Properties

Label 1840.4.a.ba.1.9
Level $1840$
Weight $4$
Character 1840.1
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $10$
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] = [1840,4,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, 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("1840.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(108.563514411\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} - 192 x^{8} + 762 x^{7} + 12246 x^{6} - 33828 x^{5} - 298243 x^{4} + 383603 x^{3} + 2423016 x^{2} + 864576 x + 57408 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-7.60722\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.60722 q^{3} +5.00000 q^{5} +20.6846 q^{7} +30.8698 q^{9} +O(q^{10})\) \(q+7.60722 q^{3} +5.00000 q^{5} +20.6846 q^{7} +30.8698 q^{9} +65.9547 q^{11} -73.2440 q^{13} +38.0361 q^{15} +70.9305 q^{17} +79.5074 q^{19} +157.352 q^{21} +23.0000 q^{23} +25.0000 q^{25} +29.4382 q^{27} -244.561 q^{29} +283.927 q^{31} +501.732 q^{33} +103.423 q^{35} -296.020 q^{37} -557.183 q^{39} +366.935 q^{41} +343.529 q^{43} +154.349 q^{45} -309.611 q^{47} +84.8511 q^{49} +539.584 q^{51} -477.489 q^{53} +329.774 q^{55} +604.830 q^{57} +251.707 q^{59} -60.6199 q^{61} +638.528 q^{63} -366.220 q^{65} -433.850 q^{67} +174.966 q^{69} +1119.80 q^{71} -583.370 q^{73} +190.180 q^{75} +1364.24 q^{77} +804.524 q^{79} -609.541 q^{81} -892.200 q^{83} +354.652 q^{85} -1860.43 q^{87} +911.783 q^{89} -1515.02 q^{91} +2159.89 q^{93} +397.537 q^{95} +53.3224 q^{97} +2036.01 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9} + 56 q^{11} + 49 q^{13} - 25 q^{15} + 240 q^{17} - 88 q^{19} + 346 q^{21} + 230 q^{23} + 250 q^{25} - 449 q^{27} + 319 q^{29} + 109 q^{31} + 504 q^{33} - 70 q^{35} + 580 q^{37} - 107 q^{39} + 259 q^{41} - 330 q^{43} + 695 q^{45} - 227 q^{47} + 630 q^{49} + 192 q^{51} - 186 q^{53} + 280 q^{55} + 1708 q^{57} - 262 q^{59} + 1000 q^{61} - 722 q^{63} + 245 q^{65} - 354 q^{67} - 115 q^{69} - 599 q^{71} + 355 q^{73} - 125 q^{75} + 1776 q^{77} + 1068 q^{79} + 3490 q^{81} - 754 q^{83} + 1200 q^{85} - 2675 q^{87} + 1740 q^{89} - 690 q^{91} + 1669 q^{93} - 440 q^{95} + 2592 q^{97} - 916 q^{99}+O(q^{100}) \) 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
\(3\) 7.60722 1.46401 0.732005 0.681299i \(-0.238585\pi\)
0.732005 + 0.681299i \(0.238585\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 20.6846 1.11686 0.558431 0.829551i \(-0.311404\pi\)
0.558431 + 0.829551i \(0.311404\pi\)
\(8\) 0 0
\(9\) 30.8698 1.14332
\(10\) 0 0
\(11\) 65.9547 1.80783 0.903914 0.427715i \(-0.140681\pi\)
0.903914 + 0.427715i \(0.140681\pi\)
\(12\) 0 0
\(13\) −73.2440 −1.56263 −0.781316 0.624135i \(-0.785451\pi\)
−0.781316 + 0.624135i \(0.785451\pi\)
\(14\) 0 0
\(15\) 38.0361 0.654725
\(16\) 0 0
\(17\) 70.9305 1.01195 0.505975 0.862548i \(-0.331133\pi\)
0.505975 + 0.862548i \(0.331133\pi\)
\(18\) 0 0
\(19\) 79.5074 0.960014 0.480007 0.877265i \(-0.340634\pi\)
0.480007 + 0.877265i \(0.340634\pi\)
\(20\) 0 0
\(21\) 157.352 1.63510
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 29.4382 0.209829
\(28\) 0 0
\(29\) −244.561 −1.56600 −0.782998 0.622024i \(-0.786311\pi\)
−0.782998 + 0.622024i \(0.786311\pi\)
\(30\) 0 0
\(31\) 283.927 1.64499 0.822496 0.568770i \(-0.192581\pi\)
0.822496 + 0.568770i \(0.192581\pi\)
\(32\) 0 0
\(33\) 501.732 2.64668
\(34\) 0 0
\(35\) 103.423 0.499476
\(36\) 0 0
\(37\) −296.020 −1.31528 −0.657639 0.753333i \(-0.728445\pi\)
−0.657639 + 0.753333i \(0.728445\pi\)
\(38\) 0 0
\(39\) −557.183 −2.28771
\(40\) 0 0
\(41\) 366.935 1.39770 0.698849 0.715270i \(-0.253696\pi\)
0.698849 + 0.715270i \(0.253696\pi\)
\(42\) 0 0
\(43\) 343.529 1.21832 0.609158 0.793049i \(-0.291507\pi\)
0.609158 + 0.793049i \(0.291507\pi\)
\(44\) 0 0
\(45\) 154.349 0.511310
\(46\) 0 0
\(47\) −309.611 −0.960881 −0.480441 0.877027i \(-0.659523\pi\)
−0.480441 + 0.877027i \(0.659523\pi\)
\(48\) 0 0
\(49\) 84.8511 0.247379
\(50\) 0 0
\(51\) 539.584 1.48151
\(52\) 0 0
\(53\) −477.489 −1.23751 −0.618756 0.785583i \(-0.712363\pi\)
−0.618756 + 0.785583i \(0.712363\pi\)
\(54\) 0 0
\(55\) 329.774 0.808485
\(56\) 0 0
\(57\) 604.830 1.40547
\(58\) 0 0
\(59\) 251.707 0.555415 0.277708 0.960666i \(-0.410425\pi\)
