Properties

Label 414.4.a.m.1.1
Level $414$
Weight $4$
Character 414.1
Self dual yes
Analytic conductor $24.427$
Analytic rank $0$
Dimension $4$
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] = [414,4,Mod(1,414)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(414, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("414.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 414.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: \(24.4267907424\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 219x^{2} - 468x + 3240 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-11.4152\) of defining polynomial
Character \(\chi\) \(=\) 414.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -13.2306 q^{5} -10.2926 q^{7} -8.00000 q^{8} +26.4612 q^{10} -14.6928 q^{11} -23.2916 q^{13} +20.5853 q^{14} +16.0000 q^{16} -119.245 q^{17} +112.461 q^{19} -52.9223 q^{20} +29.3855 q^{22} -23.0000 q^{23} +50.0484 q^{25} +46.5833 q^{26} -41.1705 q^{28} +104.646 q^{29} -27.1038 q^{31} -32.0000 q^{32} +238.490 q^{34} +136.177 q^{35} -24.7531 q^{37} -224.921 q^{38} +105.845 q^{40} +174.061 q^{41} +506.208 q^{43} -58.7710 q^{44} +46.0000 q^{46} -256.367 q^{47} -237.062 q^{49} -100.097 q^{50} -93.1665 q^{52} -200.452 q^{53} +194.394 q^{55} +82.3410 q^{56} -209.292 q^{58} +446.988 q^{59} +419.190 q^{61} +54.2077 q^{62} +64.0000 q^{64} +308.162 q^{65} -70.3675 q^{67} -476.981 q^{68} -272.355 q^{70} -190.743 q^{71} +480.337 q^{73} +49.5063 q^{74} +449.842 q^{76} +151.227 q^{77} +1321.34 q^{79} -211.689 q^{80} -348.123 q^{82} +407.398 q^{83} +1577.68 q^{85} -1012.42 q^{86} +117.542 q^{88} +1435.48 q^{89} +239.732 q^{91} -92.0000 q^{92} +512.734 q^{94} -1487.92 q^{95} -985.582 q^{97} +474.124 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} + 16 q^{4} + 18 q^{7} - 32 q^{8} - 42 q^{11} + 108 q^{13} - 36 q^{14} + 64 q^{16} - 26 q^{17} + 132 q^{19} + 84 q^{22} - 92 q^{23} + 260 q^{25} - 216 q^{26} + 72 q^{28} - 252 q^{29} + 428 q^{31}+ \cdots - 2152 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −13.2306 −1.18338 −0.591690 0.806166i \(-0.701539\pi\)
−0.591690 + 0.806166i \(0.701539\pi\)
\(6\) 0 0
\(7\) −10.2926 −0.555750 −0.277875 0.960617i \(-0.589630\pi\)
−0.277875 + 0.960617i \(0.589630\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 26.4612 0.836776
\(11\) −14.6928 −0.402730 −0.201365 0.979516i \(-0.564538\pi\)
−0.201365 + 0.979516i \(0.564538\pi\)
\(12\) 0 0
\(13\) −23.2916 −0.496918 −0.248459 0.968642i \(-0.579924\pi\)
−0.248459 + 0.968642i \(0.579924\pi\)
\(14\) 20.5853 0.392974
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −119.245 −1.70125 −0.850623 0.525775i \(-0.823775\pi\)
−0.850623 + 0.525775i \(0.823775\pi\)
\(18\) 0 0
\(19\) 112.461 1.35791 0.678953 0.734182i \(-0.262434\pi\)
0.678953 + 0.734182i \(0.262434\pi\)
\(20\) −52.9223 −0.591690
\(21\) 0 0
\(22\) 29.3855 0.284773
\(23\) −23.0000 −0.208514
\(24\) 0 0
\(25\) 50.0484 0.400387
\(26\) 46.5833 0.351374
\(27\) 0 0
\(28\) −41.1705 −0.277875
\(29\) 104.646 0.670078 0.335039 0.942204i \(-0.391251\pi\)
0.335039 + 0.942204i \(0.391251\pi\)
\(30\) 0 0
\(31\) −27.1038 −0.157032 −0.0785160 0.996913i \(-0.525018\pi\)
−0.0785160 + 0.996913i \(0.525018\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) 238.490 1.20296
\(35\) 136.177 0.657663
\(36\) 0 0
\(37\) −24.7531 −0.109983 −0.0549917 0.998487i \(-0.517513\pi\)
−0.0549917 + 0.998487i \(0.517513\pi\)
\(38\) −224.921 −0.960184
\(39\) 0 0
\(40\) 105.845 0.418388
\(41\) 174.061 0.663020 0.331510 0.943452i \(-0.392442\pi\)
0.331510 + 0.943452i \(0.392442\pi\)
\(42\) 0 0
\(43\) 506.208 1.79526 0.897628 0.440753i \(-0.145289\pi\)
0.897628 + 0.440753i \(0.145289\pi\)
\(44\) −58.7710 −0.201365
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) −256.367 −0.795637 −0.397819 0.917464i \(-0.630233\pi\)
−0.397819 + 0.917464i \(0.630233\pi\)
\(48\) 0 0
\(49\) −237.062 −0.691142
\(50\) −100.097 −0.283117
\(51\) 0 0
\(52\) −93.1665 −0.248459
\(53\) −200.452 −0.519514 −0.259757 0.965674i \(-0.583642\pi\)
−0.259757 + 0.965674i \(0.583642\pi\)
\(54\) 0 0
\(55\) 194.394 0.476583
\(56\) 82.3410 0.196487
\(57\) 0 0
\(58\) −209.292 −0.473817
\(59\) 446.988 0.986319 0.493160 0.869939i \(-0.335842\pi\)
0.493160 + 0.869939i \(0.335842\pi\)
\(60\) 0 0
\(61\) 419.190 0.879866 0.439933 0.898031i \(-0.355002\pi\)
0.439933 + 0.898031i \(0.355002\pi\)
\(62\) 54.2077 0.111038
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 308.162 0.588043
\(66\) 0 0
