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Properties

Label 1728.4.a.bc.1.1
Level $1728$
Weight $4$
Character 1728.1
Self dual yes
Analytic conductor $101.955$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1728,4,Mod(1,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,15,0,-25,0,0,0,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1728.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+15.0000 q^{5} -25.0000 q^{7} -15.0000 q^{11} -20.0000 q^{13} -72.0000 q^{17} -2.00000 q^{19} -114.000 q^{23} +100.000 q^{25} +30.0000 q^{29} +101.000 q^{31} -375.000 q^{35} +430.000 q^{37} +30.0000 q^{41} -110.000 q^{43} +330.000 q^{47} +282.000 q^{49} +621.000 q^{53} -225.000 q^{55} -660.000 q^{59} +376.000 q^{61} -300.000 q^{65} +250.000 q^{67} +360.000 q^{71} +785.000 q^{73} +375.000 q^{77} +488.000 q^{79} +489.000 q^{83} -1080.00 q^{85} +450.000 q^{89} +500.000 q^{91} -30.0000 q^{95} -1105.00 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 15.0000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −25.0000 −1.34987 −0.674937 − 0.737876i \(-0.735829\pi\)
−0.674937 + 0.737876i \(0.735829\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −15.0000 −0.411152 −0.205576 − 0.978641i \(-0.565907\pi\)
−0.205576 + 0.978641i \(0.565907\pi\)
\(12\) 0 0
\(13\) −20.0000 −0.426692 −0.213346 − 0.976977i \(-0.568436\pi\)
−0.213346 + 0.976977i \(0.568436\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −72.0000 −1.02721 −0.513605 − 0.858027i \(-0.671690\pi\)
−0.513605 + 0.858027i \(0.671690\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.0241490 −0.0120745 − 0.999927i \(-0.503844\pi\)
−0.0120745 + 0.999927i \(0.503844\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −114.000 −1.03351 −0.516753 − 0.856134i \(-0.672859\pi\)
−0.516753 + 0.856134i \(0.672859\pi\)
\(24\) 0 0
\(25\) 100.000 0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 30.0000 0.192099 0.0960493 − 0.995377i \(-0.469379\pi\)
0.0960493 + 0.995377i \(0.469379\pi\)
\(30\) 0 0
\(31\) 101.000 0.585166 0.292583 − 0.956240i \(-0.405485\pi\)
0.292583 + 0.956240i \(0.405485\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −375.000 −1.81104
\(36\) 0 0
\(37\) 430.000 1.91058 0.955291 − 0.295666i \(-0.0955415\pi\)
0.955291 + 0.295666i \(0.0955415\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 30.0000 0.114273 0.0571367 − 0.998366i \(-0.481803\pi\)
0.0571367 + 0.998366i \(0.481803\pi\)
\(42\) 0 0
\(43\) −110.000 −0.390113 −0.195056 − 0.980792i \(-0.562489\pi\)
−0.195056 + 0.980792i \(0.562489\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 330.000 1.02416 0.512079 − 0.858938i \(-0.328875\pi\)
0.512079 + 0.858938i \(0.328875\pi\)
\(48\) 0 0
\(49\) 282.000 0.822157
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 621.000 1.60945 0.804726 − 0.593647i \(-0.202312\pi\)
0.804726 + 0.593647i \(0.202312\pi\)
\(54\) 0 0
\(55\) −225.000 −0.551618
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −660.000 −1.45635 −0.728175 − 0.685391i \(-0.759631\pi\)
−0.728175 + 0.685391i \(0.759631\pi\)
\(60\) 0 0
\(61\) 376.000 0.789211 0.394605 − 0.918851i \(-0.370881\pi\)
0.394605 + 0.918851i \(0.370881\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −300.000 −0.572468
\(66\) 0 0
\(67\) 250.000 0.455856 0.227928 − 0.973678i \(-0.426805\pi\)
0.227928 + 0.973678i \(0.426805\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 360.000 0.601748 0.300874 − 0.953664i \(-0.402722\pi\)
0.300874 + 0.953664i \(0.402722\pi\)
\(72\) 0 0
\(73\) 785.000 1.25859 0.629297 − 0.777165i \(-0.283343\pi\)
0.629297 + 0.777165i \(0.283343\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 375.000 0.555003
\(78\) 0 0
\(79\) 488.000 0.694991 0.347496 − 0.937682i \(-0.387032\pi\)
0.347496 + 0.937682i \(0.387032\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 489.000 0.646683 0.323342 − 0.946282i \(-0.395194\pi\)
0.323342 + 0.946282i \(0.395194\pi\)
\(84\) 0 0
\(85\) −1080.00 −1.37815
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 450.000 0.535954 0.267977 − 0.963425i \(-0.413645\pi\)
0.267977 + 0.963425i \(0.413645\pi\)
\(90\) 0 0
\(91\) 500.000 0.575981
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −30.0000 −0.0323993
\(96\) 0 0
