Properties

Label 624.4.a.k.1.1
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
Analytic rank $0$
Dimension $2$
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] = [624,4,Mod(1,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, 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("624.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.64575\) of defining polynomial
Character \(\chi\) \(=\) 624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -7.29150 q^{5} -5.87451 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -7.29150 q^{5} -5.87451 q^{7} +9.00000 q^{9} +51.1660 q^{11} -13.0000 q^{13} +21.8745 q^{15} -73.7490 q^{17} -59.9555 q^{19} +17.6235 q^{21} -69.8301 q^{23} -71.8340 q^{25} -27.0000 q^{27} -294.826 q^{29} +334.450 q^{31} -153.498 q^{33} +42.8340 q^{35} +261.409 q^{37} +39.0000 q^{39} +222.701 q^{41} -79.2470 q^{43} -65.6235 q^{45} +584.405 q^{47} -308.490 q^{49} +221.247 q^{51} +465.158 q^{53} -373.077 q^{55} +179.867 q^{57} +530.089 q^{59} +548.332 q^{61} -52.8706 q^{63} +94.7895 q^{65} +384.959 q^{67} +209.490 q^{69} +307.004 q^{71} -844.891 q^{73} +215.502 q^{75} -300.575 q^{77} -30.1699 q^{79} +81.0000 q^{81} -19.5633 q^{83} +537.741 q^{85} +884.478 q^{87} -513.017 q^{89} +76.3686 q^{91} -1003.35 q^{93} +437.166 q^{95} +787.903 q^{97} +460.494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 4 q^{5} + 20 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 4 q^{5} + 20 q^{7} + 18 q^{9} + 60 q^{11} - 26 q^{13} + 12 q^{15} - 84 q^{17} + 60 q^{19} - 60 q^{21} + 72 q^{23} - 186 q^{25} - 54 q^{27} - 124 q^{29} + 108 q^{31} - 180 q^{33} + 128 q^{35} + 36 q^{37} + 78 q^{39} - 52 q^{41} + 32 q^{43} - 36 q^{45} + 428 q^{47} + 18 q^{49} + 252 q^{51} + 380 q^{53} - 344 q^{55} - 180 q^{57} + 1420 q^{59} + 1012 q^{61} + 180 q^{63} + 52 q^{65} + 844 q^{67} - 216 q^{69} + 868 q^{71} - 60 q^{73} + 558 q^{75} - 72 q^{77} - 272 q^{79} + 162 q^{81} + 1252 q^{83} + 504 q^{85} + 372 q^{87} + 572 q^{89} - 260 q^{91} - 324 q^{93} + 832 q^{95} + 708 q^{97} + 540 q^{99}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −7.29150 −0.652172 −0.326086 0.945340i \(-0.605730\pi\)
−0.326086 + 0.945340i \(0.605730\pi\)
\(6\) 0 0
\(7\) −5.87451 −0.317194 −0.158597 0.987343i \(-0.550697\pi\)
−0.158597 + 0.987343i \(0.550697\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 51.1660 1.40247 0.701233 0.712932i \(-0.252633\pi\)
0.701233 + 0.712932i \(0.252633\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) 21.8745 0.376532
\(16\) 0 0
\(17\) −73.7490 −1.05216 −0.526081 0.850434i \(-0.676339\pi\)
−0.526081 + 0.850434i \(0.676339\pi\)
\(18\) 0 0
\(19\) −59.9555 −0.723934 −0.361967 0.932191i \(-0.617895\pi\)
−0.361967 + 0.932191i \(0.617895\pi\)
\(20\) 0 0
\(21\) 17.6235 0.183132
\(22\) 0 0
\(23\) −69.8301 −0.633068 −0.316534 0.948581i \(-0.602519\pi\)
−0.316534 + 0.948581i \(0.602519\pi\)
\(24\) 0 0
\(25\) −71.8340 −0.574672
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −294.826 −1.88786 −0.943928 0.330151i \(-0.892900\pi\)
−0.943928 + 0.330151i \(0.892900\pi\)
\(30\) 0 0
\(31\) 334.450 1.93771 0.968854 0.247634i \(-0.0796530\pi\)
0.968854 + 0.247634i \(0.0796530\pi\)
\(32\) 0 0
\(33\) −153.498 −0.809714
\(34\) 0 0
\(35\) 42.8340 0.206865
\(36\) 0 0
\(37\) 261.409 1.16150 0.580749 0.814083i \(-0.302760\pi\)
0.580749 + 0.814083i \(0.302760\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) 222.701 0.848293 0.424146 0.905594i \(-0.360574\pi\)
0.424146 + 0.905594i \(0.360574\pi\)
\(42\) 0 0
\(43\) −79.2470 −0.281048 −0.140524 0.990077i \(-0.544879\pi\)
−0.140524 + 0.990077i \(0.544879\pi\)
\(44\) 0 0
\(45\) −65.6235 −0.217391
\(46\) 0 0
\(47\) 584.405 1.81371 0.906854 0.421445i \(-0.138477\pi\)
0.906854 + 0.421445i \(0.138477\pi\)
\(48\) 0 0
\(49\) −308.490 −0.899388
\(50\) 0 0
\(51\) 221.247 0.607466
\(52\) 0 0
\(53\) 465.158 1.20555 0.602777 0.797910i \(-0.294061\pi\)
0.602777 + 0.797910i \(0.294061\pi\)
\(54\) 0 0
\(55\) −373.077 −0.914649
\(56\) 0 0
\(57\) 179.867 0.417963
\(58\) 0 0
\(59\) 530.089 1.16969 0.584845 0.811145i \(-0.301155\pi\)
0.584845 + 0.811145i \(0.301155\pi\)
\(60\) 0 0
\(61\) 548.332 1.15093 0.575465 0.817826i \(-0.304821\pi\)
0.575465 + 0.817826i \(0.304821\pi\)
