Properties

Label 1232.4.a.bc.1.3
Level $1232$
Weight $4$
Character 1232.1
Self dual yes
Analytic conductor $72.690$
Analytic rank $0$
Dimension $7$
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] = [1232,4,Mod(1,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1232.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: \(72.6903531271\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 144x^{5} + 354x^{4} + 5172x^{3} - 6504x^{2} - 34432x + 18816 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 616)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.19046\) of defining polynomial
Character \(\chi\) \(=\) 1232.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.19046 q^{3} +8.53193 q^{5} -7.00000 q^{7} -16.8209 q^{9} +O(q^{10})\) \(q-3.19046 q^{3} +8.53193 q^{5} -7.00000 q^{7} -16.8209 q^{9} -11.0000 q^{11} +68.7212 q^{13} -27.2208 q^{15} -30.5206 q^{17} -14.5420 q^{19} +22.3332 q^{21} -76.8536 q^{23} -52.2062 q^{25} +139.809 q^{27} +149.326 q^{29} -188.846 q^{31} +35.0951 q^{33} -59.7235 q^{35} +260.827 q^{37} -219.253 q^{39} +183.545 q^{41} +354.695 q^{43} -143.515 q^{45} -420.339 q^{47} +49.0000 q^{49} +97.3747 q^{51} -468.426 q^{53} -93.8512 q^{55} +46.3956 q^{57} -160.334 q^{59} +26.0410 q^{61} +117.747 q^{63} +586.325 q^{65} +397.860 q^{67} +245.198 q^{69} +440.130 q^{71} +409.850 q^{73} +166.562 q^{75} +77.0000 q^{77} +71.6329 q^{79} +8.10998 q^{81} -701.049 q^{83} -260.399 q^{85} -476.418 q^{87} +72.3930 q^{89} -481.049 q^{91} +602.506 q^{93} -124.071 q^{95} -168.126 q^{97} +185.030 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 3 q^{3} + 11 q^{5} - 49 q^{7} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 3 q^{3} + 11 q^{5} - 49 q^{7} + 108 q^{9} - 77 q^{11} + 26 q^{13} - 67 q^{15} + 44 q^{17} - 34 q^{19} + 21 q^{21} + 29 q^{23} + 182 q^{25} - 99 q^{27} + 94 q^{29} + 173 q^{31} + 33 q^{33} - 77 q^{35} + 255 q^{37} + 60 q^{39} + 508 q^{41} - 656 q^{43} + 466 q^{45} + 18 q^{47} + 343 q^{49} - 850 q^{51} + 1806 q^{53} - 121 q^{55} + 1154 q^{57} - 665 q^{59} + 608 q^{61} - 756 q^{63} + 588 q^{65} - 669 q^{67} + 2067 q^{69} - 1169 q^{71} + 380 q^{73} - 1254 q^{75} + 539 q^{77} + 110 q^{79} + 2847 q^{81} + 496 q^{83} + 3446 q^{85} + 582 q^{87} + 1321 q^{89} - 182 q^{91} + 1493 q^{93} + 50 q^{95} + 3927 q^{97} - 1188 q^{99}+O(q^{100}) \) 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
\(3\) −3.19046 −0.614005 −0.307002 0.951709i \(-0.599326\pi\)
−0.307002 + 0.951709i \(0.599326\pi\)
\(4\) 0 0
\(5\) 8.53193 0.763119 0.381559 0.924344i \(-0.375387\pi\)
0.381559 + 0.924344i \(0.375387\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) −16.8209 −0.622998
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 68.7212 1.46614 0.733071 0.680152i \(-0.238086\pi\)
0.733071 + 0.680152i \(0.238086\pi\)
\(14\) 0 0
\(15\) −27.2208 −0.468558
\(16\) 0 0
\(17\) −30.5206 −0.435431 −0.217715 0.976012i \(-0.569860\pi\)
−0.217715 + 0.976012i \(0.569860\pi\)
\(18\) 0 0
\(19\) −14.5420 −0.175587 −0.0877935 0.996139i \(-0.527982\pi\)
−0.0877935 + 0.996139i \(0.527982\pi\)
\(20\) 0 0
\(21\) 22.3332 0.232072
\(22\) 0 0
\(23\) −76.8536 −0.696742 −0.348371 0.937357i \(-0.613265\pi\)
−0.348371 + 0.937357i \(0.613265\pi\)
\(24\) 0 0
\(25\) −52.2062 −0.417650
\(26\) 0 0
\(27\) 139.809 0.996529
\(28\) 0 0
\(29\) 149.326 0.956175 0.478088 0.878312i \(-0.341330\pi\)
0.478088 + 0.878312i \(0.341330\pi\)
\(30\) 0 0
\(31\) −188.846 −1.09412 −0.547060 0.837093i \(-0.684253\pi\)
−0.547060 + 0.837093i \(0.684253\pi\)
\(32\) 0 0
\(33\) 35.0951 0.185129
\(34\) 0 0
\(35\) −59.7235 −0.288432
\(36\) 0 0
\(37\) 260.827 1.15891 0.579456 0.815003i \(-0.303265\pi\)
0.579456 + 0.815003i \(0.303265\pi\)
\(38\) 0 0
\(39\) −219.253 −0.900218
\(40\) 0 0
\(41\) 183.545 0.699144 0.349572 0.936909i \(-0.386327\pi\)
0.349572 + 0.936909i \(0.386327\pi\)
\(42\) 0 0
\(43\) 354.695 1.25792 0.628959 0.777438i \(-0.283481\pi\)
0.628959 + 0.777438i \(0.283481\pi\)
\(44\) 0 0
\(45\) −143.515 −0.475421
\(46\) 0 0
\(47\) −420.339 −1.30453 −0.652263 0.757993i \(-0.726180\pi\)
−0.652263 + 0.757993i \(0.726180\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 97.3747 0.267357
\(52\) 0 0
\(53\) −468.426 −1.21402 −0.607012 0.794693i \(-0.707632\pi\)
