Properties

Label 960.4.a.p.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 480)
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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +5.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +5.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} +20.0000 q^{11} +58.0000 q^{13} -15.0000 q^{15} -70.0000 q^{17} +92.0000 q^{19} -36.0000 q^{21} +112.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -66.0000 q^{29} -108.000 q^{31} -60.0000 q^{33} +60.0000 q^{35} +58.0000 q^{37} -174.000 q^{39} +66.0000 q^{41} +388.000 q^{43} +45.0000 q^{45} -408.000 q^{47} -199.000 q^{49} +210.000 q^{51} -474.000 q^{53} +100.000 q^{55} -276.000 q^{57} +540.000 q^{59} -14.0000 q^{61} +108.000 q^{63} +290.000 q^{65} +276.000 q^{67} -336.000 q^{69} -96.0000 q^{71} -790.000 q^{73} -75.0000 q^{75} +240.000 q^{77} +308.000 q^{79} +81.0000 q^{81} +1036.00 q^{83} -350.000 q^{85} +198.000 q^{87} +1210.00 q^{89} +696.000 q^{91} +324.000 q^{93} +460.000 q^{95} +1426.00 q^{97} +180.000 q^{99} +O(q^{100})\)

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
\(7\) 12.0000 0.647939 0.323970 0.946068i \(-0.394982\pi\)
0.323970 + 0.946068i \(0.394982\pi\)
\(8\) 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
\(13\) 58.0000 1.23741 0.618704 0.785624i \(-0.287658\pi\)
0.618704 + 0.785624i \(0.287658\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) 0 0
\(17\) −70.0000 −0.998676 −0.499338 0.866407i \(-0.666423\pi\)
−0.499338 + 0.866407i \(0.666423\pi\)
\(18\) 0 0
\(19\) 92.0000 1.11086 0.555428 0.831565i \(-0.312555\pi\)
0.555428 + 0.831565i \(0.312555\pi\)
\(20\) 0 0
\(21\) −36.0000 −0.374088
\(22\) 0 0
\(23\) 112.000 1.01537 0.507687 0.861541i \(-0.330501\pi\)
0.507687 + 0.861541i \(0.330501\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −66.0000 −0.422617 −0.211308 0.977419i \(-0.567772\pi\)
−0.211308 + 0.977419i \(0.567772\pi\)
\(30\) 0 0
\(31\) −108.000 −0.625722 −0.312861 0.949799i \(-0.601287\pi\)
−0.312861 + 0.949799i \(0.601287\pi\)
\(32\) 0 0
\(33\) −60.0000 −0.316505
\(34\) 0 0
\(35\) 60.0000 0.289767
\(36\) 0 0
\(37\) 58.0000 0.257707 0.128853 0.991664i \(-0.458870\pi\)
0.128853 + 0.991664i \(0.458870\pi\)
\(38\) 0 0
\(39\) −174.000 −0.714418
\(40\) 0 0
\(41\) 66.0000 0.251402 0.125701 0.992068i \(-0.459882\pi\)
0.125701 + 0.992068i \(0.459882\pi\)
\(42\) 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
\(47\) −408.000 −1.26623 −0.633116 0.774057i \(-0.718224\pi\)
−0.633116 + 0.774057i \(0.718224\pi\)
\(48\) 0 0
\(49\) −199.000 −0.580175
\(50\) 0 0
\(51\) 210.000 0.576586
\(52\) 0 0
\(53\) −474.000 −1.22847 −0.614235 0.789123i \(-0.710535\pi\)
−0.614235 + 0.789123i \(0.710535\pi\)
\(54\) 0 0
\(55\) 100.000 0.245164
\(56\) 0 0
\(57\) −276.000 −0.641353
\(58\) 0 0
\(59\) 540.000 1.19156 0.595780 0.803148i \(-0.296843\pi\)
0.595780 + 0.803148i \(0.296843\pi\)
\(60\) 0 0
\(61\) −14.0000 −0.0293855 −0.0146928 0.999892i \(-0.504677\pi\)
−0.0146928 + 0.999892i \(0.504677\pi\)
\(62\) 0 0
\(63\) 108.000 0.215980
\(64\) 0 0
\(65\) 290.000 0.553386
\(66\) 0 0
\(67\) 276.000 0.503265 0.251633 0.967823i \(-0.419033\pi\)
0.251633 + 0.967823i \(0.419033\pi\)
\(68\) 0 0
\(69\) −336.000 −0.586227
\(70\) 0 0
\(71\) −96.0000 −0.160466 −0.0802331 0.996776i \(-0.525566\pi\)
−0.0802331 + 0.996776i \(0.525566\pi\)
\(72\) 0 0
\(73\) −790.000 −1.26661 −0.633305 0.773902i \(-0.718302\pi\)
−0.633305 + 0.773902i \(0.718302\pi\)
\(74\) 0 0
\(75\) −75.0000 −0.115470
\(76\) 0 0
\(77\) 240.000 0.355202
\(78\) 0 0
\(79\) 308.000 0.438642 0.219321 0.975653i \(-0.429616\pi\)
0.219321 + 0.975653i \(0.429616\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1036.00 1.37007 0.685035 0.728510i \(-0.259787\pi\)
0.685035 + 0.728510i \(0.259787\pi\)
\(84\) 0 0
\(85\) −350.000 −0.446622
\(86\) 0 0
\(87\) 198.000 0.243998
\(88\) 0 0
\(89\) 1210.00 1.44112 0.720560 0.693392i \(-0.243885\pi\)
0.720560 + 0.693392i \(0.243885\pi\)
\(90\) 0 0
\(91\) 696.000 0.801765
\(92\) 0 0
\(93\) 324.000 0.361261
\(94\) 0 0
\(95\) 460.000 0.496790
\(96\) 0 0