0.277708 + 0.960666i \(0.410425\pi\)
\(60\) 0 0
\(61\) −60.6199 −0.127239 −0.0636195 0.997974i \(-0.520264\pi\)
−0.0636195 + 0.997974i \(0.520264\pi\)
\(62\) 0 0
\(63\) 638.528 1.27694
\(64\) 0 0
\(65\) −366.220 −0.698830
\(66\) 0 0
\(67\) −433.850 −0.791093 −0.395546 0.918446i \(-0.629445\pi\)
−0.395546 + 0.918446i \(0.629445\pi\)
\(68\) 0 0
\(69\) 174.966 0.305267
\(70\) 0 0
\(71\) 1119.80 1.87178 0.935888 0.352299i \(-0.114600\pi\)
0.935888 + 0.352299i \(0.114600\pi\)
\(72\) 0 0
\(73\) −583.370 −0.935318 −0.467659 0.883909i \(-0.654903\pi\)
−0.467659 + 0.883909i \(0.654903\pi\)
\(74\) 0 0
\(75\) 190.180 0.292802
\(76\) 0 0
\(77\) 1364.24 2.01909
\(78\) 0 0
\(79\) 804.524 1.14577 0.572886 0.819635i \(-0.305824\pi\)
0.572886 + 0.819635i \(0.305824\pi\)
\(80\) 0 0
\(81\) −609.541 −0.836133
\(82\) 0 0
\(83\) −892.200 −1.17990 −0.589950 0.807440i \(-0.700852\pi\)
−0.589950 + 0.807440i \(0.700852\pi\)
\(84\) 0 0
\(85\) 354.652 0.452558
\(86\) 0 0
\(87\) −1860.43 −2.29263
\(88\) 0 0
\(89\) 911.783 1.08594 0.542971 0.839752i \(-0.317299\pi\)
0.542971 + 0.839752i \(0.317299\pi\)
\(90\) 0 0
\(91\) −1515.02 −1.74524
\(92\) 0 0
\(93\) 2159.89 2.40829
\(94\) 0 0
\(95\) 397.537 0.429331
\(96\) 0 0
\(97\) 53.3224 0.0558151 0.0279076 0.999611i \(-0.491116\pi\)
0.0279076 + 0.999611i \(0.491116\pi\)
\(98\) 0 0
\(99\) 2036.01 2.06693
\(100\) 0 0
\(101\) 134.300 0.132311 0.0661554 0.997809i \(-0.478927\pi\)
0.0661554 + 0.997809i \(0.478927\pi\)
\(102\) 0 0
\(103\) −253.628 −0.242628 −0.121314 0.992614i \(-0.538711\pi\)
−0.121314 + 0.992614i \(0.538711\pi\)
\(104\) 0 0
\(105\) 786.760 0.731237
\(106\) 0 0
\(107\) −1966.06 −1.77632 −0.888161 0.459532i \(-0.848017\pi\)
−0.888161 + 0.459532i \(0.848017\pi\)
\(108\) 0 0
\(109\) −468.098 −0.411336 −0.205668 0.978622i \(-0.565937\pi\)
−0.205668 + 0.978622i \(0.565937\pi\)
\(110\) 0 0
\(111\) −2251.89 −1.92558
\(112\) 0 0
\(113\) 62.6915 0.0521904 0.0260952 0.999659i \(-0.491693\pi\)
0.0260952 + 0.999659i \(0.491693\pi\)
\(114\) 0 0
\(115\) 115.000 0.0932505
\(116\) 0 0
\(117\) −2261.02 −1.78660
\(118\) 0 0
\(119\) 1467.17 1.13021
\(120\) 0 0
\(121\) 3019.03 2.26824
\(122\) 0 0
\(123\) 2791.35 2.04624
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −2199.61 −1.53688 −0.768442 0.639920i \(-0.778968\pi\)
−0.768442 + 0.639920i \(0.778968\pi\)
\(128\) 0 0
\(129\) 2613.30 1.78363
\(130\) 0 0
\(131\) −1375.91 −0.917662 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(132\) 0 0
\(133\) 1644.58 1.07220
\(134\) 0 0
\(135\) 147.191 0.0938383
\(136\) 0 0
\(137\) 6.01310 0.00374988 0.00187494 0.999998i \(-0.499403\pi\)
0.00187494 + 0.999998i \(0.499403\pi\)
\(138\) 0 0
\(139\) 1522.15 0.928827 0.464414 0.885618i \(-0.346265\pi\)
0.464414 + 0.885618i \(0.346265\pi\)
\(140\) 0 0
\(141\) −2355.28 −1.40674
\(142\) 0 0
\(143\) −4830.79 −2.82497
\(144\) 0 0
\(145\) −1222.81 −0.700335
\(146\) 0 0
\(147\) 645.481 0.362166
\(148\) 0 0
\(149\) −440.008 −0.241925 −0.120963 0.992657i \(-0.538598\pi\)
−0.120963 + 0.992657i \(0.538598\pi\)
\(150\) 0 0
\(151\) 2902.27 1.56413 0.782065 0.623197i \(-0.214167\pi\)
0.782065 + 0.623197i \(0.214167\pi\)
\(152\) 0 0
\(153\) 2189.61 1.15699
\(154\) 0 0
\(155\) 1419.63 0.735663
\(156\) 0 0
\(157\) 22.9338 0.0116581 0.00582904 0.999983i \(-0.498145\pi\)
0.00582904 + 0.999983i \(0.498145\pi\)
\(158\) 0 0
\(159\) −3632.36 −1.81173
\(160\) 0 0
\(161\) 475.745 0.232882
\(162\) 0 0
\(163\) −1666.89 −0.800989 −0.400494 0.916299i \(-0.631162\pi\)
−0.400494 + 0.916299i \(0.631162\pi\)
\(164\) 0 0
\(165\) 2508.66 1.18363
\(166\) 0 0
\(167\) 1048.90 0.486025 0.243012 0.970023i \(-0.421864\pi\)
0.243012 + 0.970023i \(0.421864\pi\)
\(168\) 0 0
\(169\) 3167.68 1.44182
\(170\) 0 0
\(171\) 2454.38 1.09761
\(172\) 0 0
\(173\) 1273.69 0.559752 0.279876 0.960036i \(-0.409707\pi\)
0.279876 + 0.960036i \(0.409707\pi\)
\(174\) 0 0
\(175\) 517.114 0.223372
\(176\) 0 0
\(177\) 1914.79 0.813133
\(178\) 0 0
\(179\) −2408.10 −1.00553 −0.502765 0.864423i \(-0.667684\pi\)
−0.502765 + 0.864423i \(0.667684\pi\)
\(180\) 0 0
\(181\) 1038.18 0.426339 0.213170 0.977015i \(-0.431621\pi\)
0.213170 + 0.977015i \(0.431621\pi\)
\(182\) 0 0
