Properties

Label 960.4.a.v.1.1
Level $960$
Weight $4$
Character 960.1
Self dual yes
Analytic conductor $56.642$
Analytic rank $0$
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] = [960,4,Mod(1,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, 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("960.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.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: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -5.00000 q^{5} -8.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -5.00000 q^{5} -8.00000 q^{7} +9.00000 q^{9} +20.0000 q^{11} -22.0000 q^{13} -15.0000 q^{15} -14.0000 q^{17} +76.0000 q^{19} -24.0000 q^{21} -56.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +154.000 q^{29} -160.000 q^{31} +60.0000 q^{33} +40.0000 q^{35} +162.000 q^{37} -66.0000 q^{39} -390.000 q^{41} +388.000 q^{43} -45.0000 q^{45} +544.000 q^{47} -279.000 q^{49} -42.0000 q^{51} +210.000 q^{53} -100.000 q^{55} +228.000 q^{57} -380.000 q^{59} +794.000 q^{61} -72.0000 q^{63} +110.000 q^{65} -148.000 q^{67} -168.000 q^{69} +840.000 q^{71} +858.000 q^{73} +75.0000 q^{75} -160.000 q^{77} -144.000 q^{79} +81.0000 q^{81} +316.000 q^{83} +70.0000 q^{85} +462.000 q^{87} +1098.00 q^{89} +176.000 q^{91} -480.000 q^{93} -380.000 q^{95} +994.000 q^{97} +180.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −8.00000 −0.431959 −0.215980 0.976398i \(-0.569295\pi\)
−0.215980 + 0.976398i \(0.569295\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 20.0000 0.548202 0.274101 0.961701i \(-0.411620\pi\)
0.274101 + 0.961701i \(0.411620\pi\)
\(12\) 0 0
\(13\) −22.0000 −0.469362 −0.234681 0.972072i \(-0.575405\pi\)
−0.234681 + 0.972072i \(0.575405\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) 0 0
\(17\) −14.0000 −0.199735 −0.0998676 0.995001i \(-0.531842\pi\)
−0.0998676 + 0.995001i \(0.531842\pi\)
\(18\) 0 0
\(19\) 76.0000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −24.0000 −0.249392
\(22\) 0 0
\(23\) −56.0000 −0.507687 −0.253844 0.967245i \(-0.581695\pi\)
−0.253844 + 0.967245i \(0.581695\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 154.000 0.986106 0.493053 0.869999i \(-0.335881\pi\)
0.493053 + 0.869999i \(0.335881\pi\)
\(30\) 0 0
\(31\) −160.000 −0.926995 −0.463498 0.886098i \(-0.653406\pi\)
−0.463498 + 0.886098i \(0.653406\pi\)
\(32\) 0 0
\(33\) 60.0000 0.316505
\(34\) 0 0
\(35\) 40.0000 0.193178
\(36\) 0 0
\(37\) 162.000 0.719801 0.359900 0.932991i \(-0.382811\pi\)
0.359900 + 0.932991i \(0.382811\pi\)
\(38\) 0 0
\(39\) −66.0000 −0.270986
\(40\) 0 0
\(41\) −390.000 −1.48556 −0.742778 0.669538i \(-0.766492\pi\)
−0.742778 + 0.669538i \(0.766492\pi\)
\(42\) 0 0
\(43\) 388.000 1.37603 0.688017 0.725695i \(-0.258482\pi\)
0.688017 + 0.725695i \(0.258482\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) 544.000 1.68831 0.844155 0.536099i \(-0.180103\pi\)
0.844155 + 0.536099i \(0.180103\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) 0 0
\(51\) −42.0000 −0.115317
\(52\) 0 0
\(53\) 210.000 0.544259 0.272129 0.962261i \(-0.412272\pi\)
0.272129 + 0.962261i \(0.412272\pi\)
\(54\) 0 0
\(55\) −100.000 −0.245164
\(56\) 0 0
\(57\) 228.000 0.529813
\(58\) 0 0
\(59\) −380.000 −0.838505 −0.419252 0.907870i \(-0.637708\pi\)
−0.419252 + 0.907870i \(0.637708\pi\)
\(60\) 0 0
\(61\) 794.000 1.66658 0.833289 0.552837i \(-0.186455\pi\)
0.833289 + 0.552837i \(0.186455\pi\)
\(62\) 0 0
\(63\) −72.0000 −0.143986
\(64\) 0 0
\(65\) 110.000 0.209905
\(66\) 0 0
\(67\) −148.000 −0.269867 −0.134933 0.990855i \(-0.543082\pi\)
−0.134933 + 0.990855i \(0.543082\pi\)
\(68\) 0 0
\(69\) −168.000 −0.293113
\(70\) 0 0
\(71\) 840.000 1.40408 0.702040 0.712138i \(-0.252273\pi\)
0.702040 + 0.712138i \(0.252273\pi\)
\(72\) 0 0
\(73\) 858.000 1.37563 0.687817 0.725884i \(-0.258569\pi\)
0.687817 + 0.725884i \(0.258569\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) −160.000 −0.236801
\(78\) 0 0
\(79\) −144.000 −0.205079 −0.102540 0.994729i \(-0.532697\pi\)
−0.102540 + 0.994729i \(0.532697\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 316.000 0.417898 0.208949 0.977927i \(-0.432996\pi\)
0.208949 + 0.977927i \(0.432996\pi\)
\(84\) 0 0