\(97\) −1105.00 −1.15666 −0.578329 − 0.815804i \(-0.696295\pi\)
−0.578329 + 0.815804i \(0.696295\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1425.00 1.40389 0.701945 − 0.712232i \(-0.252315\pi\)
0.701945 + 0.712232i \(0.252315\pi\)
\(102\) 0 0
\(103\) −1060.00 −1.01403 −0.507014 − 0.861938i \(-0.669251\pi\)
−0.507014 + 0.861938i \(0.669251\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1485.00 1.34169 0.670843 − 0.741600i \(-0.265933\pi\)
0.670843 + 0.741600i \(0.265933\pi\)
\(108\) 0 0
\(109\) 862.000 0.757474 0.378737 − 0.925504i \(-0.376359\pi\)
0.378737 + 0.925504i \(0.376359\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −690.000 −0.574422 −0.287211 − 0.957867i \(-0.592728\pi\)
−0.287211 + 0.957867i \(0.592728\pi\)
\(114\) 0 0
\(115\) −1710.00 −1.38659
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1800.00 1.38660
\(120\) 0 0
\(121\) −1106.00 −0.830954
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −375.000 −0.268328
\(126\) 0 0
\(127\) 1865.00 1.30309 0.651543 − 0.758611i \(-0.274122\pi\)
0.651543 + 0.758611i \(0.274122\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1155.00 −0.770327 −0.385163 − 0.922848i \(-0.625855\pi\)
−0.385163 + 0.922848i \(0.625855\pi\)
\(132\) 0 0
\(133\) 50.0000 0.0325981
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2778.00 1.73241 0.866206 − 0.499686i \(-0.166551\pi\)
0.866206 + 0.499686i \(0.166551\pi\)
\(138\) 0 0
\(139\) 1924.00 1.17404 0.587020 − 0.809572i \(-0.300301\pi\)
0.587020 + 0.809572i \(0.300301\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 300.000 0.175435
\(144\) 0 0
\(145\) 450.000 0.257727
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1455.00 0.799988 0.399994 − 0.916518i \(-0.369012\pi\)
0.399994 + 0.916518i \(0.369012\pi\)
\(150\) 0 0
\(151\) −727.000 −0.391804 −0.195902 − 0.980623i \(-0.562763\pi\)
−0.195902 + 0.980623i \(0.562763\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1515.00 0.785082
\(156\) 0 0
\(157\) −3260.00 −1.65717 −0.828587 − 0.559860i \(-0.810855\pi\)
−0.828587 + 0.559860i \(0.810855\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2850.00 1.39510
\(162\) 0 0
\(163\) −2540.00 −1.22054 −0.610270 − 0.792193i \(-0.708939\pi\)
−0.610270 + 0.792193i \(0.708939\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3498.00 −1.62086 −0.810429 − 0.585837i \(-0.800766\pi\)
−0.810429 + 0.585837i \(0.800766\pi\)
\(168\) 0 0
\(169\) −1797.00 −0.817934
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1149.00 −0.504953 −0.252476 − 0.967603i \(-0.581245\pi\)
−0.252476 + 0.967603i \(0.581245\pi\)
\(174\) 0 0
\(175\) −2500.00 −1.07990
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 315.000 0.131532 0.0657659 − 0.997835i \(-0.479051\pi\)
0.0657659 + 0.997835i \(0.479051\pi\)
\(180\) 0 0
\(181\) −1136.00 −0.466509 −0.233255 − 0.972416i \(-0.574938\pi\)
−0.233255 + 0.972416i \(0.574938\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6450.00 2.56332
\(186\) 0 0
\(187\) 1080.00 0.422339
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2460.00 −0.931934 −0.465967 − 0.884802i \(-0.654293\pi\)
−0.465967 + 0.884802i \(0.654293\pi\)
\(192\) 0 0
\(193\) 965.000 0.359908 0.179954 − 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2493.00 0.901619 0.450809 − 0.892620i \(-0.351135\pi\)
0.450809 + 0.892620i \(0.351135\pi\)
\(198\) 0 0
\(199\) −511.000 −0.182029 −0.0910146 − 0.995850i \(-0.529011\pi\)
−0.0910146 + 0.995850i \(0.529011\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −750.000 −0.259309
\(204\) 0 0
\(205\) 450.000 0.153314
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 30.0000 0.00992892
\(210\) 0 0
\(211\) 2086.00 0.680598 0.340299 − 0.940317i \(-0.389472\pi\)
0.340299 + 0.940317i \(0.389472\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1650.00 −0.523391
\(216\) 0 0
\(217\) −2525.00 −0.789899
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1440.00 0.438303
\(222\) 0 0
\(223\) 5240.00 1.57353 0.786763 − 0.617255i \(-0.211755\pi\)
0.786763 + 0.617255i \(0.211755\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2388.00 0.698225 0.349113 − 0.937081i \(-0.386483\pi\)
0.349113 + 0.937081i \(0.386483\pi\)
\(228\) 0 0
\(229\) −182.000 −0.0525192 −0.0262596 − 0.999655i \(-0.508360\pi\)