\(97\) 1426.00 1.49266 0.746332 0.665574i \(-0.231813\pi\)
0.746332 + 0.665574i \(0.231813\pi\)
\(98\) 0 0
\(99\) 180.000 0.182734
\(100\) 0 0
\(101\) −74.0000 −0.0729037 −0.0364519 0.999335i \(-0.511606\pi\)
−0.0364519 + 0.999335i \(0.511606\pi\)
\(102\) 0 0
\(103\) −1436.00 −1.37372 −0.686861 0.726789i \(-0.741012\pi\)
−0.686861 + 0.726789i \(0.741012\pi\)
\(104\) 0 0
\(105\) −180.000 −0.167297
\(106\) 0 0
\(107\) −84.0000 −0.0758933 −0.0379467 0.999280i \(-0.512082\pi\)
−0.0379467 + 0.999280i \(0.512082\pi\)
\(108\) 0 0
\(109\) 250.000 0.219685 0.109842 0.993949i \(-0.464965\pi\)
0.109842 + 0.993949i \(0.464965\pi\)
\(110\) 0 0
\(111\) −174.000 −0.148787
\(112\) 0 0
\(113\) −654.000 −0.544453 −0.272226 0.962233i \(-0.587760\pi\)
−0.272226 + 0.962233i \(0.587760\pi\)
\(114\) 0 0
\(115\) 560.000 0.454089
\(116\) 0 0
\(117\) 522.000 0.412469
\(118\) 0 0
\(119\) −840.000 −0.647081
\(120\) 0 0
\(121\) −931.000 −0.699474
\(122\) 0 0
\(123\) −198.000 −0.145147
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 2572.00 1.79707 0.898536 0.438900i \(-0.144632\pi\)
0.898536 + 0.438900i \(0.144632\pi\)
\(128\) 0 0
\(129\) −1164.00 −0.794453
\(130\) 0 0
\(131\) 836.000 0.557570 0.278785 0.960354i \(-0.410068\pi\)
0.278785 + 0.960354i \(0.410068\pi\)
\(132\) 0 0
\(133\) 1104.00 0.719766
\(134\) 0 0
\(135\) −135.000 −0.0860663
\(136\) 0 0
\(137\) 1250.00 0.779523 0.389762 0.920916i \(-0.372557\pi\)
0.389762 + 0.920916i \(0.372557\pi\)
\(138\) 0 0
\(139\) 2428.00 1.48158 0.740792 0.671734i \(-0.234450\pi\)
0.740792 + 0.671734i \(0.234450\pi\)
\(140\) 0 0
\(141\) 1224.00 0.731060
\(142\) 0 0
\(143\) 1160.00 0.678350
\(144\) 0 0
\(145\) −330.000 −0.189000
\(146\) 0 0
\(147\) 597.000 0.334964
\(148\) 0 0
\(149\) −1746.00 −0.959986 −0.479993 0.877272i \(-0.659361\pi\)
−0.479993 + 0.877272i \(0.659361\pi\)
\(150\) 0 0
\(151\) 2092.00 1.12745 0.563724 0.825963i \(-0.309368\pi\)
0.563724 + 0.825963i \(0.309368\pi\)
\(152\) 0 0
\(153\) −630.000 −0.332892
\(154\) 0 0
\(155\) −540.000 −0.279831
\(156\) 0 0
\(157\) 2162.00 1.09902 0.549511 0.835487i \(-0.314814\pi\)
0.549511 + 0.835487i \(0.314814\pi\)
\(158\) 0 0
\(159\) 1422.00 0.709257
\(160\) 0 0
\(161\) 1344.00 0.657901
\(162\) 0 0
\(163\) −932.000 −0.447852 −0.223926 0.974606i \(-0.571887\pi\)
−0.223926 + 0.974606i \(0.571887\pi\)
\(164\) 0 0
\(165\) −300.000 −0.141545
\(166\) 0 0
\(167\) 3192.00 1.47907 0.739534 0.673119i \(-0.235046\pi\)
0.739534 + 0.673119i \(0.235046\pi\)
\(168\) 0 0
\(169\) 1167.00 0.531179
\(170\) 0 0
\(171\) 828.000 0.370285
\(172\) 0 0
\(173\) −2282.00 −1.00287 −0.501437 0.865194i \(-0.667195\pi\)
−0.501437 + 0.865194i \(0.667195\pi\)
\(174\) 0 0
\(175\) 300.000 0.129588
\(176\) 0 0
\(177\) −1620.00 −0.687947
\(178\) 0 0
\(179\) 2004.00 0.836793 0.418397 0.908264i \(-0.362592\pi\)
0.418397 + 0.908264i \(0.362592\pi\)
\(180\) 0 0
\(181\) 4226.00 1.73545 0.867724 0.497046i \(-0.165582\pi\)
0.867724 + 0.497046i \(0.165582\pi\)
\(182\) 0 0
\(183\) 42.0000 0.0169657
\(184\) 0 0
\(185\) 290.000 0.115250
\(186\) 0 0
\(187\) −1400.00 −0.547477
\(188\) 0 0
\(189\) −324.000 −0.124696
\(190\) 0 0
\(191\) 2656.00 1.00619 0.503093 0.864232i \(-0.332195\pi\)
0.503093 + 0.864232i \(0.332195\pi\)
\(192\) 0 0
\(193\) 2162.00 0.806343 0.403171 0.915124i \(-0.367908\pi\)
0.403171 + 0.915124i \(0.367908\pi\)
\(194\) 0 0
\(195\) −870.000 −0.319497
\(196\) 0 0
\(197\) −3514.00 −1.27087 −0.635437 0.772153i \(-0.719180\pi\)
−0.635437 + 0.772153i \(0.719180\pi\)
\(198\) 0 0
\(199\) −988.000 −0.351947 −0.175974 0.984395i \(-0.556307\pi\)
−0.175974 + 0.984395i \(0.556307\pi\)
\(200\) 0 0
\(201\) −828.000 −0.290560
\(202\) 0 0
\(203\) −792.000 −0.273830
\(204\) 0 0
\(205\) 330.000 0.112430
\(206\) 0 0
\(207\) 1008.00 0.338458
\(208\) 0 0
\(209\) 1840.00 0.608973
\(210\) 0 0
\(211\) −3548.00 −1.15760 −0.578802 0.815468i \(-0.696480\pi\)
−0.578802 + 0.815468i \(0.696480\pi\)
\(212\) 0 0
\(213\) 288.000 0.0926452
\(214\) 0 0
\(215\) 1940.00 0.615381
\(216\) 0 0
\(217\) −1296.00 −0.405430
\(218\) 0 0
\(219\) 2370.00 0.731277
\(220\) 0 0