−0.607012 + 0.794693i \(0.707632\pi\)
\(54\) 0 0
\(55\) −93.8512 −0.230089
\(56\) 0 0
\(57\) 46.3956 0.107811
\(58\) 0 0
\(59\) −160.334 −0.353792 −0.176896 0.984230i \(-0.556606\pi\)
−0.176896 + 0.984230i \(0.556606\pi\)
\(60\) 0 0
\(61\) 26.0410 0.0546592 0.0273296 0.999626i \(-0.491300\pi\)
0.0273296 + 0.999626i \(0.491300\pi\)
\(62\) 0 0
\(63\) 117.747 0.235471
\(64\) 0 0
\(65\) 586.325 1.11884
\(66\) 0 0
\(67\) 397.860 0.725467 0.362734 0.931893i \(-0.381843\pi\)
0.362734 + 0.931893i \(0.381843\pi\)
\(68\) 0 0
\(69\) 245.198 0.427803
\(70\) 0 0
\(71\) 440.130 0.735687 0.367844 0.929888i \(-0.380096\pi\)
0.367844 + 0.929888i \(0.380096\pi\)
\(72\) 0 0
\(73\) 409.850 0.657113 0.328557 0.944484i \(-0.393438\pi\)
0.328557 + 0.944484i \(0.393438\pi\)
\(74\) 0 0
\(75\) 166.562 0.256439
\(76\) 0 0
\(77\) 77.0000 0.113961
\(78\) 0 0
\(79\) 71.6329 0.102017 0.0510084 0.998698i \(-0.483756\pi\)
0.0510084 + 0.998698i \(0.483756\pi\)
\(80\) 0 0
\(81\) 8.10998 0.0111248
\(82\) 0 0
\(83\) −701.049 −0.927110 −0.463555 0.886068i \(-0.653426\pi\)
−0.463555 + 0.886068i \(0.653426\pi\)
\(84\) 0 0
\(85\) −260.399 −0.332285
\(86\) 0 0
\(87\) −476.418 −0.587096
\(88\) 0 0
\(89\) 72.3930 0.0862207 0.0431104 0.999070i \(-0.486273\pi\)
0.0431104 + 0.999070i \(0.486273\pi\)
\(90\) 0 0
\(91\) −481.049 −0.554150
\(92\) 0 0
\(93\) 602.506 0.671795
\(94\) 0 0
\(95\) −124.071 −0.133994
\(96\) 0 0
\(97\) −168.126 −0.175985 −0.0879927 0.996121i \(-0.528045\pi\)
−0.0879927 + 0.996121i \(0.528045\pi\)
\(98\) 0 0
\(99\) 185.030 0.187841
\(100\) 0 0
\(101\) 1847.12 1.81976 0.909880 0.414873i \(-0.136174\pi\)
0.909880 + 0.414873i \(0.136174\pi\)
\(102\) 0 0
\(103\) −819.105 −0.783581 −0.391790 0.920055i \(-0.628144\pi\)
−0.391790 + 0.920055i \(0.628144\pi\)
\(104\) 0 0
\(105\) 190.546 0.177098
\(106\) 0 0
\(107\) 1226.70 1.10831 0.554156 0.832413i \(-0.313041\pi\)
0.554156 + 0.832413i \(0.313041\pi\)
\(108\) 0 0
\(109\) 1023.82 0.899676 0.449838 0.893110i \(-0.351482\pi\)
0.449838 + 0.893110i \(0.351482\pi\)
\(110\) 0 0
\(111\) −832.160 −0.711578
\(112\) 0 0
\(113\) 1786.92 1.48760 0.743802 0.668400i \(-0.233021\pi\)
0.743802 + 0.668400i \(0.233021\pi\)
\(114\) 0 0
\(115\) −655.709 −0.531697
\(116\) 0 0
\(117\) −1155.96 −0.913404
\(118\) 0 0
\(119\) 213.644 0.164577
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) −585.593 −0.429278
\(124\) 0 0
\(125\) −1511.91 −1.08184
\(126\) 0 0
\(127\) −753.691 −0.526608 −0.263304 0.964713i \(-0.584812\pi\)
−0.263304 + 0.964713i \(0.584812\pi\)
\(128\) 0 0
\(129\) −1131.64 −0.772368
\(130\) 0 0
\(131\) −757.722 −0.505362 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(132\) 0 0
\(133\) 101.794 0.0663657
\(134\) 0 0
\(135\) 1192.84 0.760470
\(136\) 0 0
\(137\) −1087.18 −0.677989 −0.338994 0.940788i \(-0.610087\pi\)
−0.338994 + 0.940788i \(0.610087\pi\)
\(138\) 0 0
\(139\) 2896.93 1.76773 0.883865 0.467742i \(-0.154932\pi\)
0.883865 + 0.467742i \(0.154932\pi\)
\(140\) 0 0
\(141\) 1341.07 0.800985
\(142\) 0 0
\(143\) −755.934 −0.442058
\(144\) 0 0
\(145\) 1274.04 0.729675
\(146\) 0 0
\(147\) −156.333 −0.0877150
\(148\) 0 0
\(149\) −2507.36 −1.37860 −0.689298 0.724478i \(-0.742081\pi\)
−0.689298 + 0.724478i \(0.742081\pi\)
\(150\) 0 0
\(151\) 606.134 0.326666 0.163333 0.986571i \(-0.447776\pi\)
0.163333 + 0.986571i \(0.447776\pi\)
\(152\) 0 0
\(153\) 513.385 0.271273
\(154\) 0 0
\(155\) −1611.22 −0.834944
\(156\) 0 0
\(157\) 1476.51 0.750565 0.375282 0.926911i \(-0.377546\pi\)
0.375282 + 0.926911i \(0.377546\pi\)
\(158\) 0 0
\(159\) 1494.50 0.745417
\(160\) 0 0
\(161\) 537.975 0.263344
\(162\) 0 0
\(163\) −190.979 −0.0917706 −0.0458853 0.998947i \(-0.514611\pi\)
−0.0458853 + 0.998947i \(0.514611\pi\)
\(164\) 0 0
\(165\) 299.429 0.141276
\(166\) 0 0
\(167\) 1349.51 0.625319 0.312660 0.949865i \(-0.398780\pi\)
0.312660 + 0.949865i \(0.398780\pi\)
\(168\) 0 0
\(169\) 2525.61 1.14957
\(170\) 0 0
\(171\) 244.610 0.109390
\(172\) 0 0
\(173\) 2263.17 0.994600 0.497300 0.867579i \(-0.334325\pi\)
0.497300 + 0.867579i \(0.334325\pi\)
\(174\) 0 0
\(175\) 365.444 0.157857
\(176\) 0 0
\(177\) 511.540 0.217230
\(178\) 0 0
\(179\) 2712.23 1.13252 0.566262 0.824225i \(-0.308389\pi\)
0.566262 + 0.824225i \(0.308389\pi\)
\(180\) 0 0