\(183\) −461.149 −0.186279
\(184\) 0 0
\(185\) −1480.10 −0.588211
\(186\) 0 0
\(187\) 4678.20 1.82943
\(188\) 0 0
\(189\) 608.916 0.234350
\(190\) 0 0
\(191\) 890.919 0.337511 0.168756 0.985658i \(-0.446025\pi\)
0.168756 + 0.985658i \(0.446025\pi\)
\(192\) 0 0
\(193\) −136.087 −0.0507552 −0.0253776 0.999678i \(-0.508079\pi\)
−0.0253776 + 0.999678i \(0.508079\pi\)
\(194\) 0 0
\(195\) −2785.91 −1.02309
\(196\) 0 0
\(197\) 1473.55 0.532926 0.266463 0.963845i \(-0.414145\pi\)
0.266463 + 0.963845i \(0.414145\pi\)
\(198\) 0 0
\(199\) 287.628 0.102459 0.0512296 0.998687i \(-0.483686\pi\)
0.0512296 + 0.998687i \(0.483686\pi\)
\(200\) 0 0
\(201\) −3300.39 −1.15817
\(202\) 0 0
\(203\) −5058.64 −1.74900
\(204\) 0 0
\(205\) 1834.67 0.625069
\(206\) 0 0
\(207\) 710.005 0.238400
\(208\) 0 0
\(209\) 5243.89 1.73554
\(210\) 0 0
\(211\) 4392.16 1.43303 0.716513 0.697573i \(-0.245737\pi\)
0.716513 + 0.697573i \(0.245737\pi\)
\(212\) 0 0
\(213\) 8518.58 2.74030
\(214\) 0 0
\(215\) 1717.64 0.544848
\(216\) 0 0
\(217\) 5872.90 1.83723
\(218\) 0 0
\(219\) −4437.82 −1.36932
\(220\) 0 0
\(221\) −5195.23 −1.58131
\(222\) 0 0
\(223\) 5397.64 1.62087 0.810433 0.585832i \(-0.199232\pi\)
0.810433 + 0.585832i \(0.199232\pi\)
\(224\) 0 0
\(225\) 771.744 0.228665
\(226\) 0 0
\(227\) 2518.70 0.736441 0.368220 0.929739i \(-0.379967\pi\)
0.368220 + 0.929739i \(0.379967\pi\)
\(228\) 0 0
\(229\) −4753.03 −1.37157 −0.685785 0.727805i \(-0.740541\pi\)
−0.685785 + 0.727805i \(0.740541\pi\)
\(230\) 0 0
\(231\) 10378.1 2.95597
\(232\) 0 0
\(233\) −1194.30 −0.335799 −0.167900 0.985804i \(-0.553698\pi\)
−0.167900 + 0.985804i \(0.553698\pi\)
\(234\) 0 0
\(235\) −1548.06 −0.429719
\(236\) 0 0
\(237\) 6120.19 1.67742
\(238\) 0 0
\(239\) −5851.02 −1.58356 −0.791780 0.610806i \(-0.790845\pi\)
−0.791780 + 0.610806i \(0.790845\pi\)
\(240\) 0 0
\(241\) 658.076 0.175894 0.0879469 0.996125i \(-0.471969\pi\)
0.0879469 + 0.996125i \(0.471969\pi\)
\(242\) 0 0
\(243\) −5431.74 −1.43394
\(244\) 0 0
\(245\) 424.255 0.110631
\(246\) 0 0
\(247\) −5823.44 −1.50015
\(248\) 0 0
\(249\) −6787.16 −1.72738
\(250\) 0 0
\(251\) 2724.99 0.685259 0.342629 0.939471i \(-0.388682\pi\)
0.342629 + 0.939471i \(0.388682\pi\)
\(252\) 0 0
\(253\) 1516.96 0.376958
\(254\) 0 0
\(255\) 2697.92 0.662550
\(256\) 0 0
\(257\) −3320.88 −0.806035 −0.403018 0.915192i \(-0.632039\pi\)
−0.403018 + 0.915192i \(0.632039\pi\)
\(258\) 0 0
\(259\) −6123.03 −1.46898
\(260\) 0 0
\(261\) −7549.55 −1.79044
\(262\) 0 0
\(263\) −432.404 −0.101381 −0.0506904 0.998714i \(-0.516142\pi\)
−0.0506904 + 0.998714i \(0.516142\pi\)
\(264\) 0 0
\(265\) −2387.44 −0.553432
\(266\) 0 0
\(267\) 6936.13 1.58983
\(268\) 0 0
\(269\) −4408.43 −0.999207 −0.499603 0.866254i \(-0.666521\pi\)
−0.499603 + 0.866254i \(0.666521\pi\)
\(270\) 0 0
\(271\) −8604.93 −1.92883 −0.964414 0.264398i \(-0.914827\pi\)
−0.964414 + 0.264398i \(0.914827\pi\)
\(272\) 0 0
\(273\) −11525.1 −2.55505
\(274\) 0 0
\(275\) 1648.87 0.361565
\(276\) 0 0
\(277\) −6830.24 −1.48155 −0.740775 0.671753i \(-0.765541\pi\)
−0.740775 + 0.671753i \(0.765541\pi\)
\(278\) 0 0
\(279\) 8764.76 1.88076
\(280\) 0 0
\(281\) −8532.12 −1.81133 −0.905665 0.423994i \(-0.860628\pi\)
−0.905665 + 0.423994i \(0.860628\pi\)
\(282\) 0 0
\(283\) 6759.86 1.41990 0.709950 0.704252i \(-0.248717\pi\)
0.709950 + 0.704252i \(0.248717\pi\)
\(284\) 0 0
\(285\) 3024.15 0.628545
\(286\) 0 0
\(287\) 7589.88 1.56103
\(288\) 0 0
\(289\) 118.130 0.0240444
\(290\) 0 0
\(291\) 405.635 0.0817139
\(292\) 0 0
\(293\) −5067.26 −1.01035 −0.505175 0.863017i \(-0.668572\pi\)
−0.505175 + 0.863017i \(0.668572\pi\)
\(294\) 0 0
\(295\) 1258.54 0.248389
\(296\) 0 0
\(297\) 1941.59 0.379334
\(298\) 0 0
\(299\) −1684.61 −0.325831
\(300\) 0 0
\(301\) 7105.74 1.36069
\(302\) 0 0
\(303\) 1021.65 0.193704
\(304\) 0 0
\(305\) −303.100 −0.0569031
\(306\) 0 0
\(307\) 4246.60 0.789467 0.394734 0.918796i \(-0.370837\pi\)
0.394734 + 0.918796i \(0.370837\pi\)
\(308\) 0 0
\(309\) −1929.40 −0.355210
\(310\) 0 0
\(311\) 918.593 0.167487 0.0837437 0.996487i \(-0.473312\pi\)
0.0837437 + 0.996487i \(0.473312\pi\)
\(312\) 0 0
\(313\) 530.478 0.0957967 0.0478984 0.998852i \(-0.484748\pi\)
0.0478984 + 0.998852i \(0.484748\pi\)