\(221\) −4060.00 −1.23577
\(222\) 0 0
\(223\) 732.000 0.219813 0.109907 0.993942i \(-0.464945\pi\)
0.109907 + 0.993942i \(0.464945\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) −5492.00 −1.60580 −0.802901 0.596113i \(-0.796711\pi\)
−0.802901 + 0.596113i \(0.796711\pi\)
\(228\) 0 0
\(229\) −798.000 −0.230277 −0.115138 0.993349i \(-0.536731\pi\)
−0.115138 + 0.993349i \(0.536731\pi\)
\(230\) 0 0
\(231\) −720.000 −0.205076
\(232\) 0 0
\(233\) −2886.00 −0.811451 −0.405726 0.913995i \(-0.632981\pi\)
−0.405726 + 0.913995i \(0.632981\pi\)
\(234\) 0 0
\(235\) −2040.00 −0.566276
\(236\) 0 0
\(237\) −924.000 −0.253250
\(238\) 0 0
\(239\) 4096.00 1.10857 0.554285 0.832327i \(-0.312992\pi\)
0.554285 + 0.832327i \(0.312992\pi\)
\(240\) 0 0
\(241\) 2354.00 0.629189 0.314594 0.949226i \(-0.398132\pi\)
0.314594 + 0.949226i \(0.398132\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −995.000 −0.259462
\(246\) 0 0
\(247\) 5336.00 1.37458
\(248\) 0 0
\(249\) −3108.00 −0.791010
\(250\) 0 0
\(251\) 2916.00 0.733292 0.366646 0.930361i \(-0.380506\pi\)
0.366646 + 0.930361i \(0.380506\pi\)
\(252\) 0 0
\(253\) 2240.00 0.556631
\(254\) 0 0
\(255\) 1050.00 0.257857
\(256\) 0 0
\(257\) 882.000 0.214076 0.107038 0.994255i \(-0.465863\pi\)
0.107038 + 0.994255i \(0.465863\pi\)
\(258\) 0 0
\(259\) 696.000 0.166978
\(260\) 0 0
\(261\) −594.000 −0.140872
\(262\) 0 0
\(263\) −4456.00 −1.04475 −0.522374 0.852716i \(-0.674954\pi\)
−0.522374 + 0.852716i \(0.674954\pi\)
\(264\) 0 0
\(265\) −2370.00 −0.549388
\(266\) 0 0
\(267\) −3630.00 −0.832031
\(268\) 0 0
\(269\) 1486.00 0.336814 0.168407 0.985718i \(-0.446138\pi\)
0.168407 + 0.985718i \(0.446138\pi\)
\(270\) 0 0
\(271\) −4676.00 −1.04814 −0.524072 0.851674i \(-0.675588\pi\)
−0.524072 + 0.851674i \(0.675588\pi\)
\(272\) 0 0
\(273\) −2088.00 −0.462899
\(274\) 0 0
\(275\) 500.000 0.109640
\(276\) 0 0
\(277\) 2898.00 0.628606 0.314303 0.949323i \(-0.398229\pi\)
0.314303 + 0.949323i \(0.398229\pi\)
\(278\) 0 0
\(279\) −972.000 −0.208574
\(280\) 0 0
\(281\) 5194.00 1.10266 0.551331 0.834287i \(-0.314120\pi\)
0.551331 + 0.834287i \(0.314120\pi\)
\(282\) 0 0
\(283\) 5420.00 1.13846 0.569232 0.822177i \(-0.307240\pi\)
0.569232 + 0.822177i \(0.307240\pi\)
\(284\) 0 0
\(285\) −1380.00 −0.286822
\(286\) 0 0
\(287\) 792.000 0.162893
\(288\) 0 0
\(289\) −13.0000 −0.00264604
\(290\) 0 0
\(291\) −4278.00 −0.861790
\(292\) 0 0
\(293\) −9130.00 −1.82041 −0.910205 0.414157i \(-0.864076\pi\)
−0.910205 + 0.414157i \(0.864076\pi\)
\(294\) 0 0
\(295\) 2700.00 0.532882
\(296\) 0 0
\(297\) −540.000 −0.105502
\(298\) 0 0
\(299\) 6496.00 1.25643
\(300\) 0 0
\(301\) 4656.00 0.891586
\(302\) 0 0
\(303\) 222.000 0.0420910
\(304\) 0 0
\(305\) −70.0000 −0.0131416
\(306\) 0 0
\(307\) −6044.00 −1.12361 −0.561807 0.827269i \(-0.689894\pi\)
−0.561807 + 0.827269i \(0.689894\pi\)
\(308\) 0 0
\(309\) 4308.00 0.793118
\(310\) 0 0
\(311\) −6120.00 −1.11586 −0.557931 0.829887i \(-0.688405\pi\)
−0.557931 + 0.829887i \(0.688405\pi\)
\(312\) 0 0
\(313\) −614.000 −0.110880 −0.0554398 0.998462i \(-0.517656\pi\)
−0.0554398 + 0.998462i \(0.517656\pi\)
\(314\) 0 0
\(315\) 540.000 0.0965891
\(316\) 0 0
\(317\) −786.000 −0.139262 −0.0696312 0.997573i \(-0.522182\pi\)
−0.0696312 + 0.997573i \(0.522182\pi\)
\(318\) 0 0
\(319\) −1320.00 −0.231680
\(320\) 0 0
\(321\) 252.000 0.0438170
\(322\) 0 0
\(323\) −6440.00 −1.10938
\(324\) 0 0
\(325\) 1450.00 0.247482
\(326\) 0 0
\(327\) −750.000 −0.126835
\(328\) 0 0
\(329\) −4896.00 −0.820441
\(330\) 0 0
\(331\) −468.000 −0.0777148 −0.0388574 0.999245i \(-0.512372\pi\)
−0.0388574 + 0.999245i \(0.512372\pi\)
\(332\) 0 0
\(333\) 522.000 0.0859022
\(334\) 0 0
\(335\) 1380.00 0.225067
\(336\) 0 0
\(337\) 1538.00 0.248606 0.124303 0.992244i \(-0.460331\pi\)
0.124303 + 0.992244i \(0.460331\pi\)
\(338\) 0 0
\(339\) 1962.00 0.314340
\(340\) 0 0
\(341\) −2160.00 −0.343022
\(342\) 0 0
\(343\) −6504.00 −1.02386
\(344\) 0 0
\(345\) −1680.00 −0.262169
\(346\) 0 0
\(347\) −8396.00 −1.29891 −0.649454 0.760401i \(-0.725002\pi\)
−0.649454 + 0.760401i \(0.725002\pi\)
\(348\) 0 0