\(181\) 2478.66 1.01789 0.508944 0.860800i \(-0.330036\pi\)
0.508944 + 0.860800i \(0.330036\pi\)
\(182\) 0 0
\(183\) −83.0829 −0.0335610
\(184\) 0 0
\(185\) 2225.36 0.884388
\(186\) 0 0
\(187\) 335.726 0.131287
\(188\) 0 0
\(189\) −978.664 −0.376652
\(190\) 0 0
\(191\) 1556.65 0.589715 0.294858 0.955541i \(-0.404728\pi\)
0.294858 + 0.955541i \(0.404728\pi\)
\(192\) 0 0
\(193\) 1298.56 0.484311 0.242156 0.970237i \(-0.422146\pi\)
0.242156 + 0.970237i \(0.422146\pi\)
\(194\) 0 0
\(195\) −1870.65 −0.686973
\(196\) 0 0
\(197\) 2357.82 0.852728 0.426364 0.904552i \(-0.359794\pi\)
0.426364 + 0.904552i \(0.359794\pi\)
\(198\) 0 0
\(199\) 2090.77 0.744776 0.372388 0.928077i \(-0.378539\pi\)
0.372388 + 0.928077i \(0.378539\pi\)
\(200\) 0 0
\(201\) −1269.36 −0.445441
\(202\) 0 0
\(203\) −1045.28 −0.361400
\(204\) 0 0
\(205\) 1565.99 0.533530
\(206\) 0 0
\(207\) 1292.75 0.434069
\(208\) 0 0
\(209\) 159.962 0.0529415
\(210\) 0 0
\(211\) 6011.00 1.96120 0.980602 0.196007i \(-0.0627976\pi\)
0.980602 + 0.196007i \(0.0627976\pi\)
\(212\) 0 0
\(213\) −1404.22 −0.451715
\(214\) 0 0
\(215\) 3026.23 0.959941
\(216\) 0 0
\(217\) 1321.92 0.413539
\(218\) 0 0
\(219\) −1307.61 −0.403471
\(220\) 0 0
\(221\) −2097.41 −0.638403
\(222\) 0 0
\(223\) 1963.16 0.589519 0.294760 0.955571i \(-0.404760\pi\)
0.294760 + 0.955571i \(0.404760\pi\)
\(224\) 0 0
\(225\) 878.159 0.260195
\(226\) 0 0
\(227\) 4697.74 1.37357 0.686784 0.726862i \(-0.259022\pi\)
0.686784 + 0.726862i \(0.259022\pi\)
\(228\) 0 0
\(229\) −1073.32 −0.309724 −0.154862 0.987936i \(-0.549493\pi\)
−0.154862 + 0.987936i \(0.549493\pi\)
\(230\) 0 0
\(231\) −245.666 −0.0699723
\(232\) 0 0
\(233\) −1521.11 −0.427688 −0.213844 0.976868i \(-0.568598\pi\)
−0.213844 + 0.976868i \(0.568598\pi\)
\(234\) 0 0
\(235\) −3586.30 −0.995508
\(236\) 0 0
\(237\) −228.542 −0.0626389
\(238\) 0 0
\(239\) 5589.90 1.51289 0.756444 0.654058i \(-0.226935\pi\)
0.756444 + 0.654058i \(0.226935\pi\)
\(240\) 0 0
\(241\) −330.180 −0.0882520 −0.0441260 0.999026i \(-0.514050\pi\)
−0.0441260 + 0.999026i \(0.514050\pi\)
\(242\) 0 0
\(243\) −3800.72 −1.00336
\(244\) 0 0
\(245\) 418.064 0.109017
\(246\) 0 0
\(247\) −999.342 −0.257436
\(248\) 0 0
\(249\) 2236.67 0.569250
\(250\) 0 0
\(251\) 3343.47 0.840788 0.420394 0.907342i \(-0.361892\pi\)
0.420394 + 0.907342i \(0.361892\pi\)
\(252\) 0 0
\(253\) 845.389 0.210076
\(254\) 0 0
\(255\) 830.794 0.204025
\(256\) 0 0
\(257\) 4255.28 1.03283 0.516414 0.856339i \(-0.327266\pi\)
0.516414 + 0.856339i \(0.327266\pi\)
\(258\) 0 0
\(259\) −1825.79 −0.438028
\(260\) 0 0
\(261\) −2511.80 −0.595696
\(262\) 0 0
\(263\) 1339.79 0.314125 0.157062 0.987589i \(-0.449798\pi\)
0.157062 + 0.987589i \(0.449798\pi\)
\(264\) 0 0
\(265\) −3996.58 −0.926445
\(266\) 0 0
\(267\) −230.967 −0.0529399
\(268\) 0 0
\(269\) 2725.81 0.617827 0.308914 0.951090i \(-0.400035\pi\)
0.308914 + 0.951090i \(0.400035\pi\)
\(270\) 0 0
\(271\) 3717.31 0.833249 0.416625 0.909079i \(-0.363213\pi\)
0.416625 + 0.909079i \(0.363213\pi\)
\(272\) 0 0
\(273\) 1534.77 0.340250
\(274\) 0 0
\(275\) 574.269 0.125926
\(276\) 0 0
\(277\) −6857.46 −1.48745 −0.743727 0.668483i \(-0.766944\pi\)
−0.743727 + 0.668483i \(0.766944\pi\)
\(278\) 0 0
\(279\) 3176.57 0.681635
\(280\) 0 0
\(281\) −2213.01 −0.469812 −0.234906 0.972018i \(-0.575478\pi\)
−0.234906 + 0.972018i \(0.575478\pi\)
\(282\) 0 0
\(283\) 6991.98 1.46866 0.734329 0.678794i \(-0.237497\pi\)
0.734329 + 0.678794i \(0.237497\pi\)
\(284\) 0 0
\(285\) 395.844 0.0822728
\(286\) 0 0
\(287\) −1284.81 −0.264252
\(288\) 0 0
\(289\) −3981.50 −0.810400
\(290\) 0 0
\(291\) 536.399 0.108056
\(292\) 0 0
\(293\) −8640.41 −1.72279 −0.861397 0.507933i \(-0.830410\pi\)
−0.861397 + 0.507933i \(0.830410\pi\)
\(294\) 0 0
\(295\) −1367.96 −0.269985
\(296\) 0 0
\(297\) −1537.90 −0.300465
\(298\) 0 0
\(299\) −5281.47 −1.02152
\(300\) 0 0
\(301\) −2482.87 −0.475448
\(302\) 0 0
\(303\) −5893.18 −1.11734
\(304\) 0 0
\(305\) 222.180 0.0417115
\(306\) 0 0
\(307\) 2848.00 0.529458 0.264729 0.964323i \(-0.414717\pi\)
0.264729 + 0.964323i \(0.414717\pi\)
\(308\) 0 0
\(309\) 2613.32 0.481122
\(310\) 0 0
\(311\) −1412.70 −0.257578 −0.128789 0.991672i \(-0.541109\pi\)
−0.128789 + 0.991672i \(0.541109\pi\)
\(312\) 0 0