\(62\) 0 0
\(63\) −52.8706 −0.105731
\(64\) 0 0
\(65\) 94.7895 0.180880
\(66\) 0 0
\(67\) 384.959 0.701945 0.350972 0.936386i \(-0.385851\pi\)
0.350972 + 0.936386i \(0.385851\pi\)
\(68\) 0 0
\(69\) 209.490 0.365502
\(70\) 0 0
\(71\) 307.004 0.513164 0.256582 0.966522i \(-0.417404\pi\)
0.256582 + 0.966522i \(0.417404\pi\)
\(72\) 0 0
\(73\) −844.891 −1.35462 −0.677309 0.735699i \(-0.736854\pi\)
−0.677309 + 0.735699i \(0.736854\pi\)
\(74\) 0 0
\(75\) 215.502 0.331787
\(76\) 0 0
\(77\) −300.575 −0.444853
\(78\) 0 0
\(79\) −30.1699 −0.0429669 −0.0214834 0.999769i \(-0.506839\pi\)
−0.0214834 + 0.999769i \(0.506839\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −19.5633 −0.0258717 −0.0129359 0.999916i \(-0.504118\pi\)
−0.0129359 + 0.999916i \(0.504118\pi\)
\(84\) 0 0
\(85\) 537.741 0.686191
\(86\) 0 0
\(87\) 884.478 1.08995
\(88\) 0 0
\(89\) −513.017 −0.611008 −0.305504 0.952191i \(-0.598825\pi\)
−0.305504 + 0.952191i \(0.598825\pi\)
\(90\) 0 0
\(91\) 76.3686 0.0879737
\(92\) 0 0
\(93\) −1003.35 −1.11874
\(94\) 0 0
\(95\) 437.166 0.472129
\(96\) 0 0
\(97\) 787.903 0.824737 0.412368 0.911017i \(-0.364702\pi\)
0.412368 + 0.911017i \(0.364702\pi\)
\(98\) 0 0
\(99\) 460.494 0.467489
\(100\) 0 0
\(101\) 1391.73 1.37111 0.685554 0.728022i \(-0.259560\pi\)
0.685554 + 0.728022i \(0.259560\pi\)
\(102\) 0 0
\(103\) −1428.58 −1.36662 −0.683309 0.730129i \(-0.739460\pi\)
−0.683309 + 0.730129i \(0.739460\pi\)
\(104\) 0 0
\(105\) −128.502 −0.119433
\(106\) 0 0
\(107\) 1363.30 1.23173 0.615867 0.787850i \(-0.288806\pi\)
0.615867 + 0.787850i \(0.288806\pi\)
\(108\) 0 0
\(109\) 979.182 0.860446 0.430223 0.902723i \(-0.358435\pi\)
0.430223 + 0.902723i \(0.358435\pi\)
\(110\) 0 0
\(111\) −784.227 −0.670591
\(112\) 0 0
\(113\) 1495.83 1.24527 0.622637 0.782511i \(-0.286062\pi\)
0.622637 + 0.782511i \(0.286062\pi\)
\(114\) 0 0
\(115\) 509.166 0.412869
\(116\) 0 0
\(117\) −117.000 −0.0924500
\(118\) 0 0
\(119\) 433.239 0.333739
\(120\) 0 0
\(121\) 1286.96 0.966913
\(122\) 0 0
\(123\) −668.102 −0.489762
\(124\) 0 0
\(125\) 1435.22 1.02696
\(126\) 0 0
\(127\) −2221.69 −1.55231 −0.776155 0.630542i \(-0.782833\pi\)
−0.776155 + 0.630542i \(0.782833\pi\)
\(128\) 0 0
\(129\) 237.741 0.162263
\(130\) 0 0
\(131\) 878.138 0.585674 0.292837 0.956162i \(-0.405401\pi\)
0.292837 + 0.956162i \(0.405401\pi\)
\(132\) 0 0
\(133\) 352.209 0.229627
\(134\) 0 0
\(135\) 196.871 0.125511
\(136\) 0 0
\(137\) −1176.62 −0.733762 −0.366881 0.930268i \(-0.619574\pi\)
−0.366881 + 0.930268i \(0.619574\pi\)
\(138\) 0 0
\(139\) −359.514 −0.219378 −0.109689 0.993966i \(-0.534986\pi\)
−0.109689 + 0.993966i \(0.534986\pi\)
\(140\) 0 0
\(141\) −1753.22 −1.04714
\(142\) 0 0
\(143\) −665.158 −0.388974
\(144\) 0 0
\(145\) 2149.73 1.23121
\(146\) 0 0
\(147\) 925.470 0.519262
\(148\) 0 0
\(149\) 1226.94 0.674594 0.337297 0.941398i \(-0.390487\pi\)
0.337297 + 0.941398i \(0.390487\pi\)
\(150\) 0 0
\(151\) 705.665 0.380306 0.190153 0.981754i \(-0.439102\pi\)
0.190153 + 0.981754i \(0.439102\pi\)
\(152\) 0 0
\(153\) −663.741 −0.350721
\(154\) 0 0
\(155\) −2438.64 −1.26372
\(156\) 0 0
\(157\) 2519.80 1.28090 0.640452 0.767998i \(-0.278747\pi\)
0.640452 + 0.767998i \(0.278747\pi\)
\(158\) 0 0
\(159\) −1395.47 −0.696027
\(160\) 0 0
\(161\) 410.217 0.200805
\(162\) 0 0
\(163\) −1221.40 −0.586919 −0.293459 0.955972i \(-0.594807\pi\)
−0.293459 + 0.955972i \(0.594807\pi\)
\(164\) 0 0
\(165\) 1119.23 0.528073
\(166\) 0 0
\(167\) −1099.52 −0.509480 −0.254740 0.967010i \(-0.581990\pi\)
−0.254740 + 0.967010i \(0.581990\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −539.600 −0.241311
\(172\) 0 0
\(173\) −1817.39 −0.798693 −0.399346 0.916800i \(-0.630763\pi\)
−0.399346 + 0.916800i \(0.630763\pi\)
\(174\) 0 0
\(175\) 421.989 0.182282
\(176\) 0 0
\(177\) −1590.27 −0.675321
\(178\) 0 0
\(179\) 4705.22 1.96472 0.982359 0.187005i \(-0.0598781\pi\)
0.982359 + 0.187005i \(0.0598781\pi\)
\(180\) 0 0
\(181\) 1694.86 0.696010 0.348005 0.937493i \(-0.386859\pi\)
0.348005 + 0.937493i \(0.386859\pi\)
\(182\) 0 0
\(183\) −1645.00 −0.664490
\(184\) 0 0
\(185\) −1906.07 −0.757496
\(186\) 0 0
\(187\) −3773.44 −1.47562
\(188\) 0 0
\(189\) 158.612 0.0610439
\(190\) 0 0