\(314\) 0 0
\(315\) 3192.64 0.571063
\(316\) 0 0
\(317\) 7854.64 1.39167 0.695837 0.718200i \(-0.255034\pi\)
0.695837 + 0.718200i \(0.255034\pi\)
\(318\) 0 0
\(319\) −16130.0 −2.83105
\(320\) 0 0
\(321\) −14956.3 −2.60055
\(322\) 0 0
\(323\) 5639.50 0.971487
\(324\) 0 0
\(325\) −1831.10 −0.312526
\(326\) 0 0
\(327\) −3560.92 −0.602200
\(328\) 0 0
\(329\) −6404.17 −1.07317
\(330\) 0 0
\(331\) 11622.4 1.92998 0.964992 0.262280i \(-0.0844746\pi\)
0.964992 + 0.262280i \(0.0844746\pi\)
\(332\) 0 0
\(333\) −9138.06 −1.50379
\(334\) 0 0
\(335\) −2169.25 −0.353787
\(336\) 0 0
\(337\) −4454.29 −0.720002 −0.360001 0.932952i \(-0.617224\pi\)
−0.360001 + 0.932952i \(0.617224\pi\)
\(338\) 0 0
\(339\) 476.908 0.0764073
\(340\) 0 0
\(341\) 18726.3 2.97386
\(342\) 0 0
\(343\) −5339.70 −0.840573
\(344\) 0 0
\(345\) 874.830 0.136520
\(346\) 0 0
\(347\) −7319.54 −1.13237 −0.566187 0.824277i \(-0.691582\pi\)
−0.566187 + 0.824277i \(0.691582\pi\)
\(348\) 0 0
\(349\) 7826.29 1.20038 0.600189 0.799858i \(-0.295092\pi\)
0.600189 + 0.799858i \(0.295092\pi\)
\(350\) 0 0
\(351\) −2156.17 −0.327885
\(352\) 0 0
\(353\) 7775.30 1.17234 0.586172 0.810187i \(-0.300634\pi\)
0.586172 + 0.810187i \(0.300634\pi\)
\(354\) 0 0
\(355\) 5599.01 0.837083
\(356\) 0 0
\(357\) 11161.0 1.65464
\(358\) 0 0
\(359\) 737.788 0.108465 0.0542325 0.998528i \(-0.482729\pi\)
0.0542325 + 0.998528i \(0.482729\pi\)
\(360\) 0 0
\(361\) −537.567 −0.0783740
\(362\) 0 0
\(363\) 22966.4 3.32073
\(364\) 0 0
\(365\) −2916.85 −0.418287
\(366\) 0 0
\(367\) 10394.4 1.47843 0.739214 0.673471i \(-0.235197\pi\)
0.739214 + 0.673471i \(0.235197\pi\)
\(368\) 0 0
\(369\) 11327.2 1.59802
\(370\) 0 0
\(371\) −9876.64 −1.38213
\(372\) 0 0
\(373\) 12595.6 1.74846 0.874232 0.485508i \(-0.161365\pi\)
0.874232 + 0.485508i \(0.161365\pi\)
\(374\) 0 0
\(375\) 950.902 0.130945
\(376\) 0 0
\(377\) 17912.6 2.44708
\(378\) 0 0
\(379\) 596.932 0.0809033 0.0404516 0.999181i \(-0.487120\pi\)
0.0404516 + 0.999181i \(0.487120\pi\)
\(380\) 0 0
\(381\) −16732.9 −2.25001
\(382\) 0 0
\(383\) −7790.02 −1.03930 −0.519650 0.854380i \(-0.673937\pi\)
−0.519650 + 0.854380i \(0.673937\pi\)
\(384\) 0 0
\(385\) 6821.22 0.902966
\(386\) 0 0
\(387\) 10604.6 1.39293
\(388\) 0 0
\(389\) 6802.65 0.886653 0.443327 0.896360i \(-0.353798\pi\)
0.443327 + 0.896360i \(0.353798\pi\)
\(390\) 0 0
\(391\) 1631.40 0.211006
\(392\) 0 0
\(393\) −10466.8 −1.34347
\(394\) 0 0
\(395\) 4022.62 0.512405
\(396\) 0 0
\(397\) −5352.36 −0.676643 −0.338322 0.941031i \(-0.609859\pi\)
−0.338322 + 0.941031i \(0.609859\pi\)
\(398\) 0 0
\(399\) 12510.7 1.56971
\(400\) 0 0
\(401\) −6144.61 −0.765205 −0.382602 0.923913i \(-0.624972\pi\)
−0.382602 + 0.923913i \(0.624972\pi\)
\(402\) 0 0
\(403\) −20795.9 −2.57052
\(404\) 0 0
\(405\) −3047.71 −0.373930
\(406\) 0 0
\(407\) −19523.9 −2.37780
\(408\) 0 0
\(409\) 9226.12 1.11541 0.557705 0.830039i \(-0.311682\pi\)
0.557705 + 0.830039i \(0.311682\pi\)
\(410\) 0 0
\(411\) 45.7430 0.00548986
\(412\) 0 0
\(413\) 5206.45 0.620322
\(414\) 0 0
\(415\) −4461.00 −0.527667
\(416\) 0 0
\(417\) 11579.3 1.35981
\(418\) 0 0
\(419\) 756.386 0.0881907 0.0440953 0.999027i \(-0.485959\pi\)
0.0440953 + 0.999027i \(0.485959\pi\)
\(420\) 0 0
\(421\) −2628.98 −0.304344 −0.152172 0.988354i \(-0.548627\pi\)
−0.152172 + 0.988354i \(0.548627\pi\)
\(422\) 0 0
\(423\) −9557.62 −1.09860
\(424\) 0 0
\(425\) 1773.26 0.202390
\(426\) 0 0
\(427\) −1253.90 −0.142108
\(428\) 0 0
\(429\) −36748.8 −4.13578
\(430\) 0 0
\(431\) 8492.79 0.949149 0.474575 0.880215i \(-0.342602\pi\)
0.474575 + 0.880215i \(0.342602\pi\)
\(432\) 0 0
\(433\) −5737.33 −0.636763 −0.318381 0.947963i \(-0.603139\pi\)
−0.318381 + 0.947963i \(0.603139\pi\)
\(434\) 0 0
\(435\) −9302.16 −1.02530
\(436\) 0 0
\(437\) 1828.67 0.200177
\(438\) 0 0
\(439\) 6974.82 0.758292 0.379146 0.925337i \(-0.376218\pi\)
0.379146 + 0.925337i \(0.376218\pi\)
\(440\) 0 0
\(441\) 2619.33 0.282835
\(442\) 0 0
\(443\) 2755.57 0.295533 0.147766 0.989022i \(-0.452792\pi\)
0.147766 + 0.989022i \(0.452792\pi\)
\(444\) 0 0
\(445\) 4558.92 0.485648
\(446\) 0 0
\(447\) −3347.23 −0.354181
\(448\) 0 0
\(449\) −2242.66 −0.235719 −0.117860 0.993030i \(-0.537603\pi\)