\(313\) 4713.31 0.851157 0.425578 0.904922i \(-0.360071\pi\)
0.425578 + 0.904922i \(0.360071\pi\)
\(314\) 0 0
\(315\) 1004.61 0.179692
\(316\) 0 0
\(317\) 709.295 0.125672 0.0628359 0.998024i \(-0.479986\pi\)
0.0628359 + 0.998024i \(0.479986\pi\)
\(318\) 0 0
\(319\) −1642.58 −0.288298
\(320\) 0 0
\(321\) −3913.74 −0.680509
\(322\) 0 0
\(323\) 443.829 0.0764560
\(324\) 0 0
\(325\) −3587.68 −0.612334
\(326\) 0 0
\(327\) −3266.47 −0.552405
\(328\) 0 0
\(329\) 2942.37 0.493064
\(330\) 0 0
\(331\) −2724.16 −0.452367 −0.226183 0.974085i \(-0.572625\pi\)
−0.226183 + 0.974085i \(0.572625\pi\)
\(332\) 0 0
\(333\) −4387.36 −0.722000
\(334\) 0 0
\(335\) 3394.51 0.553618
\(336\) 0 0
\(337\) −5644.70 −0.912422 −0.456211 0.889872i \(-0.650794\pi\)
−0.456211 + 0.889872i \(0.650794\pi\)
\(338\) 0 0
\(339\) −5701.10 −0.913396
\(340\) 0 0
\(341\) 2077.31 0.329890
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) 0 0
\(345\) 2092.01 0.326465
\(346\) 0 0
\(347\) 953.311 0.147482 0.0737412 0.997277i \(-0.476506\pi\)
0.0737412 + 0.997277i \(0.476506\pi\)
\(348\) 0 0
\(349\) −11589.4 −1.77756 −0.888779 0.458337i \(-0.848445\pi\)
−0.888779 + 0.458337i \(0.848445\pi\)
\(350\) 0 0
\(351\) 9607.86 1.46105
\(352\) 0 0
\(353\) 3423.27 0.516154 0.258077 0.966124i \(-0.416911\pi\)
0.258077 + 0.966124i \(0.416911\pi\)
\(354\) 0 0
\(355\) 3755.15 0.561417
\(356\) 0 0
\(357\) −681.623 −0.101051
\(358\) 0 0
\(359\) 2856.22 0.419904 0.209952 0.977712i \(-0.432669\pi\)
0.209952 + 0.977712i \(0.432669\pi\)
\(360\) 0 0
\(361\) −6647.53 −0.969169
\(362\) 0 0
\(363\) −386.046 −0.0558186
\(364\) 0 0
\(365\) 3496.81 0.501455
\(366\) 0 0
\(367\) 4092.10 0.582032 0.291016 0.956718i \(-0.406007\pi\)
0.291016 + 0.956718i \(0.406007\pi\)
\(368\) 0 0
\(369\) −3087.40 −0.435565
\(370\) 0 0
\(371\) 3278.98 0.458858
\(372\) 0 0
\(373\) −5886.94 −0.817196 −0.408598 0.912714i \(-0.633982\pi\)
−0.408598 + 0.912714i \(0.633982\pi\)
\(374\) 0 0
\(375\) 4823.69 0.664252
\(376\) 0 0
\(377\) 10261.8 1.40189
\(378\) 0 0
\(379\) −8729.03 −1.18306 −0.591531 0.806282i \(-0.701476\pi\)
−0.591531 + 0.806282i \(0.701476\pi\)
\(380\) 0 0
\(381\) 2404.62 0.323340
\(382\) 0 0
\(383\) −2232.54 −0.297852 −0.148926 0.988848i \(-0.547582\pi\)
−0.148926 + 0.988848i \(0.547582\pi\)
\(384\) 0 0
\(385\) 656.958 0.0869654
\(386\) 0 0
\(387\) −5966.31 −0.783681
\(388\) 0 0
\(389\) 13901.2 1.81187 0.905937 0.423413i \(-0.139168\pi\)
0.905937 + 0.423413i \(0.139168\pi\)
\(390\) 0 0
\(391\) 2345.61 0.303383
\(392\) 0 0
\(393\) 2417.48 0.310295
\(394\) 0 0
\(395\) 611.167 0.0778510
\(396\) 0 0
\(397\) −3359.38 −0.424691 −0.212345 0.977195i \(-0.568110\pi\)
−0.212345 + 0.977195i \(0.568110\pi\)
\(398\) 0 0
\(399\) −324.769 −0.0407488
\(400\) 0 0
\(401\) −82.2661 −0.0102448 −0.00512241 0.999987i \(-0.501631\pi\)
−0.00512241 + 0.999987i \(0.501631\pi\)
\(402\) 0 0
\(403\) −12977.7 −1.60414
\(404\) 0 0
\(405\) 69.1937 0.00848954
\(406\) 0 0
\(407\) −2869.10 −0.349425
\(408\) 0 0
\(409\) −9945.83 −1.20242 −0.601210 0.799091i \(-0.705315\pi\)
−0.601210 + 0.799091i \(0.705315\pi\)
\(410\) 0 0
\(411\) 3468.62 0.416288
\(412\) 0 0
\(413\) 1122.34 0.133721
\(414\) 0 0
\(415\) −5981.30 −0.707495
\(416\) 0 0
\(417\) −9242.55 −1.08539
\(418\) 0 0
\(419\) 2029.06 0.236578 0.118289 0.992979i \(-0.462259\pi\)
0.118289 + 0.992979i \(0.462259\pi\)
\(420\) 0 0
\(421\) 5169.88 0.598490 0.299245 0.954176i \(-0.403265\pi\)
0.299245 + 0.954176i \(0.403265\pi\)
\(422\) 0 0
\(423\) 7070.50 0.812717
\(424\) 0 0
\(425\) 1593.36 0.181858
\(426\) 0 0
\(427\) −182.287 −0.0206592
\(428\) 0 0
\(429\) 2411.78 0.271426
\(430\) 0 0
\(431\) −13264.9 −1.48248 −0.741241 0.671239i \(-0.765762\pi\)
−0.741241 + 0.671239i \(0.765762\pi\)
\(432\) 0 0
\(433\) −687.605 −0.0763146 −0.0381573 0.999272i \(-0.512149\pi\)
−0.0381573 + 0.999272i \(0.512149\pi\)
\(434\) 0 0
\(435\) −4064.76 −0.448024
\(436\) 0 0
\(437\) 1117.60 0.122339
\(438\) 0 0
\(439\) −1455.57 −0.158247 −0.0791234 0.996865i \(-0.525212\pi\)
−0.0791234 + 0.996865i \(0.525212\pi\)
\(440\) 0 0
\(441\) −824.227 −0.0889997
\(442\) 0 0
\(443\) −12648.9 −1.35658 −0.678291 0.734794i \(-0.737279\pi\)
−0.678291 + 0.734794i \(0.737279\pi\)
\(444\) 0 0
\(445\) 617.652 0.0657966
\(446\) 0 0