\(349\) 11090.0 1.70096 0.850479 0.526010i \(-0.176312\pi\)
0.850479 + 0.526010i \(0.176312\pi\)
\(350\) 0 0
\(351\) −1566.00 −0.238139
\(352\) 0 0
\(353\) 3298.00 0.497266 0.248633 0.968598i \(-0.420019\pi\)
0.248633 + 0.968598i \(0.420019\pi\)
\(354\) 0 0
\(355\) −480.000 −0.0717627
\(356\) 0 0
\(357\) 2520.00 0.373593
\(358\) 0 0
\(359\) −10720.0 −1.57599 −0.787994 0.615682i \(-0.788880\pi\)
−0.787994 + 0.615682i \(0.788880\pi\)
\(360\) 0 0
\(361\) 1605.00 0.233999
\(362\) 0 0
\(363\) 2793.00 0.403842
\(364\) 0 0
\(365\) −3950.00 −0.566445
\(366\) 0 0
\(367\) 4116.00 0.585432 0.292716 0.956199i \(-0.405441\pi\)
0.292716 + 0.956199i \(0.405441\pi\)
\(368\) 0 0
\(369\) 594.000 0.0838006
\(370\) 0 0
\(371\) −5688.00 −0.795974
\(372\) 0 0
\(373\) −3590.00 −0.498346 −0.249173 0.968459i \(-0.580159\pi\)
−0.249173 + 0.968459i \(0.580159\pi\)
\(374\) 0 0
\(375\) −375.000 −0.0516398
\(376\) 0 0
\(377\) −3828.00 −0.522950
\(378\) 0 0
\(379\) 12452.0 1.68764 0.843821 0.536625i \(-0.180301\pi\)
0.843821 + 0.536625i \(0.180301\pi\)
\(380\) 0 0
\(381\) −7716.00 −1.03754
\(382\) 0 0
\(383\) −12416.0 −1.65647 −0.828235 0.560381i \(-0.810655\pi\)
−0.828235 + 0.560381i \(0.810655\pi\)
\(384\) 0 0
\(385\) 1200.00 0.158851
\(386\) 0 0
\(387\) 3492.00 0.458678
\(388\) 0 0
\(389\) −2370.00 −0.308904 −0.154452 0.988000i \(-0.549361\pi\)
−0.154452 + 0.988000i \(0.549361\pi\)
\(390\) 0 0
\(391\) −7840.00 −1.01403
\(392\) 0 0
\(393\) −2508.00 −0.321913
\(394\) 0 0
\(395\) 1540.00 0.196167
\(396\) 0 0
\(397\) −9486.00 −1.19922 −0.599608 0.800294i \(-0.704677\pi\)
−0.599608 + 0.800294i \(0.704677\pi\)
\(398\) 0 0
\(399\) −3312.00 −0.415557
\(400\) 0 0
\(401\) −10630.0 −1.32378 −0.661891 0.749600i \(-0.730246\pi\)
−0.661891 + 0.749600i \(0.730246\pi\)
\(402\) 0 0
\(403\) −6264.00 −0.774273
\(404\) 0 0
\(405\) 405.000 0.0496904
\(406\) 0 0
\(407\) 1160.00 0.141275
\(408\) 0 0
\(409\) 12890.0 1.55836 0.779180 0.626800i \(-0.215636\pi\)
0.779180 + 0.626800i \(0.215636\pi\)
\(410\) 0 0
\(411\) −3750.00 −0.450058
\(412\) 0 0
\(413\) 6480.00 0.772058
\(414\) 0 0
\(415\) 5180.00 0.612714
\(416\) 0 0
\(417\) −7284.00 −0.855393
\(418\) 0 0
\(419\) 11196.0 1.30539 0.652697 0.757619i \(-0.273637\pi\)
0.652697 + 0.757619i \(0.273637\pi\)
\(420\) 0 0
\(421\) 8594.00 0.994883 0.497442 0.867497i \(-0.334273\pi\)
0.497442 + 0.867497i \(0.334273\pi\)
\(422\) 0 0
\(423\) −3672.00 −0.422077
\(424\) 0 0
\(425\) −1750.00 −0.199735
\(426\) 0 0
\(427\) −168.000 −0.0190400
\(428\) 0 0
\(429\) −3480.00 −0.391646
\(430\) 0 0
\(431\) 3544.00 0.396075 0.198038 0.980194i \(-0.436543\pi\)
0.198038 + 0.980194i \(0.436543\pi\)
\(432\) 0 0
\(433\) 5810.00 0.644829 0.322414 0.946599i \(-0.395506\pi\)
0.322414 + 0.946599i \(0.395506\pi\)
\(434\) 0 0
\(435\) 990.000 0.109119
\(436\) 0 0
\(437\) 10304.0 1.12793
\(438\) 0 0
\(439\) −9628.00 −1.04674 −0.523371 0.852105i \(-0.675326\pi\)
−0.523371 + 0.852105i \(0.675326\pi\)
\(440\) 0 0
\(441\) −1791.00 −0.193392
\(442\) 0 0
\(443\) 6100.00 0.654221 0.327110 0.944986i \(-0.393925\pi\)
0.327110 + 0.944986i \(0.393925\pi\)
\(444\) 0 0
\(445\) 6050.00 0.644489
\(446\) 0 0
\(447\) 5238.00 0.554248
\(448\) 0 0
\(449\) −7750.00 −0.814577 −0.407289 0.913300i \(-0.633526\pi\)
−0.407289 + 0.913300i \(0.633526\pi\)
\(450\) 0 0
\(451\) 1320.00 0.137819
\(452\) 0 0
\(453\) −6276.00 −0.650932
\(454\) 0 0
\(455\) 3480.00 0.358560
\(456\) 0 0
\(457\) 18314.0 1.87460 0.937301 0.348522i \(-0.113316\pi\)
0.937301 + 0.348522i \(0.113316\pi\)
\(458\) 0 0
\(459\) 1890.00 0.192195
\(460\) 0 0
\(461\) −6122.00 −0.618503 −0.309252 0.950980i \(-0.600079\pi\)
−0.309252 + 0.950980i \(0.600079\pi\)
\(462\) 0 0
\(463\) −10420.0 −1.04591 −0.522957 0.852359i \(-0.675171\pi\)
−0.522957 + 0.852359i \(0.675171\pi\)
\(464\) 0 0
\(465\) 1620.00 0.161561
\(466\) 0 0
\(467\) −12612.0 −1.24971 −0.624854 0.780742i \(-0.714842\pi\)
−0.624854 + 0.780742i \(0.714842\pi\)
\(468\) 0 0
\(469\) 3312.00 0.326085
\(470\) 0 0
\(471\) −6486.00 −0.634520
\(472\) 0 0
\(473\) 7760.00 0.754345
\(474\) 0 0
\(475\) 2300.00 0.222171