−0.117860 + 0.993030i \(0.537603\pi\)
\(450\) 0 0
\(451\) 24201.1 2.52680
\(452\) 0 0
\(453\) 22078.2 2.28990
\(454\) 0 0
\(455\) −7575.10 −0.780497
\(456\) 0 0
\(457\) −8098.67 −0.828971 −0.414486 0.910056i \(-0.636038\pi\)
−0.414486 + 0.910056i \(0.636038\pi\)
\(458\) 0 0
\(459\) 2088.06 0.212337
\(460\) 0 0
\(461\) −6133.87 −0.619703 −0.309851 0.950785i \(-0.600279\pi\)
−0.309851 + 0.950785i \(0.600279\pi\)
\(462\) 0 0
\(463\) −15671.6 −1.57305 −0.786523 0.617560i \(-0.788121\pi\)
−0.786523 + 0.617560i \(0.788121\pi\)
\(464\) 0 0
\(465\) 10799.5 1.07702
\(466\) 0 0
\(467\) 10231.3 1.01381 0.506906 0.862002i \(-0.330789\pi\)
0.506906 + 0.862002i \(0.330789\pi\)
\(468\) 0 0
\(469\) −8974.00 −0.883541
\(470\) 0 0
\(471\) 174.463 0.0170676
\(472\) 0 0
\(473\) 22657.3 2.20251
\(474\) 0 0
\(475\) 1987.69 0.192003
\(476\) 0 0
\(477\) −14740.0 −1.41488
\(478\) 0 0
\(479\) 13756.0 1.31216 0.656082 0.754690i \(-0.272213\pi\)
0.656082 + 0.754690i \(0.272213\pi\)
\(480\) 0 0
\(481\) 21681.6 2.05530
\(482\) 0 0
\(483\) 3619.10 0.340941
\(484\) 0 0
\(485\) 266.612 0.0249613
\(486\) 0 0
\(487\) 19708.6 1.83384 0.916920 0.399071i \(-0.130667\pi\)
0.916920 + 0.399071i \(0.130667\pi\)
\(488\) 0 0
\(489\) −12680.4 −1.17266
\(490\) 0 0
\(491\) 13139.1 1.20766 0.603828 0.797115i \(-0.293641\pi\)
0.603828 + 0.797115i \(0.293641\pi\)
\(492\) 0 0
\(493\) −17346.9 −1.58471
\(494\) 0 0
\(495\) 10180.0 0.924361
\(496\) 0 0
\(497\) 23162.6 2.09051
\(498\) 0 0
\(499\) −21342.8 −1.91470 −0.957350 0.288932i \(-0.906700\pi\)
−0.957350 + 0.288932i \(0.906700\pi\)
\(500\) 0 0
\(501\) 7979.19 0.711545
\(502\) 0 0
\(503\) 4725.75 0.418909 0.209454 0.977818i \(-0.432831\pi\)
0.209454 + 0.977818i \(0.432831\pi\)
\(504\) 0 0
\(505\) 671.502 0.0591712
\(506\) 0 0
\(507\) 24097.2 2.11084
\(508\) 0 0
\(509\) −16423.9 −1.43021 −0.715105 0.699017i \(-0.753621\pi\)
−0.715105 + 0.699017i \(0.753621\pi\)
\(510\) 0 0
\(511\) −12066.7 −1.04462
\(512\) 0 0
\(513\) 2340.55 0.201439
\(514\) 0 0
\(515\) −1268.14 −0.108507
\(516\) 0 0
\(517\) −20420.3 −1.73711
\(518\) 0 0
\(519\) 9689.27 0.819483
\(520\) 0 0
\(521\) 3724.80 0.313218 0.156609 0.987661i \(-0.449944\pi\)
0.156609 + 0.987661i \(0.449944\pi\)
\(522\) 0 0
\(523\) 2905.97 0.242963 0.121481 0.992594i \(-0.461236\pi\)
0.121481 + 0.992594i \(0.461236\pi\)
\(524\) 0 0
\(525\) 3933.80 0.327019
\(526\) 0 0
\(527\) 20139.1 1.66465
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) 7770.14 0.635020
\(532\) 0 0
\(533\) −26875.8 −2.18409
\(534\) 0 0
\(535\) −9830.31 −0.794395
\(536\) 0 0
\(537\) −18318.9 −1.47211
\(538\) 0 0
\(539\) 5596.33 0.447219
\(540\) 0 0
\(541\) −772.962 −0.0614274 −0.0307137 0.999528i \(-0.509778\pi\)
−0.0307137 + 0.999528i \(0.509778\pi\)
\(542\) 0 0
\(543\) 7897.67 0.624165
\(544\) 0 0
\(545\) −2340.49 −0.183955
\(546\) 0 0
\(547\) 3226.14 0.252175 0.126088 0.992019i \(-0.459758\pi\)
0.126088 + 0.992019i \(0.459758\pi\)
\(548\) 0 0
\(549\) −1871.32 −0.145476
\(550\) 0 0
\(551\) −19444.4 −1.50338
\(552\) 0 0
\(553\) 16641.2 1.27967
\(554\) 0 0
\(555\) −11259.4 −0.861146
\(556\) 0 0
\(557\) −3527.96 −0.268374 −0.134187 0.990956i \(-0.542842\pi\)
−0.134187 + 0.990956i \(0.542842\pi\)
\(558\) 0 0
\(559\) −25161.4 −1.90378
\(560\) 0 0
\(561\) 35588.1 2.67831
\(562\) 0 0
\(563\) −3414.29 −0.255586 −0.127793 0.991801i \(-0.540789\pi\)
−0.127793 + 0.991801i \(0.540789\pi\)
\(564\) 0 0
\(565\) 313.457 0.0233403
\(566\) 0 0
\(567\) −12608.1 −0.933845
\(568\) 0 0
\(569\) −9661.15 −0.711804 −0.355902 0.934523i \(-0.615826\pi\)
−0.355902 + 0.934523i \(0.615826\pi\)
\(570\) 0 0
\(571\) −16580.1 −1.21516 −0.607578 0.794260i \(-0.707859\pi\)
−0.607578 + 0.794260i \(0.707859\pi\)
\(572\) 0 0
\(573\) 6777.41 0.494120
\(574\) 0 0
\(575\) 575.000 0.0417029
\(576\) 0 0
\(577\) −4587.53 −0.330990 −0.165495 0.986211i \(-0.552922\pi\)
−0.165495 + 0.986211i \(0.552922\pi\)
\(578\) 0 0
\(579\) −1035.24 −0.0743061
\(580\) 0 0
\(581\) −18454.8 −1.31778
\(582\) 0 0
\(583\) −31492.6 −2.23721
\(584\) 0 0
\(585\) −11305.1 −0.798990
\(586\) 0 0
\(587\) 21730.5 1.52796 0.763980 0.645240i \(-0.223243\pi\)
0.763980 + 0.645240i \(0.223243\pi\)
\(588\) 0 0
\(589\) 22574.3 1.57922