\(191\) 1839.68 0.696933 0.348467 0.937321i \(-0.386702\pi\)
0.348467 + 0.937321i \(0.386702\pi\)
\(192\) 0 0
\(193\) −1776.15 −0.662437 −0.331219 0.943554i \(-0.607460\pi\)
−0.331219 + 0.943554i \(0.607460\pi\)
\(194\) 0 0
\(195\) −284.369 −0.104431
\(196\) 0 0
\(197\) 642.894 0.232509 0.116255 0.993219i \(-0.462911\pi\)
0.116255 + 0.993219i \(0.462911\pi\)
\(198\) 0 0
\(199\) 1086.88 0.387169 0.193584 0.981084i \(-0.437989\pi\)
0.193584 + 0.981084i \(0.437989\pi\)
\(200\) 0 0
\(201\) −1154.88 −0.405268
\(202\) 0 0
\(203\) 1731.96 0.598816
\(204\) 0 0
\(205\) −1623.82 −0.553232
\(206\) 0 0
\(207\) −628.470 −0.211023
\(208\) 0 0
\(209\) −3067.69 −1.01529
\(210\) 0 0
\(211\) −3275.58 −1.06872 −0.534361 0.845256i \(-0.679448\pi\)
−0.534361 + 0.845256i \(0.679448\pi\)
\(212\) 0 0
\(213\) −921.012 −0.296276
\(214\) 0 0
\(215\) 577.830 0.183292
\(216\) 0 0
\(217\) −1964.73 −0.614628
\(218\) 0 0
\(219\) 2534.67 0.782089
\(220\) 0 0
\(221\) 958.737 0.291817
\(222\) 0 0
\(223\) 2075.20 0.623164 0.311582 0.950219i \(-0.399141\pi\)
0.311582 + 0.950219i \(0.399141\pi\)
\(224\) 0 0
\(225\) −646.506 −0.191557
\(226\) 0 0
\(227\) −2673.53 −0.781712 −0.390856 0.920452i \(-0.627821\pi\)
−0.390856 + 0.920452i \(0.627821\pi\)
\(228\) 0 0
\(229\) 558.099 0.161049 0.0805245 0.996753i \(-0.474340\pi\)
0.0805245 + 0.996753i \(0.474340\pi\)
\(230\) 0 0
\(231\) 901.725 0.256836
\(232\) 0 0
\(233\) 2436.99 0.685204 0.342602 0.939481i \(-0.388692\pi\)
0.342602 + 0.939481i \(0.388692\pi\)
\(234\) 0 0
\(235\) −4261.19 −1.18285
\(236\) 0 0
\(237\) 90.5098 0.0248069
\(238\) 0 0
\(239\) 968.672 0.262168 0.131084 0.991371i \(-0.458154\pi\)
0.131084 + 0.991371i \(0.458154\pi\)
\(240\) 0 0
\(241\) −6872.52 −1.83692 −0.918460 0.395513i \(-0.870567\pi\)
−0.918460 + 0.395513i \(0.870567\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 2249.36 0.586556
\(246\) 0 0
\(247\) 779.422 0.200783
\(248\) 0 0
\(249\) 58.6900 0.0149370
\(250\) 0 0
\(251\) 3944.62 0.991960 0.495980 0.868334i \(-0.334809\pi\)
0.495980 + 0.868334i \(0.334809\pi\)
\(252\) 0 0
\(253\) −3572.93 −0.887857
\(254\) 0 0
\(255\) −1613.22 −0.396172
\(256\) 0 0
\(257\) 5755.13 1.39687 0.698434 0.715674i \(-0.253881\pi\)
0.698434 + 0.715674i \(0.253881\pi\)
\(258\) 0 0
\(259\) −1535.65 −0.368419
\(260\) 0 0
\(261\) −2653.44 −0.629285
\(262\) 0 0
\(263\) 4.67979 0.00109722 0.000548609 1.00000i \(-0.499825\pi\)
0.000548609 1.00000i \(0.499825\pi\)
\(264\) 0 0
\(265\) −3391.70 −0.786229
\(266\) 0 0
\(267\) 1539.05 0.352765
\(268\) 0 0
\(269\) 3269.87 0.741143 0.370571 0.928804i \(-0.379162\pi\)
0.370571 + 0.928804i \(0.379162\pi\)
\(270\) 0 0
\(271\) 2656.18 0.595393 0.297696 0.954661i \(-0.403782\pi\)
0.297696 + 0.954661i \(0.403782\pi\)
\(272\) 0 0
\(273\) −229.106 −0.0507916
\(274\) 0 0
\(275\) −3675.46 −0.805958
\(276\) 0 0
\(277\) −7909.63 −1.71568 −0.857840 0.513916i \(-0.828194\pi\)
−0.857840 + 0.513916i \(0.828194\pi\)
\(278\) 0 0
\(279\) 3010.05 0.645902
\(280\) 0 0
\(281\) −7720.91 −1.63911 −0.819557 0.572998i \(-0.805780\pi\)
−0.819557 + 0.572998i \(0.805780\pi\)
\(282\) 0 0
\(283\) −1318.54 −0.276957 −0.138478 0.990365i \(-0.544221\pi\)
−0.138478 + 0.990365i \(0.544221\pi\)
\(284\) 0 0
\(285\) −1311.50 −0.272584
\(286\) 0 0
\(287\) −1308.26 −0.269073
\(288\) 0 0
\(289\) 525.917 0.107046
\(290\) 0 0
\(291\) −2363.71 −0.476162
\(292\) 0 0
\(293\) −4038.77 −0.805281 −0.402641 0.915358i \(-0.631908\pi\)
−0.402641 + 0.915358i \(0.631908\pi\)
\(294\) 0 0
\(295\) −3865.14 −0.762839
\(296\) 0 0
\(297\) −1381.48 −0.269905
\(298\) 0 0
\(299\) 907.791 0.175582
\(300\) 0 0
\(301\) 465.537 0.0891466
\(302\) 0 0
\(303\) −4175.18 −0.791609
\(304\) 0 0
\(305\) −3998.16 −0.750604
\(306\) 0 0
\(307\) 9694.98 1.80235 0.901175 0.433455i \(-0.142706\pi\)
0.901175 + 0.433455i \(0.142706\pi\)
\(308\) 0 0
\(309\) 4285.73 0.789017
\(310\) 0 0
\(311\) 7949.12 1.44937 0.724684 0.689082i \(-0.241986\pi\)
0.724684 + 0.689082i \(0.241986\pi\)
\(312\) 0 0
\(313\) 5115.77 0.923835 0.461918 0.886923i \(-0.347162\pi\)
0.461918 + 0.886923i \(0.347162\pi\)
\(314\) 0 0
\(315\) 385.506 0.0689549
\(316\) 0 0
\(317\) −1218.09 −0.215820 −0.107910 0.994161i \(-0.534416\pi\)
−0.107910 + 0.994161i \(0.534416\pi\)
\(318\) 0 0