\(85\) 70.0000 0.0893243
\(86\) 0 0
\(87\) 462.000 0.569329
\(88\) 0 0
\(89\) 1098.00 1.30773 0.653864 0.756612i \(-0.273147\pi\)
0.653864 + 0.756612i \(0.273147\pi\)
\(90\) 0 0
\(91\) 176.000 0.202745
\(92\) 0 0
\(93\) −480.000 −0.535201
\(94\) 0 0
\(95\) −380.000 −0.410391
\(96\) 0 0
\(97\) 994.000 1.04047 0.520234 0.854024i \(-0.325845\pi\)
0.520234 + 0.854024i \(0.325845\pi\)
\(98\) 0 0
\(99\) 180.000 0.182734
\(100\) 0 0
\(101\) 834.000 0.821645 0.410822 0.911715i \(-0.365242\pi\)
0.410822 + 0.911715i \(0.365242\pi\)
\(102\) 0 0
\(103\) −1672.00 −1.59949 −0.799743 0.600343i \(-0.795031\pi\)
−0.799743 + 0.600343i \(0.795031\pi\)
\(104\) 0 0
\(105\) 120.000 0.111531
\(106\) 0 0
\(107\) −732.000 −0.661356 −0.330678 0.943744i \(-0.607277\pi\)
−0.330678 + 0.943744i \(0.607277\pi\)
\(108\) 0 0
\(109\) 970.000 0.852378 0.426189 0.904634i \(-0.359856\pi\)
0.426189 + 0.904634i \(0.359856\pi\)
\(110\) 0 0
\(111\) 486.000 0.415577
\(112\) 0 0
\(113\) 1938.00 1.61338 0.806689 0.590976i \(-0.201257\pi\)
0.806689 + 0.590976i \(0.201257\pi\)
\(114\) 0 0
\(115\) 280.000 0.227045
\(116\) 0 0
\(117\) −198.000 −0.156454
\(118\) 0 0
\(119\) 112.000 0.0862775
\(120\) 0 0
\(121\) −931.000 −0.699474
\(122\) 0 0
\(123\) −1170.00 −0.857686
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 528.000 0.368917 0.184458 0.982840i \(-0.440947\pi\)
0.184458 + 0.982840i \(0.440947\pi\)
\(128\) 0 0
\(129\) 1164.00 0.794453
\(130\) 0 0
\(131\) 636.000 0.424180 0.212090 0.977250i \(-0.431973\pi\)
0.212090 + 0.977250i \(0.431973\pi\)
\(132\) 0 0
\(133\) −608.000 −0.396393
\(134\) 0 0
\(135\) −135.000 −0.0860663
\(136\) 0 0
\(137\) 1754.00 1.09383 0.546914 0.837189i \(-0.315803\pi\)
0.546914 + 0.837189i \(0.315803\pi\)
\(138\) 0 0
\(139\) −2508.00 −1.53040 −0.765201 0.643792i \(-0.777360\pi\)
−0.765201 + 0.643792i \(0.777360\pi\)
\(140\) 0 0
\(141\) 1632.00 0.974746
\(142\) 0 0
\(143\) −440.000 −0.257305
\(144\) 0 0
\(145\) −770.000 −0.441000
\(146\) 0 0
\(147\) −837.000 −0.469623
\(148\) 0 0
\(149\) −1486.00 −0.817033 −0.408516 0.912751i \(-0.633954\pi\)
−0.408516 + 0.912751i \(0.633954\pi\)
\(150\) 0 0
\(151\) −2120.00 −1.14254 −0.571269 0.820763i \(-0.693549\pi\)
−0.571269 + 0.820763i \(0.693549\pi\)
\(152\) 0 0
\(153\) −126.000 −0.0665784
\(154\) 0 0
\(155\) 800.000 0.414565
\(156\) 0 0
\(157\) 1850.00 0.940421 0.470210 0.882554i \(-0.344178\pi\)
0.470210 + 0.882554i \(0.344178\pi\)
\(158\) 0 0
\(159\) 630.000 0.314228
\(160\) 0 0
\(161\) 448.000 0.219300
\(162\) 0 0
\(163\) −1172.00 −0.563179 −0.281589 0.959535i \(-0.590862\pi\)
−0.281589 + 0.959535i \(0.590862\pi\)
\(164\) 0 0
\(165\) −300.000 −0.141545
\(166\) 0 0
\(167\) 1656.00 0.767336 0.383668 0.923471i \(-0.374661\pi\)
0.383668 + 0.923471i \(0.374661\pi\)
\(168\) 0 0
\(169\) −1713.00 −0.779700
\(170\) 0 0
\(171\) 684.000 0.305888
\(172\) 0 0
\(173\) 2666.00 1.17163 0.585816 0.810444i \(-0.300774\pi\)
0.585816 + 0.810444i \(0.300774\pi\)
\(174\) 0 0
\(175\) −200.000 −0.0863919
\(176\) 0 0
\(177\) −1140.00 −0.484111
\(178\) 0 0
\(179\) 1132.00 0.472680 0.236340 0.971670i \(-0.424052\pi\)
0.236340 + 0.971670i \(0.424052\pi\)
\(180\) 0 0
\(181\) 2866.00 1.17695 0.588475 0.808515i \(-0.299728\pi\)
0.588475 + 0.808515i \(0.299728\pi\)
\(182\) 0 0
\(183\) 2382.00 0.962199
\(184\) 0 0
\(185\) −810.000 −0.321905
\(186\) 0 0
\(187\) −280.000 −0.109495
\(188\) 0 0
\(189\) −216.000 −0.0831306
\(190\) 0 0
\(191\) −1888.00 −0.715240 −0.357620 0.933867i \(-0.616412\pi\)
−0.357620 + 0.933867i \(0.616412\pi\)
\(192\) 0 0
\(193\) 1282.00 0.478137 0.239068 0.971003i \(-0.423158\pi\)
0.239068 + 0.971003i \(0.423158\pi\)
\(194\) 0 0
\(195\) 330.000 0.121189
\(196\) 0 0
\(197\) −350.000 −0.126581 −0.0632905 0.997995i \(-0.520159\pi\)
−0.0632905 + 0.997995i \(0.520159\pi\)
\(198\) 0 0
\(199\) 3400.00 1.21115 0.605577 0.795787i \(-0.292942\pi\)
0.605577 + 0.795787i \(0.292942\pi\)
\(200\) 0 0
\(201\) −444.000 −0.155808
\(202\) 0 0
\(203\) −1232.00 −0.425958
\(204\) 0 0
\(205\) 1950.00 0.664361
\(206\) 0 0
\(207\) −504.000 −0.169229
\(208\) 0 0
\(209\) 1520.00 0.503065
\(210\) 0 0
\(211\) 4652.00 1.51781 0.758903 0.651204i \(-0.225736\pi\)
0.758903 + 0.651204i \(0.225736\pi\)
