Properties

Label 1305.4.a.d.1.1
Level $1305$
Weight $4$
Character 1305.1
Self dual yes
Analytic conductor $76.997$
Analytic rank $1$
Dimension $1$
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] = [1305,4,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1305.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(76.9974925575\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -4.00000 q^{4} -5.00000 q^{5} +29.0000 q^{7} -24.0000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} -4.00000 q^{4} -5.00000 q^{5} +29.0000 q^{7} -24.0000 q^{8} -10.0000 q^{10} +15.0000 q^{11} +3.00000 q^{13} +58.0000 q^{14} -16.0000 q^{16} -121.000 q^{17} -40.0000 q^{19} +20.0000 q^{20} +30.0000 q^{22} +116.000 q^{23} +25.0000 q^{25} +6.00000 q^{26} -116.000 q^{28} -29.0000 q^{29} -116.000 q^{31} +160.000 q^{32} -242.000 q^{34} -145.000 q^{35} +36.0000 q^{37} -80.0000 q^{38} +120.000 q^{40} +170.000 q^{41} +230.000 q^{43} -60.0000 q^{44} +232.000 q^{46} -231.000 q^{47} +498.000 q^{49} +50.0000 q^{50} -12.0000 q^{52} -456.000 q^{53} -75.0000 q^{55} -696.000 q^{56} -58.0000 q^{58} -576.000 q^{59} +342.000 q^{61} -232.000 q^{62} +448.000 q^{64} -15.0000 q^{65} -269.000 q^{67} +484.000 q^{68} -290.000 q^{70} -302.000 q^{71} -372.000 q^{73} +72.0000 q^{74} +160.000 q^{76} +435.000 q^{77} -348.000 q^{79} +80.0000 q^{80} +340.000 q^{82} +512.000 q^{83} +605.000 q^{85} +460.000 q^{86} -360.000 q^{88} -1525.00 q^{89} +87.0000 q^{91} -464.000 q^{92} -462.000 q^{94} +200.000 q^{95} -560.000 q^{97} +996.000 q^{98} +O(q^{100})\)

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 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −4.00000 −0.500000
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 29.0000 1.56585 0.782926 0.622114i \(-0.213726\pi\)
0.782926 + 0.622114i \(0.213726\pi\)
\(8\) −24.0000 −1.06066
\(9\) 0 0
\(10\) −10.0000 −0.316228
\(11\) 15.0000 0.411152 0.205576 0.978641i \(-0.434093\pi\)
0.205576 + 0.978641i \(0.434093\pi\)
\(12\) 0 0
\(13\) 3.00000 0.0640039 0.0320019 0.999488i \(-0.489812\pi\)
0.0320019 + 0.999488i \(0.489812\pi\)
\(14\) 58.0000 1.10723
\(15\) 0 0
\(16\) −16.0000 −0.250000
\(17\) −121.000 −1.72628 −0.863141 0.504962i \(-0.831506\pi\)
−0.863141 + 0.504962i \(0.831506\pi\)
\(18\) 0 0
\(19\) −40.0000 −0.482980 −0.241490 0.970403i \(-0.577636\pi\)
−0.241490 + 0.970403i \(0.577636\pi\)
\(20\) 20.0000 0.223607
\(21\) 0 0
\(22\) 30.0000 0.290728
\(23\) 116.000 1.05164 0.525819 0.850597i \(-0.323759\pi\)
0.525819 + 0.850597i \(0.323759\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 6.00000 0.0452576
\(27\) 0 0
\(28\) −116.000 −0.782926
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) −116.000 −0.672071 −0.336036 0.941849i \(-0.609086\pi\)
−0.336036 + 0.941849i \(0.609086\pi\)
\(32\) 160.000 0.883883
\(33\) 0 0
\(34\) −242.000 −1.22067
\(35\) −145.000 −0.700271
\(36\) 0 0
\(37\) 36.0000 0.159956 0.0799779 0.996797i \(-0.474515\pi\)
0.0799779 + 0.996797i \(0.474515\pi\)
\(38\) −80.0000 −0.341519
\(39\) 0 0
\(40\) 120.000 0.474342
\(41\) 170.000 0.647550 0.323775 0.946134i \(-0.395048\pi\)
0.323775 + 0.946134i \(0.395048\pi\)
\(42\) 0 0
\(43\) 230.000 0.815690 0.407845 0.913051i \(-0.366280\pi\)
0.407845 + 0.913051i \(0.366280\pi\)
\(44\) −60.0000 −0.205576
\(45\) 0 0
\(46\) 232.000 0.743620
\(47\) −231.000 −0.716911 −0.358455 0.933547i \(-0.616697\pi\)
−0.358455 + 0.933547i \(0.616697\pi\)
\(48\) 0 0
\(49\) 498.000 1.45190
\(50\) 50.0000 0.141421
\(51\) 0 0
\(52\) −12.0000 −0.0320019
\(53\) −456.000 −1.18182 −0.590910 0.806738i \(-0.701231\pi\)
−0.590910 + 0.806738i \(0.701231\pi\)
\(54\) 0 0
\(55\) −75.0000 −0.183873
\(56\) −696.000 −1.66084
\(57\) 0 0
\(58\) −58.0000 −0.131306
\(59\) −576.000 −1.27100 −0.635498 0.772102i \(-0.719205\pi\)
−0.635498 + 0.772102i \(0.719205\pi\)
\(60\) 0 0
\(61\) 342.000 0.717846 0.358923 0.933367i \(-0.383144\pi\)
0.358923 + 0.933367i \(0.383144\pi\)
\(62\) −232.000 −0.475226
\(63\) 0 0
\(64\) 448.000 0.875000
\(65\) −15.0000 −0.0286234
\(66\) 0 0
\(67\) −269.000 −0.490501 −0.245251 0.969460i \(-0.578870\pi\)
−0.245251 + 0.969460i \(0.578870\pi\)
\(68\) 484.000 0.863141
\(69\) 0 0
\(70\) −290.000 −0.495166
\(71\) −302.000 −0.504800 −0.252400 0.967623i \(-0.581220\pi\)
−0.252400 + 0.967623i \(0.581220\pi\)
\(72\) 0 0
\(73\) −372.000 −0.596429 −0.298214 0.954499i \(-0.596391\pi\)
−0.298214 + 0.954499i \(0.596391\pi\)
\(74\) 72.0000 0.113106
\(75\) 0 0