\(447\) 7999.64 0.846465
\(448\) 0 0
\(449\) −13112.4 −1.37820 −0.689100 0.724666i \(-0.741994\pi\)
−0.689100 + 0.724666i \(0.741994\pi\)
\(450\) 0 0
\(451\) −2018.99 −0.210800
\(452\) 0 0
\(453\) −1933.85 −0.200574
\(454\) 0 0
\(455\) −4104.27 −0.422882
\(456\) 0 0
\(457\) 8153.14 0.834547 0.417273 0.908781i \(-0.362986\pi\)
0.417273 + 0.908781i \(0.362986\pi\)
\(458\) 0 0
\(459\) −4267.05 −0.433919
\(460\) 0 0
\(461\) −10774.4 −1.08853 −0.544267 0.838912i \(-0.683192\pi\)
−0.544267 + 0.838912i \(0.683192\pi\)
\(462\) 0 0
\(463\) −4799.27 −0.481730 −0.240865 0.970559i \(-0.577431\pi\)
−0.240865 + 0.970559i \(0.577431\pi\)
\(464\) 0 0
\(465\) 5140.54 0.512660
\(466\) 0 0
\(467\) −13503.2 −1.33802 −0.669008 0.743255i \(-0.733281\pi\)
−0.669008 + 0.743255i \(0.733281\pi\)
\(468\) 0 0
\(469\) −2785.02 −0.274201
\(470\) 0 0
\(471\) −4710.76 −0.460850
\(472\) 0 0
\(473\) −3901.65 −0.379277
\(474\) 0 0
\(475\) 759.181 0.0733339
\(476\) 0 0
\(477\) 7879.37 0.756335
\(478\) 0 0
\(479\) −9189.09 −0.876535 −0.438267 0.898845i \(-0.644408\pi\)
−0.438267 + 0.898845i \(0.644408\pi\)
\(480\) 0 0
\(481\) 17924.4 1.69913
\(482\) 0 0
\(483\) −1716.39 −0.161694
\(484\) 0 0
\(485\) −1434.44 −0.134298
\(486\) 0 0
\(487\) 16306.7 1.51731 0.758654 0.651494i \(-0.225857\pi\)
0.758654 + 0.651494i \(0.225857\pi\)
\(488\) 0 0
\(489\) 609.311 0.0563476
\(490\) 0 0
\(491\) −10336.0 −0.950012 −0.475006 0.879982i \(-0.657554\pi\)
−0.475006 + 0.879982i \(0.657554\pi\)
\(492\) 0 0
\(493\) −4557.50 −0.416348
\(494\) 0 0
\(495\) 1578.67 0.143345
\(496\) 0 0
\(497\) −3080.91 −0.278064
\(498\) 0 0
\(499\) −11993.1 −1.07592 −0.537961 0.842970i \(-0.680805\pi\)
−0.537961 + 0.842970i \(0.680805\pi\)
\(500\) 0 0
\(501\) −4305.56 −0.383949
\(502\) 0 0
\(503\) 14930.0 1.32346 0.661728 0.749744i \(-0.269824\pi\)
0.661728 + 0.749744i \(0.269824\pi\)
\(504\) 0 0
\(505\) 15759.5 1.38869
\(506\) 0 0
\(507\) −8057.86 −0.705843
\(508\) 0 0
\(509\) −9086.79 −0.791287 −0.395643 0.918404i \(-0.629478\pi\)
−0.395643 + 0.918404i \(0.629478\pi\)
\(510\) 0 0
\(511\) −2868.95 −0.248365
\(512\) 0 0
\(513\) −2033.10 −0.174978
\(514\) 0 0
\(515\) −6988.54 −0.597965
\(516\) 0 0
\(517\) 4623.73 0.393329
\(518\) 0 0
\(519\) −7220.56 −0.610689
\(520\) 0 0
\(521\) 21210.9 1.78362 0.891811 0.452408i \(-0.149435\pi\)
0.891811 + 0.452408i \(0.149435\pi\)
\(522\) 0 0
\(523\) −10266.9 −0.858392 −0.429196 0.903211i \(-0.641203\pi\)
−0.429196 + 0.903211i \(0.641203\pi\)
\(524\) 0 0
\(525\) −1165.93 −0.0969249
\(526\) 0 0
\(527\) 5763.69 0.476414
\(528\) 0 0
\(529\) −6260.53 −0.514550
\(530\) 0 0
\(531\) 2696.97 0.220412
\(532\) 0 0
\(533\) 12613.4 1.02504
\(534\) 0 0
\(535\) 10466.1 0.845774
\(536\) 0 0
\(537\) −8653.28 −0.695376
\(538\) 0 0
\(539\) −539.000 −0.0430730
\(540\) 0 0
\(541\) 20174.0 1.60323 0.801616 0.597839i \(-0.203974\pi\)
0.801616 + 0.597839i \(0.203974\pi\)
\(542\) 0 0
\(543\) −7908.09 −0.624988
\(544\) 0 0
\(545\) 8735.20 0.686559
\(546\) 0 0
\(547\) −6521.92 −0.509794 −0.254897 0.966968i \(-0.582041\pi\)
−0.254897 + 0.966968i \(0.582041\pi\)
\(548\) 0 0
\(549\) −438.035 −0.0340526
\(550\) 0 0
\(551\) −2171.49 −0.167892
\(552\) 0 0
\(553\) −501.431 −0.0385588
\(554\) 0 0
\(555\) −7099.93 −0.543018
\(556\) 0 0
\(557\) −5459.87 −0.415336 −0.207668 0.978199i \(-0.566587\pi\)
−0.207668 + 0.978199i \(0.566587\pi\)
\(558\) 0 0
\(559\) 24375.1 1.84429
\(560\) 0 0
\(561\) −1071.12 −0.0806110
\(562\) 0 0
\(563\) 4928.99 0.368973 0.184487 0.982835i \(-0.440938\pi\)
0.184487 + 0.982835i \(0.440938\pi\)
\(564\) 0 0
\(565\) 15245.9 1.13522
\(566\) 0 0
\(567\) −56.7699 −0.00420478
\(568\) 0 0
\(569\) 8485.96 0.625220 0.312610 0.949882i \(-0.398797\pi\)
0.312610 + 0.949882i \(0.398797\pi\)
\(570\) 0 0
\(571\) 25006.4 1.83273 0.916363 0.400348i \(-0.131111\pi\)
0.916363 + 0.400348i \(0.131111\pi\)
\(572\) 0 0
\(573\) −4966.45 −0.362088
\(574\) 0 0
\(575\) 4012.24 0.290994
\(576\) 0 0
\(577\) −24688.3 −1.78126 −0.890630 0.454730i \(-0.849736\pi\)
−0.890630 + 0.454730i \(0.849736\pi\)
\(578\) 0 0
\(579\) −4142.99 −0.297369
\(580\) 0 0
\(581\) 4907.34 0.350415
\(582\) 0 0
\(583\) 5152.69 0.366042
\(584\) 0 0
\(585\) −9862.54 −0.697035
\(586\) 0 0
\(587\) 6033.27 0.424224 0.212112 0.977245i \(-0.431966\pi\)