\(476\) 0 0
\(477\) −4266.00 −0.409490
\(478\) 0 0
\(479\) −3352.00 −0.319743 −0.159871 0.987138i \(-0.551108\pi\)
−0.159871 + 0.987138i \(0.551108\pi\)
\(480\) 0 0
\(481\) 3364.00 0.318888
\(482\) 0 0
\(483\) −4032.00 −0.379839
\(484\) 0 0
\(485\) 7130.00 0.667539
\(486\) 0 0
\(487\) −17108.0 −1.59186 −0.795932 0.605386i \(-0.793019\pi\)
−0.795932 + 0.605386i \(0.793019\pi\)
\(488\) 0 0
\(489\) 2796.00 0.258567
\(490\) 0 0
\(491\) 11388.0 1.04671 0.523354 0.852116i \(-0.324681\pi\)
0.523354 + 0.852116i \(0.324681\pi\)
\(492\) 0 0
\(493\) 4620.00 0.422057
\(494\) 0 0
\(495\) 900.000 0.0817212
\(496\) 0 0
\(497\) −1152.00 −0.103972
\(498\) 0 0
\(499\) 8996.00 0.807047 0.403523 0.914969i \(-0.367785\pi\)
0.403523 + 0.914969i \(0.367785\pi\)
\(500\) 0 0
\(501\) −9576.00 −0.853940
\(502\) 0 0
\(503\) −19504.0 −1.72891 −0.864454 0.502713i \(-0.832335\pi\)
−0.864454 + 0.502713i \(0.832335\pi\)
\(504\) 0 0
\(505\) −370.000 −0.0326035
\(506\) 0 0
\(507\) −3501.00 −0.306676
\(508\) 0 0
\(509\) −8306.00 −0.723295 −0.361647 0.932315i \(-0.617786\pi\)
−0.361647 + 0.932315i \(0.617786\pi\)
\(510\) 0 0
\(511\) −9480.00 −0.820686
\(512\) 0 0
\(513\) −2484.00 −0.213784
\(514\) 0 0
\(515\) −7180.00 −0.614347
\(516\) 0 0
\(517\) −8160.00 −0.694152
\(518\) 0 0
\(519\) 6846.00 0.579010
\(520\) 0 0
\(521\) 14850.0 1.24873 0.624367 0.781131i \(-0.285357\pi\)
0.624367 + 0.781131i \(0.285357\pi\)
\(522\) 0 0
\(523\) −20044.0 −1.67584 −0.837919 0.545795i \(-0.816228\pi\)
−0.837919 + 0.545795i \(0.816228\pi\)
\(524\) 0 0
\(525\) −900.000 −0.0748176
\(526\) 0 0
\(527\) 7560.00 0.624893
\(528\) 0 0
\(529\) 377.000 0.0309855
\(530\) 0 0
\(531\) 4860.00 0.397187
\(532\) 0 0
\(533\) 3828.00 0.311086
\(534\) 0 0
\(535\) −420.000 −0.0339405
\(536\) 0 0
\(537\) −6012.00 −0.483123
\(538\) 0 0
\(539\) −3980.00 −0.318053
\(540\) 0 0
\(541\) 21930.0 1.74278 0.871390 0.490590i \(-0.163219\pi\)
0.871390 + 0.490590i \(0.163219\pi\)
\(542\) 0 0
\(543\) −12678.0 −1.00196
\(544\) 0 0
\(545\) 1250.00 0.0982461
\(546\) 0 0
\(547\) −19988.0 −1.56239 −0.781193 0.624290i \(-0.785389\pi\)
−0.781193 + 0.624290i \(0.785389\pi\)
\(548\) 0 0
\(549\) −126.000 −0.00979517
\(550\) 0 0
\(551\) −6072.00 −0.469466
\(552\) 0 0
\(553\) 3696.00 0.284213
\(554\) 0 0
\(555\) −870.000 −0.0665395
\(556\) 0 0
\(557\) 11718.0 0.891396 0.445698 0.895183i \(-0.352956\pi\)
0.445698 + 0.895183i \(0.352956\pi\)
\(558\) 0 0
\(559\) 22504.0 1.70272
\(560\) 0 0
\(561\) 4200.00 0.316086
\(562\) 0 0
\(563\) −12756.0 −0.954887 −0.477443 0.878662i \(-0.658436\pi\)
−0.477443 + 0.878662i \(0.658436\pi\)
\(564\) 0 0
\(565\) −3270.00 −0.243487
\(566\) 0 0
\(567\) 972.000 0.0719932
\(568\) 0 0
\(569\) −15790.0 −1.16336 −0.581679 0.813418i \(-0.697604\pi\)
−0.581679 + 0.813418i \(0.697604\pi\)
\(570\) 0 0
\(571\) −2500.00 −0.183225 −0.0916127 0.995795i \(-0.529202\pi\)
−0.0916127 + 0.995795i \(0.529202\pi\)
\(572\) 0 0
\(573\) −7968.00 −0.580921
\(574\) 0 0
\(575\) 2800.00 0.203075
\(576\) 0 0
\(577\) 13778.0 0.994083 0.497041 0.867727i \(-0.334420\pi\)
0.497041 + 0.867727i \(0.334420\pi\)
\(578\) 0 0
\(579\) −6486.00 −0.465542
\(580\) 0 0
\(581\) 12432.0 0.887722
\(582\) 0 0
\(583\) −9480.00 −0.673450
\(584\) 0 0
\(585\) 2610.00 0.184462
\(586\) 0 0
\(587\) 11724.0 0.824363 0.412182 0.911102i \(-0.364767\pi\)
0.412182 + 0.911102i \(0.364767\pi\)
\(588\) 0 0
\(589\) −9936.00 −0.695086
\(590\) 0 0
\(591\) 10542.0 0.733739
\(592\) 0 0
\(593\) −27054.0 −1.87348 −0.936741 0.350024i \(-0.886174\pi\)
−0.936741 + 0.350024i \(0.886174\pi\)
\(594\) 0 0
\(595\) −4200.00 −0.289384
\(596\) 0 0
\(597\) 2964.00 0.203197
\(598\) 0 0
\(599\) 15328.0 1.04555 0.522776 0.852470i \(-0.324897\pi\)
0.522776 + 0.852470i \(0.324897\pi\)
\(600\) 0 0
\(601\) −25286.0 −1.71620 −0.858101 0.513481i \(-0.828356\pi\)
−0.858101 + 0.513481i \(0.828356\pi\)
\(602\) 0 0
\(603\) 2484.00 0.167755
\(604\) 0 0
\(605\) −4655.00 −0.312814
\(606\) 0 0
\(607\) 22060.0 1.47510 0.737552 0.675291i \(-0.235982\pi\)
0.737552 + 0.675291i \(0.235982\pi\)
\(608\) 0 0
\(609\) 2376.00 0.158096
\(610\) 0 0