\(212\) 0 0
\(213\) 2520.00 0.810646
\(214\) 0 0
\(215\) −1940.00 −0.615381
\(216\) 0 0
\(217\) 1280.00 0.400424
\(218\) 0 0
\(219\) 2574.00 0.794223
\(220\) 0 0
\(221\) 308.000 0.0937481
\(222\) 0 0
\(223\) −4016.00 −1.20597 −0.602985 0.797753i \(-0.706022\pi\)
−0.602985 + 0.797753i \(0.706022\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) 2316.00 0.677173 0.338587 0.940935i \(-0.390051\pi\)
0.338587 + 0.940935i \(0.390051\pi\)
\(228\) 0 0
\(229\) −94.0000 −0.0271253 −0.0135627 0.999908i \(-0.504317\pi\)
−0.0135627 + 0.999908i \(0.504317\pi\)
\(230\) 0 0
\(231\) −480.000 −0.136717
\(232\) 0 0
\(233\) −4230.00 −1.18934 −0.594671 0.803969i \(-0.702718\pi\)
−0.594671 + 0.803969i \(0.702718\pi\)
\(234\) 0 0
\(235\) −2720.00 −0.755035
\(236\) 0 0
\(237\) −432.000 −0.118403
\(238\) 0 0
\(239\) −2064.00 −0.558615 −0.279308 0.960202i \(-0.590105\pi\)
−0.279308 + 0.960202i \(0.590105\pi\)
\(240\) 0 0
\(241\) 4562.00 1.21935 0.609677 0.792650i \(-0.291299\pi\)
0.609677 + 0.792650i \(0.291299\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 1395.00 0.363768
\(246\) 0 0
\(247\) −1672.00 −0.430716
\(248\) 0 0
\(249\) 948.000 0.241273
\(250\) 0 0
\(251\) 2532.00 0.636727 0.318363 0.947969i \(-0.396867\pi\)
0.318363 + 0.947969i \(0.396867\pi\)
\(252\) 0 0
\(253\) −1120.00 −0.278315
\(254\) 0 0
\(255\) 210.000 0.0515714
\(256\) 0 0
\(257\) 3522.00 0.854850 0.427425 0.904051i \(-0.359421\pi\)
0.427425 + 0.904051i \(0.359421\pi\)
\(258\) 0 0
\(259\) −1296.00 −0.310925
\(260\) 0 0
\(261\) 1386.00 0.328702
\(262\) 0 0
\(263\) 2232.00 0.523312 0.261656 0.965161i \(-0.415731\pi\)
0.261656 + 0.965161i \(0.415731\pi\)
\(264\) 0 0
\(265\) −1050.00 −0.243400
\(266\) 0 0
\(267\) 3294.00 0.755017
\(268\) 0 0
\(269\) −2806.00 −0.636003 −0.318002 0.948090i \(-0.603012\pi\)
−0.318002 + 0.948090i \(0.603012\pi\)
\(270\) 0 0
\(271\) −4848.00 −1.08670 −0.543349 0.839507i \(-0.682844\pi\)
−0.543349 + 0.839507i \(0.682844\pi\)
\(272\) 0 0
\(273\) 528.000 0.117055
\(274\) 0 0
\(275\) 500.000 0.109640
\(276\) 0 0
\(277\) −7790.00 −1.68973 −0.844866 0.534978i \(-0.820320\pi\)
−0.844866 + 0.534978i \(0.820320\pi\)
\(278\) 0 0
\(279\) −1440.00 −0.308998
\(280\) 0 0
\(281\) −118.000 −0.0250509 −0.0125254 0.999922i \(-0.503987\pi\)
−0.0125254 + 0.999922i \(0.503987\pi\)
\(282\) 0 0
\(283\) −6508.00 −1.36700 −0.683499 0.729951i \(-0.739543\pi\)
−0.683499 + 0.729951i \(0.739543\pi\)
\(284\) 0 0
\(285\) −1140.00 −0.236940
\(286\) 0 0
\(287\) 3120.00 0.641700
\(288\) 0 0
\(289\) −4717.00 −0.960106
\(290\) 0 0
\(291\) 2982.00 0.600715
\(292\) 0 0
\(293\) 8770.00 1.74863 0.874315 0.485358i \(-0.161311\pi\)
0.874315 + 0.485358i \(0.161311\pi\)
\(294\) 0 0
\(295\) 1900.00 0.374991
\(296\) 0 0
\(297\) 540.000 0.105502
\(298\) 0 0
\(299\) 1232.00 0.238289
\(300\) 0 0
\(301\) −3104.00 −0.594391
\(302\) 0 0
\(303\) 2502.00 0.474377
\(304\) 0 0
\(305\) −3970.00 −0.745317
\(306\) 0 0
\(307\) −4292.00 −0.797907 −0.398953 0.916971i \(-0.630626\pi\)
−0.398953 + 0.916971i \(0.630626\pi\)
\(308\) 0 0
\(309\) −5016.00 −0.923464
\(310\) 0 0
\(311\) 9464.00 1.72558 0.862788 0.505566i \(-0.168716\pi\)
0.862788 + 0.505566i \(0.168716\pi\)
\(312\) 0 0
\(313\) 9578.00 1.72965 0.864825 0.502073i \(-0.167429\pi\)
0.864825 + 0.502073i \(0.167429\pi\)
\(314\) 0 0
\(315\) 360.000 0.0643927
\(316\) 0 0
\(317\) 186.000 0.0329552 0.0164776 0.999864i \(-0.494755\pi\)
0.0164776 + 0.999864i \(0.494755\pi\)
\(318\) 0 0
\(319\) 3080.00 0.540586
\(320\) 0 0
\(321\) −2196.00 −0.381834
\(322\) 0 0
\(323\) −1064.00 −0.183290
\(324\) 0 0
\(325\) −550.000 −0.0938723
\(326\) 0 0
\(327\) 2910.00 0.492120
\(328\) 0 0
\(329\) −4352.00 −0.729281
\(330\) 0 0
\(331\) −492.000 −0.0817002 −0.0408501 0.999165i \(-0.513007\pi\)
−0.0408501 + 0.999165i \(0.513007\pi\)
\(332\) 0 0
\(333\) 1458.00 0.239934
\(334\) 0 0
\(335\) 740.000 0.120688
\(336\) 0 0
\(337\) 2290.00 0.370161 0.185080 0.982723i \(-0.440745\pi\)
0.185080 + 0.982723i \(0.440745\pi\)
\(338\) 0 0
\(339\) 5814.00 0.931484
\(340\) 0 0
\(341\) −3200.00 −0.508181
\(342\) 0 0
\(343\) 4976.00 0.783320
\(344\) 0 0
\(345\) 840.000 0.131084
\(346\) 0 0
\(347\) −6092.00 −0.942466 −0.471233 0.882009i \(-0.656191\pi\)