\(319\) −15085.1 −2.64766
\(320\) 0 0
\(321\) −4089.91 −0.711142
\(322\) 0 0
\(323\) 4421.66 0.761696
\(324\) 0 0
\(325\) 933.842 0.159385
\(326\) 0 0
\(327\) −2937.55 −0.496779
\(328\) 0 0
\(329\) −3433.09 −0.575296
\(330\) 0 0
\(331\) 9888.33 1.64203 0.821015 0.570907i \(-0.193408\pi\)
0.821015 + 0.570907i \(0.193408\pi\)
\(332\) 0 0
\(333\) 2352.68 0.387166
\(334\) 0 0
\(335\) −2806.93 −0.457788
\(336\) 0 0
\(337\) −9544.60 −1.54281 −0.771406 0.636344i \(-0.780446\pi\)
−0.771406 + 0.636344i \(0.780446\pi\)
\(338\) 0 0
\(339\) −4487.49 −0.718959
\(340\) 0 0
\(341\) 17112.5 2.71757
\(342\) 0 0
\(343\) 3827.18 0.602474
\(344\) 0 0
\(345\) −1527.50 −0.238370
\(346\) 0 0
\(347\) −4547.76 −0.703563 −0.351782 0.936082i \(-0.614424\pi\)
−0.351782 + 0.936082i \(0.614424\pi\)
\(348\) 0 0
\(349\) 1513.12 0.232079 0.116039 0.993245i \(-0.462980\pi\)
0.116039 + 0.993245i \(0.462980\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) 0 0
\(353\) 61.1114 0.00921425 0.00460713 0.999989i \(-0.498534\pi\)
0.00460713 + 0.999989i \(0.498534\pi\)
\(354\) 0 0
\(355\) −2238.52 −0.334671
\(356\) 0 0
\(357\) −1299.72 −0.192684
\(358\) 0 0
\(359\) 1065.57 0.156653 0.0783265 0.996928i \(-0.475042\pi\)
0.0783265 + 0.996928i \(0.475042\pi\)
\(360\) 0 0
\(361\) −3264.33 −0.475920
\(362\) 0 0
\(363\) −3860.88 −0.558247
\(364\) 0 0
\(365\) 6160.53 0.883443
\(366\) 0 0
\(367\) 8962.02 1.27470 0.637348 0.770576i \(-0.280031\pi\)
0.637348 + 0.770576i \(0.280031\pi\)
\(368\) 0 0
\(369\) 2004.31 0.282764
\(370\) 0 0
\(371\) −2732.58 −0.382394
\(372\) 0 0
\(373\) 1250.29 0.173560 0.0867798 0.996228i \(-0.472342\pi\)
0.0867798 + 0.996228i \(0.472342\pi\)
\(374\) 0 0
\(375\) −4305.65 −0.592914
\(376\) 0 0
\(377\) 3832.74 0.523597
\(378\) 0 0
\(379\) −8367.05 −1.13400 −0.567001 0.823717i \(-0.691896\pi\)
−0.567001 + 0.823717i \(0.691896\pi\)
\(380\) 0 0
\(381\) 6665.08 0.896227
\(382\) 0 0
\(383\) 6241.47 0.832700 0.416350 0.909204i \(-0.363309\pi\)
0.416350 + 0.909204i \(0.363309\pi\)
\(384\) 0 0
\(385\) 2191.64 0.290121
\(386\) 0 0
\(387\) −713.223 −0.0936826
\(388\) 0 0
\(389\) 2651.76 0.345628 0.172814 0.984954i \(-0.444714\pi\)
0.172814 + 0.984954i \(0.444714\pi\)
\(390\) 0 0
\(391\) 5149.90 0.666091
\(392\) 0 0
\(393\) −2634.42 −0.338139
\(394\) 0 0
\(395\) 219.984 0.0280218
\(396\) 0 0
\(397\) −11460.4 −1.44882 −0.724408 0.689372i \(-0.757887\pi\)
−0.724408 + 0.689372i \(0.757887\pi\)
\(398\) 0 0
\(399\) −1056.63 −0.132575
\(400\) 0 0
\(401\) −6271.25 −0.780976 −0.390488 0.920608i \(-0.627694\pi\)
−0.390488 + 0.920608i \(0.627694\pi\)
\(402\) 0 0
\(403\) −4347.85 −0.537423
\(404\) 0 0
\(405\) −590.612 −0.0724635
\(406\) 0 0
\(407\) 13375.3 1.62896
\(408\) 0 0
\(409\) −4432.82 −0.535915 −0.267957 0.963431i \(-0.586349\pi\)
−0.267957 + 0.963431i \(0.586349\pi\)
\(410\) 0 0
\(411\) 3529.86 0.423638
\(412\) 0 0
\(413\) −3114.01 −0.371018
\(414\) 0 0
\(415\) 142.646 0.0168728
\(416\) 0 0
\(417\) 1078.54 0.126658
\(418\) 0 0
\(419\) −11383.5 −1.32726 −0.663630 0.748061i \(-0.730985\pi\)
−0.663630 + 0.748061i \(0.730985\pi\)
\(420\) 0 0
\(421\) 7422.33 0.859245 0.429623 0.903009i \(-0.358647\pi\)
0.429623 + 0.903009i \(0.358647\pi\)
\(422\) 0 0
\(423\) 5259.65 0.604569
\(424\) 0 0
\(425\) 5297.69 0.604648
\(426\) 0 0
\(427\) −3221.18 −0.365068
\(428\) 0 0
\(429\) 1995.47 0.224574
\(430\) 0 0
\(431\) −17065.8 −1.90727 −0.953633 0.300971i \(-0.902689\pi\)
−0.953633 + 0.300971i \(0.902689\pi\)
\(432\) 0 0
\(433\) −13703.0 −1.52084 −0.760420 0.649432i \(-0.775007\pi\)
−0.760420 + 0.649432i \(0.775007\pi\)
\(434\) 0 0
\(435\) −6449.18 −0.710838
\(436\) 0 0
\(437\) 4186.70 0.458300
\(438\) 0 0
\(439\) −8990.23 −0.977404 −0.488702 0.872451i \(-0.662529\pi\)
−0.488702 + 0.872451i \(0.662529\pi\)
\(440\) 0 0
\(441\) −2776.41 −0.299796
\(442\) 0 0
\(443\) −11047.9 −1.18488 −0.592438 0.805616i \(-0.701835\pi\)
−0.592438 + 0.805616i \(0.701835\pi\)
\(444\) 0 0
\(445\) 3740.66 0.398482
\(446\) 0 0
\(447\) −3680.81 −0.389477
\(448\) 0 0
\(449\) 14780.2 1.55350 0.776750 0.629809i \(-0.216867\pi\)
0.776750 + 0.629809i \(0.216867\pi\)
\(450\) 0 0
\(451\) 11394.7 1.18970
\(452\) 0 0
\(453\) −2117.00 −0.219570
\(454\) 0 0
\(455\) −556.842 −0.0573740
\(456\) 0 0