\(611\) −23664.0 −1.56685
\(612\) 0 0
\(613\) 8810.00 0.580477 0.290239 0.956954i \(-0.406265\pi\)
0.290239 + 0.956954i \(0.406265\pi\)
\(614\) 0 0
\(615\) −990.000 −0.0649116
\(616\) 0 0
\(617\) −11766.0 −0.767717 −0.383858 0.923392i \(-0.625405\pi\)
−0.383858 + 0.923392i \(0.625405\pi\)
\(618\) 0 0
\(619\) 28316.0 1.83864 0.919318 0.393515i \(-0.128741\pi\)
0.919318 + 0.393515i \(0.128741\pi\)
\(620\) 0 0
\(621\) −3024.00 −0.195409
\(622\) 0 0
\(623\) 14520.0 0.933758
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −5520.00 −0.351591
\(628\) 0 0
\(629\) −4060.00 −0.257365
\(630\) 0 0
\(631\) 1388.00 0.0875680 0.0437840 0.999041i \(-0.486059\pi\)
0.0437840 + 0.999041i \(0.486059\pi\)
\(632\) 0 0
\(633\) 10644.0 0.668343
\(634\) 0 0
\(635\) 12860.0 0.803675
\(636\) 0 0
\(637\) −11542.0 −0.717913
\(638\) 0 0
\(639\) −864.000 −0.0534888
\(640\) 0 0
\(641\) −15974.0 −0.984298 −0.492149 0.870511i \(-0.663788\pi\)
−0.492149 + 0.870511i \(0.663788\pi\)
\(642\) 0 0
\(643\) −13044.0 −0.800008 −0.400004 0.916513i \(-0.630991\pi\)
−0.400004 + 0.916513i \(0.630991\pi\)
\(644\) 0 0
\(645\) −5820.00 −0.355290
\(646\) 0 0
\(647\) −18016.0 −1.09472 −0.547359 0.836898i \(-0.684367\pi\)
−0.547359 + 0.836898i \(0.684367\pi\)
\(648\) 0 0
\(649\) 10800.0 0.653216
\(650\) 0 0
\(651\) 3888.00 0.234075
\(652\) 0 0
\(653\) 17830.0 1.06852 0.534259 0.845321i \(-0.320591\pi\)
0.534259 + 0.845321i \(0.320591\pi\)
\(654\) 0 0
\(655\) 4180.00 0.249353
\(656\) 0 0
\(657\) −7110.00 −0.422203
\(658\) 0 0
\(659\) 20740.0 1.22597 0.612986 0.790094i \(-0.289968\pi\)
0.612986 + 0.790094i \(0.289968\pi\)
\(660\) 0 0
\(661\) −12070.0 −0.710240 −0.355120 0.934821i \(-0.615560\pi\)
−0.355120 + 0.934821i \(0.615560\pi\)
\(662\) 0 0
\(663\) 12180.0 0.713472
\(664\) 0 0
\(665\) 5520.00 0.321889
\(666\) 0 0
\(667\) −7392.00 −0.429115
\(668\) 0 0
\(669\) −2196.00 −0.126909
\(670\) 0 0
\(671\) −280.000 −0.0161092
\(672\) 0 0
\(673\) 20514.0 1.17497 0.587486 0.809234i \(-0.300118\pi\)
0.587486 + 0.809234i \(0.300118\pi\)
\(674\) 0 0
\(675\) −675.000 −0.0384900
\(676\) 0 0
\(677\) 13326.0 0.756514 0.378257 0.925701i \(-0.376524\pi\)
0.378257 + 0.925701i \(0.376524\pi\)
\(678\) 0 0
\(679\) 17112.0 0.967155
\(680\) 0 0
\(681\) 16476.0 0.927110
\(682\) 0 0
\(683\) −2988.00 −0.167398 −0.0836989 0.996491i \(-0.526673\pi\)
−0.0836989 + 0.996491i \(0.526673\pi\)
\(684\) 0 0
\(685\) 6250.00 0.348613
\(686\) 0 0
\(687\) 2394.00 0.132950
\(688\) 0 0
\(689\) −27492.0 −1.52012
\(690\) 0 0
\(691\) −16628.0 −0.915425 −0.457713 0.889100i \(-0.651331\pi\)
−0.457713 + 0.889100i \(0.651331\pi\)
\(692\) 0 0
\(693\) 2160.00 0.118401
\(694\) 0 0
\(695\) 12140.0 0.662585
\(696\) 0 0
\(697\) −4620.00 −0.251069
\(698\) 0 0
\(699\) 8658.00 0.468492
\(700\) 0 0
\(701\) −28082.0 −1.51304 −0.756521 0.653969i \(-0.773103\pi\)
−0.756521 + 0.653969i \(0.773103\pi\)
\(702\) 0 0
\(703\) 5336.00 0.286275
\(704\) 0 0
\(705\) 6120.00 0.326940
\(706\) 0 0
\(707\) −888.000 −0.0472372
\(708\) 0 0
\(709\) −20158.0 −1.06777 −0.533885 0.845557i \(-0.679269\pi\)
−0.533885 + 0.845557i \(0.679269\pi\)
\(710\) 0 0
\(711\) 2772.00 0.146214
\(712\) 0 0
\(713\) −12096.0 −0.635342
\(714\) 0 0
\(715\) 5800.00 0.303367
\(716\) 0 0
\(717\) −12288.0 −0.640033
\(718\) 0 0
\(719\) 30536.0 1.58387 0.791934 0.610607i \(-0.209075\pi\)
0.791934 + 0.610607i \(0.209075\pi\)
\(720\) 0 0
\(721\) −17232.0 −0.890088
\(722\) 0 0
\(723\) −7062.00 −0.363262
\(724\) 0 0
\(725\) −1650.00 −0.0845234
\(726\) 0 0
\(727\) −7204.00 −0.367512 −0.183756 0.982972i \(-0.558826\pi\)
−0.183756 + 0.982972i \(0.558826\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −27160.0 −1.37421
\(732\) 0 0
\(733\) −5398.00 −0.272005 −0.136003 0.990708i \(-0.543426\pi\)
−0.136003 + 0.990708i \(0.543426\pi\)
\(734\) 0 0
\(735\) 2985.00 0.149801
\(736\) 0 0
\(737\) 5520.00 0.275891
\(738\) 0 0
\(739\) −5660.00 −0.281741 −0.140870 0.990028i \(-0.544990\pi\)
−0.140870 + 0.990028i \(0.544990\pi\)
\(740\) 0 0
\(741\) −16008.0 −0.793615
\(742\) 0 0
\(743\) 3624.00 0.178939 0.0894695 0.995990i \(-0.471483\pi\)