\(76\) 160.000 0.241490
\(77\) 435.000 0.643803
\(78\) 0 0
\(79\) −348.000 −0.495608 −0.247804 0.968810i \(-0.579709\pi\)
−0.247804 + 0.968810i \(0.579709\pi\)
\(80\) 80.0000 0.111803
\(81\) 0 0
\(82\) 340.000 0.457887
\(83\) 512.000 0.677100 0.338550 0.940948i \(-0.390064\pi\)
0.338550 + 0.940948i \(0.390064\pi\)
\(84\) 0 0
\(85\) 605.000 0.772017
\(86\) 460.000 0.576780
\(87\) 0 0
\(88\) −360.000 −0.436092
\(89\) −1525.00 −1.81629 −0.908144 0.418657i \(-0.862501\pi\)
−0.908144 + 0.418657i \(0.862501\pi\)
\(90\) 0 0
\(91\) 87.0000 0.100221
\(92\) −464.000 −0.525819
\(93\) 0 0
\(94\) −462.000 −0.506933
\(95\) 200.000 0.215995
\(96\) 0 0
\(97\) −560.000 −0.586179 −0.293090 0.956085i \(-0.594683\pi\)
−0.293090 + 0.956085i \(0.594683\pi\)
\(98\) 996.000 1.02664
\(99\) 0 0
\(100\) −100.000 −0.100000
\(101\) −1447.00 −1.42556 −0.712782 0.701386i \(-0.752565\pi\)
−0.712782 + 0.701386i \(0.752565\pi\)
\(102\) 0 0
\(103\) 556.000 0.531886 0.265943 0.963989i \(-0.414317\pi\)
0.265943 + 0.963989i \(0.414317\pi\)
\(104\) −72.0000 −0.0678864
\(105\) 0 0
\(106\) −912.000 −0.835672
\(107\) −1558.00 −1.40764 −0.703820 0.710378i \(-0.748524\pi\)
−0.703820 + 0.710378i \(0.748524\pi\)
\(108\) 0 0
\(109\) −881.000 −0.774170 −0.387085 0.922044i \(-0.626518\pi\)
−0.387085 + 0.922044i \(0.626518\pi\)
\(110\) −150.000 −0.130018
\(111\) 0 0
\(112\) −464.000 −0.391463
\(113\) −313.000 −0.260571 −0.130286 0.991476i \(-0.541589\pi\)
−0.130286 + 0.991476i \(0.541589\pi\)
\(114\) 0 0
\(115\) −580.000 −0.470307
\(116\) 116.000 0.0928477
\(117\) 0 0
\(118\) −1152.00 −0.898730
\(119\) −3509.00 −2.70311
\(120\) 0 0
\(121\) −1106.00 −0.830954
\(122\) 684.000 0.507594
\(123\) 0 0
\(124\) 464.000 0.336036
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −494.000 −0.345161 −0.172580 0.984995i \(-0.555210\pi\)
−0.172580 + 0.984995i \(0.555210\pi\)
\(128\) −384.000 −0.265165
\(129\) 0 0
\(130\) −30.0000 −0.0202398
\(131\) 2007.00 1.33857 0.669284 0.743007i \(-0.266601\pi\)
0.669284 + 0.743007i \(0.266601\pi\)
\(132\) 0 0
\(133\) −1160.00 −0.756276
\(134\) −538.000 −0.346837
\(135\) 0 0
\(136\) 2904.00 1.83100
\(137\) −918.000 −0.572482 −0.286241 0.958158i \(-0.592406\pi\)
−0.286241 + 0.958158i \(0.592406\pi\)
\(138\) 0 0
\(139\) 1619.00 0.987927 0.493963 0.869483i \(-0.335548\pi\)
0.493963 + 0.869483i \(0.335548\pi\)
\(140\) 580.000 0.350135
\(141\) 0 0
\(142\) −604.000 −0.356948
\(143\) 45.0000 0.0263153
\(144\) 0 0
\(145\) 145.000 0.0830455
\(146\) −744.000 −0.421739
\(147\) 0 0
\(148\) −144.000 −0.0799779
\(149\) −3046.00 −1.67475 −0.837376 0.546627i \(-0.815911\pi\)
−0.837376 + 0.546627i \(0.815911\pi\)
\(150\) 0 0
\(151\) 1576.00 0.849358 0.424679 0.905344i \(-0.360387\pi\)
0.424679 + 0.905344i \(0.360387\pi\)
\(152\) 960.000 0.512278
\(153\) 0 0
\(154\) 870.000 0.455238
\(155\) 580.000 0.300559
\(156\) 0 0
\(157\) −3544.00 −1.80154 −0.900771 0.434295i \(-0.856998\pi\)
−0.900771 + 0.434295i \(0.856998\pi\)
\(158\) −696.000 −0.350448
\(159\) 0 0
\(160\) −800.000 −0.395285
\(161\) 3364.00 1.64671
\(162\) 0 0
\(163\) −768.000 −0.369045 −0.184523 0.982828i \(-0.559074\pi\)
−0.184523 + 0.982828i \(0.559074\pi\)
\(164\) −680.000 −0.323775
\(165\) 0 0
\(166\) 1024.00 0.478782
\(167\) 1624.00 0.752508 0.376254 0.926516i \(-0.377212\pi\)
0.376254 + 0.926516i \(0.377212\pi\)
\(168\) 0 0
\(169\) −2188.00 −0.995904
\(170\) 1210.00 0.545899
\(171\) 0 0
\(172\) −920.000 −0.407845
\(173\) −1700.00 −0.747102 −0.373551 0.927610i \(-0.621860\pi\)
−0.373551 + 0.927610i \(0.621860\pi\)
\(174\) 0 0
\(175\) 725.000 0.313171
\(176\) −240.000 −0.102788
\(177\) 0 0
\(178\) −3050.00 −1.28431
\(179\) −3870.00 −1.61596 −0.807982 0.589208i \(-0.799440\pi\)
−0.807982 + 0.589208i \(0.799440\pi\)
\(180\) 0 0
\(181\) 1757.00 0.721529 0.360765 0.932657i \(-0.382516\pi\)
0.360765 + 0.932657i \(0.382516\pi\)
\(182\) 174.000 0.0708667
\(183\) 0 0
\(184\) −2784.00 −1.11543
\(185\) −180.000 −0.0715344
\(186\) 0 0
\(187\) −1815.00 −0.709764
\(188\) 924.000 0.358455
\(189\) 0 0
\(190\) 400.000 0.152732
\(191\) 2048.00 0.775854 0.387927 0.921690i \(-0.373191\pi\)
0.387927 + 0.921690i \(0.373191\pi\)
\(192\) 0 0
\(193\) −2398.00 −0.894362 −0.447181 0.894444i \(-0.647572\pi\)
−0.447181 + 0.894444i \(0.647572\pi\)
\(194\) −1120.00 −0.414491
\(195\) 0 0
\(196\) −1992.00 −0.725948
\(197\) −3966.00 −1.43434 −0.717172 0.696896i \(-0.754564\pi\)
−0.717172 + 0.696896i \(0.754564\pi\)
\(198\) 0 0