−0.471233 + 0.882009i \(0.656191\pi\)
\(348\) 0 0
\(349\) −5766.00 −0.884375 −0.442188 0.896923i \(-0.645797\pi\)
−0.442188 + 0.896923i \(0.645797\pi\)
\(350\) 0 0
\(351\) −594.000 −0.0903287
\(352\) 0 0
\(353\) −9374.00 −1.41339 −0.706696 0.707517i \(-0.749815\pi\)
−0.706696 + 0.707517i \(0.749815\pi\)
\(354\) 0 0
\(355\) −4200.00 −0.627924
\(356\) 0 0
\(357\) 336.000 0.0498123
\(358\) 0 0
\(359\) 3528.00 0.518665 0.259332 0.965788i \(-0.416497\pi\)
0.259332 + 0.965788i \(0.416497\pi\)
\(360\) 0 0
\(361\) −1083.00 −0.157895
\(362\) 0 0
\(363\) −2793.00 −0.403842
\(364\) 0 0
\(365\) −4290.00 −0.615202
\(366\) 0 0
\(367\) −7616.00 −1.08325 −0.541624 0.840621i \(-0.682190\pi\)
−0.541624 + 0.840621i \(0.682190\pi\)
\(368\) 0 0
\(369\) −3510.00 −0.495185
\(370\) 0 0
\(371\) −1680.00 −0.235098
\(372\) 0 0
\(373\) −3406.00 −0.472804 −0.236402 0.971655i \(-0.575968\pi\)
−0.236402 + 0.971655i \(0.575968\pi\)
\(374\) 0 0
\(375\) −375.000 −0.0516398
\(376\) 0 0
\(377\) −3388.00 −0.462841
\(378\) 0 0
\(379\) −12284.0 −1.66487 −0.832436 0.554121i \(-0.813055\pi\)
−0.832436 + 0.554121i \(0.813055\pi\)
\(380\) 0 0
\(381\) 1584.00 0.212994
\(382\) 0 0
\(383\) −5424.00 −0.723638 −0.361819 0.932248i \(-0.617844\pi\)
−0.361819 + 0.932248i \(0.617844\pi\)
\(384\) 0 0
\(385\) 800.000 0.105901
\(386\) 0 0
\(387\) 3492.00 0.458678
\(388\) 0 0
\(389\) −3486.00 −0.454363 −0.227182 0.973852i \(-0.572951\pi\)
−0.227182 + 0.973852i \(0.572951\pi\)
\(390\) 0 0
\(391\) 784.000 0.101403
\(392\) 0 0
\(393\) 1908.00 0.244900
\(394\) 0 0
\(395\) 720.000 0.0917143
\(396\) 0 0
\(397\) 3626.00 0.458397 0.229199 0.973380i \(-0.426389\pi\)
0.229199 + 0.973380i \(0.426389\pi\)
\(398\) 0 0
\(399\) −1824.00 −0.228858
\(400\) 0 0
\(401\) 5874.00 0.731505 0.365753 0.930712i \(-0.380812\pi\)
0.365753 + 0.930712i \(0.380812\pi\)
\(402\) 0 0
\(403\) 3520.00 0.435096
\(404\) 0 0
\(405\) −405.000 −0.0496904
\(406\) 0 0
\(407\) 3240.00 0.394597
\(408\) 0 0
\(409\) −12662.0 −1.53080 −0.765398 0.643557i \(-0.777458\pi\)
−0.765398 + 0.643557i \(0.777458\pi\)
\(410\) 0 0
\(411\) 5262.00 0.631521
\(412\) 0 0
\(413\) 3040.00 0.362200
\(414\) 0 0
\(415\) −1580.00 −0.186890
\(416\) 0 0
\(417\) −7524.00 −0.883578
\(418\) 0 0
\(419\) 6396.00 0.745740 0.372870 0.927884i \(-0.378374\pi\)
0.372870 + 0.927884i \(0.378374\pi\)
\(420\) 0 0
\(421\) −8286.00 −0.959228 −0.479614 0.877480i \(-0.659223\pi\)
−0.479614 + 0.877480i \(0.659223\pi\)
\(422\) 0 0
\(423\) 4896.00 0.562770
\(424\) 0 0
\(425\) −350.000 −0.0399470
\(426\) 0 0
\(427\) −6352.00 −0.719894
\(428\) 0 0
\(429\) −1320.00 −0.148555
\(430\) 0 0
\(431\) 4112.00 0.459555 0.229777 0.973243i \(-0.426200\pi\)
0.229777 + 0.973243i \(0.426200\pi\)
\(432\) 0 0
\(433\) 5330.00 0.591555 0.295778 0.955257i \(-0.404421\pi\)
0.295778 + 0.955257i \(0.404421\pi\)
\(434\) 0 0
\(435\) −2310.00 −0.254612
\(436\) 0 0
\(437\) −4256.00 −0.465886
\(438\) 0 0
\(439\) −11272.0 −1.22547 −0.612737 0.790287i \(-0.709932\pi\)
−0.612737 + 0.790287i \(0.709932\pi\)
\(440\) 0 0
\(441\) −2511.00 −0.271137
\(442\) 0 0
\(443\) 14196.0 1.52251 0.761255 0.648452i \(-0.224583\pi\)
0.761255 + 0.648452i \(0.224583\pi\)
\(444\) 0 0
\(445\) −5490.00 −0.584834
\(446\) 0 0
\(447\) −4458.00 −0.471714
\(448\) 0 0
\(449\) −5886.00 −0.618658 −0.309329 0.950955i \(-0.600104\pi\)
−0.309329 + 0.950955i \(0.600104\pi\)
\(450\) 0 0
\(451\) −7800.00 −0.814385
\(452\) 0 0
\(453\) −6360.00 −0.659644
\(454\) 0 0
\(455\) −880.000 −0.0906704
\(456\) 0 0
\(457\) −7526.00 −0.770353 −0.385177 0.922843i \(-0.625859\pi\)
−0.385177 + 0.922843i \(0.625859\pi\)
\(458\) 0 0
\(459\) −378.000 −0.0384391
\(460\) 0 0
\(461\) −8502.00 −0.858954 −0.429477 0.903078i \(-0.641302\pi\)
−0.429477 + 0.903078i \(0.641302\pi\)
\(462\) 0 0
\(463\) −12672.0 −1.27196 −0.635980 0.771705i \(-0.719404\pi\)
−0.635980 + 0.771705i \(0.719404\pi\)
\(464\) 0 0
\(465\) 2400.00 0.239349
\(466\) 0 0
\(467\) 16540.0 1.63893 0.819465 0.573130i \(-0.194271\pi\)
0.819465 + 0.573130i \(0.194271\pi\)
\(468\) 0 0
\(469\) 1184.00 0.116572
\(470\) 0 0
\(471\) 5550.00 0.542952
\(472\) 0 0
\(473\) 7760.00 0.754345
\(474\) 0 0
\(475\) 1900.00 0.183533
\(476\) 0 0