0.0894695 + 0.995990i \(0.471483\pi\)
\(744\) 0 0
\(745\) −8730.00 −0.429319
\(746\) 0 0
\(747\) 9324.00 0.456690
\(748\) 0 0
\(749\) −1008.00 −0.0491743
\(750\) 0 0
\(751\) 12532.0 0.608920 0.304460 0.952525i \(-0.401524\pi\)
0.304460 + 0.952525i \(0.401524\pi\)
\(752\) 0 0
\(753\) −8748.00 −0.423366
\(754\) 0 0
\(755\) 10460.0 0.504210
\(756\) 0 0
\(757\) 9026.00 0.433363 0.216681 0.976242i \(-0.430477\pi\)
0.216681 + 0.976242i \(0.430477\pi\)
\(758\) 0 0
\(759\) −6720.00 −0.321371
\(760\) 0 0
\(761\) 4674.00 0.222644 0.111322 0.993784i \(-0.464491\pi\)
0.111322 + 0.993784i \(0.464491\pi\)
\(762\) 0 0
\(763\) 3000.00 0.142342
\(764\) 0 0
\(765\) −3150.00 −0.148874
\(766\) 0 0
\(767\) 31320.0 1.47445
\(768\) 0 0
\(769\) 38386.0 1.80004 0.900022 0.435843i \(-0.143550\pi\)
0.900022 + 0.435843i \(0.143550\pi\)
\(770\) 0 0
\(771\) −2646.00 −0.123597
\(772\) 0 0
\(773\) 16774.0 0.780490 0.390245 0.920711i \(-0.372390\pi\)
0.390245 + 0.920711i \(0.372390\pi\)
\(774\) 0 0
\(775\) −2700.00 −0.125144
\(776\) 0 0
\(777\) −2088.00 −0.0964049
\(778\) 0 0
\(779\) 6072.00 0.279271
\(780\) 0 0
\(781\) −1920.00 −0.0879680
\(782\) 0 0
\(783\) 1782.00 0.0813327
\(784\) 0 0
\(785\) 10810.0 0.491497
\(786\) 0 0
\(787\) 27116.0 1.22818 0.614092 0.789234i \(-0.289522\pi\)
0.614092 + 0.789234i \(0.289522\pi\)
\(788\) 0 0
\(789\) 13368.0 0.603186
\(790\) 0 0
\(791\) −7848.00 −0.352772
\(792\) 0 0
\(793\) −812.000 −0.0363619
\(794\) 0 0
\(795\) 7110.00 0.317190
\(796\) 0 0
\(797\) 17494.0 0.777502 0.388751 0.921343i \(-0.372907\pi\)
0.388751 + 0.921343i \(0.372907\pi\)
\(798\) 0 0
\(799\) 28560.0 1.26456
\(800\) 0 0
\(801\) 10890.0 0.480374
\(802\) 0 0
\(803\) −15800.0 −0.694359
\(804\) 0 0
\(805\) 6720.00 0.294222
\(806\) 0 0
\(807\) −4458.00 −0.194460
\(808\) 0 0
\(809\) 9298.00 0.404079 0.202040 0.979377i \(-0.435243\pi\)
0.202040 + 0.979377i \(0.435243\pi\)
\(810\) 0 0
\(811\) 21252.0 0.920171 0.460085 0.887875i \(-0.347819\pi\)
0.460085 + 0.887875i \(0.347819\pi\)
\(812\) 0 0
\(813\) 14028.0 0.605146
\(814\) 0 0
\(815\) −4660.00 −0.200285
\(816\) 0 0
\(817\) 35696.0 1.52857
\(818\) 0 0
\(819\) 6264.00 0.267255
\(820\) 0 0
\(821\) −1578.00 −0.0670799 −0.0335399 0.999437i \(-0.510678\pi\)
−0.0335399 + 0.999437i \(0.510678\pi\)
\(822\) 0 0
\(823\) −9652.00 −0.408806 −0.204403 0.978887i \(-0.565525\pi\)
−0.204403 + 0.978887i \(0.565525\pi\)
\(824\) 0 0
\(825\) −1500.00 −0.0633010
\(826\) 0 0
\(827\) 15612.0 0.656448 0.328224 0.944600i \(-0.393550\pi\)
0.328224 + 0.944600i \(0.393550\pi\)
\(828\) 0 0
\(829\) 13194.0 0.552770 0.276385 0.961047i \(-0.410863\pi\)
0.276385 + 0.961047i \(0.410863\pi\)
\(830\) 0 0
\(831\) −8694.00 −0.362926
\(832\) 0 0
\(833\) 13930.0 0.579407
\(834\) 0 0
\(835\) 15960.0 0.661459
\(836\) 0 0
\(837\) 2916.00 0.120420
\(838\) 0 0
\(839\) 22512.0 0.926342 0.463171 0.886269i \(-0.346712\pi\)
0.463171 + 0.886269i \(0.346712\pi\)
\(840\) 0 0
\(841\) −20033.0 −0.821395
\(842\) 0 0
\(843\) −15582.0 −0.636622
\(844\) 0 0
\(845\) 5835.00 0.237550
\(846\) 0 0
\(847\) −11172.0 −0.453217
\(848\) 0 0
\(849\) −16260.0 −0.657293
\(850\) 0 0
\(851\) 6496.00 0.261669
\(852\) 0 0
\(853\) 3114.00 0.124996 0.0624978 0.998045i \(-0.480093\pi\)
0.0624978 + 0.998045i \(0.480093\pi\)
\(854\) 0 0
\(855\) 4140.00 0.165597
\(856\) 0 0
\(857\) −36254.0 −1.44506 −0.722528 0.691342i \(-0.757020\pi\)
−0.722528 + 0.691342i \(0.757020\pi\)
\(858\) 0 0
\(859\) −27644.0 −1.09802 −0.549011 0.835815i \(-0.684996\pi\)
−0.549011 + 0.835815i \(0.684996\pi\)
\(860\) 0 0
\(861\) −2376.00 −0.0940463
\(862\) 0 0
\(863\) 8608.00 0.339536 0.169768 0.985484i \(-0.445698\pi\)
0.169768 + 0.985484i \(0.445698\pi\)
\(864\) 0 0
\(865\) −11410.0 −0.448499
\(866\) 0 0
\(867\) 39.0000 0.00152769
\(868\) 0 0
\(869\) 6160.00 0.240465
\(870\) 0 0
\(871\) 16008.0 0.622744
\(872\) 0 0
\(873\) 12834.0 0.497555
\(874\) 0 0
\(875\) 1500.00 0.0579534
\(876\) 0 0
\(877\) −37294.0 −1.43595 −0.717975 0.696068i \(-0.754931\pi\)
−0.717975 + 0.696068i \(0.754931\pi\)
\(878\) 0 0
\(879\) 27390.0 1.05101
\(880\) 0 0
