 0 0
\(479\) 11.7410 0.536461 0.268230 0.963355i \(-0.413561\pi\)
0.268230 + 0.963355i \(0.413561\pi\)
\(480\) 0 0
\(481\) 4.42996 0.201989
\(482\) 0 0
\(483\) 3.85882 0.175582
\(484\) 0 0
\(485\) −2.03749 −0.0925178
\(486\) 0 0
\(487\) −11.5378 −0.522826 −0.261413 0.965227i \(-0.584188\pi\)
−0.261413 + 0.965227i \(0.584188\pi\)
\(488\) 0 0
\(489\) 7.49807 0.339075
\(490\) 0 0
\(491\) 25.5592 1.15347 0.576735 0.816931i \(-0.304327\pi\)
0.576735 + 0.816931i \(0.304327\pi\)
\(492\) 0 0
\(493\) 56.7317 2.55507
\(494\) 0 0
\(495\) 0.331373 0.0148941
\(496\) 0 0
\(497\) 6.99897 0.313947
\(498\) 0 0
\(499\) −10.8716 −0.486678 −0.243339 0.969941i \(-0.578243\pi\)
−0.243339 + 0.969941i \(0.578243\pi\)
\(500\) 0 0
\(501\) −4.66494 −0.208414
\(502\) 0 0
\(503\) −39.0853 −1.74273 −0.871363 0.490639i \(-0.836763\pi\)
−0.871363 + 0.490639i \(0.836763\pi\)
\(504\) 0 0
\(505\) 1.54295 0.0686602
\(506\) 0 0
\(507\) 1.32285 0.0587496
\(508\) 0 0
\(509\) −26.8541 −1.19029 −0.595144 0.803619i \(-0.702905\pi\)
−0.595144 + 0.803619i \(0.702905\pi\)
\(510\) 0 0
\(511\) −14.5064 −0.641726
\(512\) 0 0
\(513\) 5.09404 0.224907
\(514\) 0 0
\(515\) −4.73325 −0.208572
\(516\) 0 0
\(517\) −7.57057 −0.332953
\(518\) 0 0
\(519\) −31.2078 −1.36987
\(520\) 0 0
\(521\) −8.50250 −0.372501 −0.186251 0.982502i \(-0.559634\pi\)
−0.186251 + 0.982502i \(0.559634\pi\)
\(522\) 0 0
\(523\) 36.3165 1.58801 0.794005 0.607911i \(-0.207992\pi\)
0.794005 + 0.607911i \(0.207992\pi\)
\(524\) 0 0
\(525\) −6.52127 −0.284612
\(526\) 0 0
\(527\) −11.9315 −0.519742
\(528\) 0 0
\(529\) −14.4908 −0.630033
\(530\) 0 0
\(531\) −13.7291 −0.595791
\(532\) 0 0
\(533\) −10.9830 −0.475728
\(534\) 0 0
\(535\) −4.92536 −0.212942
\(536\) 0 0
\(537\) −6.46647 −0.279049
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 4.05641 0.174398 0.0871992 0.996191i \(-0.472208\pi\)
0.0871992 + 0.996191i \(0.472208\pi\)
\(542\) 0 0
\(543\) 8.87608 0.380909
\(544\) 0 0
\(545\) 0.730877 0.0313073
\(546\) 0 0
\(547\) −15.9673 −0.682712 −0.341356 0.939934i \(-0.610886\pi\)
−0.341356 + 0.939934i \(0.610886\pi\)
\(548\) 0 0
\(549\) −13.1265 −0.560226
\(550\) 0 0
\(551\) −8.20187 −0.349411
\(552\) 0 0
\(553\) 5.79013 0.246221
\(554\) 0 0
\(555\) 1.55341 0.0659388
\(556\) 0 0
\(557\) −9.45131 −0.400465 −0.200232 0.979748i \(-0.564170\pi\)
−0.200232 + 0.979748i \(0.564170\pi\)
\(558\) 0 0
\(559\) −11.7925 −0.498769
\(560\) 0 0
\(561\) 8.29046 0.350024
\(562\) 0 0
\(563\) 22.6366 0.954019 0.477010 0.878898i \(-0.341721\pi\)
0.477010 + 0.878898i \(0.341721\pi\)
\(564\) 0 0
\(565\) 2.70790 0.113922
\(566\) 0 0
\(567\) −3.68706 −0.154842
\(568\) 0 0