\(199\) 641.000 0.228338 0.114169 0.993461i \(-0.463579\pi\)
0.114169 + 0.993461i \(0.463579\pi\)
\(200\) −600.000 −0.212132
\(201\) 0 0
\(202\) −2894.00 −1.00803
\(203\) −841.000 −0.290772
\(204\) 0 0
\(205\) −850.000 −0.289593
\(206\) 1112.00 0.376101
\(207\) 0 0
\(208\) −48.0000 −0.0160010
\(209\) −600.000 −0.198578
\(210\) 0 0
\(211\) 5438.00 1.77425 0.887126 0.461526i \(-0.152698\pi\)
0.887126 + 0.461526i \(0.152698\pi\)
\(212\) 1824.00 0.590910
\(213\) 0 0
\(214\) −3116.00 −0.995352
\(215\) −1150.00 −0.364788
\(216\) 0 0
\(217\) −3364.00 −1.05236
\(218\) −1762.00 −0.547421
\(219\) 0 0
\(220\) 300.000 0.0919363
\(221\) −363.000 −0.110489
\(222\) 0 0
\(223\) 2799.00 0.840515 0.420258 0.907405i \(-0.361940\pi\)
0.420258 + 0.907405i \(0.361940\pi\)
\(224\) 4640.00 1.38403
\(225\) 0 0
\(226\) −626.000 −0.184252
\(227\) −1492.00 −0.436245 −0.218122 0.975921i \(-0.569993\pi\)
−0.218122 + 0.975921i \(0.569993\pi\)
\(228\) 0 0
\(229\) 4622.00 1.33376 0.666879 0.745166i \(-0.267630\pi\)
0.666879 + 0.745166i \(0.267630\pi\)
\(230\) −1160.00 −0.332557
\(231\) 0 0
\(232\) 696.000 0.196960
\(233\) −4170.00 −1.17247 −0.586236 0.810141i \(-0.699391\pi\)
−0.586236 + 0.810141i \(0.699391\pi\)
\(234\) 0 0
\(235\) 1155.00 0.320612
\(236\) 2304.00 0.635498
\(237\) 0 0
\(238\) −7018.00 −1.91138
\(239\) −1686.00 −0.456311 −0.228155 0.973625i \(-0.573269\pi\)
−0.228155 + 0.973625i \(0.573269\pi\)
\(240\) 0 0
\(241\) −3925.00 −1.04909 −0.524547 0.851382i \(-0.675765\pi\)
−0.524547 + 0.851382i \(0.675765\pi\)
\(242\) −2212.00 −0.587573
\(243\) 0 0
\(244\) −1368.00 −0.358923
\(245\) −2490.00 −0.649307
\(246\) 0 0
\(247\) −120.000 −0.0309126
\(248\) 2784.00 0.712839
\(249\) 0 0
\(250\) −250.000 −0.0632456
\(251\) 5775.00 1.45225 0.726125 0.687563i \(-0.241319\pi\)
0.726125 + 0.687563i \(0.241319\pi\)
\(252\) 0 0
\(253\) 1740.00 0.432383
\(254\) −988.000 −0.244065
\(255\) 0 0
\(256\) −4352.00 −1.06250
\(257\) 3146.00 0.763588 0.381794 0.924247i \(-0.375306\pi\)
0.381794 + 0.924247i \(0.375306\pi\)
\(258\) 0 0
\(259\) 1044.00 0.250467
\(260\) 60.0000 0.0143117
\(261\) 0 0
\(262\) 4014.00 0.946510
\(263\) 5768.00 1.35236 0.676179 0.736737i \(-0.263635\pi\)
0.676179 + 0.736737i \(0.263635\pi\)
\(264\) 0 0
\(265\) 2280.00 0.528526
\(266\) −2320.00 −0.534768
\(267\) 0 0
\(268\) 1076.00 0.245251
\(269\) 7341.00 1.66390 0.831949 0.554852i \(-0.187225\pi\)
0.831949 + 0.554852i \(0.187225\pi\)
\(270\) 0 0
\(271\) −14.0000 −0.00313815 −0.00156908 0.999999i \(-0.500499\pi\)
−0.00156908 + 0.999999i \(0.500499\pi\)
\(272\) 1936.00 0.431571
\(273\) 0 0
\(274\) −1836.00 −0.404806
\(275\) 375.000 0.0822304
\(276\) 0 0
\(277\) 3721.00 0.807124 0.403562 0.914952i \(-0.367772\pi\)
0.403562 + 0.914952i \(0.367772\pi\)
\(278\) 3238.00 0.698570
\(279\) 0 0
\(280\) 3480.00 0.742749
\(281\) −924.000 −0.196161 −0.0980805 0.995178i \(-0.531270\pi\)
−0.0980805 + 0.995178i \(0.531270\pi\)
\(282\) 0 0
\(283\) 1940.00 0.407495 0.203747 0.979023i \(-0.434688\pi\)
0.203747 + 0.979023i \(0.434688\pi\)
\(284\) 1208.00 0.252400
\(285\) 0 0
\(286\) 90.0000 0.0186077
\(287\) 4930.00 1.01397
\(288\) 0 0
\(289\) 9728.00 1.98005
\(290\) 290.000 0.0587220
\(291\) 0 0
\(292\) 1488.00 0.298214
\(293\) −5267.00 −1.05018 −0.525088 0.851048i \(-0.675968\pi\)
−0.525088 + 0.851048i \(0.675968\pi\)
\(294\) 0 0
\(295\) 2880.00 0.568407
\(296\) −864.000 −0.169659
\(297\) 0 0
\(298\) −6092.00 −1.18423
\(299\) 348.000 0.0673089
\(300\) 0 0
\(301\) 6670.00 1.27725
\(302\) 3152.00 0.600587
\(303\) 0 0
\(304\) 640.000 0.120745
\(305\) −1710.00 −0.321031
\(306\) 0 0
\(307\) 6856.00 1.27457 0.637284 0.770629i \(-0.280058\pi\)
0.637284 + 0.770629i \(0.280058\pi\)
\(308\) −1740.00 −0.321902
\(309\) 0 0
\(310\) 1160.00 0.212528
\(311\) 2447.00 0.446163 0.223081 0.974800i \(-0.428388\pi\)
0.223081 + 0.974800i \(0.428388\pi\)
\(312\) 0 0
\(313\) 511.000 0.0922793 0.0461397 0.998935i \(-0.485308\pi\)
0.0461397 + 0.998935i \(0.485308\pi\)
\(314\) −7088.00 −1.27388
\(315\) 0 0
\(316\) 1392.00 0.247804
\(317\) −7167.00 −1.26984 −0.634919 0.772578i \(-0.718967\pi\)
−0.634919 + 0.772578i \(0.718967\pi\)
\(318\) 0 0
\(319\) −435.000 −0.0763490
\(320\) −2240.00 −0.391312
\(321\) 0 0
\(322\) 6728.00 1.16440
\(323\) 4840.00 0.833761
\(324\) 0 0
\(325\) 75.0000 0.0128008
\(326\) −1536.00 −0.260955
\(327\) 0 0
\(328\) −4080.00 −0.686830
\(329\) −6699.00 −1.12258
\(330\) 0 0
\(331\) −8962.00 −1.48821 −0.744103 0.668065i \(-0.767123\pi\)