\(477\) 1890.00 0.181420
\(478\) 0 0
\(479\) −8864.00 −0.845525 −0.422763 0.906241i \(-0.638940\pi\)
−0.422763 + 0.906241i \(0.638940\pi\)
\(480\) 0 0
\(481\) −3564.00 −0.337847
\(482\) 0 0
\(483\) 1344.00 0.126613
\(484\) 0 0
\(485\) −4970.00 −0.465311
\(486\) 0 0
\(487\) −3688.00 −0.343161 −0.171580 0.985170i \(-0.554887\pi\)
−0.171580 + 0.985170i \(0.554887\pi\)
\(488\) 0 0
\(489\) −3516.00 −0.325151
\(490\) 0 0
\(491\) −16140.0 −1.48348 −0.741739 0.670688i \(-0.765999\pi\)
−0.741739 + 0.670688i \(0.765999\pi\)
\(492\) 0 0
\(493\) −2156.00 −0.196960
\(494\) 0 0
\(495\) −900.000 −0.0817212
\(496\) 0 0
\(497\) −6720.00 −0.606505
\(498\) 0 0
\(499\) 1580.00 0.141745 0.0708723 0.997485i \(-0.477422\pi\)
0.0708723 + 0.997485i \(0.477422\pi\)
\(500\) 0 0
\(501\) 4968.00 0.443022
\(502\) 0 0
\(503\) −15000.0 −1.32966 −0.664828 0.746996i \(-0.731495\pi\)
−0.664828 + 0.746996i \(0.731495\pi\)
\(504\) 0 0
\(505\) −4170.00 −0.367451
\(506\) 0 0
\(507\) −5139.00 −0.450160
\(508\) 0 0
\(509\) −20486.0 −1.78394 −0.891971 0.452094i \(-0.850677\pi\)
−0.891971 + 0.452094i \(0.850677\pi\)
\(510\) 0 0
\(511\) −6864.00 −0.594218
\(512\) 0 0
\(513\) 2052.00 0.176604
\(514\) 0 0
\(515\) 8360.00 0.715312
\(516\) 0 0
\(517\) 10880.0 0.925535
\(518\) 0 0
\(519\) 7998.00 0.676442
\(520\) 0 0
\(521\) 7706.00 0.647996 0.323998 0.946058i \(-0.394973\pi\)
0.323998 + 0.946058i \(0.394973\pi\)
\(522\) 0 0
\(523\) −3932.00 −0.328746 −0.164373 0.986398i \(-0.552560\pi\)
−0.164373 + 0.986398i \(0.552560\pi\)
\(524\) 0 0
\(525\) −600.000 −0.0498784
\(526\) 0 0
\(527\) 2240.00 0.185154
\(528\) 0 0
\(529\) −9031.00 −0.742254
\(530\) 0 0
\(531\) −3420.00 −0.279502
\(532\) 0 0
\(533\) 8580.00 0.697263
\(534\) 0 0
\(535\) 3660.00 0.295767
\(536\) 0 0
\(537\) 3396.00 0.272902
\(538\) 0 0
\(539\) −5580.00 −0.445914
\(540\) 0 0
\(541\) 23930.0 1.90172 0.950860 0.309620i \(-0.100202\pi\)
0.950860 + 0.309620i \(0.100202\pi\)
\(542\) 0 0
\(543\) 8598.00 0.679513
\(544\) 0 0
\(545\) −4850.00 −0.381195
\(546\) 0 0
\(547\) 11468.0 0.896410 0.448205 0.893931i \(-0.352063\pi\)
0.448205 + 0.893931i \(0.352063\pi\)
\(548\) 0 0
\(549\) 7146.00 0.555526
\(550\) 0 0
\(551\) 11704.0 0.904913
\(552\) 0 0
\(553\) 1152.00 0.0885859
\(554\) 0 0
\(555\) −2430.00 −0.185852
\(556\) 0 0
\(557\) 11498.0 0.874660 0.437330 0.899301i \(-0.355924\pi\)
0.437330 + 0.899301i \(0.355924\pi\)
\(558\) 0 0
\(559\) −8536.00 −0.645857
\(560\) 0 0
\(561\) −840.000 −0.0632172
\(562\) 0 0
\(563\) 16988.0 1.27169 0.635843 0.771819i \(-0.280653\pi\)
0.635843 + 0.771819i \(0.280653\pi\)
\(564\) 0 0
\(565\) −9690.00 −0.721525
\(566\) 0 0
\(567\) −648.000 −0.0479955
\(568\) 0 0
\(569\) −17366.0 −1.27947 −0.639737 0.768594i \(-0.720957\pi\)
−0.639737 + 0.768594i \(0.720957\pi\)
\(570\) 0 0
\(571\) −24860.0 −1.82199 −0.910997 0.412413i \(-0.864686\pi\)
−0.910997 + 0.412413i \(0.864686\pi\)
\(572\) 0 0
\(573\) −5664.00 −0.412944
\(574\) 0 0
\(575\) −1400.00 −0.101537
\(576\) 0 0
\(577\) −26302.0 −1.89769 −0.948845 0.315744i \(-0.897746\pi\)
−0.948845 + 0.315744i \(0.897746\pi\)
\(578\) 0 0
\(579\) 3846.00 0.276052
\(580\) 0 0
\(581\) −2528.00 −0.180515
\(582\) 0 0
\(583\) 4200.00 0.298364
\(584\) 0 0
\(585\) 990.000 0.0699683
\(586\) 0 0
\(587\) 7812.00 0.549294 0.274647 0.961545i \(-0.411439\pi\)
0.274647 + 0.961545i \(0.411439\pi\)
\(588\) 0 0
\(589\) −12160.0 −0.850669
\(590\) 0 0
\(591\) −1050.00 −0.0730816
\(592\) 0 0
\(593\) 7986.00 0.553028 0.276514 0.961010i \(-0.410821\pi\)
0.276514 + 0.961010i \(0.410821\pi\)
\(594\) 0 0
\(595\) −560.000 −0.0385845
\(596\) 0 0
\(597\) 10200.0 0.699260
\(598\) 0 0
\(599\) 21048.0 1.43572 0.717861 0.696186i \(-0.245121\pi\)
0.717861 + 0.696186i \(0.245121\pi\)
\(600\) 0 0
\(601\) 1738.00 0.117961 0.0589804 0.998259i \(-0.481215\pi\)
0.0589804 + 0.998259i \(0.481215\pi\)
\(602\) 0 0
\(603\) −1332.00 −0.0899556
\(604\) 0 0
\(605\) 4655.00 0.312814
\(606\) 0 0
\(607\) −18576.0 −1.24214 −0.621068 0.783757i \(-0.713301\pi\)
−0.621068 + 0.783757i \(0.713301\pi\)
\(608\) 0 0
\(609\) −3696.00 −0.245927
\(610\) 0 0
\(611\) −11968.0 −0.792428
\(612\) 0 0
\(613\) 13602.0 0.896215 0.448107 0.893980i \(-0.352098\pi\)