\(569\) 12.0891 0.506803 0.253401 0.967361i \(-0.418451\pi\)
0.253401 + 0.967361i \(0.418451\pi\)
\(570\) 0 0
\(571\) −23.8055 −0.996231 −0.498115 0.867111i \(-0.665974\pi\)
−0.498115 + 0.867111i \(0.665974\pi\)
\(572\) 0 0
\(573\) −11.5784 −0.483695
\(574\) 0 0
\(575\) −14.3803 −0.599701
\(576\) 0 0
\(577\) 32.5932 1.35687 0.678436 0.734660i \(-0.262658\pi\)
0.678436 + 0.734660i \(0.262658\pi\)
\(578\) 0 0
\(579\) −13.1733 −0.547463
\(580\) 0 0
\(581\) −10.5193 −0.436413
\(582\) 0 0
\(583\) −7.67340 −0.317800
\(584\) 0 0
\(585\) −0.331373 −0.0137006
\(586\) 0 0
\(587\) 0.932270 0.0384789 0.0192394 0.999815i \(-0.493876\pi\)
0.0192394 + 0.999815i \(0.493876\pi\)
\(588\) 0 0
\(589\) 1.72496 0.0710759
\(590\) 0 0
\(591\) 10.0005 0.411365
\(592\) 0 0
\(593\) 27.2128 1.11750 0.558748 0.829338i \(-0.311282\pi\)
0.558748 + 0.829338i \(0.311282\pi\)
\(594\) 0 0
\(595\) −1.66130 −0.0681067
\(596\) 0 0
\(597\) −3.31979 −0.135870
\(598\) 0 0
\(599\) −20.7953 −0.849674 −0.424837 0.905270i \(-0.639669\pi\)
−0.424837 + 0.905270i \(0.639669\pi\)
\(600\) 0 0
\(601\) 35.5409 1.44974 0.724871 0.688884i \(-0.241899\pi\)
0.724871 + 0.688884i \(0.241899\pi\)
\(602\) 0 0
\(603\) 6.87388 0.279926
\(604\) 0 0
\(605\) −0.265081 −0.0107771
\(606\) 0 0
\(607\) −26.6990 −1.08368 −0.541839 0.840482i \(-0.682272\pi\)
−0.541839 + 0.840482i \(0.682272\pi\)
\(608\) 0 0
\(609\) 11.9747 0.485240
\(610\) 0 0
\(611\) 7.57057 0.306272
\(612\) 0 0
\(613\) −36.0041 −1.45419 −0.727095 0.686537i \(-0.759130\pi\)
−0.727095 + 0.686537i \(0.759130\pi\)
\(614\) 0 0
\(615\) −3.85133 −0.155301
\(616\) 0 0
\(617\) −17.5314 −0.705790 −0.352895 0.935663i \(-0.614803\pi\)
−0.352895 + 0.935663i \(0.614803\pi\)
\(618\) 0 0
\(619\) −19.7846 −0.795211 −0.397606 0.917556i \(-0.630159\pi\)
−0.397606 + 0.917556i \(0.630159\pi\)
\(620\) 0 0
\(621\) −16.4003 −0.658121
\(622\) 0 0
\(623\) 17.0323 0.682384
\(624\) 0 0
\(625\) 23.9509 0.958037
\(626\) 0 0
\(627\) −1.19858 −0.0478665
\(628\) 0 0
\(629\) −27.7632 −1.10699
\(630\) 0 0
\(631\) 28.2437 1.12437 0.562183 0.827013i \(-0.309962\pi\)
0.562183 + 0.827013i \(0.309962\pi\)
\(632\) 0 0
\(633\) 28.4243 1.12976
\(634\) 0 0
\(635\) 1.32294 0.0524992
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −8.74928 −0.346116
\(640\) 0 0
\(641\) 31.9757 1.26296 0.631482 0.775391i \(-0.282447\pi\)
0.631482 + 0.775391i \(0.282447\pi\)
\(642\) 0 0
\(643\) −22.5812 −0.890514 −0.445257 0.895403i \(-0.646888\pi\)
−0.445257 + 0.895403i \(0.646888\pi\)
\(644\) 0 0
\(645\) −4.13517 −0.162822
\(646\) 0 0
\(647\) −3.39460 −0.133456 −0.0667278 0.997771i \(-0.521256\pi\)
−0.0667278 + 0.997771i \(0.521256\pi\)
\(648\) 0 0
\(649\) 10.9826 0.431103