−0.744103 + 0.668065i \(0.767123\pi\)
\(332\) −2048.00 −0.338550
\(333\) 0 0
\(334\) 3248.00 0.532104
\(335\) 1345.00 0.219359
\(336\) 0 0
\(337\) 2706.00 0.437404 0.218702 0.975792i \(-0.429818\pi\)
0.218702 + 0.975792i \(0.429818\pi\)
\(338\) −4376.00 −0.704210
\(339\) 0 0
\(340\) −2420.00 −0.386009
\(341\) −1740.00 −0.276323
\(342\) 0 0
\(343\) 4495.00 0.707601
\(344\) −5520.00 −0.865170
\(345\) 0 0
\(346\) −3400.00 −0.528281
\(347\) −2232.00 −0.345303 −0.172651 0.984983i \(-0.555233\pi\)
−0.172651 + 0.984983i \(0.555233\pi\)
\(348\) 0 0
\(349\) 9742.00 1.49420 0.747102 0.664709i \(-0.231445\pi\)
0.747102 + 0.664709i \(0.231445\pi\)
\(350\) 1450.00 0.221445
\(351\) 0 0
\(352\) 2400.00 0.363410
\(353\) 2694.00 0.406196 0.203098 0.979158i \(-0.434899\pi\)
0.203098 + 0.979158i \(0.434899\pi\)
\(354\) 0 0
\(355\) 1510.00 0.225753
\(356\) 6100.00 0.908144
\(357\) 0 0
\(358\) −7740.00 −1.14266
\(359\) 960.000 0.141133 0.0705667 0.997507i \(-0.477519\pi\)
0.0705667 + 0.997507i \(0.477519\pi\)
\(360\) 0 0
\(361\) −5259.00 −0.766730
\(362\) 3514.00 0.510198
\(363\) 0 0
\(364\) −348.000 −0.0501103
\(365\) 1860.00 0.266731
\(366\) 0 0
\(367\) −4632.00 −0.658824 −0.329412 0.944186i \(-0.606851\pi\)
−0.329412 + 0.944186i \(0.606851\pi\)
\(368\) −1856.00 −0.262909
\(369\) 0 0
\(370\) −360.000 −0.0505825
\(371\) −13224.0 −1.85055
\(372\) 0 0
\(373\) −6682.00 −0.927563 −0.463781 0.885950i \(-0.653508\pi\)
−0.463781 + 0.885950i \(0.653508\pi\)
\(374\) −3630.00 −0.501879
\(375\) 0 0
\(376\) 5544.00 0.760399
\(377\) −87.0000 −0.0118852
\(378\) 0 0
\(379\) 11270.0 1.52744 0.763722 0.645546i \(-0.223370\pi\)
0.763722 + 0.645546i \(0.223370\pi\)
\(380\) −800.000 −0.107998
\(381\) 0 0
\(382\) 4096.00 0.548611
\(383\) −1016.00 −0.135549 −0.0677744 0.997701i \(-0.521590\pi\)
−0.0677744 + 0.997701i \(0.521590\pi\)
\(384\) 0 0
\(385\) −2175.00 −0.287918
\(386\) −4796.00 −0.632409
\(387\) 0 0
\(388\) 2240.00 0.293090
\(389\) 3727.00 0.485775 0.242887 0.970054i \(-0.421905\pi\)
0.242887 + 0.970054i \(0.421905\pi\)
\(390\) 0 0
\(391\) −14036.0 −1.81542
\(392\) −11952.0 −1.53997
\(393\) 0 0
\(394\) −7932.00 −1.01423
\(395\) 1740.00 0.221643
\(396\) 0 0
\(397\) 1990.00 0.251575 0.125787 0.992057i \(-0.459854\pi\)
0.125787 + 0.992057i \(0.459854\pi\)
\(398\) 1282.00 0.161459
\(399\) 0 0
\(400\) −400.000 −0.0500000
\(401\) −6696.00 −0.833871 −0.416936 0.908936i \(-0.636896\pi\)
−0.416936 + 0.908936i \(0.636896\pi\)
\(402\) 0 0
\(403\) −348.000 −0.0430152
\(404\) 5788.00 0.712782
\(405\) 0 0
\(406\) −1682.00 −0.205607
\(407\) 540.000 0.0657661
\(408\) 0 0
\(409\) 252.000 0.0304660 0.0152330 0.999884i \(-0.495151\pi\)
0.0152330 + 0.999884i \(0.495151\pi\)
\(410\) −1700.00 −0.204773
\(411\) 0 0
\(412\) −2224.00 −0.265943
\(413\) −16704.0 −1.99019
\(414\) 0 0
\(415\) −2560.00 −0.302808
\(416\) 480.000 0.0565720
\(417\) 0 0
\(418\) −1200.00 −0.140416
\(419\) −7418.00 −0.864900 −0.432450 0.901658i \(-0.642351\pi\)
−0.432450 + 0.901658i \(0.642351\pi\)
\(420\) 0 0
\(421\) 11268.0 1.30444 0.652219 0.758030i \(-0.273838\pi\)
0.652219 + 0.758030i \(0.273838\pi\)
\(422\) 10876.0 1.25459
\(423\) 0 0
\(424\) 10944.0 1.25351
\(425\) −3025.00 −0.345257
\(426\) 0 0
\(427\) 9918.00 1.12404
\(428\) 6232.00 0.703820
\(429\) 0 0
\(430\) −2300.00 −0.257944
\(431\) 11600.0 1.29641 0.648205 0.761466i \(-0.275520\pi\)
0.648205 + 0.761466i \(0.275520\pi\)
\(432\) 0 0
\(433\) 1072.00 0.118977 0.0594885 0.998229i \(-0.481053\pi\)
0.0594885 + 0.998229i \(0.481053\pi\)
\(434\) −6728.00 −0.744134
\(435\) 0 0
\(436\) 3524.00 0.387085
\(437\) −4640.00 −0.507921
\(438\) 0 0
\(439\) −12339.0 −1.34148 −0.670738 0.741694i \(-0.734023\pi\)
−0.670738 + 0.741694i \(0.734023\pi\)
\(440\) 1800.00 0.195026
\(441\) 0 0
\(442\) −726.000 −0.0781274
\(443\) 15263.0 1.63695 0.818473 0.574545i \(-0.194821\pi\)
0.818473 + 0.574545i \(0.194821\pi\)
\(444\) 0 0
\(445\) 7625.00 0.812269
\(446\) 5598.00 0.594334
\(447\) 0 0
\(448\) 12992.0 1.37012
\(449\) −11019.0 −1.15817 −0.579085 0.815267i \(-0.696590\pi\)
−0.579085 + 0.815267i \(0.696590\pi\)
\(450\) 0 0
\(451\) 2550.00 0.266241
\(452\) 1252.00 0.130286
\(453\) 0 0
\(454\) −2984.00 −0.308471
\(455\) −435.000 −0.0448200
\(456\) 0 0
\(457\) −6769.00 −0.692868 −0.346434 0.938074i \(-0.612607\pi\)
−0.346434 + 0.938074i \(0.612607\pi\)
\(458\) 9244.00 0.943109
\(459\) 0 0
\(460\) 2320.00 0.235153
\(461\) 5514.00 0.557077 0.278539 0.960425i \(-0.410150\pi\)