\(67\) −70.3675 −0.128310 −0.0641549 0.997940i \(-0.520435\pi\)
−0.0641549 + 0.997940i \(0.520435\pi\)
\(68\) −476.981 −0.850623
\(69\) 0 0
\(70\) −272.355 −0.465038
\(71\) −190.743 −0.318832 −0.159416 0.987211i \(-0.550961\pi\)
−0.159416 + 0.987211i \(0.550961\pi\)
\(72\) 0 0
\(73\) 480.337 0.770125 0.385063 0.922890i \(-0.374180\pi\)
0.385063 + 0.922890i \(0.374180\pi\)
\(74\) 49.5063 0.0777701
\(75\) 0 0
\(76\) 449.842 0.678953
\(77\) 151.227 0.223817
\(78\) 0 0
\(79\) 1321.34 1.88180 0.940901 0.338681i \(-0.109981\pi\)
0.940901 + 0.338681i \(0.109981\pi\)
\(80\) −211.689 −0.295845
\(81\) 0 0
\(82\) −348.123 −0.468826
\(83\) 407.398 0.538767 0.269384 0.963033i \(-0.413180\pi\)
0.269384 + 0.963033i \(0.413180\pi\)
\(84\) 0 0
\(85\) 1577.68 2.01322
\(86\) −1012.42 −1.26944
\(87\) 0 0
\(88\) 117.542 0.142387
\(89\) 1435.48 1.70967 0.854835 0.518899i \(-0.173658\pi\)
0.854835 + 0.518899i \(0.173658\pi\)
\(90\) 0 0
\(91\) 239.732 0.276162
\(92\) −92.0000 −0.104257
\(93\) 0 0
\(94\) 512.734 0.562601
\(95\) −1487.92 −1.60692
\(96\) 0 0
\(97\) −985.582 −1.03166 −0.515828 0.856692i \(-0.672516\pi\)
−0.515828 + 0.856692i \(0.672516\pi\)
\(98\) 474.124 0.488711
\(99\) 0 0
\(100\) 200.194 0.200194
\(101\) 1107.79 1.09138 0.545689 0.837988i \(-0.316268\pi\)
0.545689 + 0.837988i \(0.316268\pi\)
\(102\) 0 0
\(103\) 225.199 0.215432 0.107716 0.994182i \(-0.465646\pi\)
0.107716 + 0.994182i \(0.465646\pi\)
\(104\) 186.333 0.175687
\(105\) 0 0
\(106\) 400.904 0.367352
\(107\) 1020.49 0.922005 0.461002 0.887399i \(-0.347490\pi\)
0.461002 + 0.887399i \(0.347490\pi\)
\(108\) 0 0
\(109\) −706.032 −0.620418 −0.310209 0.950668i \(-0.600399\pi\)
−0.310209 + 0.950668i \(0.600399\pi\)
\(110\) −388.788 −0.336995
\(111\) 0 0
\(112\) −164.682 −0.138937
\(113\) 1639.57 1.36494 0.682468 0.730915i \(-0.260906\pi\)
0.682468 + 0.730915i \(0.260906\pi\)
\(114\) 0 0
\(115\) 304.303 0.246752
\(116\) 418.584 0.335039
\(117\) 0 0
\(118\) −893.975 −0.697433
\(119\) 1227.35 0.945467
\(120\) 0 0
\(121\) −1115.12 −0.837808
\(122\) −838.381 −0.622159
\(123\) 0 0
\(124\) −108.415 −0.0785160
\(125\) 991.653 0.709569
\(126\) 0 0
\(127\) −1639.53 −1.14555 −0.572774 0.819714i \(-0.694133\pi\)
−0.572774 + 0.819714i \(0.694133\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) −616.324 −0.415809
\(131\) −629.817 −0.420056 −0.210028 0.977695i \(-0.567356\pi\)
−0.210028 + 0.977695i \(0.567356\pi\)
\(132\) 0 0
\(133\) −1157.51 −0.754656
\(134\) 140.735 0.0907287
\(135\) 0 0
\(136\) 953.961 0.601482
\(137\) −1140.97 −0.711528 −0.355764 0.934576i \(-0.615779\pi\)
−0.355764 + 0.934576i \(0.615779\pi\)
\(138\) 0 0
\(139\) −136.295 −0.0831682 −0.0415841 0.999135i \(-0.513240\pi\)
−0.0415841 + 0.999135i \(0.513240\pi\)
\(140\) 544.710 0.328831
\(141\) 0 0
\(142\) 381.487 0.225448
\(143\) 342.218 0.200124
\(144\) 0 0
\(145\) −1384.53 −0.792957
\(146\) −960.673 −0.544561
\(147\) 0 0
\(148\) −99.0125 −0.0549917
\(149\) −2087.11 −1.14754 −0.573768 0.819018i \(-0.694519\pi\)
−0.573768 + 0.819018i \(0.694519\pi\)
\(150\) 0 0
\(151\) 1944.71 1.04807 0.524034 0.851697i \(-0.324427\pi\)
0.524034 + 0.851697i \(0.324427\pi\)
\(152\) −899.684 −0.480092
\(153\) 0 0
\(154\) −302.454 −0.158263
\(155\) 358.600 0.185828
\(156\) 0 0
\(157\) −734.564 −0.373405 −0.186702 0.982417i \(-0.559780\pi\)
−0.186702 + 0.982417i \(0.559780\pi\)
\(158\) −2642.68 −1.33064
\(159\) 0 0
\(160\) 423.379 0.209194
\(161\) 236.730 0.115882
\(162\) 0 0
\(163\) −2737.81 −1.31559 −0.657796 0.753196i \(-0.728511\pi\)
−0.657796 + 0.753196i \(0.728511\pi\)
\(164\) 696.245 0.331510
\(165\) 0 0
\(166\) −814.795 −0.380966
\(167\) 1702.98 0.789103 0.394551 0.918874i \(-0.370900\pi\)
0.394551 + 0.918874i \(0.370900\pi\)
\(168\) 0 0
\(169\) −1654.50 −0.753072
\(170\) −3155.37 −1.42356
\(171\) 0 0
\(172\) 2024.83 0.897628
\(173\) 139.362 0.0612456 0.0306228 0.999531i \(-0.490251\pi\)
0.0306228 + 0.999531i \(0.490251\pi\)
\(174\) 0 0
\(175\) −515.130 −0.222515
\(176\) −235.084 −0.100683
\(177\) 0 0
\(178\) −2870.96 −1.20892
\(179\) −982.417 −0.410219 −0.205110 0.978739i \(-0.565755\pi\)
−0.205110 + 0.978739i \(0.565755\pi\)
\(180\) 0 0
\(181\) −3593.96 −1.47590 −0.737948 0.674858i \(-0.764205\pi\)
−0.737948 + 0.674858i \(0.764205\pi\)
\(182\) −479.464 −0.195276
\(183\) 0 0
\(184\) 184.000 0.0737210
\(185\) 327.498 0.130152
\(186\) 0 0
\(187\) 1752.04 0.685144
\(188\) −1025.47 −0.397819
\(189\) 0 0
\(190\) 2975.84 1.13626
\(191\) −51.6405 −0.0195632 −0.00978161 0.999952i \(-0.503114\pi\)