\(650\) 0 0
\(651\) −2.51845 −0.0987057
\(652\) 0 0
\(653\) 7.23714 0.283211 0.141606 0.989923i \(-0.454774\pi\)
0.141606 + 0.989923i \(0.454774\pi\)
\(654\) 0 0
\(655\) −2.67631 −0.104572
\(656\) 0 0
\(657\) 18.1342 0.707482
\(658\) 0 0
\(659\) 21.5392 0.839047 0.419524 0.907744i \(-0.362197\pi\)
0.419524 + 0.907744i \(0.362197\pi\)
\(660\) 0 0
\(661\) −37.8960 −1.47398 −0.736991 0.675903i \(-0.763754\pi\)
−0.736991 + 0.675903i \(0.763754\pi\)
\(662\) 0 0
\(663\) −8.29046 −0.321975
\(664\) 0 0
\(665\) 0.240179 0.00931375
\(666\) 0 0
\(667\) 26.4060 1.02244
\(668\) 0 0
\(669\) 21.4591 0.829657
\(670\) 0 0
\(671\) 10.5005 0.405369
\(672\) 0 0
\(673\) −16.3029 −0.628430 −0.314215 0.949352i \(-0.601741\pi\)
−0.314215 + 0.949352i \(0.601741\pi\)
\(674\) 0 0
\(675\) 27.7159 1.06679
\(676\) 0 0
\(677\) 24.1809 0.929346 0.464673 0.885482i \(-0.346172\pi\)
0.464673 + 0.885482i \(0.346172\pi\)
\(678\) 0 0
\(679\) 7.68630 0.294973
\(680\) 0 0
\(681\) −26.1972 −1.00388
\(682\) 0 0
\(683\) −25.9657 −0.993548 −0.496774 0.867880i \(-0.665482\pi\)
−0.496774 + 0.867880i \(0.665482\pi\)
\(684\) 0 0
\(685\) −1.85434 −0.0708507
\(686\) 0 0
\(687\) 18.7219 0.714285
\(688\) 0 0
\(689\) 7.67340 0.292333
\(690\) 0 0
\(691\) −15.5423 −0.591256 −0.295628 0.955303i \(-0.595529\pi\)
−0.295628 + 0.955303i \(0.595529\pi\)
\(692\) 0 0
\(693\) −1.25008 −0.0474866
\(694\) 0 0
\(695\) 2.57088 0.0975190
\(696\) 0 0
\(697\) 68.8323 2.60721
\(698\) 0 0
\(699\) −12.6587 −0.478795
\(700\) 0 0
\(701\) 47.6404 1.79935 0.899677 0.436556i \(-0.143802\pi\)
0.899677 + 0.436556i \(0.143802\pi\)
\(702\) 0 0
\(703\) 4.01380 0.151383
\(704\) 0 0
\(705\) 2.65471 0.0999820
\(706\) 0 0
\(707\) −5.82065 −0.218908
\(708\) 0 0
\(709\) 2.13047 0.0800114 0.0400057 0.999199i \(-0.487262\pi\)
0.0400057 + 0.999199i \(0.487262\pi\)
\(710\) 0 0
\(711\) −7.23812 −0.271451
\(712\) 0 0
\(713\) −5.55353 −0.207981
\(714\) 0 0
\(715\) 0.265081 0.00991347
\(716\) 0 0
\(717\) −29.4234 −1.09884
\(718\) 0 0
\(719\) 38.8042 1.44715 0.723576 0.690245i \(-0.242497\pi\)
0.723576 + 0.690245i \(0.242497\pi\)
\(720\) 0 0
\(721\) 17.8559 0.664987
\(722\) 0 0
\(723\) −13.1614 −0.489476
\(724\) 0 0
\(725\) −44.6251 −1.65734
\(726\) 0 0
\(727\) −39.4060 −1.46149 −0.730744 0.682652i \(-0.760827\pi\)
−0.730744 + 0.682652i \(0.760827\pi\)
\(728\) 0 0
\(729\) 26.9210 0.997075
\(730\) 0 0
\(731\) 73.9052 2.73348
\(732\) 0 0
\(733\) −11.5509 −0.426641 −0.213320 0.976982i \(-0.568428\pi\)
−0.213320 + 0.976982i \(0.568428\pi\)
\(734\) 0 0
\(735\) −0.350661 −0.0129343
\(736\) 0 0
\(737\) −5.49875 −0.202549
\(738\) 0 0
\(739\) −35.9578 −1.32273 −0.661365 0.750064i \(-0.730023\pi\)