0.278539 + 0.960425i \(0.410150\pi\)
\(462\) 0 0
\(463\) −14205.0 −1.42584 −0.712918 0.701247i \(-0.752627\pi\)
−0.712918 + 0.701247i \(0.752627\pi\)
\(464\) 464.000 0.0464238
\(465\) 0 0
\(466\) −8340.00 −0.829062
\(467\) 9660.00 0.957198 0.478599 0.878034i \(-0.341145\pi\)
0.478599 + 0.878034i \(0.341145\pi\)
\(468\) 0 0
\(469\) −7801.00 −0.768053
\(470\) 2310.00 0.226707
\(471\) 0 0
\(472\) 13824.0 1.34810
\(473\) 3450.00 0.335372
\(474\) 0 0
\(475\) −1000.00 −0.0965961
\(476\) 14036.0 1.35155
\(477\) 0 0
\(478\) −3372.00 −0.322660
\(479\) 7656.00 0.730296 0.365148 0.930950i \(-0.381018\pi\)
0.365148 + 0.930950i \(0.381018\pi\)
\(480\) 0 0
\(481\) 108.000 0.0102378
\(482\) −7850.00 −0.741821
\(483\) 0 0
\(484\) 4424.00 0.415477
\(485\) 2800.00 0.262147
\(486\) 0 0
\(487\) −15968.0 −1.48579 −0.742894 0.669409i \(-0.766548\pi\)
−0.742894 + 0.669409i \(0.766548\pi\)
\(488\) −8208.00 −0.761391
\(489\) 0 0
\(490\) −4980.00 −0.459130
\(491\) −4236.00 −0.389344 −0.194672 0.980868i \(-0.562364\pi\)
−0.194672 + 0.980868i \(0.562364\pi\)
\(492\) 0 0
\(493\) 3509.00 0.320563
\(494\) −240.000 −0.0218585
\(495\) 0 0
\(496\) 1856.00 0.168018
\(497\) −8758.00 −0.790443
\(498\) 0 0
\(499\) −16941.0 −1.51981 −0.759903 0.650036i \(-0.774754\pi\)
−0.759903 + 0.650036i \(0.774754\pi\)
\(500\) 500.000 0.0447214
\(501\) 0 0
\(502\) 11550.0 1.02690
\(503\) 1857.00 0.164611 0.0823057 0.996607i \(-0.473772\pi\)
0.0823057 + 0.996607i \(0.473772\pi\)
\(504\) 0 0
\(505\) 7235.00 0.637531
\(506\) 3480.00 0.305741
\(507\) 0 0
\(508\) 1976.00 0.172580
\(509\) 3096.00 0.269603 0.134801 0.990873i \(-0.456960\pi\)
0.134801 + 0.990873i \(0.456960\pi\)
\(510\) 0 0
\(511\) −10788.0 −0.933920
\(512\) −5632.00 −0.486136
\(513\) 0 0
\(514\) 6292.00 0.539938
\(515\) −2780.00 −0.237867
\(516\) 0 0
\(517\) −3465.00 −0.294759
\(518\) 2088.00 0.177107
\(519\) 0 0
\(520\) 360.000 0.0303597
\(521\) −7530.00 −0.633196 −0.316598 0.948560i \(-0.602541\pi\)
−0.316598 + 0.948560i \(0.602541\pi\)
\(522\) 0 0
\(523\) −5767.00 −0.482167 −0.241083 0.970504i \(-0.577503\pi\)
−0.241083 + 0.970504i \(0.577503\pi\)
\(524\) −8028.00 −0.669284
\(525\) 0 0
\(526\) 11536.0 0.956261
\(527\) 14036.0 1.16019
\(528\) 0 0
\(529\) 1289.00 0.105942
\(530\) 4560.00 0.373724
\(531\) 0 0
\(532\) 4640.00 0.378138
\(533\) 510.000 0.0414457
\(534\) 0 0
\(535\) 7790.00 0.629516
\(536\) 6456.00 0.520255
\(537\) 0 0
\(538\) 14682.0 1.17655
\(539\) 7470.00 0.596949
\(540\) 0 0
\(541\) −21122.0 −1.67857 −0.839284 0.543693i \(-0.817026\pi\)
−0.839284 + 0.543693i \(0.817026\pi\)
\(542\) −28.0000 −0.00221901
\(543\) 0 0
\(544\) −19360.0 −1.52583
\(545\) 4405.00 0.346219
\(546\) 0 0
\(547\) 17857.0 1.39581 0.697907 0.716188i \(-0.254115\pi\)
0.697907 + 0.716188i \(0.254115\pi\)
\(548\) 3672.00 0.286241
\(549\) 0 0
\(550\) 750.000 0.0581456
\(551\) 1160.00 0.0896872
\(552\) 0 0
\(553\) −10092.0 −0.776050
\(554\) 7442.00 0.570723
\(555\) 0 0
\(556\) −6476.00 −0.493963
\(557\) −6040.00 −0.459467 −0.229733 0.973254i \(-0.573785\pi\)
−0.229733 + 0.973254i \(0.573785\pi\)
\(558\) 0 0
\(559\) 690.000 0.0522073
\(560\) 2320.00 0.175068
\(561\) 0 0
\(562\) −1848.00 −0.138707
\(563\) 15371.0 1.15064 0.575320 0.817928i \(-0.304877\pi\)
0.575320 + 0.817928i \(0.304877\pi\)
\(564\) 0 0
\(565\) 1565.00 0.116531
\(566\) 3880.00 0.288142
\(567\) 0 0
\(568\) 7248.00 0.535421
\(569\) −17901.0 −1.31889 −0.659445 0.751752i \(-0.729209\pi\)
−0.659445 + 0.751752i \(0.729209\pi\)
\(570\) 0 0
\(571\) −10056.0 −0.737006 −0.368503 0.929627i \(-0.620130\pi\)
−0.368503 + 0.929627i \(0.620130\pi\)
\(572\) −180.000 −0.0131577
\(573\) 0 0
\(574\) 9860.00 0.716983
\(575\) 2900.00 0.210328
\(576\) 0 0
\(577\) 3068.00 0.221356 0.110678 0.993856i \(-0.464698\pi\)
0.110678 + 0.993856i \(0.464698\pi\)
\(578\) 19456.0 1.40011
\(579\) 0 0
\(580\) −580.000 −0.0415227
\(581\) 14848.0 1.06024
\(582\) 0 0
\(583\) −6840.00 −0.485907
\(584\) 8928.00 0.632608
\(585\) 0 0
\(586\) −10534.0 −0.742586
\(587\) 2984.00 0.209817 0.104909 0.994482i \(-0.466545\pi\)
0.104909 + 0.994482i \(0.466545\pi\)
\(588\) 0 0
\(589\) 4640.00 0.324597
\(590\) 5760.00 0.401924
\(591\) 0 0
\(592\) −576.000 −0.0399889
\(593\) −5952.00 −0.412174 −0.206087 0.978534i \(-0.566073\pi\)
−0.206087 + 0.978534i \(0.566073\pi\)
\(594\) 0 0
\(595\) 17545.0 1.20887
\(596\) 12184.0 0.837376
\(597\) 0 0
\(598\) 696.000 0.0475946
\(599\) 12999.0 0.886686 0.443343 0.896352i \(-0.353792\pi\)