\(457\) 407.385 0.0416995 0.0208498 0.999783i \(-0.493363\pi\)
0.0208498 + 0.999783i \(0.493363\pi\)
\(458\) 0 0
\(459\) 1991.22 0.202489
\(460\) 0 0
\(461\) 13198.6 1.33345 0.666725 0.745303i \(-0.267695\pi\)
0.666725 + 0.745303i \(0.267695\pi\)
\(462\) 0 0
\(463\) 534.058 0.0536064 0.0268032 0.999641i \(-0.491467\pi\)
0.0268032 + 0.999641i \(0.491467\pi\)
\(464\) 0 0
\(465\) 7315.92 0.729608
\(466\) 0 0
\(467\) −5150.44 −0.510351 −0.255176 0.966895i \(-0.582133\pi\)
−0.255176 + 0.966895i \(0.582133\pi\)
\(468\) 0 0
\(469\) −2261.45 −0.222652
\(470\) 0 0
\(471\) −7559.40 −0.739530
\(472\) 0 0
\(473\) −4054.76 −0.394160
\(474\) 0 0
\(475\) 4306.85 0.416025
\(476\) 0 0
\(477\) 4186.42 0.401851
\(478\) 0 0
\(479\) 20668.2 1.97151 0.985756 0.168179i \(-0.0537888\pi\)
0.985756 + 0.168179i \(0.0537888\pi\)
\(480\) 0 0
\(481\) −3398.32 −0.322141
\(482\) 0 0
\(483\) −1230.65 −0.115935
\(484\) 0 0
\(485\) −5745.00 −0.537870
\(486\) 0 0
\(487\) −3239.32 −0.301412 −0.150706 0.988579i \(-0.548155\pi\)
−0.150706 + 0.988579i \(0.548155\pi\)
\(488\) 0 0
\(489\) 3664.21 0.338858
\(490\) 0 0
\(491\) 4704.78 0.432432 0.216216 0.976346i \(-0.430629\pi\)
0.216216 + 0.976346i \(0.430629\pi\)
\(492\) 0 0
\(493\) 21743.1 1.98633
\(494\) 0 0
\(495\) −3357.69 −0.304883
\(496\) 0 0
\(497\) −1803.50 −0.162772
\(498\) 0 0
\(499\) 12173.8 1.09213 0.546065 0.837743i \(-0.316125\pi\)
0.546065 + 0.837743i \(0.316125\pi\)
\(500\) 0 0
\(501\) 3298.55 0.294148
\(502\) 0 0
\(503\) 10090.3 0.894439 0.447219 0.894424i \(-0.352414\pi\)
0.447219 + 0.894424i \(0.352414\pi\)
\(504\) 0 0
\(505\) −10147.8 −0.894198
\(506\) 0 0
\(507\) −507.000 −0.0444116
\(508\) 0 0
\(509\) 12946.5 1.12739 0.563696 0.825982i \(-0.309379\pi\)
0.563696 + 0.825982i \(0.309379\pi\)
\(510\) 0 0
\(511\) 4963.32 0.429676
\(512\) 0 0
\(513\) 1618.80 0.139321
\(514\) 0 0
\(515\) 10416.5 0.891270
\(516\) 0 0
\(517\) 29901.7 2.54366
\(518\) 0 0
\(519\) 5452.18 0.461126
\(520\) 0 0
\(521\) −2267.25 −0.190653 −0.0953265 0.995446i \(-0.530390\pi\)
−0.0953265 + 0.995446i \(0.530390\pi\)
\(522\) 0 0
\(523\) −8850.80 −0.739997 −0.369998 0.929032i \(-0.620642\pi\)
−0.369998 + 0.929032i \(0.620642\pi\)
\(524\) 0 0
\(525\) −1265.97 −0.105241
\(526\) 0 0
\(527\) −24665.3 −2.03878
\(528\) 0 0
\(529\) −7290.76 −0.599224
\(530\) 0 0
\(531\) 4770.80 0.389897
\(532\) 0 0
\(533\) −2895.11 −0.235274
\(534\) 0 0
\(535\) −9940.54 −0.803303
\(536\) 0 0
\(537\) −14115.7 −1.13433
\(538\) 0 0
\(539\) −15784.2 −1.26136
\(540\) 0 0
\(541\) −6720.57 −0.534085 −0.267042 0.963685i \(-0.586046\pi\)
−0.267042 + 0.963685i \(0.586046\pi\)
\(542\) 0 0
\(543\) −5084.57 −0.401841
\(544\) 0 0
\(545\) −7139.71 −0.561159
\(546\) 0 0
\(547\) 3251.85 0.254185 0.127093 0.991891i \(-0.459435\pi\)
0.127093 + 0.991891i \(0.459435\pi\)
\(548\) 0 0
\(549\) 4934.99 0.383643
\(550\) 0 0
\(551\) 17676.5 1.36668
\(552\) 0 0
\(553\) 177.234 0.0136288
\(554\) 0 0
\(555\) 5718.20 0.437340
\(556\) 0 0
\(557\) 4215.22 0.320655 0.160327 0.987064i \(-0.448745\pi\)
0.160327 + 0.987064i \(0.448745\pi\)
\(558\) 0 0
\(559\) 1030.21 0.0779487
\(560\) 0 0
\(561\) 11320.3 0.851951
\(562\) 0 0
\(563\) −15802.9 −1.18297 −0.591486 0.806316i \(-0.701458\pi\)
−0.591486 + 0.806316i \(0.701458\pi\)
\(564\) 0 0
\(565\) −10906.8 −0.812132
\(566\) 0 0
\(567\) −475.835 −0.0352437
\(568\) 0 0
\(569\) 12964.1 0.955155 0.477578 0.878590i \(-0.341515\pi\)
0.477578 + 0.878590i \(0.341515\pi\)
\(570\) 0 0
\(571\) 13555.7 0.993499 0.496750 0.867894i \(-0.334527\pi\)
0.496750 + 0.867894i \(0.334527\pi\)
\(572\) 0 0
\(573\) −5519.03 −0.402375
\(574\) 0 0
\(575\) 5016.17 0.363807
\(576\) 0 0
\(577\) 25349.5 1.82897 0.914483 0.404624i \(-0.132598\pi\)
0.914483 + 0.404624i \(0.132598\pi\)
\(578\) 0 0
\(579\) 5328.46 0.382458
\(580\) 0 0
\(581\) 114.925 0.00820635
\(582\) 0 0
\(583\) 23800.3 1.69075
\(584\) 0 0
\(585\) 853.106 0.0602933
\(586\) 0 0
\(587\) 18013.3 1.26659 0.633296 0.773910i \(-0.281702\pi\)
0.633296 + 0.773910i \(0.281702\pi\)
\(588\) 0 0
\(589\) −20052.1 −1.40277
\(590\) 0 0
\(591\) −1928.68 −0.134239
\(592\) 0 0
\(593\) 21521.6 1.49036 0.745181 0.666862i \(-0.232363\pi\)
0.745181 + 0.666862i \(0.232363\pi\)
\(594\) 0 0
\(595\) −3158.96 −0.217655