−0.661365 + 0.750064i \(0.730023\pi\)
\(740\) 0 0
\(741\) 1.19858 0.0440308
\(742\) 0 0
\(743\) −27.7002 −1.01622 −0.508111 0.861292i \(-0.669656\pi\)
−0.508111 + 0.861292i \(0.669656\pi\)
\(744\) 0 0
\(745\) −5.56461 −0.203872
\(746\) 0 0
\(747\) 13.1499 0.481131
\(748\) 0 0
\(749\) 18.5806 0.678919
\(750\) 0 0
\(751\) −31.4638 −1.14813 −0.574066 0.818809i \(-0.694635\pi\)
−0.574066 + 0.818809i \(0.694635\pi\)
\(752\) 0 0
\(753\) −15.1319 −0.551437
\(754\) 0 0
\(755\) −3.93404 −0.143174
\(756\) 0 0
\(757\) −13.6973 −0.497837 −0.248919 0.968524i \(-0.580075\pi\)
−0.248919 + 0.968524i \(0.580075\pi\)
\(758\) 0 0
\(759\) 3.85882 0.140066
\(760\) 0 0
\(761\) 21.3047 0.772296 0.386148 0.922437i \(-0.373805\pi\)
0.386148 + 0.922437i \(0.373805\pi\)
\(762\) 0 0
\(763\) −2.75718 −0.0998167
\(764\) 0 0
\(765\) 2.07676 0.0750854
\(766\) 0 0
\(767\) −10.9826 −0.396557
\(768\) 0 0
\(769\) −14.9356 −0.538590 −0.269295 0.963058i \(-0.586791\pi\)
−0.269295 + 0.963058i \(0.586791\pi\)
\(770\) 0 0
\(771\) 22.3271 0.804090
\(772\) 0 0
\(773\) −16.6355 −0.598337 −0.299168 0.954200i \(-0.596709\pi\)
−0.299168 + 0.954200i \(0.596709\pi\)
\(774\) 0 0
\(775\) 9.38528 0.337129
\(776\) 0 0
\(777\) −5.86015 −0.210232
\(778\) 0 0
\(779\) −9.95128 −0.356542
\(780\) 0 0
\(781\) 6.99897 0.250443
\(782\) 0 0
\(783\) −50.8935 −1.81879
\(784\) 0 0
\(785\) 2.36645 0.0844624
\(786\) 0 0
\(787\) 7.08534 0.252565 0.126283 0.991994i \(-0.459695\pi\)
0.126283 + 0.991994i \(0.459695\pi\)
\(788\) 0 0
\(789\) −32.4928 −1.15678
\(790\) 0 0
\(791\) −10.2154 −0.363217
\(792\) 0 0
\(793\) −10.5005 −0.372885
\(794\) 0 0
\(795\) 2.69076 0.0954315
\(796\) 0 0
\(797\) −34.8957 −1.23607 −0.618035 0.786151i \(-0.712071\pi\)
−0.618035 + 0.786151i \(0.712071\pi\)
\(798\) 0 0
\(799\) −47.4458 −1.67851
\(800\) 0 0
\(801\) −21.2917 −0.752306
\(802\) 0 0
\(803\) −14.5064 −0.511920
\(804\) 0 0
\(805\) −0.773258 −0.0272538
\(806\) 0 0
\(807\) 34.4021 1.21101
\(808\) 0 0
\(809\) 7.58554 0.266693 0.133347 0.991069i \(-0.457428\pi\)
0.133347 + 0.991069i \(0.457428\pi\)
\(810\) 0 0
\(811\) −26.4792 −0.929809 −0.464904 0.885361i \(-0.653911\pi\)
−0.464904 + 0.885361i \(0.653911\pi\)
\(812\) 0 0
\(813\) −8.56994 −0.300561
\(814\) 0 0
\(815\) −1.50252 −0.0526309
\(816\) 0 0
\(817\) −10.6847 −0.373810
\(818\) 0 0
\(819\) 1.25008 0.0436813
\(820\) 0 0
\(821\) −43.6089 −1.52196 −0.760981 0.648775i \(-0.775282\pi\)
−0.760981 + 0.648775i \(0.775282\pi\)
\(822\) 0 0
\(823\) −31.0230 −1.08139 −0.540697 0.841217i \(-0.681840\pi\)
−0.540697 + 0.841217i \(0.681840\pi\)
\(824\) 0 0
\(825\) −6.52127 −0.227042
\(826\) 0 0