−0.00978161 + 0.999952i \(0.503114\pi\)
\(192\) 0 0
\(193\) 1176.44 0.438768 0.219384 0.975639i \(-0.429595\pi\)
0.219384 + 0.975639i \(0.429595\pi\)
\(194\) 1971.16 0.729492
\(195\) 0 0
\(196\) −948.247 −0.345571
\(197\) −1092.50 −0.395113 −0.197557 0.980291i \(-0.563301\pi\)
−0.197557 + 0.980291i \(0.563301\pi\)
\(198\) 0 0
\(199\) 155.379 0.0553495 0.0276747 0.999617i \(-0.491190\pi\)
0.0276747 + 0.999617i \(0.491190\pi\)
\(200\) −400.387 −0.141558
\(201\) 0 0
\(202\) −2215.58 −0.771721
\(203\) −1077.08 −0.372396
\(204\) 0 0
\(205\) −2302.93 −0.784604
\(206\) −450.398 −0.152334
\(207\) 0 0
\(208\) −372.666 −0.124230
\(209\) −1652.36 −0.546870
\(210\) 0 0
\(211\) −5365.96 −1.75075 −0.875374 0.483447i \(-0.839385\pi\)
−0.875374 + 0.483447i \(0.839385\pi\)
\(212\) −801.809 −0.259757
\(213\) 0 0
\(214\) −2040.98 −0.651956
\(215\) −6697.43 −2.12447
\(216\) 0 0
\(217\) 278.970 0.0872705
\(218\) 1412.06 0.438702
\(219\) 0 0
\(220\) 777.575 0.238291
\(221\) 2777.41 0.845380
\(222\) 0 0
\(223\) 446.221 0.133996 0.0669982 0.997753i \(-0.478658\pi\)
0.0669982 + 0.997753i \(0.478658\pi\)
\(224\) 329.364 0.0982436
\(225\) 0 0
\(226\) −3279.14 −0.965156
\(227\) 2263.94 0.661950 0.330975 0.943639i \(-0.392622\pi\)
0.330975 + 0.943639i \(0.392622\pi\)
\(228\) 0 0
\(229\) 1722.70 0.497116 0.248558 0.968617i \(-0.420043\pi\)
0.248558 + 0.968617i \(0.420043\pi\)
\(230\) −608.607 −0.174480
\(231\) 0 0
\(232\) −837.167 −0.236908
\(233\) 4122.28 1.15905 0.579527 0.814953i \(-0.303237\pi\)
0.579527 + 0.814953i \(0.303237\pi\)
\(234\) 0 0
\(235\) 3391.88 0.941541
\(236\) 1787.95 0.493160
\(237\) 0 0
\(238\) −2454.69 −0.668546
\(239\) −1199.74 −0.324707 −0.162353 0.986733i \(-0.551908\pi\)
−0.162353 + 0.986733i \(0.551908\pi\)
\(240\) 0 0
\(241\) 7262.37 1.94112 0.970561 0.240855i \(-0.0774278\pi\)
0.970561 + 0.240855i \(0.0774278\pi\)
\(242\) 2230.25 0.592420
\(243\) 0 0
\(244\) 1676.76 0.439933
\(245\) 3136.47 0.817884
\(246\) 0 0
\(247\) −2619.39 −0.674768
\(248\) 216.831 0.0555192
\(249\) 0 0
\(250\) −1983.31 −0.501741
\(251\) 1662.11 0.417974 0.208987 0.977918i \(-0.432983\pi\)
0.208987 + 0.977918i \(0.432983\pi\)
\(252\) 0 0
\(253\) 337.933 0.0839751
\(254\) 3279.05 0.810024
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 3216.11 0.780605 0.390302 0.920687i \(-0.372370\pi\)
0.390302 + 0.920687i \(0.372370\pi\)
\(258\) 0 0
\(259\) 254.775 0.0611233
\(260\) 1232.65 0.294021
\(261\) 0 0
\(262\) 1259.63 0.297025
\(263\) 1552.10 0.363903 0.181951 0.983308i \(-0.441759\pi\)
0.181951 + 0.983308i \(0.441759\pi\)
\(264\) 0 0
\(265\) 2652.10 0.614782
\(266\) 2315.03 0.533622
\(267\) 0 0
\(268\) −281.470 −0.0641549
\(269\) −4575.50 −1.03707 −0.518537 0.855055i \(-0.673523\pi\)
−0.518537 + 0.855055i \(0.673523\pi\)
\(270\) 0 0
\(271\) −2277.78 −0.510574 −0.255287 0.966865i \(-0.582170\pi\)
−0.255287 + 0.966865i \(0.582170\pi\)
\(272\) −1907.92 −0.425312
\(273\) 0 0
\(274\) 2281.93 0.503127
\(275\) −735.350 −0.161248
\(276\) 0 0
\(277\) 780.955 0.169397 0.0846986 0.996407i \(-0.473007\pi\)
0.0846986 + 0.996407i \(0.473007\pi\)
\(278\) 272.590 0.0588088
\(279\) 0 0
\(280\) −1089.42 −0.232519
\(281\) −52.7391 −0.0111963 −0.00559813 0.999984i \(-0.501782\pi\)
−0.00559813 + 0.999984i \(0.501782\pi\)
\(282\) 0 0
\(283\) −6200.42 −1.30239 −0.651196 0.758910i \(-0.725732\pi\)
−0.651196 + 0.758910i \(0.725732\pi\)
\(284\) −762.973 −0.159416
\(285\) 0 0
\(286\) −684.437 −0.141509
\(287\) −1791.55 −0.368473
\(288\) 0 0
\(289\) 9306.41 1.89424
\(290\) 2769.05 0.560705
\(291\) 0 0
\(292\) 1921.35 0.385063
\(293\) 6131.11 1.22247 0.611234 0.791450i \(-0.290673\pi\)
0.611234 + 0.791450i \(0.290673\pi\)
\(294\) 0 0
\(295\) −5913.91 −1.16719
\(296\) 198.025 0.0388850
\(297\) 0 0
\(298\) 4174.23 0.811431
\(299\) 535.707 0.103615
\(300\) 0 0
\(301\) −5210.21 −0.997713
\(302\) −3889.42 −0.741096
\(303\) 0 0
\(304\) 1799.37 0.339476
\(305\) −5546.13 −1.04122
\(306\) 0 0
\(307\) −7056.18 −1.31178 −0.655892 0.754855i \(-0.727707\pi\)
−0.655892 + 0.754855i \(0.727707\pi\)
\(308\) 604.908 0.111909
\(309\) 0 0
\(310\) −717.199 −0.131401
\(311\) −7368.26 −1.34346 −0.671730 0.740796i \(-0.734448\pi\)
−0.671730 + 0.740796i \(0.734448\pi\)
\(312\) 0 0
\(313\) 9068.44 1.63763 0.818816 0.574056i \(-0.194631\pi\)
0.818816 + 0.574056i \(0.194631\pi\)
\(314\) 1469.13 0.264037
\(315\) 0 0
\(316\) 5285.36 0.940901
\(317\) 1952.28 0.345902 0.172951 0.984930i \(-0.444670\pi\)
0.172951 + 0.984930i \(0.444670\pi\)
\(318\) 0 0
\(319\) −1537.54 −0.269861