\(590\) 0 0
\(591\) 11209.6 0.780209
\(592\) 0 0
\(593\) −3557.48 −0.246354 −0.123177 0.992385i \(-0.539308\pi\)
−0.123177 + 0.992385i \(0.539308\pi\)
\(594\) 0 0
\(595\) 7335.83 0.505445
\(596\) 0 0
\(597\) 2188.05 0.150001
\(598\) 0 0
\(599\) 4940.46 0.336998 0.168499 0.985702i \(-0.446108\pi\)
0.168499 + 0.985702i \(0.446108\pi\)
\(600\) 0 0
\(601\) −24459.7 −1.66012 −0.830060 0.557675i \(-0.811694\pi\)
−0.830060 + 0.557675i \(0.811694\pi\)
\(602\) 0 0
\(603\) −13392.8 −0.904476
\(604\) 0 0
\(605\) 15095.1 1.01439
\(606\) 0 0
\(607\) −4098.17 −0.274035 −0.137018 0.990569i \(-0.543752\pi\)
−0.137018 + 0.990569i \(0.543752\pi\)
\(608\) 0 0
\(609\) −38482.2 −2.56056
\(610\) 0 0
\(611\) 22677.1 1.50150
\(612\) 0 0
\(613\) −10486.4 −0.690936 −0.345468 0.938431i \(-0.612280\pi\)
−0.345468 + 0.938431i \(0.612280\pi\)
\(614\) 0 0
\(615\) 13956.8 0.915107
\(616\) 0 0
\(617\) −6366.64 −0.415415 −0.207708 0.978191i \(-0.566600\pi\)
−0.207708 + 0.978191i \(0.566600\pi\)
\(618\) 0 0
\(619\) −21614.5 −1.40349 −0.701746 0.712427i \(-0.747596\pi\)
−0.701746 + 0.712427i \(0.747596\pi\)
\(620\) 0 0
\(621\) 677.078 0.0437524
\(622\) 0 0
\(623\) 18859.8 1.21285
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 39891.4 2.54085
\(628\) 0 0
\(629\) −20996.8 −1.33100
\(630\) 0 0
\(631\) −20515.5 −1.29431 −0.647154 0.762359i \(-0.724041\pi\)
−0.647154 + 0.762359i \(0.724041\pi\)
\(632\) 0 0
\(633\) 33412.1 2.09797
\(634\) 0 0
\(635\) −10998.1 −0.687315
\(636\) 0 0
\(637\) −6214.83 −0.386563
\(638\) 0 0
\(639\) 34568.0 2.14005
\(640\) 0 0
\(641\) 14561.7 0.897275 0.448637 0.893714i \(-0.351909\pi\)
0.448637 + 0.893714i \(0.351909\pi\)
\(642\) 0 0
\(643\) −14571.0 −0.893659 −0.446830 0.894619i \(-0.647447\pi\)
−0.446830 + 0.894619i \(0.647447\pi\)
\(644\) 0 0
\(645\) 13066.5 0.797662
\(646\) 0 0
\(647\) −9034.71 −0.548981 −0.274491 0.961590i \(-0.588509\pi\)
−0.274491 + 0.961590i \(0.588509\pi\)
\(648\) 0 0
\(649\) 16601.3 1.00409
\(650\) 0 0
\(651\) 44676.5 2.68972
\(652\) 0 0
\(653\) 10412.8 0.624018 0.312009 0.950079i \(-0.398998\pi\)
0.312009 + 0.950079i \(0.398998\pi\)
\(654\) 0 0
\(655\) −6879.55 −0.410391
\(656\) 0 0
\(657\) −18008.5 −1.06937
\(658\) 0 0
\(659\) −19212.5 −1.13568 −0.567840 0.823139i \(-0.692221\pi\)
−0.567840 + 0.823139i \(0.692221\pi\)
\(660\) 0 0
\(661\) 17563.2 1.03348 0.516739 0.856143i \(-0.327146\pi\)
0.516739 + 0.856143i \(0.327146\pi\)
\(662\) 0 0
\(663\) −39521.2 −2.31505
\(664\) 0 0
\(665\) 8222.88 0.479503
\(666\) 0 0
\(667\) −5624.91 −0.326533
\(668\) 0 0
\(669\) 41061.1 2.37296
\(670\) 0 0
\(671\) −3998.17 −0.230026
\(672\) 0 0
\(673\) −11984.7 −0.686445 −0.343222 0.939254i \(-0.611518\pi\)
−0.343222 + 0.939254i \(0.611518\pi\)
\(674\) 0 0
\(675\) 735.955 0.0419658
\(676\) 0 0
\(677\) 23516.9 1.33505 0.667525 0.744587i \(-0.267354\pi\)
0.667525 + 0.744587i \(0.267354\pi\)
\(678\) 0 0
\(679\) 1102.95 0.0623378
\(680\) 0 0
\(681\) 19160.3 1.07816
\(682\) 0 0
\(683\) 7449.33 0.417336 0.208668 0.977987i \(-0.433087\pi\)
0.208668 + 0.977987i \(0.433087\pi\)
\(684\) 0 0
\(685\) 30.0655 0.00167700
\(686\) 0 0
\(687\) −36157.4 −2.00799
\(688\) 0 0
\(689\) 34973.2 1.93378
\(690\) 0 0
\(691\) 21369.7 1.17647 0.588236 0.808689i \(-0.299823\pi\)
0.588236 + 0.808689i \(0.299823\pi\)
\(692\) 0 0
\(693\) 42113.9 2.30848
\(694\) 0 0
\(695\) 7610.75 0.415384
\(696\) 0 0
\(697\) 26026.9 1.41440
\(698\) 0 0
\(699\) −9085.30 −0.491613
\(700\) 0 0
\(701\) 1166.89 0.0628713 0.0314357 0.999506i \(-0.489992\pi\)
0.0314357 + 0.999506i \(0.489992\pi\)
\(702\) 0 0
\(703\) −23535.8 −1.26269
\(704\) 0 0
\(705\) −11776.4 −0.629113
\(706\) 0 0
\(707\) 2777.95 0.147773
\(708\) 0 0
\(709\) −31306.3 −1.65830 −0.829149 0.559028i \(-0.811174\pi\)
−0.829149 + 0.559028i \(0.811174\pi\)
\(710\) 0 0
\(711\) 24835.5 1.30999
\(712\) 0 0
\(713\) 6530.32 0.343005
\(714\) 0 0
\(715\) −24153.9 −1.26336
\(716\) 0 0
\(717\) −44510.0 −2.31835
\(718\) 0 0
\(719\) 25017.0 1.29760 0.648802 0.760957i \(-0.275270\pi\)
0.648802 + 0.760957i \(0.275270\pi\)
\(720\) 0 0
\(721\) −5246.18 −0.270982
\(722\) 0 0
\(723\) 5006.13 0.257510
\(724\) 0 0
\(725\) −6114.03 −0.313199
\(726\) 0 0
\(727\) −22380.1 −1.14172 −0.570861 0.821047i \(-0.693391\pi\)