\(596\) 0 0
\(597\) −3260.63 −0.223532
\(598\) 0 0
\(599\) −10081.8 −0.687700 −0.343850 0.939025i \(-0.611731\pi\)
−0.343850 + 0.939025i \(0.611731\pi\)
\(600\) 0 0
\(601\) −5180.24 −0.351591 −0.175796 0.984427i \(-0.556250\pi\)
−0.175796 + 0.984427i \(0.556250\pi\)
\(602\) 0 0
\(603\) 3464.64 0.233982
\(604\) 0 0
\(605\) −9383.88 −0.630593
\(606\) 0 0
\(607\) 24166.4 1.61595 0.807976 0.589215i \(-0.200563\pi\)
0.807976 + 0.589215i \(0.200563\pi\)
\(608\) 0 0
\(609\) −5195.88 −0.345727
\(610\) 0 0
\(611\) −7597.27 −0.503032
\(612\) 0 0
\(613\) 23768.7 1.56608 0.783040 0.621971i \(-0.213668\pi\)
0.783040 + 0.621971i \(0.213668\pi\)
\(614\) 0 0
\(615\) 4871.47 0.319409
\(616\) 0 0
\(617\) −20306.1 −1.32495 −0.662475 0.749084i \(-0.730494\pi\)
−0.662475 + 0.749084i \(0.730494\pi\)
\(618\) 0 0
\(619\) 16062.5 1.04299 0.521493 0.853256i \(-0.325375\pi\)
0.521493 + 0.853256i \(0.325375\pi\)
\(620\) 0 0
\(621\) 1885.41 0.121834
\(622\) 0 0
\(623\) 3013.72 0.193808
\(624\) 0 0
\(625\) −1485.63 −0.0950803
\(626\) 0 0
\(627\) 9203.06 0.586180
\(628\) 0 0
\(629\) −19278.7 −1.22208
\(630\) 0 0
\(631\) −6958.72 −0.439021 −0.219511 0.975610i \(-0.570446\pi\)
−0.219511 + 0.975610i \(0.570446\pi\)
\(632\) 0 0
\(633\) 9826.74 0.617027
\(634\) 0 0
\(635\) 16199.5 1.01237
\(636\) 0 0
\(637\) 4010.37 0.249445
\(638\) 0 0
\(639\) 2763.04 0.171055
\(640\) 0 0
\(641\) −15200.3 −0.936625 −0.468312 0.883563i \(-0.655138\pi\)
−0.468312 + 0.883563i \(0.655138\pi\)
\(642\) 0 0
\(643\) 27963.3 1.71503 0.857516 0.514457i \(-0.172006\pi\)
0.857516 + 0.514457i \(0.172006\pi\)
\(644\) 0 0
\(645\) −1733.49 −0.105823
\(646\) 0 0
\(647\) −3952.07 −0.240142 −0.120071 0.992765i \(-0.538312\pi\)
−0.120071 + 0.992765i \(0.538312\pi\)
\(648\) 0 0
\(649\) 27122.5 1.64045
\(650\) 0 0
\(651\) 5894.18 0.354856
\(652\) 0 0
\(653\) −3772.15 −0.226058 −0.113029 0.993592i \(-0.536055\pi\)
−0.113029 + 0.993592i \(0.536055\pi\)
\(654\) 0 0
\(655\) −6402.95 −0.381960
\(656\) 0 0
\(657\) −7604.02 −0.451539
\(658\) 0 0
\(659\) −7305.22 −0.431822 −0.215911 0.976413i \(-0.569272\pi\)
−0.215911 + 0.976413i \(0.569272\pi\)
\(660\) 0 0
\(661\) −10985.6 −0.646431 −0.323216 0.946325i \(-0.604764\pi\)
−0.323216 + 0.946325i \(0.604764\pi\)
\(662\) 0 0
\(663\) −2876.21 −0.168481
\(664\) 0 0
\(665\) −2568.14 −0.149756
\(666\) 0 0
\(667\) 20587.7 1.19514
\(668\) 0 0
\(669\) −6225.59 −0.359784
\(670\) 0 0
\(671\) 28056.0 1.61414
\(672\) 0 0
\(673\) −9236.54 −0.529038 −0.264519 0.964380i \(-0.585213\pi\)
−0.264519 + 0.964380i \(0.585213\pi\)
\(674\) 0 0
\(675\) 1939.52 0.110596
\(676\) 0 0
\(677\) −1763.78 −0.100129 −0.0500646 0.998746i \(-0.515943\pi\)
−0.0500646 + 0.998746i \(0.515943\pi\)
\(678\) 0 0
\(679\) −4628.54 −0.261601
\(680\) 0 0
\(681\) 8020.60 0.451322
\(682\) 0 0
\(683\) 9580.46 0.536729 0.268365 0.963317i \(-0.413517\pi\)
0.268365 + 0.963317i \(0.413517\pi\)
\(684\) 0 0
\(685\) 8579.32 0.478539
\(686\) 0 0
\(687\) −1674.30 −0.0929817
\(688\) 0 0
\(689\) −6047.06 −0.334361
\(690\) 0 0
\(691\) −10720.2 −0.590179 −0.295090 0.955470i \(-0.595349\pi\)
−0.295090 + 0.955470i \(0.595349\pi\)
\(692\) 0 0
\(693\) −2705.18 −0.148284
\(694\) 0 0
\(695\) 2621.40 0.143072
\(696\) 0 0
\(697\) −16424.0 −0.892542
\(698\) 0 0
\(699\) −7310.97 −0.395603
\(700\) 0 0
\(701\) −9060.31 −0.488164 −0.244082 0.969755i \(-0.578487\pi\)
−0.244082 + 0.969755i \(0.578487\pi\)
\(702\) 0 0
\(703\) −15672.9 −0.840847
\(704\) 0 0
\(705\) 12783.6 0.682918
\(706\) 0 0
\(707\) −8175.70 −0.434907
\(708\) 0 0
\(709\) 26681.4 1.41332 0.706658 0.707556i \(-0.250202\pi\)
0.706658 + 0.707556i \(0.250202\pi\)
\(710\) 0 0
\(711\) −271.530 −0.0143223
\(712\) 0 0
\(713\) −23354.6 −1.22670
\(714\) 0 0
\(715\) 4850.00 0.253678
\(716\) 0 0
\(717\) −2906.02 −0.151363
\(718\) 0 0
\(719\) 491.820 0.0255102 0.0127551 0.999919i \(-0.495940\pi\)
0.0127551 + 0.999919i \(0.495940\pi\)
\(720\) 0 0
\(721\) 8392.18 0.433483
\(722\) 0 0
\(723\) 20617.6 1.06055
\(724\) 0 0
\(725\) 21178.5 1.08490
\(726\) 0 0
\(727\) −12358.2 −0.630452 −0.315226 0.949017i \(-0.602080\pi\)
−0.315226 + 0.949017i \(0.602080\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 5844.39 0.295708
\(732\) 0 0
\(733\) −24037.3 −1.21124 −0.605620 0.795754i \(-0.707075\pi\)