\(320\) −846.758 −0.147922
\(321\) 0 0
\(322\) −473.461 −0.0819408
\(323\) −13410.4 −2.31013
\(324\) 0 0
\(325\) −1165.71 −0.198960
\(326\) 5475.61 0.930264
\(327\) 0 0
\(328\) −1392.49 −0.234413
\(329\) 2638.69 0.442175
\(330\) 0 0
\(331\) 11508.2 1.91102 0.955511 0.294956i \(-0.0953051\pi\)
0.955511 + 0.294956i \(0.0953051\pi\)
\(332\) 1629.59 0.269384
\(333\) 0 0
\(334\) −3405.95 −0.557980
\(335\) 931.003 0.151839
\(336\) 0 0
\(337\) −425.493 −0.0687776 −0.0343888 0.999409i \(-0.510948\pi\)
−0.0343888 + 0.999409i \(0.510948\pi\)
\(338\) 3309.00 0.532503
\(339\) 0 0
\(340\) 6310.73 1.00661
\(341\) 398.230 0.0632416
\(342\) 0 0
\(343\) 5970.36 0.939852
\(344\) −4049.67 −0.634719
\(345\) 0 0
\(346\) −278.724 −0.0433072
\(347\) −1438.94 −0.222612 −0.111306 0.993786i \(-0.535503\pi\)
−0.111306 + 0.993786i \(0.535503\pi\)
\(348\) 0 0
\(349\) −681.032 −0.104455 −0.0522275 0.998635i \(-0.516632\pi\)
−0.0522275 + 0.998635i \(0.516632\pi\)
\(350\) 1030.26 0.157342
\(351\) 0 0
\(352\) 470.168 0.0711933
\(353\) 6608.06 0.996350 0.498175 0.867076i \(-0.334004\pi\)
0.498175 + 0.867076i \(0.334004\pi\)
\(354\) 0 0
\(355\) 2523.65 0.377299
\(356\) 5741.92 0.854835
\(357\) 0 0
\(358\) 1964.83 0.290069
\(359\) 2801.68 0.411886 0.205943 0.978564i \(-0.433974\pi\)
0.205943 + 0.978564i \(0.433974\pi\)
\(360\) 0 0
\(361\) 5788.36 0.843908
\(362\) 7187.93 1.04362
\(363\) 0 0
\(364\) 958.928 0.138081
\(365\) −6355.14 −0.911351
\(366\) 0 0
\(367\) 8642.79 1.22929 0.614645 0.788803i \(-0.289299\pi\)
0.614645 + 0.788803i \(0.289299\pi\)
\(368\) −368.000 −0.0521286
\(369\) 0 0
\(370\) −654.997 −0.0920315
\(371\) 2063.18 0.288720
\(372\) 0 0
\(373\) −9923.81 −1.37757 −0.688787 0.724963i \(-0.741857\pi\)
−0.688787 + 0.724963i \(0.741857\pi\)
\(374\) −3504.08 −0.484470
\(375\) 0 0
\(376\) 2050.94 0.281300
\(377\) −2437.37 −0.332974
\(378\) 0 0
\(379\) 5030.01 0.681727 0.340863 0.940113i \(-0.389281\pi\)
0.340863 + 0.940113i \(0.389281\pi\)
\(380\) −5951.67 −0.803459
\(381\) 0 0
\(382\) 103.281 0.0138333
\(383\) −2494.31 −0.332776 −0.166388 0.986060i \(-0.553210\pi\)
−0.166388 + 0.986060i \(0.553210\pi\)
\(384\) 0 0
\(385\) −2000.82 −0.264861
\(386\) −2352.89 −0.310256
\(387\) 0 0
\(388\) −3942.33 −0.515828
\(389\) 6948.47 0.905659 0.452830 0.891597i \(-0.350415\pi\)
0.452830 + 0.891597i \(0.350415\pi\)
\(390\) 0 0
\(391\) 2742.64 0.354735
\(392\) 1896.49 0.244356
\(393\) 0 0
\(394\) 2185.00 0.279387
\(395\) −17482.1 −2.22689
\(396\) 0 0
\(397\) 7921.85 1.00148 0.500738 0.865599i \(-0.333062\pi\)
0.500738 + 0.865599i \(0.333062\pi\)
\(398\) −310.759 −0.0391380
\(399\) 0 0
\(400\) 800.775 0.100097
\(401\) 10902.0 1.35765 0.678826 0.734299i \(-0.262489\pi\)
0.678826 + 0.734299i \(0.262489\pi\)
\(402\) 0 0
\(403\) 631.292 0.0780320
\(404\) 4431.16 0.545689
\(405\) 0 0
\(406\) 2154.16 0.263323
\(407\) 363.692 0.0442937
\(408\) 0 0
\(409\) −3070.16 −0.371172 −0.185586 0.982628i \(-0.559418\pi\)
−0.185586 + 0.982628i \(0.559418\pi\)
\(410\) 4605.87 0.554799
\(411\) 0 0
\(412\) 900.796 0.107716
\(413\) −4600.68 −0.548146
\(414\) 0 0
\(415\) −5390.11 −0.637566
\(416\) 745.332 0.0878435
\(417\) 0 0
\(418\) 3304.71 0.386695
\(419\) 9682.14 1.12889 0.564443 0.825472i \(-0.309091\pi\)
0.564443 + 0.825472i \(0.309091\pi\)
\(420\) 0 0
\(421\) 7481.37 0.866080 0.433040 0.901375i \(-0.357441\pi\)
0.433040 + 0.901375i \(0.357441\pi\)
\(422\) 10731.9 1.23797
\(423\) 0 0
\(424\) 1603.62 0.183676
\(425\) −5968.03 −0.681158
\(426\) 0 0
\(427\) −4314.57 −0.488985
\(428\) 4081.96 0.461002
\(429\) 0 0
\(430\) 13394.9 1.50223
\(431\) −16513.5 −1.84554 −0.922768 0.385356i \(-0.874079\pi\)
−0.922768 + 0.385356i \(0.874079\pi\)
\(432\) 0 0
\(433\) 11765.8 1.30583 0.652917 0.757429i \(-0.273545\pi\)
0.652917 + 0.757429i \(0.273545\pi\)
\(434\) −557.939 −0.0617095
\(435\) 0 0
\(436\) −2824.13 −0.310209
\(437\) −2586.59 −0.283143
\(438\) 0 0
\(439\) 16715.0 1.81722 0.908612 0.417640i \(-0.137143\pi\)
0.908612 + 0.417640i \(0.137143\pi\)
\(440\) −1555.15 −0.168498
\(441\) 0 0
\(442\) −5554.83 −0.597774
\(443\) −12687.1 −1.36068 −0.680341 0.732895i \(-0.738168\pi\)
−0.680341 + 0.732895i \(0.738168\pi\)
\(444\) 0 0
\(445\) −18992.3 −2.02319
\(446\) −892.442 −0.0947497
\(447\) 0 0
\(448\) −658.728 −0.0694687
\(449\) 16077.7 1.68987 0.844937 0.534866i \(-0.179638\pi\)
0.844937 + 0.534866i \(0.179638\pi\)
\(450\) 0 0
\(451\) −2557.44 −0.267018
\(452\) 6558.28 0.682468
\(453\) 0 0
\(454\) −4527.87 −0.468070
\(455\) −3171.80 −0.326805