\(827\) 36.6638 1.27493 0.637463 0.770481i \(-0.279984\pi\)
0.637463 + 0.770481i \(0.279984\pi\)
\(828\) 0 0
\(829\) −3.77612 −0.131150 −0.0655750 0.997848i \(-0.520888\pi\)
−0.0655750 + 0.997848i \(0.520888\pi\)
\(830\) 0 0
\(831\) 5.79270 0.200946
\(832\) 0 0
\(833\) 6.26714 0.217144
\(834\) 0 0
\(835\) 0.934793 0.0323499
\(836\) 0 0
\(837\) 10.7036 0.369971
\(838\) 0 0
\(839\) −7.89129 −0.272437 −0.136219 0.990679i \(-0.543495\pi\)
−0.136219 + 0.990679i \(0.543495\pi\)
\(840\) 0 0
\(841\) 52.9431 1.82563
\(842\) 0 0
\(843\) −28.0915 −0.967523
\(844\) 0 0
\(845\) −0.265081 −0.00911907
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) −5.63087 −0.193251
\(850\) 0 0
\(851\) −12.9225 −0.442976
\(852\) 0 0
\(853\) 0.998656 0.0341933 0.0170967 0.999854i \(-0.494558\pi\)
0.0170967 + 0.999854i \(0.494558\pi\)
\(854\) 0 0
\(855\) −0.300243 −0.0102681
\(856\) 0 0
\(857\) 12.4632 0.425735 0.212868 0.977081i \(-0.431720\pi\)
0.212868 + 0.977081i \(0.431720\pi\)
\(858\) 0 0
\(859\) 20.6856 0.705782 0.352891 0.935664i \(-0.385199\pi\)
0.352891 + 0.935664i \(0.385199\pi\)
\(860\) 0 0
\(861\) 14.5289 0.495142
\(862\) 0 0
\(863\) 7.26678 0.247364 0.123682 0.992322i \(-0.460530\pi\)
0.123682 + 0.992322i \(0.460530\pi\)
\(864\) 0 0
\(865\) 6.25365 0.212630
\(866\) 0 0
\(867\) 29.4691 1.00082
\(868\) 0 0
\(869\) 5.79013 0.196417
\(870\) 0 0
\(871\) 5.49875 0.186318
\(872\) 0 0
\(873\) −9.60849 −0.325198
\(874\) 0 0
\(875\) 2.63218 0.0889841
\(876\) 0 0
\(877\) 13.7870 0.465555 0.232777 0.972530i \(-0.425219\pi\)
0.232777 + 0.972530i \(0.425219\pi\)
\(878\) 0 0
\(879\) −20.7121 −0.698602
\(880\) 0 0
\(881\) 23.8073 0.802090 0.401045 0.916058i \(-0.368647\pi\)
0.401045 + 0.916058i \(0.368647\pi\)
\(882\) 0 0
\(883\) −17.5886 −0.591904 −0.295952 0.955203i \(-0.595637\pi\)
−0.295952 + 0.955203i \(0.595637\pi\)
\(884\) 0 0
\(885\) −3.85116 −0.129455
\(886\) 0 0
\(887\) −31.6100 −1.06136 −0.530681 0.847572i \(-0.678064\pi\)
−0.530681 + 0.847572i \(0.678064\pi\)
\(888\) 0 0
\(889\) −4.99069 −0.167382
\(890\) 0 0
\(891\) −3.68706 −0.123521
\(892\) 0 0
\(893\) 6.85938 0.229540
\(894\) 0 0
\(895\) 1.29580 0.0433137
\(896\) 0 0
\(897\) −3.85882 −0.128842
\(898\) 0 0
\(899\) −17.2338 −0.574778
\(900\) 0 0
\(901\) −48.0903 −1.60212
\(902\) 0 0
\(903\) 15.5996 0.519123
\(904\) 0 0
\(905\) −1.77865 −0.0591244
\(906\) 0 0
\(907\) −27.9021 −0.926475 −0.463238 0.886234i \(-0.653312\pi\)
−0.463238 + 0.886234i \(0.653312\pi\)
\(908\) 0 0
\(909\) 7.27628 0.241339
\(910\) 0 0
\(911\) −39.3855 −1.30490 −0.652451 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(912\) 0 0
\(913\) −10.5193 −0.348137
\(914\) 0 0
\(915\) −3.68213 −0.121728