0.212112 + 0.977245i \(0.431966\pi\)
\(588\) 0 0
\(589\) 2746.19 0.192113
\(590\) 0 0
\(591\) −7522.52 −0.523579
\(592\) 0 0
\(593\) −8105.88 −0.561330 −0.280665 0.959806i \(-0.590555\pi\)
−0.280665 + 0.959806i \(0.590555\pi\)
\(594\) 0 0
\(595\) 1822.79 0.125592
\(596\) 0 0
\(597\) −6670.51 −0.457296
\(598\) 0 0
\(599\) −13393.5 −0.913595 −0.456798 0.889571i \(-0.651004\pi\)
−0.456798 + 0.889571i \(0.651004\pi\)
\(600\) 0 0
\(601\) 3006.70 0.204069 0.102035 0.994781i \(-0.467465\pi\)
0.102035 + 0.994781i \(0.467465\pi\)
\(602\) 0 0
\(603\) −6692.38 −0.451965
\(604\) 0 0
\(605\) 1032.36 0.0693744
\(606\) 0 0
\(607\) 7506.67 0.501955 0.250977 0.967993i \(-0.419248\pi\)
0.250977 + 0.967993i \(0.419248\pi\)
\(608\) 0 0
\(609\) 3334.93 0.221902
\(610\) 0 0
\(611\) −28886.2 −1.91262
\(612\) 0 0
\(613\) −13157.8 −0.866950 −0.433475 0.901166i \(-0.642713\pi\)
−0.433475 + 0.901166i \(0.642713\pi\)
\(614\) 0 0
\(615\) −4996.24 −0.327590
\(616\) 0 0
\(617\) 23420.9 1.52818 0.764092 0.645107i \(-0.223187\pi\)
0.764092 + 0.645107i \(0.223187\pi\)
\(618\) 0 0
\(619\) 904.750 0.0587479 0.0293740 0.999568i \(-0.490649\pi\)
0.0293740 + 0.999568i \(0.490649\pi\)
\(620\) 0 0
\(621\) −10744.8 −0.694324
\(622\) 0 0
\(623\) −506.751 −0.0325884
\(624\) 0 0
\(625\) −6373.73 −0.407919
\(626\) 0 0
\(627\) −510.351 −0.0325063
\(628\) 0 0
\(629\) −7960.60 −0.504626
\(630\) 0 0
\(631\) 22840.2 1.44097 0.720486 0.693469i \(-0.243919\pi\)
0.720486 + 0.693469i \(0.243919\pi\)
\(632\) 0 0
\(633\) −19177.9 −1.20419
\(634\) 0 0
\(635\) −6430.43 −0.401865
\(636\) 0 0
\(637\) 3367.34 0.209449
\(638\) 0 0
\(639\) −7403.40 −0.458332
\(640\) 0 0
\(641\) −23658.3 −1.45780 −0.728899 0.684621i \(-0.759968\pi\)
−0.728899 + 0.684621i \(0.759968\pi\)
\(642\) 0 0
\(643\) −1184.17 −0.0726268 −0.0363134 0.999340i \(-0.511561\pi\)
−0.0363134 + 0.999340i \(0.511561\pi\)
\(644\) 0 0
\(645\) −9655.08 −0.589408
\(646\) 0 0
\(647\) 8378.23 0.509091 0.254546 0.967061i \(-0.418074\pi\)
0.254546 + 0.967061i \(0.418074\pi\)
\(648\) 0 0
\(649\) 1763.68 0.106672
\(650\) 0 0
\(651\) −4217.54 −0.253915
\(652\) 0 0
\(653\) 29112.3 1.74464 0.872322 0.488932i \(-0.162613\pi\)
0.872322 + 0.488932i \(0.162613\pi\)
\(654\) 0 0
\(655\) −6464.83 −0.385651
\(656\) 0 0
\(657\) −6894.06 −0.409380
\(658\) 0 0
\(659\) −21401.7 −1.26509 −0.632544 0.774525i \(-0.717989\pi\)
−0.632544 + 0.774525i \(0.717989\pi\)
\(660\) 0 0
\(661\) −1209.18 −0.0711522 −0.0355761 0.999367i \(-0.511327\pi\)
−0.0355761 + 0.999367i \(0.511327\pi\)
\(662\) 0 0
\(663\) 6691.71 0.391983
\(664\) 0 0
\(665\) 868.496 0.0506449
\(666\) 0 0
\(667\) −11476.2 −0.666208
\(668\) 0 0
\(669\) −6263.38 −0.361968
\(670\) 0 0
\(671\) −286.451 −0.0164804
\(672\) 0 0
\(673\) −12981.0 −0.743506 −0.371753 0.928332i \(-0.621243\pi\)
−0.371753 + 0.928332i \(0.621243\pi\)
\(674\) 0 0
\(675\) −7298.91 −0.416200
\(676\) 0 0
\(677\) 32237.3 1.83010 0.915051 0.403339i \(-0.132150\pi\)
0.915051 + 0.403339i \(0.132150\pi\)
\(678\) 0 0
\(679\) 1176.88 0.0665162
\(680\) 0 0
\(681\) −14988.0 −0.843377
\(682\) 0 0
\(683\) −23482.5 −1.31557 −0.657784 0.753207i \(-0.728506\pi\)
−0.657784 + 0.753207i \(0.728506\pi\)
\(684\) 0 0
\(685\) −9275.78 −0.517386
\(686\) 0 0
\(687\) 3424.38 0.190172
\(688\) 0 0
\(689\) −32190.8 −1.77993
\(690\) 0 0
\(691\) 34349.6 1.89106 0.945529 0.325538i \(-0.105545\pi\)
0.945529 + 0.325538i \(0.105545\pi\)
\(692\) 0 0
\(693\) −1295.21 −0.0709972
\(694\) 0 0
\(695\) 24716.4 1.34899
\(696\) 0 0
\(697\) −5601.89 −0.304429
\(698\) 0 0
\(699\) 4853.04 0.262602
\(700\) 0 0
\(701\) 11366.2 0.612406 0.306203 0.951966i \(-0.400941\pi\)
0.306203 + 0.951966i \(0.400941\pi\)
\(702\) 0 0
\(703\) −3792.94 −0.203490
\(704\) 0 0
\(705\) 11442.0 0.611247
\(706\) 0 0
\(707\) −12929.9 −0.687804
\(708\) 0 0
\(709\) 25488.2 1.35011 0.675057 0.737766i \(-0.264119\pi\)
0.675057 + 0.737766i \(0.264119\pi\)
\(710\) 0 0
\(711\) −1204.93 −0.0635563
\(712\) 0 0
\(713\) 14513.5 0.762320
\(714\) 0 0
\(715\) −6449.57 −0.337343
\(716\) 0 0
\(717\) −17834.4 −0.928921
\(718\) 0 0
\(719\) −15240.6 −0.790513 −0.395257 0.918571i \(-0.629344\pi\)
−0.395257 + 0.918571i \(0.629344\pi\)
\(720\) 0 0
\(721\) 5733.73 0.296166
\(722\) 0 0
\(723\) 1053.43 0.0541872
\(724\) 0 0