\(456\) 0 0
\(457\) 7930.19 0.811726 0.405863 0.913934i \(-0.366971\pi\)
0.405863 + 0.913934i \(0.366971\pi\)
\(458\) −3445.41 −0.351514
\(459\) 0 0
\(460\) 1217.21 0.123376
\(461\) −11530.0 −1.16487 −0.582436 0.812877i \(-0.697900\pi\)
−0.582436 + 0.812877i \(0.697900\pi\)
\(462\) 0 0
\(463\) −7590.41 −0.761893 −0.380947 0.924597i \(-0.624402\pi\)
−0.380947 + 0.924597i \(0.624402\pi\)
\(464\) 1674.33 0.167519
\(465\) 0 0
\(466\) −8244.56 −0.819575
\(467\) −6686.26 −0.662534 −0.331267 0.943537i \(-0.607476\pi\)
−0.331267 + 0.943537i \(0.607476\pi\)
\(468\) 0 0
\(469\) 724.266 0.0713081
\(470\) −6783.77 −0.665770
\(471\) 0 0
\(472\) −3575.90 −0.348716
\(473\) −7437.60 −0.723004
\(474\) 0 0
\(475\) 5628.47 0.543688
\(476\) 4909.38 0.472734
\(477\) 0 0
\(478\) 2399.49 0.229602
\(479\) −12072.0 −1.15153 −0.575766 0.817614i \(-0.695296\pi\)
−0.575766 + 0.817614i \(0.695296\pi\)
\(480\) 0 0
\(481\) 576.541 0.0546528
\(482\) −14524.7 −1.37258
\(483\) 0 0
\(484\) −4460.49 −0.418904
\(485\) 13039.8 1.22084
\(486\) 0 0
\(487\) −1155.32 −0.107500 −0.0537499 0.998554i \(-0.517117\pi\)
−0.0537499 + 0.998554i \(0.517117\pi\)
\(488\) −3353.52 −0.311080
\(489\) 0 0
\(490\) −6272.93 −0.578331
\(491\) −7251.93 −0.666548 −0.333274 0.942830i \(-0.608153\pi\)
−0.333274 + 0.942830i \(0.608153\pi\)
\(492\) 0 0
\(493\) −12478.5 −1.13997
\(494\) 5238.78 0.477133
\(495\) 0 0
\(496\) −433.661 −0.0392580
\(497\) 1963.25 0.177191
\(498\) 0 0
\(499\) 11437.9 1.02612 0.513059 0.858354i \(-0.328512\pi\)
0.513059 + 0.858354i \(0.328512\pi\)
\(500\) 3966.61 0.354785
\(501\) 0 0
\(502\) −3324.22 −0.295552
\(503\) −5624.43 −0.498570 −0.249285 0.968430i \(-0.580196\pi\)
−0.249285 + 0.968430i \(0.580196\pi\)
\(504\) 0 0
\(505\) −14656.7 −1.29152
\(506\) −675.867 −0.0593794
\(507\) 0 0
\(508\) −6558.11 −0.572774
\(509\) 14983.3 1.30476 0.652381 0.757891i \(-0.273770\pi\)
0.652381 + 0.757891i \(0.273770\pi\)
\(510\) 0 0
\(511\) −4943.93 −0.427997
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) −6432.22 −0.551971
\(515\) −2979.51 −0.254938
\(516\) 0 0
\(517\) 3766.74 0.320427
\(518\) −509.549 −0.0432207
\(519\) 0 0
\(520\) −2465.30 −0.207904
\(521\) −11536.9 −0.970139 −0.485069 0.874476i \(-0.661206\pi\)
−0.485069 + 0.874476i \(0.661206\pi\)
\(522\) 0 0
\(523\) 12093.5 1.01111 0.505555 0.862794i \(-0.331288\pi\)
0.505555 + 0.862794i \(0.331288\pi\)
\(524\) −2519.27 −0.210028
\(525\) 0 0
\(526\) −3104.20 −0.257318
\(527\) 3232.00 0.267150
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) −5304.20 −0.434716
\(531\) 0 0
\(532\) −4630.06 −0.377328
\(533\) −4054.17 −0.329467
\(534\) 0 0
\(535\) −13501.7 −1.09108
\(536\) 562.940 0.0453644
\(537\) 0 0
\(538\) 9150.99 0.733322
\(539\) 3483.09 0.278344
\(540\) 0 0
\(541\) 5566.83 0.442397 0.221198 0.975229i \(-0.429003\pi\)
0.221198 + 0.975229i \(0.429003\pi\)
\(542\) 4555.57 0.361030
\(543\) 0 0
\(544\) 3815.85 0.300741
\(545\) 9341.21 0.734190
\(546\) 0 0
\(547\) −19574.6 −1.53007 −0.765034 0.643989i \(-0.777278\pi\)
−0.765034 + 0.643989i \(0.777278\pi\)
\(548\) −4563.87 −0.355764
\(549\) 0 0
\(550\) 1470.70 0.114020
\(551\) 11768.5 0.909903
\(552\) 0 0
\(553\) −13600.1 −1.04581
\(554\) −1561.91 −0.119782
\(555\) 0 0
\(556\) −545.180 −0.0415841
\(557\) 18416.6 1.40096 0.700481 0.713671i \(-0.252969\pi\)
0.700481 + 0.713671i \(0.252969\pi\)
\(558\) 0 0
\(559\) −11790.4 −0.892095
\(560\) 2178.84 0.164416
\(561\) 0 0
\(562\) 105.478 0.00791695
\(563\) 3944.28 0.295260 0.147630 0.989043i \(-0.452835\pi\)
0.147630 + 0.989043i \(0.452835\pi\)
\(564\) 0 0
\(565\) −21692.5 −1.61524
\(566\) 12400.8 0.920930
\(567\) 0 0
\(568\) 1525.95 0.112724
\(569\) 22904.6 1.68754 0.843769 0.536707i \(-0.180332\pi\)
0.843769 + 0.536707i \(0.180332\pi\)
\(570\) 0 0
\(571\) 9613.06 0.704543 0.352272 0.935898i \(-0.385409\pi\)
0.352272 + 0.935898i \(0.385409\pi\)
\(572\) 1368.87 0.100062
\(573\) 0 0
\(574\) 3583.10 0.260550
\(575\) −1151.11 −0.0834865
\(576\) 0 0
\(577\) 7910.77 0.570762 0.285381 0.958414i \(-0.407880\pi\)
0.285381 + 0.958414i \(0.407880\pi\)
\(578\) −18612.8 −1.33943
\(579\) 0 0
\(580\) −5538.11 −0.396478
\(581\) −4193.19 −0.299420
\(582\) 0 0
\(583\) 2945.20 0.209224
\(584\) −3842.69 −0.272280
\(585\) 0 0
\(586\) −12262.2 −0.864416
\(587\) 3351.56 0.235662 0.117831 0.993034i \(-0.462406\pi\)
0.117831 + 0.993034i \(0.462406\pi\)
\(588\) 0 0
\(589\) −3048.11 −0.213235
\(590\) 11827.8 0.825328
\(591\) 0 0
\(592\) −396.050 −0.0274959
\(593\) 12072.3 0.836002 0.418001 0.908447i \(-0.362731\pi\)