0.448107 + 0.893980i \(0.352098\pi\)
\(614\) 0 0
\(615\) 5850.00 0.383569
\(616\) 0 0
\(617\) 19578.0 1.27744 0.638720 0.769439i \(-0.279464\pi\)
0.638720 + 0.769439i \(0.279464\pi\)
\(618\) 0 0
\(619\) 12308.0 0.799193 0.399596 0.916691i \(-0.369150\pi\)
0.399596 + 0.916691i \(0.369150\pi\)
\(620\) 0 0
\(621\) −1512.00 −0.0977045
\(622\) 0 0
\(623\) −8784.00 −0.564885
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 4560.00 0.290445
\(628\) 0 0
\(629\) −2268.00 −0.143770
\(630\) 0 0
\(631\) 8600.00 0.542568 0.271284 0.962499i \(-0.412552\pi\)
0.271284 + 0.962499i \(0.412552\pi\)
\(632\) 0 0
\(633\) 13956.0 0.876305
\(634\) 0 0
\(635\) −2640.00 −0.164985
\(636\) 0 0
\(637\) 6138.00 0.381784
\(638\) 0 0
\(639\) 7560.00 0.468027
\(640\) 0 0
\(641\) 6978.00 0.429976 0.214988 0.976617i \(-0.431029\pi\)
0.214988 + 0.976617i \(0.431029\pi\)
\(642\) 0 0
\(643\) −7668.00 −0.470290 −0.235145 0.971960i \(-0.575556\pi\)
−0.235145 + 0.971960i \(0.575556\pi\)
\(644\) 0 0
\(645\) −5820.00 −0.355290
\(646\) 0 0
\(647\) 15384.0 0.934787 0.467394 0.884049i \(-0.345193\pi\)
0.467394 + 0.884049i \(0.345193\pi\)
\(648\) 0 0
\(649\) −7600.00 −0.459670
\(650\) 0 0
\(651\) 3840.00 0.231185
\(652\) 0 0
\(653\) 2186.00 0.131003 0.0655014 0.997852i \(-0.479135\pi\)
0.0655014 + 0.997852i \(0.479135\pi\)
\(654\) 0 0
\(655\) −3180.00 −0.189699
\(656\) 0 0
\(657\) 7722.00 0.458545
\(658\) 0 0
\(659\) −1524.00 −0.0900859 −0.0450430 0.998985i \(-0.514342\pi\)
−0.0450430 + 0.998985i \(0.514342\pi\)
\(660\) 0 0
\(661\) 4242.00 0.249614 0.124807 0.992181i \(-0.460169\pi\)
0.124807 + 0.992181i \(0.460169\pi\)
\(662\) 0 0
\(663\) 924.000 0.0541255
\(664\) 0 0
\(665\) 3040.00 0.177272
\(666\) 0 0
\(667\) −8624.00 −0.500634
\(668\) 0 0
\(669\) −12048.0 −0.696267
\(670\) 0 0
\(671\) 15880.0 0.913622
\(672\) 0 0
\(673\) 24354.0 1.39491 0.697457 0.716626i \(-0.254315\pi\)
0.697457 + 0.716626i \(0.254315\pi\)
\(674\) 0 0
\(675\) 675.000 0.0384900
\(676\) 0 0
\(677\) 322.000 0.0182799 0.00913993 0.999958i \(-0.497091\pi\)
0.00913993 + 0.999958i \(0.497091\pi\)
\(678\) 0 0
\(679\) −7952.00 −0.449440
\(680\) 0 0
\(681\) 6948.00 0.390966
\(682\) 0 0
\(683\) −7932.00 −0.444377 −0.222189 0.975004i \(-0.571320\pi\)
−0.222189 + 0.975004i \(0.571320\pi\)
\(684\) 0 0
\(685\) −8770.00 −0.489174
\(686\) 0 0
\(687\) −282.000 −0.0156608
\(688\) 0 0
\(689\) −4620.00 −0.255454
\(690\) 0 0
\(691\) 20684.0 1.13872 0.569361 0.822088i \(-0.307191\pi\)
0.569361 + 0.822088i \(0.307191\pi\)
\(692\) 0 0
\(693\) −1440.00 −0.0789337
\(694\) 0 0
\(695\) 12540.0 0.684416
\(696\) 0 0
\(697\) 5460.00 0.296718
\(698\) 0 0
\(699\) −12690.0 −0.686666
\(700\) 0 0
\(701\) −25222.0 −1.35895 −0.679473 0.733700i \(-0.737792\pi\)
−0.679473 + 0.733700i \(0.737792\pi\)
\(702\) 0 0
\(703\) 12312.0 0.660535
\(704\) 0 0
\(705\) −8160.00 −0.435920
\(706\) 0 0
\(707\) −6672.00 −0.354917
\(708\) 0 0
\(709\) −23678.0 −1.25423 −0.627113 0.778928i \(-0.715764\pi\)
−0.627113 + 0.778928i \(0.715764\pi\)
\(710\) 0 0
\(711\) −1296.00 −0.0683598
\(712\) 0 0
\(713\) 8960.00 0.470624
\(714\) 0 0
\(715\) 2200.00 0.115070
\(716\) 0 0
\(717\) −6192.00 −0.322517
\(718\) 0 0
\(719\) −8432.00 −0.437358 −0.218679 0.975797i \(-0.570175\pi\)
−0.218679 + 0.975797i \(0.570175\pi\)
\(720\) 0 0
\(721\) 13376.0 0.690913
\(722\) 0 0
\(723\) 13686.0 0.703994
\(724\) 0 0
\(725\) 3850.00 0.197221
\(726\) 0 0
\(727\) −8312.00 −0.424037 −0.212019 0.977266i \(-0.568004\pi\)
−0.212019 + 0.977266i \(0.568004\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −5432.00 −0.274842
\(732\) 0 0
\(733\) 26298.0 1.32516 0.662578 0.748993i \(-0.269462\pi\)
0.662578 + 0.748993i \(0.269462\pi\)
\(734\) 0 0
\(735\) 4185.00 0.210022
\(736\) 0 0
\(737\) −2960.00 −0.147942
\(738\) 0 0
\(739\) 16956.0 0.844028 0.422014 0.906589i \(-0.361323\pi\)
0.422014 + 0.906589i \(0.361323\pi\)
\(740\) 0 0
\(741\) −5016.00 −0.248674
\(742\) 0 0
\(743\) 17880.0 0.882845 0.441422 0.897299i \(-0.354474\pi\)
0.441422 + 0.897299i \(0.354474\pi\)
\(744\) 0 0
\(745\) 7430.00 0.365388
\(746\) 0 0
\(747\) 2844.00 0.139299
\(748\) 0 0
\(749\) 5856.00 0.285679
\(750\) 0 0
\(751\) −22032.0 −1.07052 −0.535259 0.844688i \(-0.679786\pi\)