\(916\) 0 0
\(917\) 10.0962 0.333405
\(918\) 0 0
\(919\) −5.67607 −0.187236 −0.0936182 0.995608i \(-0.529843\pi\)
−0.0936182 + 0.995608i \(0.529843\pi\)
\(920\) 0 0
\(921\) 34.3339 1.13134
\(922\) 0 0
\(923\) −6.99897 −0.230374
\(924\) 0 0
\(925\) 21.8385 0.718045
\(926\) 0 0
\(927\) −22.3212 −0.733126
\(928\) 0 0
\(929\) 27.0701 0.888142 0.444071 0.895992i \(-0.353534\pi\)
0.444071 + 0.895992i \(0.353534\pi\)
\(930\) 0 0
\(931\) −0.906059 −0.0296949
\(932\) 0 0
\(933\) −26.1029 −0.854571
\(934\) 0 0
\(935\) −1.66130 −0.0543304
\(936\) 0 0
\(937\) 35.0768 1.14591 0.572954 0.819588i \(-0.305797\pi\)
0.572954 + 0.819588i \(0.305797\pi\)
\(938\) 0 0
\(939\) 13.9773 0.456131
\(940\) 0 0
\(941\) −16.8702 −0.549954 −0.274977 0.961451i \(-0.588670\pi\)
−0.274977 + 0.961451i \(0.588670\pi\)
\(942\) 0 0
\(943\) 32.0382 1.04331
\(944\) 0 0
\(945\) 1.49034 0.0484807
\(946\) 0 0
\(947\) −19.6718 −0.639247 −0.319623 0.947545i \(-0.603556\pi\)
−0.319623 + 0.947545i \(0.603556\pi\)
\(948\) 0 0
\(949\) 14.5064 0.470898
\(950\) 0 0
\(951\) 33.0040 1.07023
\(952\) 0 0
\(953\) −50.7375 −1.64355 −0.821775 0.569813i \(-0.807016\pi\)
−0.821775 + 0.569813i \(0.807016\pi\)
\(954\) 0 0
\(955\) 2.32016 0.0750787
\(956\) 0 0
\(957\) 11.9747 0.387088
\(958\) 0 0
\(959\) 6.99537 0.225892
\(960\) 0 0
\(961\) −27.3755 −0.883081
\(962\) 0 0
\(963\) −23.2272 −0.748486
\(964\) 0 0
\(965\) 2.63975 0.0849767
\(966\) 0 0
\(967\) −35.5690 −1.14382 −0.571911 0.820316i \(-0.693798\pi\)
−0.571911 + 0.820316i \(0.693798\pi\)
\(968\) 0 0
\(969\) −7.51164 −0.241309
\(970\) 0 0
\(971\) 37.3057 1.19720 0.598598 0.801050i \(-0.295725\pi\)
0.598598 + 0.801050i \(0.295725\pi\)
\(972\) 0 0
\(973\) −9.69846 −0.310918
\(974\) 0 0
\(975\) 6.52127 0.208848
\(976\) 0 0
\(977\) −57.4095 −1.83669 −0.918346 0.395778i \(-0.870475\pi\)
−0.918346 + 0.395778i \(0.870475\pi\)
\(978\) 0 0
\(979\) 17.0323 0.544354
\(980\) 0 0
\(981\) 3.44670 0.110045
\(982\) 0 0
\(983\) −3.91259 −0.124792 −0.0623961 0.998051i \(-0.519874\pi\)
−0.0623961 + 0.998051i \(0.519874\pi\)
\(984\) 0 0
\(985\) −2.00397 −0.0638517
\(986\) 0 0
\(987\) −10.0147 −0.318771
\(988\) 0 0
\(989\) 34.3994 1.09384
\(990\) 0 0
\(991\) 39.0959 1.24192 0.620960 0.783842i \(-0.286743\pi\)
0.620960 + 0.783842i \(0.286743\pi\)
\(992\) 0 0
\(993\) −25.9757 −0.824314
\(994\) 0 0
\(995\) 0.665243 0.0210896
\(996\) 0 0
\(997\) 46.0451 1.45826 0.729132 0.684373i \(-0.239924\pi\)
0.729132 + 0.684373i \(0.239924\pi\)
\(998\) 0 0
\(999\) 24.9061 0.787994
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 4004.2.a.h.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4004.2.a.h.1.6 9 1.1 even 1 trivial