0.418001 + 0.908447i \(0.362731\pi\)
\(594\) 0 0
\(595\) −16238.5 −1.11885
\(596\) −8348.45 −0.573768
\(597\) 0 0
\(598\) −1071.41 −0.0732666
\(599\) −13998.0 −0.954826 −0.477413 0.878679i \(-0.658425\pi\)
−0.477413 + 0.878679i \(0.658425\pi\)
\(600\) 0 0
\(601\) 7796.28 0.529146 0.264573 0.964366i \(-0.414769\pi\)
0.264573 + 0.964366i \(0.414769\pi\)
\(602\) 10420.4 0.705490
\(603\) 0 0
\(604\) 7778.84 0.524034
\(605\) 14753.7 0.991445
\(606\) 0 0
\(607\) −5202.79 −0.347899 −0.173950 0.984755i \(-0.555653\pi\)
−0.173950 + 0.984755i \(0.555653\pi\)
\(608\) −3598.74 −0.240046
\(609\) 0 0
\(610\) 11092.3 0.736251
\(611\) 5971.20 0.395367
\(612\) 0 0
\(613\) −15697.6 −1.03429 −0.517144 0.855898i \(-0.673005\pi\)
−0.517144 + 0.855898i \(0.673005\pi\)
\(614\) 14112.4 0.927571
\(615\) 0 0
\(616\) −1209.82 −0.0791313
\(617\) −26268.4 −1.71398 −0.856991 0.515331i \(-0.827669\pi\)
−0.856991 + 0.515331i \(0.827669\pi\)
\(618\) 0 0
\(619\) 1486.34 0.0965124 0.0482562 0.998835i \(-0.484634\pi\)
0.0482562 + 0.998835i \(0.484634\pi\)
\(620\) 1434.40 0.0929142
\(621\) 0 0
\(622\) 14736.5 0.949969
\(623\) −14774.9 −0.950149
\(624\) 0 0
\(625\) −19376.2 −1.24008
\(626\) −18136.9 −1.15798
\(627\) 0 0
\(628\) −2938.25 −0.186702
\(629\) 2951.69 0.187109
\(630\) 0 0
\(631\) 9554.18 0.602767 0.301384 0.953503i \(-0.402552\pi\)
0.301384 + 0.953503i \(0.402552\pi\)
\(632\) −10570.7 −0.665318
\(633\) 0 0
\(634\) −3904.55 −0.244589
\(635\) 21691.9 1.35562
\(636\) 0 0
\(637\) 5521.56 0.343441
\(638\) 3075.07 0.190820
\(639\) 0 0
\(640\) 1693.52 0.104597
\(641\) 28526.3 1.75776 0.878879 0.477045i \(-0.158292\pi\)
0.878879 + 0.477045i \(0.158292\pi\)
\(642\) 0 0
\(643\) −21360.4 −1.31006 −0.655032 0.755601i \(-0.727345\pi\)
−0.655032 + 0.755601i \(0.727345\pi\)
\(644\) 946.922 0.0579409
\(645\) 0 0
\(646\) 26820.7 1.63351
\(647\) 17343.0 1.05382 0.526910 0.849921i \(-0.323350\pi\)
0.526910 + 0.849921i \(0.323350\pi\)
\(648\) 0 0
\(649\) −6567.48 −0.397221
\(650\) 2331.42 0.140686
\(651\) 0 0
\(652\) −10951.2 −0.657796
\(653\) −6044.46 −0.362233 −0.181116 0.983462i \(-0.557971\pi\)
−0.181116 + 0.983462i \(0.557971\pi\)
\(654\) 0 0
\(655\) 8332.85 0.497086
\(656\) 2784.98 0.165755
\(657\) 0 0
\(658\) −5277.38 −0.312665
\(659\) 29374.0 1.73634 0.868172 0.496264i \(-0.165295\pi\)
0.868172 + 0.496264i \(0.165295\pi\)
\(660\) 0 0
\(661\) 16562.1 0.974572 0.487286 0.873242i \(-0.337987\pi\)
0.487286 + 0.873242i \(0.337987\pi\)
\(662\) −23016.4 −1.35130
\(663\) 0 0
\(664\) −3259.18 −0.190483
\(665\) 15314.6 0.893044
\(666\) 0 0
\(667\) −2406.86 −0.139721
\(668\) 6811.90 0.394551
\(669\) 0 0
\(670\) −1862.01 −0.107367
\(671\) −6159.06 −0.354349
\(672\) 0 0
\(673\) −1304.30 −0.0747059 −0.0373529 0.999302i \(-0.511893\pi\)
−0.0373529 + 0.999302i \(0.511893\pi\)
\(674\) 850.985 0.0486331
\(675\) 0 0
\(676\) −6618.00 −0.376536
\(677\) −8890.43 −0.504708 −0.252354 0.967635i \(-0.581205\pi\)
−0.252354 + 0.967635i \(0.581205\pi\)
\(678\) 0 0
\(679\) 10144.2 0.573343
\(680\) −12621.5 −0.711781
\(681\) 0 0
\(682\) −796.460 −0.0447185
\(683\) 16635.8 0.931991 0.465995 0.884787i \(-0.345696\pi\)
0.465995 + 0.884787i \(0.345696\pi\)
\(684\) 0 0
\(685\) 15095.7 0.842008
\(686\) −11940.7 −0.664576
\(687\) 0 0
\(688\) 8099.33 0.448814
\(689\) 4668.86 0.258156
\(690\) 0 0
\(691\) 22690.2 1.24917 0.624585 0.780957i \(-0.285268\pi\)
0.624585 + 0.780957i \(0.285268\pi\)
\(692\) 557.448 0.0306228
\(693\) 0 0
\(694\) 2877.88 0.157411
\(695\) 1803.26 0.0984196
\(696\) 0 0
\(697\) −20756.0 −1.12796
\(698\) 1362.06 0.0738608
\(699\) 0 0
\(700\) −2060.52 −0.111258
\(701\) −32623.6 −1.75774 −0.878872 0.477059i \(-0.841703\pi\)
−0.878872 + 0.477059i \(0.841703\pi\)
\(702\) 0 0
\(703\) −2783.75 −0.149347
\(704\) −940.337 −0.0503413
\(705\) 0 0
\(706\) −13216.1 −0.704526
\(707\) −11402.1 −0.606533
\(708\) 0 0
\(709\) −26739.1 −1.41637 −0.708187 0.706025i \(-0.750486\pi\)
−0.708187 + 0.706025i \(0.750486\pi\)
\(710\) −5047.29 −0.266791
\(711\) 0 0
\(712\) −11483.8 −0.604460
\(713\) 623.388 0.0327434
\(714\) 0 0
\(715\) −4527.75 −0.236823
\(716\) −3929.67 −0.205110
\(717\) 0 0
\(718\) −5603.36 −0.291248
\(719\) 20628.0 1.06995 0.534976 0.844867i \(-0.320321\pi\)
0.534976 + 0.844867i \(0.320321\pi\)
\(720\) 0 0
\(721\) −2317.89 −0.119726
\(722\) −11576.7 −0.596733
\(723\) 0 0
\(724\) −14375.9 −0.737948
\(725\) 5237.36 0.268291
\(726\) 0 0
\(727\) −35027.0 −1.78691 −0.893453 0.449158i \(-0.851724\pi\)
−0.893453 + 0.449158i \(0.851724\pi\)