−0.535259 + 0.844688i \(0.679786\pi\)
\(752\) 0 0
\(753\) 7596.00 0.367614
\(754\) 0 0
\(755\) 10600.0 0.510958
\(756\) 0 0
\(757\) −11534.0 −0.553779 −0.276889 0.960902i \(-0.589304\pi\)
−0.276889 + 0.960902i \(0.589304\pi\)
\(758\) 0 0
\(759\) −3360.00 −0.160685
\(760\) 0 0
\(761\) 38250.0 1.82203 0.911013 0.412378i \(-0.135302\pi\)
0.911013 + 0.412378i \(0.135302\pi\)
\(762\) 0 0
\(763\) −7760.00 −0.368192
\(764\) 0 0
\(765\) 630.000 0.0297748
\(766\) 0 0
\(767\) 8360.00 0.393562
\(768\) 0 0
\(769\) 19330.0 0.906447 0.453223 0.891397i \(-0.350274\pi\)
0.453223 + 0.891397i \(0.350274\pi\)
\(770\) 0 0
\(771\) 10566.0 0.493548
\(772\) 0 0
\(773\) 40674.0 1.89255 0.946276 0.323361i \(-0.104813\pi\)
0.946276 + 0.323361i \(0.104813\pi\)
\(774\) 0 0
\(775\) −4000.00 −0.185399
\(776\) 0 0
\(777\) −3888.00 −0.179513
\(778\) 0 0
\(779\) −29640.0 −1.36324
\(780\) 0 0
\(781\) 16800.0 0.769720
\(782\) 0 0
\(783\) 4158.00 0.189776
\(784\) 0 0
\(785\) −9250.00 −0.420569
\(786\) 0 0
\(787\) 7004.00 0.317237 0.158619 0.987340i \(-0.449296\pi\)
0.158619 + 0.987340i \(0.449296\pi\)
\(788\) 0 0
\(789\) 6696.00 0.302134
\(790\) 0 0
\(791\) −15504.0 −0.696914
\(792\) 0 0
\(793\) −17468.0 −0.782228
\(794\) 0 0
\(795\) −3150.00 −0.140527
\(796\) 0 0
\(797\) −12198.0 −0.542127 −0.271064 0.962561i \(-0.587375\pi\)
−0.271064 + 0.962561i \(0.587375\pi\)
\(798\) 0 0
\(799\) −7616.00 −0.337215
\(800\) 0 0
\(801\) 9882.00 0.435909
\(802\) 0 0
\(803\) 17160.0 0.754126
\(804\) 0 0
\(805\) −2240.00 −0.0980741
\(806\) 0 0
\(807\) −8418.00 −0.367197
\(808\) 0 0
\(809\) −25734.0 −1.11837 −0.559184 0.829044i \(-0.688885\pi\)
−0.559184 + 0.829044i \(0.688885\pi\)
\(810\) 0 0
\(811\) 15668.0 0.678394 0.339197 0.940715i \(-0.389845\pi\)
0.339197 + 0.940715i \(0.389845\pi\)
\(812\) 0 0
\(813\) −14544.0 −0.627405
\(814\) 0 0
\(815\) 5860.00 0.251861
\(816\) 0 0
\(817\) 29488.0 1.26274
\(818\) 0 0
\(819\) 1584.00 0.0675817
\(820\) 0 0
\(821\) 34450.0 1.46445 0.732225 0.681063i \(-0.238482\pi\)
0.732225 + 0.681063i \(0.238482\pi\)
\(822\) 0 0
\(823\) 38792.0 1.64302 0.821509 0.570195i \(-0.193132\pi\)
0.821509 + 0.570195i \(0.193132\pi\)
\(824\) 0 0
\(825\) 1500.00 0.0633010
\(826\) 0 0
\(827\) −20460.0 −0.860295 −0.430147 0.902759i \(-0.641538\pi\)
−0.430147 + 0.902759i \(0.641538\pi\)
\(828\) 0 0
\(829\) −5542.00 −0.232185 −0.116093 0.993238i \(-0.537037\pi\)
−0.116093 + 0.993238i \(0.537037\pi\)
\(830\) 0 0
\(831\) −23370.0 −0.975567
\(832\) 0 0
\(833\) 3906.00 0.162467
\(834\) 0 0
\(835\) −8280.00 −0.343163
\(836\) 0 0
\(837\) −4320.00 −0.178400
\(838\) 0 0
\(839\) −25240.0 −1.03860 −0.519298 0.854593i \(-0.673806\pi\)
−0.519298 + 0.854593i \(0.673806\pi\)
\(840\) 0 0
\(841\) −673.000 −0.0275944
\(842\) 0 0
\(843\) −354.000 −0.0144631
\(844\) 0 0
\(845\) 8565.00 0.348692
\(846\) 0 0
\(847\) 7448.00 0.302144
\(848\) 0 0
\(849\) −19524.0 −0.789237
\(850\) 0 0
\(851\) −9072.00 −0.365434
\(852\) 0 0
\(853\) 37330.0 1.49842 0.749212 0.662331i \(-0.230433\pi\)
0.749212 + 0.662331i \(0.230433\pi\)
\(854\) 0 0
\(855\) −3420.00 −0.136797
\(856\) 0 0
\(857\) −3894.00 −0.155212 −0.0776059 0.996984i \(-0.524728\pi\)
−0.0776059 + 0.996984i \(0.524728\pi\)
\(858\) 0 0
\(859\) 20324.0 0.807271 0.403636 0.914920i \(-0.367746\pi\)
0.403636 + 0.914920i \(0.367746\pi\)
\(860\) 0 0
\(861\) 9360.00 0.370485
\(862\) 0 0
\(863\) −6288.00 −0.248026 −0.124013 0.992281i \(-0.539576\pi\)
−0.124013 + 0.992281i \(0.539576\pi\)
\(864\) 0 0
\(865\) −13330.0 −0.523969
\(866\) 0 0
\(867\) −14151.0 −0.554317
\(868\) 0 0
\(869\) −2880.00 −0.112425
\(870\) 0 0
\(871\) 3256.00 0.126665
\(872\) 0 0
\(873\) 8946.00 0.346823
\(874\) 0 0
\(875\) 1000.00 0.0386356
\(876\) 0 0
\(877\) 24650.0 0.949112 0.474556 0.880225i \(-0.342609\pi\)
0.474556 + 0.880225i \(0.342609\pi\)
\(878\) 0 0
\(879\) 26310.0 1.00957
\(880\) 0 0
\(881\) 9426.00 0.360465 0.180233 0.983624i \(-0.442315\pi\)
0.180233 + 0.983624i \(0.442315\pi\)
\(882\) 0 0
\(883\) −9316.00 −0.355049 −0.177525 0.984116i \(-0.556809\pi\)
−0.177525 + 0.984116i \(0.556809\pi\)
\(884\) 0 0
\(885\) 5700.00 0.216501
\(886\) 0 0
\(887\) −6968.00 −0.263768 −0.131884 0.991265i \(-0.542103\pi\)