−0.605620 + 0.795754i \(0.707075\pi\)
\(734\) 0 0
\(735\) −6748.07 −0.338648
\(736\) 0 0
\(737\) 19696.8 0.984454
\(738\) 0 0
\(739\) −21651.5 −1.07776 −0.538879 0.842383i \(-0.681152\pi\)
−0.538879 + 0.842383i \(0.681152\pi\)
\(740\) 0 0
\(741\) −2338.27 −0.115922
\(742\) 0 0
\(743\) 2580.05 0.127393 0.0636964 0.997969i \(-0.479711\pi\)
0.0636964 + 0.997969i \(0.479711\pi\)
\(744\) 0 0
\(745\) −8946.21 −0.439951
\(746\) 0 0
\(747\) −176.070 −0.00862391
\(748\) 0 0
\(749\) −8008.74 −0.390698
\(750\) 0 0
\(751\) −38542.3 −1.87274 −0.936371 0.351012i \(-0.885838\pi\)
−0.936371 + 0.351012i \(0.885838\pi\)
\(752\) 0 0
\(753\) −11833.9 −0.572708
\(754\) 0 0
\(755\) −5145.36 −0.248025
\(756\) 0 0
\(757\) 997.651 0.0478999 0.0239500 0.999713i \(-0.492376\pi\)
0.0239500 + 0.999713i \(0.492376\pi\)
\(758\) 0 0
\(759\) 10718.8 0.512605
\(760\) 0 0
\(761\) −29051.7 −1.38387 −0.691933 0.721962i \(-0.743241\pi\)
−0.691933 + 0.721962i \(0.743241\pi\)
\(762\) 0 0
\(763\) −5752.21 −0.272928
\(764\) 0 0
\(765\) 4839.67 0.228730
\(766\) 0 0
\(767\) −6891.16 −0.324414
\(768\) 0 0
\(769\) −8935.69 −0.419024 −0.209512 0.977806i \(-0.567188\pi\)
−0.209512 + 0.977806i \(0.567188\pi\)
\(770\) 0 0
\(771\) −17265.4 −0.806482
\(772\) 0 0
\(773\) −22239.3 −1.03479 −0.517395 0.855747i \(-0.673098\pi\)
−0.517395 + 0.855747i \(0.673098\pi\)
\(774\) 0 0
\(775\) −24024.9 −1.11355
\(776\) 0 0
\(777\) 4606.95 0.212707
\(778\) 0 0
\(779\) −13352.1 −0.614108
\(780\) 0 0
\(781\) 15708.2 0.719696
\(782\) 0 0
\(783\) 7960.31 0.363318
\(784\) 0 0
\(785\) −18373.1 −0.835369
\(786\) 0 0
\(787\) −36698.1 −1.66219 −0.831096 0.556129i \(-0.812286\pi\)
−0.831096 + 0.556129i \(0.812286\pi\)
\(788\) 0 0
\(789\) −14.0394 −0.000633479 0
\(790\) 0 0
\(791\) −8787.27 −0.394993
\(792\) 0 0
\(793\) −7128.32 −0.319211
\(794\) 0 0
\(795\) 10175.1 0.453929
\(796\) 0 0
\(797\) −35811.1 −1.59159 −0.795794 0.605568i \(-0.792946\pi\)
−0.795794 + 0.605568i \(0.792946\pi\)
\(798\) 0 0
\(799\) −43099.3 −1.90832
\(800\) 0 0
\(801\) −4617.15 −0.203669
\(802\) 0 0
\(803\) −43229.7 −1.89981
\(804\) 0 0
\(805\) −2991.10 −0.130960
\(806\) 0 0
\(807\) −9809.61 −0.427899
\(808\) 0 0
\(809\) 28481.2 1.23776 0.618879 0.785486i \(-0.287587\pi\)
0.618879 + 0.785486i \(0.287587\pi\)
\(810\) 0 0
\(811\) 18290.5 0.791944 0.395972 0.918263i \(-0.370408\pi\)
0.395972 + 0.918263i \(0.370408\pi\)
\(812\) 0 0
\(813\) −7968.54 −0.343750
\(814\) 0 0
\(815\) 8905.87 0.382772
\(816\) 0 0
\(817\) 4751.30 0.203460
\(818\) 0 0
\(819\) 687.317 0.0293246
\(820\) 0 0
\(821\) −21577.5 −0.917248 −0.458624 0.888631i \(-0.651657\pi\)
−0.458624 + 0.888631i \(0.651657\pi\)
\(822\) 0 0
\(823\) 25940.2 1.09869 0.549344 0.835596i \(-0.314878\pi\)
0.549344 + 0.835596i \(0.314878\pi\)
\(824\) 0 0
\(825\) 11026.4 0.465320
\(826\) 0 0
\(827\) 3112.03 0.130853 0.0654267 0.997857i \(-0.479159\pi\)
0.0654267 + 0.997857i \(0.479159\pi\)
\(828\) 0 0
\(829\) −3729.80 −0.156262 −0.0781312 0.996943i \(-0.524895\pi\)
−0.0781312 + 0.996943i \(0.524895\pi\)
\(830\) 0 0
\(831\) 23728.9 0.990549
\(832\) 0 0
\(833\) 22750.8 0.946303
\(834\) 0 0
\(835\) 8017.12 0.332268
\(836\) 0 0
\(837\) −9030.14 −0.372912
\(838\) 0 0
\(839\) 22919.3 0.943102 0.471551 0.881839i \(-0.343694\pi\)
0.471551 + 0.881839i \(0.343694\pi\)
\(840\) 0 0
\(841\) 62533.4 2.56400
\(842\) 0 0
\(843\) 23162.7 0.946343
\(844\) 0 0
\(845\) −1232.26 −0.0501671
\(846\) 0 0
\(847\) −7560.26 −0.306698
\(848\) 0 0
\(849\) 3955.61 0.159901
\(850\) 0 0
\(851\) −18254.2 −0.735307
\(852\) 0 0
\(853\) 10092.6 0.405115 0.202557 0.979270i \(-0.435075\pi\)
0.202557 + 0.979270i \(0.435075\pi\)
\(854\) 0 0
\(855\) 3934.49 0.157376
\(856\) 0 0
\(857\) −14803.0 −0.590036 −0.295018 0.955492i \(-0.595326\pi\)
−0.295018 + 0.955492i \(0.595326\pi\)
\(858\) 0 0
\(859\) −7164.74 −0.284584 −0.142292 0.989825i \(-0.545447\pi\)
−0.142292 + 0.989825i \(0.545447\pi\)
\(860\) 0 0
\(861\) 3924.77 0.155349
\(862\) 0 0
\(863\) −20504.4 −0.808782 −0.404391 0.914586i \(-0.632517\pi\)
−0.404391 + 0.914586i \(0.632517\pi\)
\(864\) 0 0
\(865\) 13251.5 0.520885
\(866\) 0 0
\(867\) −1577.75 −0.0618031
\(868\) 0 0
\(869\) −1543.68 −0.0602596
\(870\) 0 0
\(871\) −5004.47 −0.194684