\(728\) −1917.86 −0.0976380
\(729\) 0 0
\(730\) 12710.3 0.644422
\(731\) −60362.9 −3.05417
\(732\) 0 0
\(733\) −17949.2 −0.904460 −0.452230 0.891901i \(-0.649371\pi\)
−0.452230 + 0.891901i \(0.649371\pi\)
\(734\) −17285.6 −0.869240
\(735\) 0 0
\(736\) 736.000 0.0368605
\(737\) 1033.89 0.0516742
\(738\) 0 0
\(739\) −9531.78 −0.474469 −0.237234 0.971452i \(-0.576241\pi\)
−0.237234 + 0.971452i \(0.576241\pi\)
\(740\) 1309.99 0.0650761
\(741\) 0 0
\(742\) −4126.36 −0.204156
\(743\) −12341.6 −0.609378 −0.304689 0.952452i \(-0.598552\pi\)
−0.304689 + 0.952452i \(0.598552\pi\)
\(744\) 0 0
\(745\) 27613.7 1.35797
\(746\) 19847.6 0.974092
\(747\) 0 0
\(748\) 7008.16 0.342572
\(749\) −10503.5 −0.512404
\(750\) 0 0
\(751\) 20118.7 0.977553 0.488777 0.872409i \(-0.337443\pi\)
0.488777 + 0.872409i \(0.337443\pi\)
\(752\) −4101.87 −0.198909
\(753\) 0 0
\(754\) 4874.75 0.235448
\(755\) −25729.6 −1.24026
\(756\) 0 0
\(757\) 40167.2 1.92854 0.964269 0.264926i \(-0.0853476\pi\)
0.964269 + 0.264926i \(0.0853476\pi\)
\(758\) −10060.0 −0.482054
\(759\) 0 0
\(760\) 11903.3 0.568131
\(761\) 21242.6 1.01188 0.505941 0.862568i \(-0.331145\pi\)
0.505941 + 0.862568i \(0.331145\pi\)
\(762\) 0 0
\(763\) 7266.92 0.344797
\(764\) −206.562 −0.00978161
\(765\) 0 0
\(766\) 4988.62 0.235308
\(767\) −10411.1 −0.490120
\(768\) 0 0
\(769\) −40261.8 −1.88801 −0.944003 0.329937i \(-0.892973\pi\)
−0.944003 + 0.329937i \(0.892973\pi\)
\(770\) 4001.65 0.187285
\(771\) 0 0
\(772\) 4705.77 0.219384
\(773\) 3861.68 0.179683 0.0898415 0.995956i \(-0.471364\pi\)
0.0898415 + 0.995956i \(0.471364\pi\)
\(774\) 0 0
\(775\) −1356.50 −0.0628736
\(776\) 7884.66 0.364746
\(777\) 0 0
\(778\) −13896.9 −0.640398
\(779\) 19575.0 0.900318
\(780\) 0 0
\(781\) 2802.55 0.128403
\(782\) −5485.28 −0.250835
\(783\) 0 0
\(784\) −3792.99 −0.172786
\(785\) 9718.71 0.441880
\(786\) 0 0
\(787\) 40478.2 1.83341 0.916704 0.399568i \(-0.130840\pi\)
0.916704 + 0.399568i \(0.130840\pi\)
\(788\) −4370.00 −0.197557
\(789\) 0 0
\(790\) 34964.2 1.57465
\(791\) −16875.5 −0.758563
\(792\) 0 0
\(793\) −9763.62 −0.437221
\(794\) −15843.7 −0.708151
\(795\) 0 0
\(796\) 621.517 0.0276747
\(797\) 7456.91 0.331415 0.165707 0.986175i \(-0.447009\pi\)
0.165707 + 0.986175i \(0.447009\pi\)
\(798\) 0 0
\(799\) 30570.5 1.35358
\(800\) −1601.55 −0.0707792
\(801\) 0 0
\(802\) −21804.0 −0.960005
\(803\) −7057.47 −0.310153
\(804\) 0 0
\(805\) −3132.08 −0.137132
\(806\) −1262.58 −0.0551770
\(807\) 0 0
\(808\) −8862.32 −0.385861
\(809\) −4767.72 −0.207199 −0.103600 0.994619i \(-0.533036\pi\)
−0.103600 + 0.994619i \(0.533036\pi\)
\(810\) 0 0
\(811\) 19199.4 0.831298 0.415649 0.909525i \(-0.363554\pi\)
0.415649 + 0.909525i \(0.363554\pi\)
\(812\) −4308.33 −0.186198
\(813\) 0 0
\(814\) −727.384 −0.0313204
\(815\) 36222.8 1.55685
\(816\) 0 0
\(817\) 56928.4 2.43779
\(818\) 6140.31 0.262458
\(819\) 0 0
\(820\) −9211.73 −0.392302
\(821\) 1595.59 0.0678277 0.0339138 0.999425i \(-0.489203\pi\)
0.0339138 + 0.999425i \(0.489203\pi\)
\(822\) 0 0
\(823\) 2041.80 0.0864795 0.0432398 0.999065i \(-0.486232\pi\)
0.0432398 + 0.999065i \(0.486232\pi\)
\(824\) −1801.59 −0.0761668
\(825\) 0 0
\(826\) 9201.35 0.387598
\(827\) 13117.5 0.551559 0.275780 0.961221i \(-0.411064\pi\)
0.275780 + 0.961221i \(0.411064\pi\)
\(828\) 0 0
\(829\) 20788.1 0.870930 0.435465 0.900206i \(-0.356584\pi\)
0.435465 + 0.900206i \(0.356584\pi\)
\(830\) 10780.2 0.450828
\(831\) 0 0
\(832\) −1490.66 −0.0621148
\(833\) 28268.5 1.17580
\(834\) 0 0
\(835\) −22531.4 −0.933808
\(836\) −6609.42 −0.273435
\(837\) 0 0
\(838\) −19364.3 −0.798243
\(839\) −27202.7 −1.11936 −0.559680 0.828709i \(-0.689076\pi\)
−0.559680 + 0.828709i \(0.689076\pi\)
\(840\) 0 0
\(841\) −13438.2 −0.550996
\(842\) −14962.7 −0.612411
\(843\) 0 0
\(844\) −21463.8 −0.875374
\(845\) 21890.0 0.891171
\(846\) 0 0
\(847\) 11477.5 0.465612
\(848\) −3207.24 −0.129878
\(849\) 0 0
\(850\) 11936.1 0.481651
\(851\) 569.322 0.0229331
\(852\) 0 0
\(853\) 21562.1 0.865501 0.432750 0.901514i \(-0.357543\pi\)
0.432750 + 0.901514i \(0.357543\pi\)
\(854\) 8629.14 0.345765
\(855\) 0 0
\(856\) −8163.92 −0.325978
\(857\) 15694.8 0.625581 0.312790 0.949822i \(-0.398736\pi\)
0.312790 + 0.949822i \(0.398736\pi\)
\(858\) 0 0
\(859\) 12667.3 0.503144 0.251572 0.967839i \(-0.419052\pi\)
0.251572 + 0.967839i \(0.419052\pi\)
\(860\) −26789.7 −1.06224
\(861\) 0 0
\(862\) 33026.9 1.30499
\(863\) −5118.43 −0.201893 −0.100946 0.994892i \(-0.532187\pi\)