−0.131884 + 0.991265i \(0.542103\pi\)
\(888\) 0 0
\(889\) −4224.00 −0.159357
\(890\) 0 0
\(891\) 1620.00 0.0609114
\(892\) 0 0
\(893\) 41344.0 1.54930
\(894\) 0 0
\(895\) −5660.00 −0.211389
\(896\) 0 0
\(897\) 3696.00 0.137576
\(898\) 0 0
\(899\) −24640.0 −0.914116
\(900\) 0 0
\(901\) −2940.00 −0.108708
\(902\) 0 0
\(903\) −9312.00 −0.343172
\(904\) 0 0
\(905\) −14330.0 −0.526348
\(906\) 0 0
\(907\) −35964.0 −1.31661 −0.658305 0.752751i \(-0.728726\pi\)
−0.658305 + 0.752751i \(0.728726\pi\)
\(908\) 0 0
\(909\) 7506.00 0.273882
\(910\) 0 0
\(911\) −47888.0 −1.74160 −0.870801 0.491635i \(-0.836399\pi\)
−0.870801 + 0.491635i \(0.836399\pi\)
\(912\) 0 0
\(913\) 6320.00 0.229093
\(914\) 0 0
\(915\) −11910.0 −0.430309
\(916\) 0 0
\(917\) −5088.00 −0.183229
\(918\) 0 0
\(919\) 12760.0 0.458013 0.229006 0.973425i \(-0.426452\pi\)
0.229006 + 0.973425i \(0.426452\pi\)
\(920\) 0 0
\(921\) −12876.0 −0.460672
\(922\) 0 0
\(923\) −18480.0 −0.659021
\(924\) 0 0
\(925\) 4050.00 0.143960
\(926\) 0 0
\(927\) −15048.0 −0.533162
\(928\) 0 0
\(929\) −25054.0 −0.884817 −0.442409 0.896814i \(-0.645876\pi\)
−0.442409 + 0.896814i \(0.645876\pi\)
\(930\) 0 0
\(931\) −21204.0 −0.746437
\(932\) 0 0
\(933\) 28392.0 0.996262
\(934\) 0 0
\(935\) 1400.00 0.0489678
\(936\) 0 0
\(937\) 28282.0 0.986054 0.493027 0.870014i \(-0.335890\pi\)
0.493027 + 0.870014i \(0.335890\pi\)
\(938\) 0 0
\(939\) 28734.0 0.998614
\(940\) 0 0
\(941\) 30634.0 1.06125 0.530627 0.847605i \(-0.321957\pi\)
0.530627 + 0.847605i \(0.321957\pi\)
\(942\) 0 0
\(943\) 21840.0 0.754198
\(944\) 0 0
\(945\) 1080.00 0.0371771
\(946\) 0 0
\(947\) 48572.0 1.66671 0.833357 0.552735i \(-0.186416\pi\)
0.833357 + 0.552735i \(0.186416\pi\)
\(948\) 0 0
\(949\) −18876.0 −0.645670
\(950\) 0 0
\(951\) 558.000 0.0190267
\(952\) 0 0
\(953\) 12906.0 0.438685 0.219342 0.975648i \(-0.429609\pi\)
0.219342 + 0.975648i \(0.429609\pi\)
\(954\) 0 0
\(955\) 9440.00 0.319865
\(956\) 0 0
\(957\) 9240.00 0.312107
\(958\) 0 0
\(959\) −14032.0 −0.472489
\(960\) 0 0
\(961\) −4191.00 −0.140680
\(962\) 0 0
\(963\) −6588.00 −0.220452
\(964\) 0 0
\(965\) −6410.00 −0.213829
\(966\) 0 0
\(967\) 5880.00 0.195541 0.0977705 0.995209i \(-0.468829\pi\)
0.0977705 + 0.995209i \(0.468829\pi\)
\(968\) 0 0
\(969\) −3192.00 −0.105822
\(970\) 0 0
\(971\) 55444.0 1.83242 0.916211 0.400695i \(-0.131231\pi\)
0.916211 + 0.400695i \(0.131231\pi\)
\(972\) 0 0
\(973\) 20064.0 0.661071
\(974\) 0 0
\(975\) −1650.00 −0.0541972
\(976\) 0 0
\(977\) 32050.0 1.04951 0.524755 0.851254i \(-0.324157\pi\)
0.524755 + 0.851254i \(0.324157\pi\)
\(978\) 0 0
\(979\) 21960.0 0.716900
\(980\) 0 0
\(981\) 8730.00 0.284126
\(982\) 0 0
\(983\) −29880.0 −0.969506 −0.484753 0.874651i \(-0.661090\pi\)
−0.484753 + 0.874651i \(0.661090\pi\)
\(984\) 0 0
\(985\) 1750.00 0.0566088
\(986\) 0 0
\(987\) −13056.0 −0.421051
\(988\) 0 0
\(989\) −21728.0 −0.698595
\(990\) 0 0
\(991\) −5216.00 −0.167196 −0.0835982 0.996500i \(-0.526641\pi\)
−0.0835982 + 0.996500i \(0.526641\pi\)
\(992\) 0 0
\(993\) −1476.00 −0.0471696
\(994\) 0 0
\(995\) −17000.0 −0.541644
\(996\) 0 0
\(997\) −6750.00 −0.214418 −0.107209 0.994237i \(-0.534191\pi\)
−0.107209 + 0.994237i \(0.534191\pi\)
\(998\) 0 0
\(999\) 4374.00 0.138526
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 960.4.a.v.1.1 1
4.3 odd 2 960.4.a.g.1.1 1
8.3 odd 2 120.4.a.f.1.1 1
8.5 even 2 240.4.a.d.1.1 1
24.5 odd 2 720.4.a.f.1.1 1
24.11 even 2 360.4.a.e.1.1 1
40.3 even 4 600.4.f.g.49.2 2
40.13 odd 4 1200.4.f.g.49.1 2
40.19 odd 2 600.4.a.d.1.1 1
40.27 even 4 600.4.f.g.49.1 2
40.29 even 2 1200.4.a.bf.1.1 1
40.37 odd 4 1200.4.f.g.49.2 2
120.59 even 2 1800.4.a.k.1.1 1
120.83 odd 4 1800.4.f.h.649.1 2
120.107 odd 4 1800.4.f.h.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.f.1.1 1 8.3 odd 2
240.4.a.d.1.1 1 8.5 even 2
360.4.a.e.1.1 1 24.11 even 2
600.4.a.d.1.1 1 40.19 odd 2
600.4.f.g.49.1 2 40.27 even 4
600.4.f.g.49.2 2 40.3 even 4
720.4.a.f.1.1 1 24.5 odd 2
960.4.a.g.1.1 1 4.3 odd 2
960.4.a.v.1.1 1 1.1 even 1 trivial
1200.4.a.bf.1.1 1 40.29 even 2
1200.4.f.g.49.1 2 40.13 odd 4
1200.4.f.g.49.2 2 40.37 odd 4
1800.4.a.k.1.1 1 120.59 even 2
1800.4.f.h.649.1 2 120.83 odd 4
1800.4.f.h.649.2 2 120.107 odd 4