−0.570861 + 0.821047i \(0.693391\pi\)
\(728\) 0 0
\(729\) −24862.8 −1.26316
\(730\) 0 0
\(731\) 24366.6 1.23288
\(732\) 0 0
\(733\) −11655.7 −0.587333 −0.293666 0.955908i \(-0.594875\pi\)
−0.293666 + 0.955908i \(0.594875\pi\)
\(734\) 0 0
\(735\) 3227.40 0.161965
\(736\) 0 0
\(737\) −28614.5 −1.43016
\(738\) 0 0
\(739\) −11983.0 −0.596484 −0.298242 0.954490i \(-0.596400\pi\)
−0.298242 + 0.954490i \(0.596400\pi\)
\(740\) 0 0
\(741\) −44300.2 −2.19623
\(742\) 0 0
\(743\) 8516.73 0.420523 0.210262 0.977645i \(-0.432568\pi\)
0.210262 + 0.977645i \(0.432568\pi\)
\(744\) 0 0
\(745\) −2200.04 −0.108192
\(746\) 0 0
\(747\) −27542.0 −1.34901
\(748\) 0 0
\(749\) −40667.1 −1.98391
\(750\) 0 0
\(751\) 3530.23 0.171531 0.0857655 0.996315i \(-0.472666\pi\)
0.0857655 + 0.996315i \(0.472666\pi\)
\(752\) 0 0
\(753\) 20729.6 1.00323
\(754\) 0 0
\(755\) 14511.4 0.699500
\(756\) 0 0
\(757\) −28421.2 −1.36458 −0.682290 0.731081i \(-0.739016\pi\)
−0.682290 + 0.731081i \(0.739016\pi\)
\(758\) 0 0
\(759\) 11539.8 0.551870
\(760\) 0 0
\(761\) −15456.9 −0.736286 −0.368143 0.929769i \(-0.620006\pi\)
−0.368143 + 0.929769i \(0.620006\pi\)
\(762\) 0 0
\(763\) −9682.40 −0.459405
\(764\) 0 0
\(765\) 10948.0 0.517421
\(766\) 0 0
\(767\) −18436.0 −0.867910
\(768\) 0 0
\(769\) −10820.0 −0.507385 −0.253692 0.967285i \(-0.581645\pi\)
−0.253692 + 0.967285i \(0.581645\pi\)
\(770\) 0 0
\(771\) −25262.7 −1.18004
\(772\) 0 0
\(773\) −29863.4 −1.38954 −0.694769 0.719233i \(-0.744494\pi\)
−0.694769 + 0.719233i \(0.744494\pi\)
\(774\) 0 0
\(775\) 7098.17 0.328999
\(776\) 0 0
\(777\) −46579.3 −2.15061
\(778\) 0 0
\(779\) 29174.0 1.34181
\(780\) 0 0
\(781\) 73856.2 3.38385
\(782\) 0 0
\(783\) −7199.44 −0.328591
\(784\) 0 0
\(785\) 114.669 0.00521365
\(786\) 0 0
\(787\) 1588.27 0.0719388 0.0359694 0.999353i \(-0.488548\pi\)
0.0359694 + 0.999353i \(0.488548\pi\)
\(788\) 0 0
\(789\) −3289.39 −0.148423
\(790\) 0 0
\(791\) 1296.75 0.0582895
\(792\) 0 0
\(793\) 4440.04 0.198828
\(794\) 0 0
\(795\) −18161.8 −0.810230
\(796\) 0 0
\(797\) −38280.0 −1.70131 −0.850656 0.525723i \(-0.823795\pi\)
−0.850656 + 0.525723i \(0.823795\pi\)
\(798\) 0 0
\(799\) −21960.9 −0.972365
\(800\) 0 0
\(801\) 28146.5 1.24158
\(802\) 0 0
\(803\) −38476.0 −1.69089
\(804\) 0 0
\(805\) 2378.72 0.104148
\(806\) 0 0
\(807\) −33535.9 −1.46285
\(808\) 0 0
\(809\) −21369.6 −0.928694 −0.464347 0.885653i \(-0.653711\pi\)
−0.464347 + 0.885653i \(0.653711\pi\)
\(810\) 0 0
\(811\) −16291.7 −0.705401 −0.352700 0.935736i \(-0.614736\pi\)
−0.352700 + 0.935736i \(0.614736\pi\)
\(812\) 0 0
\(813\) −65459.6 −2.82382
\(814\) 0 0
\(815\) −8334.46 −0.358213
\(816\) 0 0
\(817\) 27313.1 1.16960
\(818\) 0 0
\(819\) −46768.3 −1.99538
\(820\) 0 0
\(821\) −45046.6 −1.91491 −0.957454 0.288587i \(-0.906814\pi\)
−0.957454 + 0.288587i \(0.906814\pi\)
\(822\) 0 0
\(823\) −14521.8 −0.615064 −0.307532 0.951538i \(-0.599503\pi\)
−0.307532 + 0.951538i \(0.599503\pi\)
\(824\) 0 0
\(825\) 12543.3 0.529335
\(826\) 0 0
\(827\) −27039.3 −1.13694 −0.568470 0.822704i \(-0.692465\pi\)
−0.568470 + 0.822704i \(0.692465\pi\)
\(828\) 0 0
\(829\) −8102.50 −0.339459 −0.169730 0.985491i \(-0.554289\pi\)
−0.169730 + 0.985491i \(0.554289\pi\)
\(830\) 0 0
\(831\) −51959.1 −2.16900
\(832\) 0 0
\(833\) 6018.53 0.250336
\(834\) 0 0
\(835\) 5244.49 0.217357
\(836\) 0 0
\(837\) 8358.29 0.345167
\(838\) 0 0
\(839\) −868.168 −0.0357241 −0.0178620 0.999840i \(-0.505686\pi\)
−0.0178620 + 0.999840i \(0.505686\pi\)
\(840\) 0 0
\(841\) 35421.3 1.45235
\(842\) 0 0
\(843\) −64905.7 −2.65181
\(844\) 0 0
\(845\) 15838.4 0.644802
\(846\) 0 0
\(847\) 62447.3 2.53331
\(848\) 0 0
\(849\) 51423.7 2.07875
\(850\) 0 0
\(851\) −6808.45 −0.274255
\(852\) 0 0
\(853\) 14481.6 0.581291 0.290645 0.956831i \(-0.406130\pi\)
0.290645 + 0.956831i \(0.406130\pi\)
\(854\) 0 0
\(855\) 12271.9 0.490865
\(856\) 0 0
\(857\) 45840.2 1.82716 0.913578 0.406664i \(-0.133308\pi\)
0.913578 + 0.406664i \(0.133308\pi\)
\(858\) 0 0
\(859\) −12096.4 −0.480469 −0.240235 0.970715i \(-0.577224\pi\)
−0.240235 + 0.970715i \(0.577224\pi\)
\(860\) 0 0
\(861\) 57737.9 2.28537
\(862\) 0 0
\(863\) 3013.77 0.118876 0.0594380 0.998232i \(-0.481069\pi\)