−0.100946 + 0.994892i \(0.532187\pi\)
\(864\) 0 0
\(865\) −1843.84 −0.0724768
\(866\) −23531.5 −0.923364
\(867\) 0 0
\(868\) 1115.88 0.0436352
\(869\) −19414.1 −0.757859
\(870\) 0 0
\(871\) 1638.97 0.0637595
\(872\) 5648.25 0.219351
\(873\) 0 0
\(874\) 5173.18 0.200212
\(875\) −10206.7 −0.394343
\(876\) 0 0
\(877\) 46908.3 1.80614 0.903068 0.429498i \(-0.141309\pi\)
0.903068 + 0.429498i \(0.141309\pi\)
\(878\) −33429.9 −1.28497
\(879\) 0 0
\(880\) 3110.30 0.119146
\(881\) −8248.69 −0.315443 −0.157722 0.987484i \(-0.550415\pi\)
−0.157722 + 0.987484i \(0.550415\pi\)
\(882\) 0 0
\(883\) 12437.1 0.473999 0.237000 0.971510i \(-0.423836\pi\)
0.237000 + 0.971510i \(0.423836\pi\)
\(884\) 11109.7 0.422690
\(885\) 0 0
\(886\) 25374.2 0.962148
\(887\) 43105.2 1.63172 0.815858 0.578253i \(-0.196265\pi\)
0.815858 + 0.578253i \(0.196265\pi\)
\(888\) 0 0
\(889\) 16875.0 0.636637
\(890\) 37984.5 1.43061
\(891\) 0 0
\(892\) 1784.88 0.0669982
\(893\) −28831.1 −1.08040
\(894\) 0 0
\(895\) 12997.9 0.485445
\(896\) 1317.46 0.0491218
\(897\) 0 0
\(898\) −32155.4 −1.19492
\(899\) −2836.31 −0.105224
\(900\) 0 0
\(901\) 23903.0 0.883821
\(902\) 5114.88 0.188810
\(903\) 0 0
\(904\) −13116.6 −0.482578
\(905\) 47550.2 1.74655
\(906\) 0 0
\(907\) −9236.78 −0.338150 −0.169075 0.985603i \(-0.554078\pi\)
−0.169075 + 0.985603i \(0.554078\pi\)
\(908\) 9055.74 0.330975
\(909\) 0 0
\(910\) 6343.59 0.231086
\(911\) 34284.3 1.24686 0.623431 0.781879i \(-0.285738\pi\)
0.623431 + 0.781879i \(0.285738\pi\)
\(912\) 0 0
\(913\) −5985.80 −0.216978
\(914\) −15860.4 −0.573977
\(915\) 0 0
\(916\) 6890.82 0.248558
\(917\) 6482.47 0.233446
\(918\) 0 0
\(919\) −47474.8 −1.70408 −0.852039 0.523478i \(-0.824634\pi\)
−0.852039 + 0.523478i \(0.824634\pi\)
\(920\) −2434.43 −0.0872399
\(921\) 0 0
\(922\) 23060.0 0.823689
\(923\) 4442.72 0.158433
\(924\) 0 0
\(925\) −1238.86 −0.0440360
\(926\) 15180.8 0.538740
\(927\) 0 0
\(928\) −3348.67 −0.118454
\(929\) −10436.4 −0.368577 −0.184288 0.982872i \(-0.558998\pi\)
−0.184288 + 0.982872i \(0.558998\pi\)
\(930\) 0 0
\(931\) −26660.1 −0.938506
\(932\) 16489.1 0.579527
\(933\) 0 0
\(934\) 13372.5 0.468482
\(935\) −23180.5 −0.810785
\(936\) 0 0
\(937\) 24205.7 0.843935 0.421967 0.906611i \(-0.361340\pi\)
0.421967 + 0.906611i \(0.361340\pi\)
\(938\) −1448.53 −0.0504224
\(939\) 0 0
\(940\) 13567.5 0.470771
\(941\) 50113.7 1.73609 0.868045 0.496485i \(-0.165376\pi\)
0.868045 + 0.496485i \(0.165376\pi\)
\(942\) 0 0
\(943\) −4003.41 −0.138249
\(944\) 7151.80 0.246580
\(945\) 0 0
\(946\) 14875.2 0.511241
\(947\) −30633.5 −1.05117 −0.525583 0.850742i \(-0.676153\pi\)
−0.525583 + 0.850742i \(0.676153\pi\)
\(948\) 0 0
\(949\) −11187.8 −0.382689
\(950\) −11256.9 −0.384446
\(951\) 0 0
\(952\) −9818.77 −0.334273
\(953\) −15558.4 −0.528842 −0.264421 0.964407i \(-0.585181\pi\)
−0.264421 + 0.964407i \(0.585181\pi\)
\(954\) 0 0
\(955\) 683.234 0.0231507
\(956\) −4798.97 −0.162353
\(957\) 0 0
\(958\) 24144.0 0.814256
\(959\) 11743.5 0.395432
\(960\) 0 0
\(961\) −29056.4 −0.975341
\(962\) −1153.08 −0.0386454
\(963\) 0 0
\(964\) 29049.5 0.970561
\(965\) −15565.0 −0.519229
\(966\) 0 0
\(967\) 6567.58 0.218406 0.109203 0.994019i \(-0.465170\pi\)
0.109203 + 0.994019i \(0.465170\pi\)
\(968\) 8920.98 0.296210
\(969\) 0 0
\(970\) −26079.7 −0.863266
\(971\) −41067.3 −1.35727 −0.678636 0.734474i \(-0.737429\pi\)
−0.678636 + 0.734474i \(0.737429\pi\)
\(972\) 0 0
\(973\) 1402.83 0.0462207
\(974\) 2310.63 0.0760139
\(975\) 0 0
\(976\) 6707.05 0.219966
\(977\) −55639.7 −1.82198 −0.910989 0.412430i \(-0.864680\pi\)
−0.910989 + 0.412430i \(0.864680\pi\)
\(978\) 0 0
\(979\) −21091.2 −0.688536
\(980\) 12545.9 0.408942
\(981\) 0 0
\(982\) 14503.9 0.471320
\(983\) 60200.2 1.95329 0.976646 0.214854i \(-0.0689274\pi\)
0.976646 + 0.214854i \(0.0689274\pi\)
\(984\) 0 0
\(985\) 14454.4 0.467569
\(986\) 24957.0 0.806079
\(987\) 0 0
\(988\) −10477.6 −0.337384
\(989\) −11642.8 −0.374337
\(990\) 0 0
\(991\) −12965.4 −0.415599 −0.207800 0.978171i \(-0.566630\pi\)
−0.207800 + 0.978171i \(0.566630\pi\)
\(992\) 867.323 0.0277596
\(993\) 0 0
\(994\) −3926.50 −0.125293
\(995\) −2055.76 −0.0654995
\(996\) 0 0
\(997\) −5425.12 −0.172332 −0.0861661 0.996281i \(-0.527462\pi\)
−0.0861661 + 0.996281i \(0.527462\pi\)
\(998\) −22875.9 −0.725574
\(999\) 0 0
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 414.4.a.m.1.1 4
3.2 odd 2 414.4.a.n.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
414.4.a.m.1.1 4 1.1 even 1 trivial
414.4.a.n.1.4 yes 4 3.2 odd 2