\(725\) −7795.73 −0.399347
\(726\) 0 0
\(727\) −25465.8 −1.29914 −0.649569 0.760302i \(-0.725051\pi\)
−0.649569 + 0.760302i \(0.725051\pi\)
\(728\) 0 0
\(729\) 11907.1 0.604943
\(730\) 0 0
\(731\) −10825.5 −0.547736
\(732\) 0 0
\(733\) 28441.7 1.43318 0.716588 0.697497i \(-0.245703\pi\)
0.716588 + 0.697497i \(0.245703\pi\)
\(734\) 0 0
\(735\) −1333.82 −0.0669369
\(736\) 0 0
\(737\) −4376.46 −0.218737
\(738\) 0 0
\(739\) 16740.8 0.833315 0.416657 0.909064i \(-0.363201\pi\)
0.416657 + 0.909064i \(0.363201\pi\)
\(740\) 0 0
\(741\) 3188.36 0.158067
\(742\) 0 0
\(743\) −10929.1 −0.539638 −0.269819 0.962911i \(-0.586964\pi\)
−0.269819 + 0.962911i \(0.586964\pi\)
\(744\) 0 0
\(745\) −21392.6 −1.05203
\(746\) 0 0
\(747\) 11792.3 0.577588
\(748\) 0 0
\(749\) −8586.89 −0.418903
\(750\) 0 0
\(751\) −17745.8 −0.862256 −0.431128 0.902291i \(-0.641884\pi\)
−0.431128 + 0.902291i \(0.641884\pi\)
\(752\) 0 0
\(753\) −10667.2 −0.516248
\(754\) 0 0
\(755\) 5171.49 0.249285
\(756\) 0 0
\(757\) −7640.51 −0.366842 −0.183421 0.983034i \(-0.558717\pi\)
−0.183421 + 0.983034i \(0.558717\pi\)
\(758\) 0 0
\(759\) −2697.18 −0.128988
\(760\) 0 0
\(761\) −2633.59 −0.125450 −0.0627250 0.998031i \(-0.519979\pi\)
−0.0627250 + 0.998031i \(0.519979\pi\)
\(762\) 0 0
\(763\) −7166.77 −0.340045
\(764\) 0 0
\(765\) 4380.16 0.207013
\(766\) 0 0
\(767\) −11018.4 −0.518709
\(768\) 0 0
\(769\) 10669.3 0.500317 0.250158 0.968205i \(-0.419517\pi\)
0.250158 + 0.968205i \(0.419517\pi\)
\(770\) 0 0
\(771\) −13576.3 −0.634162
\(772\) 0 0
\(773\) −17023.0 −0.792076 −0.396038 0.918234i \(-0.629615\pi\)
−0.396038 + 0.918234i \(0.629615\pi\)
\(774\) 0 0
\(775\) 9858.94 0.456959
\(776\) 0 0
\(777\) 5825.12 0.268951
\(778\) 0 0
\(779\) −2669.10 −0.122761
\(780\) 0 0
\(781\) −4841.43 −0.221818
\(782\) 0 0
\(783\) 20877.1 0.952856
\(784\) 0 0
\(785\) 12597.5 0.572770
\(786\) 0 0
\(787\) −20986.2 −0.950543 −0.475271 0.879839i \(-0.657650\pi\)
−0.475271 + 0.879839i \(0.657650\pi\)
\(788\) 0 0
\(789\) −4274.54 −0.192874
\(790\) 0 0
\(791\) −12508.4 −0.562261
\(792\) 0 0
\(793\) 1789.57 0.0801382
\(794\) 0 0
\(795\) 12750.9 0.568841
\(796\) 0 0
\(797\) −24611.2 −1.09382 −0.546910 0.837191i \(-0.684196\pi\)
−0.546910 + 0.837191i \(0.684196\pi\)
\(798\) 0 0
\(799\) 12829.0 0.568031
\(800\) 0 0
\(801\) −1217.72 −0.0537153
\(802\) 0 0
\(803\) −4508.34 −0.198127
\(804\) 0 0
\(805\) 4589.96 0.200963
\(806\) 0 0
\(807\) −8696.59 −0.379349
\(808\) 0 0
\(809\) −21256.6 −0.923787 −0.461893 0.886935i \(-0.652830\pi\)
−0.461893 + 0.886935i \(0.652830\pi\)
\(810\) 0 0
\(811\) −19012.5 −0.823203 −0.411601 0.911364i \(-0.635030\pi\)
−0.411601 + 0.911364i \(0.635030\pi\)
\(812\) 0 0
\(813\) −11859.9 −0.511619
\(814\) 0 0
\(815\) −1629.42 −0.0700319
\(816\) 0 0
\(817\) −5157.96 −0.220874
\(818\) 0 0
\(819\) 8091.70 0.345234
\(820\) 0 0
\(821\) 2422.77 0.102991 0.0514953 0.998673i \(-0.483601\pi\)
0.0514953 + 0.998673i \(0.483601\pi\)
\(822\) 0 0
\(823\) 32860.8 1.39181 0.695903 0.718136i \(-0.255004\pi\)
0.695903 + 0.718136i \(0.255004\pi\)
\(824\) 0 0
\(825\) −1832.18 −0.0773193
\(826\) 0 0
\(827\) 9714.43 0.408469 0.204234 0.978922i \(-0.434530\pi\)
0.204234 + 0.978922i \(0.434530\pi\)
\(828\) 0 0
\(829\) 25958.6 1.08755 0.543775 0.839231i \(-0.316994\pi\)
0.543775 + 0.839231i \(0.316994\pi\)
\(830\) 0 0
\(831\) 21878.5 0.913304
\(832\) 0 0
\(833\) −1495.51 −0.0622044
\(834\) 0 0
\(835\) 11513.9 0.477193
\(836\) 0 0
\(837\) −26402.4 −1.09032
\(838\) 0 0
\(839\) 15023.6 0.618203 0.309102 0.951029i \(-0.399972\pi\)
0.309102 + 0.951029i \(0.399972\pi\)
\(840\) 0 0
\(841\) −2090.83 −0.0857285
\(842\) 0 0
\(843\) 7060.53 0.288467
\(844\) 0 0
\(845\) 21548.3 0.877260
\(846\) 0 0
\(847\) −847.000 −0.0343604
\(848\) 0 0
\(849\) −22307.6 −0.901762
\(850\) 0 0
\(851\) −20045.5 −0.807463
\(852\) 0 0
\(853\) 8558.51 0.343538 0.171769 0.985137i \(-0.445052\pi\)
0.171769 + 0.985137i \(0.445052\pi\)
\(854\) 0 0
\(855\) 2086.99 0.0834779
\(856\) 0 0
\(857\) 24270.5 0.967403 0.483702 0.875233i \(-0.339292\pi\)
0.483702 + 0.875233i \(0.339292\pi\)
\(858\) 0 0
\(859\) 33934.7 1.34789 0.673945 0.738782i \(-0.264599\pi\)
0.673945 + 0.738782i \(0.264599\pi\)
\(860\) 0 0
\(861\) 4099.15 0.162252
\(862\) 0 0