0.0594380 + 0.998232i \(0.481069\pi\)
\(864\) 0 0
\(865\) 6368.47 0.250329
\(866\) 0 0
\(867\) 898.643 0.0352013
\(868\) 0 0
\(869\) 53062.2 2.07136
\(870\) 0 0
\(871\) 31776.9 1.23619
\(872\) 0 0
\(873\) 1646.05 0.0638148
\(874\) 0 0
\(875\) 2585.57 0.0998951
\(876\) 0 0
\(877\) 1553.17 0.0598027 0.0299013 0.999553i \(-0.490481\pi\)
0.0299013 + 0.999553i \(0.490481\pi\)
\(878\) 0 0
\(879\) −38547.8 −1.47916
\(880\) 0 0
\(881\) −12466.6 −0.476743 −0.238372 0.971174i \(-0.576614\pi\)
−0.238372 + 0.971174i \(0.576614\pi\)
\(882\) 0 0
\(883\) −12283.0 −0.468126 −0.234063 0.972221i \(-0.575202\pi\)
−0.234063 + 0.972221i \(0.575202\pi\)
\(884\) 0 0
\(885\) 9573.96 0.363644
\(886\) 0 0
\(887\) 5256.94 0.198997 0.0994987 0.995038i \(-0.468276\pi\)
0.0994987 + 0.995038i \(0.468276\pi\)
\(888\) 0 0
\(889\) −45498.1 −1.71649
\(890\) 0 0
\(891\) −40202.1 −1.51158
\(892\) 0 0
\(893\) −24616.4 −0.922459
\(894\) 0 0
\(895\) −12040.5 −0.449687
\(896\) 0 0
\(897\) −12815.2 −0.477020
\(898\) 0 0
\(899\) −69437.6 −2.57605
\(900\) 0 0
\(901\) −33868.5 −1.25230
\(902\) 0 0
\(903\) 54054.9 1.99206
\(904\) 0 0
\(905\) 5190.91 0.190665
\(906\) 0 0
\(907\) 28669.8 1.04958 0.524788 0.851233i \(-0.324144\pi\)
0.524788 + 0.851233i \(0.324144\pi\)
\(908\) 0 0
\(909\) 4145.82 0.151274
\(910\) 0 0
\(911\) 10069.9 0.366225 0.183113 0.983092i \(-0.441383\pi\)
0.183113 + 0.983092i \(0.441383\pi\)
\(912\) 0 0
\(913\) −58844.8 −2.13305
\(914\) 0 0
\(915\) −2305.74 −0.0833066
\(916\) 0 0
\(917\) −28460.1 −1.02490
\(918\) 0 0
\(919\) −49898.4 −1.79108 −0.895538 0.444986i \(-0.853209\pi\)
−0.895538 + 0.444986i \(0.853209\pi\)
\(920\) 0 0
\(921\) 32304.8 1.15579
\(922\) 0 0
\(923\) −82018.7 −2.92490
\(924\) 0 0
\(925\) −7400.49 −0.263056
\(926\) 0 0
\(927\) −7829.43 −0.277403
\(928\) 0 0
\(929\) −8573.05 −0.302769 −0.151385 0.988475i \(-0.548373\pi\)
−0.151385 + 0.988475i \(0.548373\pi\)
\(930\) 0 0
\(931\) 6746.29 0.237487
\(932\) 0 0
\(933\) 6987.93 0.245203
\(934\) 0 0
\(935\) 23391.0 0.818147
\(936\) 0 0
\(937\) 3952.76 0.137813 0.0689066 0.997623i \(-0.478049\pi\)
0.0689066 + 0.997623i \(0.478049\pi\)
\(938\) 0 0
\(939\) 4035.46 0.140247
\(940\) 0 0
\(941\) −1708.81 −0.0591982 −0.0295991 0.999562i \(-0.509423\pi\)
−0.0295991 + 0.999562i \(0.509423\pi\)
\(942\) 0 0
\(943\) 8439.50 0.291440
\(944\) 0 0
\(945\) 3044.58 0.104804
\(946\) 0 0
\(947\) −25389.4 −0.871219 −0.435609 0.900136i \(-0.643467\pi\)
−0.435609 + 0.900136i \(0.643467\pi\)
\(948\) 0 0
\(949\) 42728.3 1.46156
\(950\) 0 0
\(951\) 59752.0 2.03742
\(952\) 0 0
\(953\) −11245.0 −0.382228 −0.191114 0.981568i \(-0.561210\pi\)
−0.191114 + 0.981568i \(0.561210\pi\)
\(954\) 0 0
\(955\) 4454.59 0.150940
\(956\) 0 0
\(957\) −122704. −4.14469
\(958\) 0 0
\(959\) 124.378 0.00418810
\(960\) 0 0
\(961\) 50823.5 1.70600
\(962\) 0 0
\(963\) −60691.9 −2.03091
\(964\) 0 0
\(965\) −680.435 −0.0226984
\(966\) 0 0
\(967\) 3022.48 0.100513 0.0502566 0.998736i \(-0.483996\pi\)
0.0502566 + 0.998736i \(0.483996\pi\)
\(968\) 0 0
\(969\) 42900.9 1.42227
\(970\) 0 0
\(971\) −50953.9 −1.68403 −0.842013 0.539457i \(-0.818630\pi\)
−0.842013 + 0.539457i \(0.818630\pi\)
\(972\) 0 0
\(973\) 31485.0 1.03737
\(974\) 0 0
\(975\) −13929.6 −0.457542
\(976\) 0 0
\(977\) 38184.7 1.25040 0.625199 0.780466i \(-0.285018\pi\)
0.625199 + 0.780466i \(0.285018\pi\)
\(978\) 0 0
\(979\) 60136.4 1.96320
\(980\) 0 0
\(981\) −14450.1 −0.470291
\(982\) 0 0
\(983\) 13378.9 0.434099 0.217050 0.976161i \(-0.430357\pi\)
0.217050 + 0.976161i \(0.430357\pi\)
\(984\) 0 0
\(985\) 7367.77 0.238332
\(986\) 0 0
\(987\) −48717.9 −1.57113
\(988\) 0 0
\(989\) 7901.16 0.254037
\(990\) 0 0
\(991\) −22011.7 −0.705576 −0.352788 0.935703i \(-0.614766\pi\)
−0.352788 + 0.935703i \(0.614766\pi\)
\(992\) 0 0
\(993\) 88414.1 2.82551
\(994\) 0 0
\(995\) 1438.14 0.0458212
\(996\) 0 0
\(997\) 6203.59 0.197061 0.0985304 0.995134i \(-0.468586\pi\)
0.0985304 + 0.995134i \(0.468586\pi\)
\(998\) 0 0
\(999\) −8714.28 −0.275984
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 1840.4.a.ba.1.9 10
4.3 odd 2 920.4.a.h.1.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.4.a.h.1.2 10 4.3 odd 2
1840.4.a.ba.1.9 10 1.1 even 1 trivial