0.443343 + 0.896352i \(0.353792\pi\)
\(600\) 0 0
\(601\) 23398.0 1.58806 0.794030 0.607878i \(-0.207979\pi\)
0.794030 + 0.607878i \(0.207979\pi\)
\(602\) 13340.0 0.903153
\(603\) 0 0
\(604\) −6304.00 −0.424679
\(605\) 5530.00 0.371614
\(606\) 0 0
\(607\) 26116.0 1.74632 0.873160 0.487434i \(-0.162067\pi\)
0.873160 + 0.487434i \(0.162067\pi\)
\(608\) −6400.00 −0.426898
\(609\) 0 0
\(610\) −3420.00 −0.227003
\(611\) −693.000 −0.0458851
\(612\) 0 0
\(613\) −24185.0 −1.59351 −0.796756 0.604301i \(-0.793452\pi\)
−0.796756 + 0.604301i \(0.793452\pi\)
\(614\) 13712.0 0.901256
\(615\) 0 0
\(616\) −10440.0 −0.682856
\(617\) −2214.00 −0.144461 −0.0722304 0.997388i \(-0.523012\pi\)
−0.0722304 + 0.997388i \(0.523012\pi\)
\(618\) 0 0
\(619\) −19586.0 −1.27177 −0.635887 0.771782i \(-0.719365\pi\)
−0.635887 + 0.771782i \(0.719365\pi\)
\(620\) −2320.00 −0.150280
\(621\) 0 0
\(622\) 4894.00 0.315485
\(623\) −44225.0 −2.84404
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 1022.00 0.0652513
\(627\) 0 0
\(628\) 14176.0 0.900771
\(629\) −4356.00 −0.276129
\(630\) 0 0
\(631\) 4377.00 0.276142 0.138071 0.990422i \(-0.455910\pi\)
0.138071 + 0.990422i \(0.455910\pi\)
\(632\) 8352.00 0.525672
\(633\) 0 0
\(634\) −14334.0 −0.897911
\(635\) 2470.00 0.154361
\(636\) 0 0
\(637\) 1494.00 0.0929269
\(638\) −870.000 −0.0539869
\(639\) 0 0
\(640\) 1920.00 0.118585
\(641\) 24015.0 1.47977 0.739887 0.672731i \(-0.234879\pi\)
0.739887 + 0.672731i \(0.234879\pi\)
\(642\) 0 0
\(643\) −19465.0 −1.19382 −0.596909 0.802309i \(-0.703605\pi\)
−0.596909 + 0.802309i \(0.703605\pi\)
\(644\) −13456.0 −0.823355
\(645\) 0 0
\(646\) 9680.00 0.589558
\(647\) 19954.0 1.21248 0.606239 0.795283i \(-0.292678\pi\)
0.606239 + 0.795283i \(0.292678\pi\)
\(648\) 0 0
\(649\) −8640.00 −0.522573
\(650\) 150.000 0.00905151
\(651\) 0 0
\(652\) 3072.00 0.184523
\(653\) 22597.0 1.35419 0.677097 0.735894i \(-0.263238\pi\)
0.677097 + 0.735894i \(0.263238\pi\)
\(654\) 0 0
\(655\) −10035.0 −0.598626
\(656\) −2720.00 −0.161887
\(657\) 0 0
\(658\) −13398.0 −0.793782
\(659\) −7467.00 −0.441385 −0.220693 0.975343i \(-0.570832\pi\)
−0.220693 + 0.975343i \(0.570832\pi\)
\(660\) 0 0
\(661\) 16601.0 0.976859 0.488430 0.872603i \(-0.337570\pi\)
0.488430 + 0.872603i \(0.337570\pi\)
\(662\) −17924.0 −1.05232
\(663\) 0 0
\(664\) −12288.0 −0.718173
\(665\) 5800.00 0.338217
\(666\) 0 0
\(667\) −3364.00 −0.195284
\(668\) −6496.00 −0.376254
\(669\) 0 0
\(670\) 2690.00 0.155110
\(671\) 5130.00 0.295144
\(672\) 0 0
\(673\) 23571.0 1.35007 0.675034 0.737787i \(-0.264129\pi\)
0.675034 + 0.737787i \(0.264129\pi\)
\(674\) 5412.00 0.309291
\(675\) 0 0
\(676\) 8752.00 0.497952
\(677\) 16963.0 0.962985 0.481493 0.876450i \(-0.340095\pi\)
0.481493 + 0.876450i \(0.340095\pi\)
\(678\) 0 0
\(679\) −16240.0 −0.917870
\(680\) −14520.0 −0.818848
\(681\) 0 0
\(682\) −3480.00 −0.195390
\(683\) −6144.00 −0.344207 −0.172104 0.985079i \(-0.555056\pi\)
−0.172104 + 0.985079i \(0.555056\pi\)
\(684\) 0 0
\(685\) 4590.00 0.256022
\(686\) 8990.00 0.500350
\(687\) 0 0
\(688\) −3680.00 −0.203923
\(689\) −1368.00 −0.0756410
\(690\) 0 0
\(691\) 18461.0 1.01634 0.508169 0.861257i \(-0.330323\pi\)
0.508169 + 0.861257i \(0.330323\pi\)
\(692\) 6800.00 0.373551
\(693\) 0 0
\(694\) −4464.00 −0.244166
\(695\) −8095.00 −0.441814
\(696\) 0 0
\(697\) −20570.0 −1.11785
\(698\) 19484.0 1.05656
\(699\) 0 0
\(700\) −2900.00 −0.156585
\(701\) 7550.00 0.406790 0.203395 0.979097i \(-0.434802\pi\)
0.203395 + 0.979097i \(0.434802\pi\)
\(702\) 0 0
\(703\) −1440.00 −0.0772555
\(704\) 6720.00 0.359758
\(705\) 0 0
\(706\) 5388.00 0.287224
\(707\) −41963.0 −2.23222
\(708\) 0 0
\(709\) 29126.0 1.54281 0.771403 0.636347i \(-0.219555\pi\)
0.771403 + 0.636347i \(0.219555\pi\)
\(710\) 3020.00 0.159632
\(711\) 0 0
\(712\) 36600.0 1.92646
\(713\) −13456.0 −0.706776
\(714\) 0 0
\(715\) −225.000 −0.0117686
\(716\) 15480.0 0.807982
\(717\) 0 0
\(718\) 1920.00 0.0997963
\(719\) 31670.0 1.64269 0.821343 0.570434i \(-0.193225\pi\)
0.821343 + 0.570434i \(0.193225\pi\)
\(720\) 0 0
\(721\) 16124.0 0.832856
\(722\) −10518.0 −0.542160
\(723\) 0 0
\(724\) −7028.00 −0.360765
\(725\) −725.000 −0.0371391
\(726\) 0 0
\(727\) 21080.0 1.07540 0.537699 0.843137i \(-0.319294\pi\)
0.537699 + 0.843137i \(0.319294\pi\)
\(728\) −2088.00 −0.106300
\(729\) 0 0
\(730\) 3720.00 0.188607
\(731\) −27830.0 −1.40811
\(732\) 0 0
\(733\) −31730.0 −1.59887 −0.799437 0.600750i \(-0.794869\pi\)