\(863\) −23863.8 −0.941291 −0.470646 0.882322i \(-0.655979\pi\)
−0.470646 + 0.882322i \(0.655979\pi\)
\(864\) 0 0
\(865\) 19309.2 0.758998
\(866\) 0 0
\(867\) 12702.8 0.497590
\(868\) 0 0
\(869\) −787.962 −0.0307593
\(870\) 0 0
\(871\) 27341.4 1.06364
\(872\) 0 0
\(873\) 2828.03 0.109639
\(874\) 0 0
\(875\) 10583.4 0.408895
\(876\) 0 0
\(877\) −44421.4 −1.71038 −0.855191 0.518312i \(-0.826560\pi\)
−0.855191 + 0.518312i \(0.826560\pi\)
\(878\) 0 0
\(879\) 27566.9 1.05780
\(880\) 0 0
\(881\) −23584.1 −0.901893 −0.450947 0.892551i \(-0.648914\pi\)
−0.450947 + 0.892551i \(0.648914\pi\)
\(882\) 0 0
\(883\) 40074.6 1.52731 0.763657 0.645622i \(-0.223402\pi\)
0.763657 + 0.645622i \(0.223402\pi\)
\(884\) 0 0
\(885\) 4364.42 0.165772
\(886\) 0 0
\(887\) 47624.2 1.80278 0.901388 0.433012i \(-0.142549\pi\)
0.901388 + 0.433012i \(0.142549\pi\)
\(888\) 0 0
\(889\) 5275.84 0.199039
\(890\) 0 0
\(891\) −89.2098 −0.00335425
\(892\) 0 0
\(893\) 6112.55 0.229058
\(894\) 0 0
\(895\) 23140.6 0.864251
\(896\) 0 0
\(897\) 16850.3 0.627220
\(898\) 0 0
\(899\) −28199.6 −1.04617
\(900\) 0 0
\(901\) 14296.6 0.528623
\(902\) 0 0
\(903\) 7921.49 0.291928
\(904\) 0 0
\(905\) 21147.8 0.776769
\(906\) 0 0
\(907\) 17445.9 0.638680 0.319340 0.947640i \(-0.396539\pi\)
0.319340 + 0.947640i \(0.396539\pi\)
\(908\) 0 0
\(909\) −31070.4 −1.13371
\(910\) 0 0
\(911\) −44006.4 −1.60044 −0.800218 0.599709i \(-0.795283\pi\)
−0.800218 + 0.599709i \(0.795283\pi\)
\(912\) 0 0
\(913\) 7711.54 0.279534
\(914\) 0 0
\(915\) −708.857 −0.0256110
\(916\) 0 0
\(917\) 5304.05 0.191009
\(918\) 0 0
\(919\) 5259.05 0.188771 0.0943853 0.995536i \(-0.469911\pi\)
0.0943853 + 0.995536i \(0.469911\pi\)
\(920\) 0 0
\(921\) −9086.42 −0.325090
\(922\) 0 0
\(923\) 30246.3 1.07862
\(924\) 0 0
\(925\) −13616.8 −0.484020
\(926\) 0 0
\(927\) 13778.1 0.488169
\(928\) 0 0
\(929\) −10092.4 −0.356428 −0.178214 0.983992i \(-0.557032\pi\)
−0.178214 + 0.983992i \(0.557032\pi\)
\(930\) 0 0
\(931\) −712.556 −0.0250839
\(932\) 0 0
\(933\) 4507.17 0.158154
\(934\) 0 0
\(935\) 2864.39 0.100188
\(936\) 0 0
\(937\) −50066.0 −1.74555 −0.872777 0.488119i \(-0.837683\pi\)
−0.872777 + 0.488119i \(0.837683\pi\)
\(938\) 0 0
\(939\) −15037.6 −0.522614
\(940\) 0 0
\(941\) −10595.4 −0.367057 −0.183529 0.983014i \(-0.558752\pi\)
−0.183529 + 0.983014i \(0.558752\pi\)
\(942\) 0 0
\(943\) −14106.1 −0.487123
\(944\) 0 0
\(945\) −8349.89 −0.287430
\(946\) 0 0
\(947\) −48636.9 −1.66894 −0.834471 0.551052i \(-0.814227\pi\)
−0.834471 + 0.551052i \(0.814227\pi\)
\(948\) 0 0
\(949\) 28165.4 0.963421
\(950\) 0 0
\(951\) −2262.98 −0.0771631
\(952\) 0 0
\(953\) 29709.9 1.00986 0.504931 0.863160i \(-0.331518\pi\)
0.504931 + 0.863160i \(0.331518\pi\)
\(954\) 0 0
\(955\) 13281.3 0.450023
\(956\) 0 0
\(957\) 5240.60 0.177016
\(958\) 0 0
\(959\) 7610.29 0.256256
\(960\) 0 0
\(961\) 5871.81 0.197100
\(962\) 0 0
\(963\) −20634.2 −0.690477
\(964\) 0 0
\(965\) 11079.2 0.369587
\(966\) 0 0
\(967\) 23662.3 0.786897 0.393448 0.919347i \(-0.371282\pi\)
0.393448 + 0.919347i \(0.371282\pi\)
\(968\) 0 0
\(969\) −1416.02 −0.0469443
\(970\) 0 0
\(971\) 6841.96 0.226127 0.113063 0.993588i \(-0.463934\pi\)
0.113063 + 0.993588i \(0.463934\pi\)
\(972\) 0 0
\(973\) −20278.5 −0.668139
\(974\) 0 0
\(975\) 11446.4 0.375976
\(976\) 0 0
\(977\) −35439.8 −1.16051 −0.580256 0.814434i \(-0.697047\pi\)
−0.580256 + 0.814434i \(0.697047\pi\)
\(978\) 0 0
\(979\) −796.323 −0.0259965
\(980\) 0 0
\(981\) −17221.7 −0.560496
\(982\) 0 0
\(983\) −20807.5 −0.675133 −0.337566 0.941302i \(-0.609604\pi\)
−0.337566 + 0.941302i \(0.609604\pi\)
\(984\) 0 0
\(985\) 20116.7 0.650733
\(986\) 0 0
\(987\) −9387.52 −0.302744
\(988\) 0 0
\(989\) −27259.6 −0.876445
\(990\) 0 0
\(991\) −2744.82 −0.0879840 −0.0439920 0.999032i \(-0.514008\pi\)
−0.0439920 + 0.999032i \(0.514008\pi\)
\(992\) 0 0
\(993\) 8691.33 0.277755
\(994\) 0 0
\(995\) 17838.3 0.568353
\(996\) 0 0
\(997\) 8624.49 0.273962 0.136981 0.990574i \(-0.456260\pi\)
0.136981 + 0.990574i \(0.456260\pi\)
\(998\) 0 0
\(999\) 36466.0 1.15489
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 1232.4.a.bc.1.3 7
4.3 odd 2 616.4.a.k.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
616.4.a.k.1.5 7 4.3 odd 2
1232.4.a.bc.1.3 7 1.1 even 1 trivial