\(881\) −5742.00 −0.219583 −0.109792 0.993955i \(-0.535018\pi\)
−0.109792 + 0.993955i \(0.535018\pi\)
\(882\) 0 0
\(883\) 46028.0 1.75421 0.877104 0.480301i \(-0.159472\pi\)
0.877104 + 0.480301i \(0.159472\pi\)
\(884\) 0 0
\(885\) −8100.00 −0.307659
\(886\) 0 0
\(887\) −10136.0 −0.383691 −0.191845 0.981425i \(-0.561447\pi\)
−0.191845 + 0.981425i \(0.561447\pi\)
\(888\) 0 0
\(889\) 30864.0 1.16439
\(890\) 0 0
\(891\) 1620.00 0.0609114
\(892\) 0 0
\(893\) −37536.0 −1.40660
\(894\) 0 0
\(895\) 10020.0 0.374225
\(896\) 0 0
\(897\) −19488.0 −0.725402
\(898\) 0 0
\(899\) 7128.00 0.264441
\(900\) 0 0
\(901\) 33180.0 1.22684
\(902\) 0 0
\(903\) −13968.0 −0.514757
\(904\) 0 0
\(905\) 21130.0 0.776116
\(906\) 0 0
\(907\) 30900.0 1.13122 0.565611 0.824672i \(-0.308641\pi\)
0.565611 + 0.824672i \(0.308641\pi\)
\(908\) 0 0
\(909\) −666.000 −0.0243012
\(910\) 0 0
\(911\) −45152.0 −1.64210 −0.821050 0.570857i \(-0.806611\pi\)
−0.821050 + 0.570857i \(0.806611\pi\)
\(912\) 0 0
\(913\) 20720.0 0.751075
\(914\) 0 0
\(915\) 210.000 0.00758731
\(916\) 0 0
\(917\) 10032.0 0.361271
\(918\) 0 0
\(919\) 30044.0 1.07841 0.539206 0.842174i \(-0.318725\pi\)
0.539206 + 0.842174i \(0.318725\pi\)
\(920\) 0 0
\(921\) 18132.0 0.648718
\(922\) 0 0
\(923\) −5568.00 −0.198562
\(924\) 0 0
\(925\) 1450.00 0.0515413
\(926\) 0 0
\(927\) −12924.0 −0.457907
\(928\) 0 0
\(929\) −11382.0 −0.401971 −0.200986 0.979594i \(-0.564414\pi\)
−0.200986 + 0.979594i \(0.564414\pi\)
\(930\) 0 0
\(931\) −18308.0 −0.644490
\(932\) 0 0
\(933\) 18360.0 0.644244
\(934\) 0 0
\(935\) −7000.00 −0.244839
\(936\) 0 0
\(937\) −34758.0 −1.21184 −0.605920 0.795525i \(-0.707195\pi\)
−0.605920 + 0.795525i \(0.707195\pi\)
\(938\) 0 0
\(939\) 1842.00 0.0640164
\(940\) 0 0
\(941\) 13270.0 0.459713 0.229856 0.973225i \(-0.426174\pi\)
0.229856 + 0.973225i \(0.426174\pi\)
\(942\) 0 0
\(943\) 7392.00 0.255267
\(944\) 0 0
\(945\) −1620.00 −0.0557657
\(946\) 0 0
\(947\) 35620.0 1.22228 0.611138 0.791524i \(-0.290712\pi\)
0.611138 + 0.791524i \(0.290712\pi\)
\(948\) 0 0
\(949\) −45820.0 −1.56731
\(950\) 0 0
\(951\) 2358.00 0.0804031
\(952\) 0 0
\(953\) −16926.0 −0.575327 −0.287664 0.957731i \(-0.592879\pi\)
−0.287664 + 0.957731i \(0.592879\pi\)
\(954\) 0 0
\(955\) 13280.0 0.449980
\(956\) 0 0
\(957\) 3960.00 0.133760
\(958\) 0 0
\(959\) 15000.0 0.505084
\(960\) 0 0
\(961\) −18127.0 −0.608472
\(962\) 0 0
\(963\) −756.000 −0.0252978
\(964\) 0 0
\(965\) 10810.0 0.360607
\(966\) 0 0
\(967\) −27676.0 −0.920372 −0.460186 0.887822i \(-0.652217\pi\)
−0.460186 + 0.887822i \(0.652217\pi\)
\(968\) 0 0
\(969\) 19320.0 0.640503
\(970\) 0 0
\(971\) 46916.0 1.55057 0.775286 0.631610i \(-0.217606\pi\)
0.775286 + 0.631610i \(0.217606\pi\)
\(972\) 0 0
\(973\) 29136.0 0.959977
\(974\) 0 0
\(975\) −4350.00 −0.142884
\(976\) 0 0
\(977\) 27594.0 0.903593 0.451796 0.892121i \(-0.350783\pi\)
0.451796 + 0.892121i \(0.350783\pi\)
\(978\) 0 0
\(979\) 24200.0 0.790026
\(980\) 0 0
\(981\) 2250.00 0.0732283
\(982\) 0 0
\(983\) −33016.0 −1.07126 −0.535629 0.844453i \(-0.679925\pi\)
−0.535629 + 0.844453i \(0.679925\pi\)
\(984\) 0 0
\(985\) −17570.0 −0.568352
\(986\) 0 0
\(987\) 14688.0 0.473682
\(988\) 0 0
\(989\) 43456.0 1.39719
\(990\) 0 0
\(991\) 2276.00 0.0729561 0.0364781 0.999334i \(-0.488386\pi\)
0.0364781 + 0.999334i \(0.488386\pi\)
\(992\) 0 0
\(993\) 1404.00 0.0448687
\(994\) 0 0
\(995\) −4940.00 −0.157396
\(996\) 0 0
\(997\) −57654.0 −1.83141 −0.915707 0.401846i \(-0.868369\pi\)
−0.915707 + 0.401846i \(0.868369\pi\)
\(998\) 0 0
\(999\) −1566.00 −0.0495956
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.p.1.1 1
4.3 odd 2 960.4.a.be.1.1 1
8.3 odd 2 480.4.a.a.1.1 1
8.5 even 2 480.4.a.h.1.1 yes 1
24.5 odd 2 1440.4.a.q.1.1 1
24.11 even 2 1440.4.a.l.1.1 1
40.19 odd 2 2400.4.a.u.1.1 1
40.29 even 2 2400.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.a.1.1 1 8.3 odd 2
480.4.a.h.1.1 yes 1 8.5 even 2
960.4.a.p.1.1 1 1.1 even 1 trivial
960.4.a.be.1.1 1 4.3 odd 2
1440.4.a.l.1.1 1 24.11 even 2
1440.4.a.q.1.1 1 24.5 odd 2
2400.4.a.b.1.1 1 40.29 even 2
2400.4.a.u.1.1 1 40.19 odd 2