−0.799437 + 0.600750i \(0.794869\pi\)
\(734\) −9264.00 −0.465859
\(735\) 0 0
\(736\) 18560.0 0.929525
\(737\) −4035.00 −0.201670
\(738\) 0 0
\(739\) −35010.0 −1.74271 −0.871356 0.490652i \(-0.836759\pi\)
−0.871356 + 0.490652i \(0.836759\pi\)
\(740\) 720.000 0.0357672
\(741\) 0 0
\(742\) −26448.0 −1.30854
\(743\) 36625.0 1.80840 0.904200 0.427110i \(-0.140468\pi\)
0.904200 + 0.427110i \(0.140468\pi\)
\(744\) 0 0
\(745\) 15230.0 0.748972
\(746\) −13364.0 −0.655886
\(747\) 0 0
\(748\) 7260.00 0.354882
\(749\) −45182.0 −2.20416
\(750\) 0 0
\(751\) 8420.00 0.409121 0.204561 0.978854i \(-0.434423\pi\)
0.204561 + 0.978854i \(0.434423\pi\)
\(752\) 3696.00 0.179228
\(753\) 0 0
\(754\) −174.000 −0.00840412
\(755\) −7880.00 −0.379844
\(756\) 0 0
\(757\) −14404.0 −0.691575 −0.345788 0.938313i \(-0.612388\pi\)
−0.345788 + 0.938313i \(0.612388\pi\)
\(758\) 22540.0 1.08007
\(759\) 0 0
\(760\) −4800.00 −0.229098
\(761\) −7208.00 −0.343351 −0.171675 0.985154i \(-0.554918\pi\)
−0.171675 + 0.985154i \(0.554918\pi\)
\(762\) 0 0
\(763\) −25549.0 −1.21224
\(764\) −8192.00 −0.387927
\(765\) 0 0
\(766\) −2032.00 −0.0958474
\(767\) −1728.00 −0.0813487
\(768\) 0 0
\(769\) −31638.0 −1.48361 −0.741805 0.670616i \(-0.766030\pi\)
−0.741805 + 0.670616i \(0.766030\pi\)
\(770\) −4350.00 −0.203588
\(771\) 0 0
\(772\) 9592.00 0.447181
\(773\) 930.000 0.0432727 0.0216363 0.999766i \(-0.493112\pi\)
0.0216363 + 0.999766i \(0.493112\pi\)
\(774\) 0 0
\(775\) −2900.00 −0.134414
\(776\) 13440.0 0.621737
\(777\) 0 0
\(778\) 7454.00 0.343495
\(779\) −6800.00 −0.312754
\(780\) 0 0
\(781\) −4530.00 −0.207549
\(782\) −28072.0 −1.28370
\(783\) 0 0
\(784\) −7968.00 −0.362974
\(785\) 17720.0 0.805674
\(786\) 0 0
\(787\) 7188.00 0.325571 0.162786 0.986661i \(-0.447952\pi\)
0.162786 + 0.986661i \(0.447952\pi\)
\(788\) 15864.0 0.717172
\(789\) 0 0
\(790\) 3480.00 0.156725
\(791\) −9077.00 −0.408016
\(792\) 0 0
\(793\) 1026.00 0.0459449
\(794\) 3980.00 0.177890
\(795\) 0 0
\(796\) −2564.00 −0.114169
\(797\) 31986.0 1.42158 0.710792 0.703402i \(-0.248337\pi\)
0.710792 + 0.703402i \(0.248337\pi\)
\(798\) 0 0
\(799\) 27951.0 1.23759
\(800\) 4000.00 0.176777
\(801\) 0 0
\(802\) −13392.0 −0.589636
\(803\) −5580.00 −0.245223
\(804\) 0 0
\(805\) −16820.0 −0.736431
\(806\) −696.000 −0.0304163
\(807\) 0 0
\(808\) 34728.0 1.51204
\(809\) −8235.00 −0.357883 −0.178941 0.983860i \(-0.557267\pi\)
−0.178941 + 0.983860i \(0.557267\pi\)
\(810\) 0 0
\(811\) −9031.00 −0.391025 −0.195513 0.980701i \(-0.562637\pi\)
−0.195513 + 0.980701i \(0.562637\pi\)
\(812\) 3364.00 0.145386
\(813\) 0 0
\(814\) 1080.00 0.0465037
\(815\) 3840.00 0.165042
\(816\) 0 0
\(817\) −9200.00 −0.393962
\(818\) 504.000 0.0215427
\(819\) 0 0
\(820\) 3400.00 0.144797
\(821\) −41318.0 −1.75640 −0.878202 0.478289i \(-0.841257\pi\)
−0.878202 + 0.478289i \(0.841257\pi\)
\(822\) 0 0
\(823\) 31150.0 1.31934 0.659672 0.751553i \(-0.270695\pi\)
0.659672 + 0.751553i \(0.270695\pi\)
\(824\) −13344.0 −0.564151
\(825\) 0 0
\(826\) −33408.0 −1.40728
\(827\) −7584.00 −0.318889 −0.159445 0.987207i \(-0.550970\pi\)
−0.159445 + 0.987207i \(0.550970\pi\)
\(828\) 0 0
\(829\) −28316.0 −1.18632 −0.593158 0.805086i \(-0.702119\pi\)
−0.593158 + 0.805086i \(0.702119\pi\)
\(830\) −5120.00 −0.214118
\(831\) 0 0
\(832\) 1344.00 0.0560034
\(833\) −60258.0 −2.50638
\(834\) 0 0
\(835\) −8120.00 −0.336532
\(836\) 2400.00 0.0992892
\(837\) 0 0
\(838\) −14836.0 −0.611577
\(839\) 26615.0 1.09518 0.547588 0.836748i \(-0.315546\pi\)
0.547588 + 0.836748i \(0.315546\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 22536.0 0.922377
\(843\) 0 0
\(844\) −21752.0 −0.887126
\(845\) 10940.0 0.445382
\(846\) 0 0
\(847\) −32074.0 −1.30115
\(848\) 7296.00 0.295455
\(849\) 0 0
\(850\) −6050.00 −0.244133
\(851\) 4176.00 0.168216
\(852\) 0 0
\(853\) 33538.0 1.34621 0.673106 0.739546i \(-0.264960\pi\)
0.673106 + 0.739546i \(0.264960\pi\)
\(854\) 19836.0 0.794817
\(855\) 0 0
\(856\) 37392.0 1.49303
\(857\) 26394.0 1.05204 0.526022 0.850471i \(-0.323683\pi\)
0.526022 + 0.850471i \(0.323683\pi\)
\(858\) 0 0
\(859\) −12076.0 −0.479660 −0.239830 0.970815i \(-0.577092\pi\)
−0.239830 + 0.970815i \(0.577092\pi\)
\(860\) 4600.00 0.182394
\(861\) 0 0
\(862\) 23200.0 0.916700
\(863\) 2450.00 0.0966384 0.0483192 0.998832i \(-0.484614\pi\)
0.0483192 + 0.998832i \(0.484614\pi\)
\(864\) 0 0
\(865\) 8500.00 0.334114
\(866\) 2144.00 0.0841294
\(867\) 0 0