\(872\) 0 0
\(873\) 7091.13 0.274912
\(874\) 0 0
\(875\) −8431.19 −0.325744
\(876\) 0 0
\(877\) −23980.1 −0.923318 −0.461659 0.887057i \(-0.652746\pi\)
−0.461659 + 0.887057i \(0.652746\pi\)
\(878\) 0 0
\(879\) 12116.3 0.464929
\(880\) 0 0
\(881\) 32577.9 1.24583 0.622915 0.782289i \(-0.285948\pi\)
0.622915 + 0.782289i \(0.285948\pi\)
\(882\) 0 0
\(883\) 5201.59 0.198242 0.0991208 0.995075i \(-0.468397\pi\)
0.0991208 + 0.995075i \(0.468397\pi\)
\(884\) 0 0
\(885\) 11595.4 0.440425
\(886\) 0 0
\(887\) −10719.5 −0.405779 −0.202890 0.979202i \(-0.565033\pi\)
−0.202890 + 0.979202i \(0.565033\pi\)
\(888\) 0 0
\(889\) 13051.4 0.492383
\(890\) 0 0
\(891\) 4144.45 0.155830
\(892\) 0 0
\(893\) −35038.3 −1.31300
\(894\) 0 0
\(895\) −34308.1 −1.28133
\(896\) 0 0
\(897\) −2723.37 −0.101372
\(898\) 0 0
\(899\) −98604.5 −3.65811
\(900\) 0 0
\(901\) −34305.0 −1.26844
\(902\) 0 0
\(903\) −1396.61 −0.0514688
\(904\) 0 0
\(905\) −12358.1 −0.453918
\(906\) 0 0
\(907\) 18995.9 0.695424 0.347712 0.937601i \(-0.386959\pi\)
0.347712 + 0.937601i \(0.386959\pi\)
\(908\) 0 0
\(909\) 12525.5 0.457036
\(910\) 0 0
\(911\) 8475.13 0.308226 0.154113 0.988053i \(-0.450748\pi\)
0.154113 + 0.988053i \(0.450748\pi\)
\(912\) 0 0
\(913\) −1000.98 −0.0362842
\(914\) 0 0
\(915\) 11994.5 0.433361
\(916\) 0 0
\(917\) −5158.63 −0.185772
\(918\) 0 0
\(919\) 41161.3 1.47746 0.738731 0.674000i \(-0.235425\pi\)
0.738731 + 0.674000i \(0.235425\pi\)
\(920\) 0 0
\(921\) −29084.9 −1.04059
\(922\) 0 0
\(923\) −3991.05 −0.142326
\(924\) 0 0
\(925\) −18778.1 −0.667480
\(926\) 0 0
\(927\) −12857.2 −0.455539
\(928\) 0 0
\(929\) −24181.6 −0.854007 −0.427004 0.904250i \(-0.640431\pi\)
−0.427004 + 0.904250i \(0.640431\pi\)
\(930\) 0 0
\(931\) 18495.7 0.651098
\(932\) 0 0
\(933\) −23847.4 −0.836792
\(934\) 0 0
\(935\) 27514.1 0.962360
\(936\) 0 0
\(937\) 3815.36 0.133023 0.0665114 0.997786i \(-0.478813\pi\)
0.0665114 + 0.997786i \(0.478813\pi\)
\(938\) 0 0
\(939\) −15347.3 −0.533377
\(940\) 0 0
\(941\) 34381.2 1.19107 0.595534 0.803330i \(-0.296940\pi\)
0.595534 + 0.803330i \(0.296940\pi\)
\(942\) 0 0
\(943\) −15551.2 −0.537027
\(944\) 0 0
\(945\) −1156.52 −0.0398111
\(946\) 0 0
\(947\) 20897.6 0.717087 0.358543 0.933513i \(-0.383274\pi\)
0.358543 + 0.933513i \(0.383274\pi\)
\(948\) 0 0
\(949\) 10983.6 0.375703
\(950\) 0 0
\(951\) 3654.28 0.124604
\(952\) 0 0
\(953\) −21656.9 −0.736136 −0.368068 0.929799i \(-0.619981\pi\)
−0.368068 + 0.929799i \(0.619981\pi\)
\(954\) 0 0
\(955\) −13414.0 −0.454520
\(956\) 0 0
\(957\) 45255.2 1.52862
\(958\) 0 0
\(959\) 6912.06 0.232745
\(960\) 0 0
\(961\) 82065.6 2.75471
\(962\) 0 0
\(963\) 12269.7 0.410578
\(964\) 0 0
\(965\) 12950.8 0.432023
\(966\) 0 0
\(967\) −38712.2 −1.28738 −0.643691 0.765285i \(-0.722598\pi\)
−0.643691 + 0.765285i \(0.722598\pi\)
\(968\) 0 0
\(969\) −13265.0 −0.439766
\(970\) 0 0
\(971\) −11532.4 −0.381147 −0.190573 0.981673i \(-0.561035\pi\)
−0.190573 + 0.981673i \(0.561035\pi\)
\(972\) 0 0
\(973\) 2111.97 0.0695853
\(974\) 0 0
\(975\) −2801.53 −0.0920212
\(976\) 0 0
\(977\) 26477.6 0.867036 0.433518 0.901145i \(-0.357272\pi\)
0.433518 + 0.901145i \(0.357272\pi\)
\(978\) 0 0
\(979\) −26249.0 −0.856918
\(980\) 0 0
\(981\) 8812.64 0.286815
\(982\) 0 0
\(983\) 23993.5 0.778508 0.389254 0.921131i \(-0.372733\pi\)
0.389254 + 0.921131i \(0.372733\pi\)
\(984\) 0 0
\(985\) −4687.66 −0.151636
\(986\) 0 0
\(987\) 10299.3 0.332148
\(988\) 0 0
\(989\) 5533.83 0.177923
\(990\) 0 0
\(991\) 9436.25 0.302475 0.151237 0.988497i \(-0.451674\pi\)
0.151237 + 0.988497i \(0.451674\pi\)
\(992\) 0 0
\(993\) −29665.0 −0.948026
\(994\) 0 0
\(995\) −7924.96 −0.252501
\(996\) 0 0
\(997\) −31197.1 −0.990995 −0.495498 0.868609i \(-0.665014\pi\)
−0.495498 + 0.868609i \(0.665014\pi\)
\(998\) 0 0
\(999\) −7058.05 −0.223530
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 624.4.a.k.1.1 2
3.2 odd 2 1872.4.a.bf.1.2 2
4.3 odd 2 312.4.a.e.1.1 2
8.3 odd 2 2496.4.a.y.1.2 2
8.5 even 2 2496.4.a.bh.1.2 2
12.11 even 2 936.4.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.e.1.1 2 4.3 odd 2
624.4.a.k.1.1 2 1.1 even 1 trivial
936.4.a.f.1.2 2 12.11 even 2
1872.4.a.bf.1.2 2 3.2 odd 2
2496.4.a.y.1.2 2 8.3 odd 2
2496.4.a.bh.1.2 2 8.5 even 2