\(868\) 13456.0 0.526182
\(869\) −5220.00 −0.203770
\(870\) 0 0
\(871\) −807.000 −0.0313940
\(872\) 21144.0 0.821131
\(873\) 0 0
\(874\) −9280.00 −0.359154
\(875\) −3625.00 −0.140054
\(876\) 0 0
\(877\) −15566.0 −0.599346 −0.299673 0.954042i \(-0.596878\pi\)
−0.299673 + 0.954042i \(0.596878\pi\)
\(878\) −24678.0 −0.948567
\(879\) 0 0
\(880\) 1200.00 0.0459682
\(881\) −34497.0 −1.31922 −0.659610 0.751608i \(-0.729279\pi\)
−0.659610 + 0.751608i \(0.729279\pi\)
\(882\) 0 0
\(883\) 17044.0 0.649577 0.324788 0.945787i \(-0.394707\pi\)
0.324788 + 0.945787i \(0.394707\pi\)
\(884\) 1452.00 0.0552444
\(885\) 0 0
\(886\) 30526.0 1.15750
\(887\) 16903.0 0.639850 0.319925 0.947443i \(-0.396342\pi\)
0.319925 + 0.947443i \(0.396342\pi\)
\(888\) 0 0
\(889\) −14326.0 −0.540471
\(890\) 15250.0 0.574361
\(891\) 0 0
\(892\) −11196.0 −0.420258
\(893\) 9240.00 0.346254
\(894\) 0 0
\(895\) 19350.0 0.722681
\(896\) −11136.0 −0.415209
\(897\) 0 0
\(898\) −22038.0 −0.818951
\(899\) 3364.00 0.124801
\(900\) 0 0
\(901\) 55176.0 2.04015
\(902\) 5100.00 0.188261
\(903\) 0 0
\(904\) 7512.00 0.276378
\(905\) −8785.00 −0.322678
\(906\) 0 0
\(907\) 40504.0 1.48282 0.741408 0.671055i \(-0.234159\pi\)
0.741408 + 0.671055i \(0.234159\pi\)
\(908\) 5968.00 0.218122
\(909\) 0 0
\(910\) −870.000 −0.0316925
\(911\) 14783.0 0.537632 0.268816 0.963192i \(-0.413368\pi\)
0.268816 + 0.963192i \(0.413368\pi\)
\(912\) 0 0
\(913\) 7680.00 0.278391
\(914\) −13538.0 −0.489931
\(915\) 0 0
\(916\) −18488.0 −0.666879
\(917\) 58203.0 2.09600
\(918\) 0 0
\(919\) 42241.0 1.51622 0.758108 0.652129i \(-0.226124\pi\)
0.758108 + 0.652129i \(0.226124\pi\)
\(920\) 13920.0 0.498836
\(921\) 0 0
\(922\) 11028.0 0.393913
\(923\) −906.000 −0.0323092
\(924\) 0 0
\(925\) 900.000 0.0319912
\(926\) −28410.0 −1.00822
\(927\) 0 0
\(928\) −4640.00 −0.164133
\(929\) −28686.0 −1.01309 −0.506543 0.862215i \(-0.669077\pi\)
−0.506543 + 0.862215i \(0.669077\pi\)
\(930\) 0 0
\(931\) −19920.0 −0.701237
\(932\) 16680.0 0.586236
\(933\) 0 0
\(934\) 19320.0 0.676841
\(935\) 9075.00 0.317416
\(936\) 0 0
\(937\) −53063.0 −1.85005 −0.925023 0.379912i \(-0.875954\pi\)
−0.925023 + 0.379912i \(0.875954\pi\)
\(938\) −15602.0 −0.543095
\(939\) 0 0
\(940\) −4620.00 −0.160306
\(941\) 16542.0 0.573065 0.286532 0.958071i \(-0.407497\pi\)
0.286532 + 0.958071i \(0.407497\pi\)
\(942\) 0 0
\(943\) 19720.0 0.680988
\(944\) 9216.00 0.317749
\(945\) 0 0
\(946\) 6900.00 0.237144
\(947\) −30839.0 −1.05822 −0.529109 0.848554i \(-0.677474\pi\)
−0.529109 + 0.848554i \(0.677474\pi\)
\(948\) 0 0
\(949\) −1116.00 −0.0381738
\(950\) −2000.00 −0.0683038
\(951\) 0 0
\(952\) 84216.0 2.86708
\(953\) −46314.0 −1.57425 −0.787124 0.616795i \(-0.788431\pi\)
−0.787124 + 0.616795i \(0.788431\pi\)
\(954\) 0 0
\(955\) −10240.0 −0.346972
\(956\) 6744.00 0.228155
\(957\) 0 0
\(958\) 15312.0 0.516397
\(959\) −26622.0 −0.896423
\(960\) 0 0
\(961\) −16335.0 −0.548320
\(962\) 216.000 0.00723921
\(963\) 0 0
\(964\) 15700.0 0.524547
\(965\) 11990.0 0.399971
\(966\) 0 0
\(967\) −12904.0 −0.429126 −0.214563 0.976710i \(-0.568833\pi\)
−0.214563 + 0.976710i \(0.568833\pi\)
\(968\) 26544.0 0.881360
\(969\) 0 0
\(970\) 5600.00 0.185366
\(971\) 900.000 0.0297450 0.0148725 0.999889i \(-0.495266\pi\)
0.0148725 + 0.999889i \(0.495266\pi\)
\(972\) 0 0
\(973\) 46951.0 1.54695
\(974\) −31936.0 −1.05061
\(975\) 0 0
\(976\) −5472.00 −0.179462
\(977\) 20376.0 0.667232 0.333616 0.942709i \(-0.391731\pi\)
0.333616 + 0.942709i \(0.391731\pi\)
\(978\) 0 0
\(979\) −22875.0 −0.746770
\(980\) 9960.00 0.324654
\(981\) 0 0
\(982\) −8472.00 −0.275308
\(983\) −13456.0 −0.436602 −0.218301 0.975881i \(-0.570051\pi\)
−0.218301 + 0.975881i \(0.570051\pi\)
\(984\) 0 0
\(985\) 19830.0 0.641458
\(986\) 7018.00 0.226672
\(987\) 0 0
\(988\) 480.000 0.0154563
\(989\) 26680.0 0.857811
\(990\) 0 0
\(991\) −27245.0 −0.873326 −0.436663 0.899625i \(-0.643840\pi\)
−0.436663 + 0.899625i \(0.643840\pi\)
\(992\) −18560.0 −0.594033
\(993\) 0 0
\(994\) −17516.0 −0.558927
\(995\) −3205.00 −0.102116
\(996\) 0 0
\(997\) −1552.00 −0.0493002 −0.0246501 0.999696i \(-0.507847\pi\)
−0.0246501 + 0.999696i \(0.507847\pi\)
\(998\) −33882.0 −1.07467
\(999\) 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 1305.4.a.d.1.1 1
3.2 odd 2 435.4.a.a.1.1 1
15.14 odd 2 2175.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.a.1.1 1 3.2 odd 2
1305.4.a.d.1.1 1 1.1 even 1 trivial
2175.4.a.c.1.1 1 15.14 odd 2