−0.0262596 + 0.999655i \(0.508360\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −450.000 −0.126526 −0.0632628 − 0.997997i \(-0.520151\pi\)
−0.0632628 + 0.997997i \(0.520151\pi\)
\(234\) 0 0
\(235\) 4950.00 1.37405
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −5190.00 −1.40466 −0.702329 − 0.711853i \(-0.747856\pi\)
−0.702329 + 0.711853i \(0.747856\pi\)
\(240\) 0 0
\(241\) −2266.00 −0.605668 −0.302834 − 0.953043i \(-0.597933\pi\)
−0.302834 + 0.953043i \(0.597933\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4230.00 1.10304
\(246\) 0 0
\(247\) 40.0000 0.0103042
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2880.00 −0.724239 −0.362119 − 0.932132i \(-0.617947\pi\)
−0.362119 + 0.932132i \(0.617947\pi\)
\(252\) 0 0
\(253\) 1710.00 0.424928
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4188.00 1.01650 0.508250 − 0.861210i \(-0.330293\pi\)
0.508250 + 0.861210i \(0.330293\pi\)
\(258\) 0 0
\(259\) −10750.0 −2.57904
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3030.00 0.710410 0.355205 − 0.934788i \(-0.384411\pi\)
0.355205 + 0.934788i \(0.384411\pi\)
\(264\) 0 0
\(265\) 9315.00 2.15931
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3510.00 0.795571 0.397785 − 0.917479i \(-0.369779\pi\)
0.397785 + 0.917479i \(0.369779\pi\)
\(270\) 0 0
\(271\) 2999.00 0.672237 0.336119 − 0.941820i \(-0.390886\pi\)
0.336119 + 0.941820i \(0.390886\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1500.00 −0.328921
\(276\) 0 0
\(277\) 7720.00 1.67455 0.837274 − 0.546783i \(-0.184148\pi\)
0.837274 + 0.546783i \(0.184148\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7440.00 1.57948 0.789739 − 0.613443i \(-0.210216\pi\)
0.789739 + 0.613443i \(0.210216\pi\)
\(282\) 0 0
\(283\) −830.000 −0.174341 −0.0871703 − 0.996193i \(-0.527782\pi\)
−0.0871703 + 0.996193i \(0.527782\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −750.000 −0.154255
\(288\) 0 0
\(289\) 271.000 0.0551598
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 546.000 0.108866 0.0544329 − 0.998517i \(-0.482665\pi\)
0.0544329 + 0.998517i \(0.482665\pi\)
\(294\) 0 0
\(295\) −9900.00 −1.95390
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2280.00 0.440989
\(300\) 0 0
\(301\) 2750.00 0.526603
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5640.00 1.05884
\(306\) 0 0
\(307\) 5560.00 1.03364 0.516818 − 0.856096i \(-0.327117\pi\)
0.516818 + 0.856096i \(0.327117\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8670.00 1.58081 0.790403 − 0.612587i \(-0.209871\pi\)
0.790403 + 0.612587i \(0.209871\pi\)
\(312\) 0 0
\(313\) 4565.00 0.824374 0.412187 − 0.911099i \(-0.364765\pi\)
0.412187 + 0.911099i \(0.364765\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4233.00 0.749997 0.374998 − 0.927025i \(-0.377643\pi\)
0.374998 + 0.927025i \(0.377643\pi\)
\(318\) 0 0
\(319\) −450.000 −0.0789817
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 144.000 0.0248061
\(324\) 0 0
\(325\) −2000.00 −0.341354
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8250.00 −1.38248
\(330\) 0 0
\(331\) −542.000 −0.0900031 −0.0450015 − 0.998987i \(-0.514329\pi\)
−0.0450015 + 0.998987i \(0.514329\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3750.00 0.611595
\(336\) 0 0
\(337\) 5690.00 0.919745 0.459872 − 0.887985i \(-0.347895\pi\)
0.459872 + 0.887985i \(0.347895\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1515.00 −0.240592
\(342\) 0 0
\(343\) 1525.00 0.240065
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5055.00 0.782036 0.391018 − 0.920383i \(-0.372123\pi\)
0.391018 + 0.920383i \(0.372123\pi\)
\(348\) 0 0
\(349\) −1622.00 −0.248778 −0.124389 − 0.992234i \(-0.539697\pi\)
−0.124389 + 0.992234i \(0.539697\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −30.0000 −0.00452334 −0.00226167 − 0.999997i \(-0.500720\pi\)
−0.00226167 + 0.999997i \(0.500720\pi\)
\(354\) 0 0
\(355\) 5400.00 0.807330
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7470.00 1.09819 0.549097 − 0.835759i \(-0.314972\pi\)
0.549097 + 0.835759i \(0.314972\pi\)
\(360\) 0 0
\(361\) −6855.00 −0.999417
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11775.0 1.68858
\(366\) 0 0
\(367\) −1375.00 −0.195571 −0.0977853 − 0.995208i \(-0.531176\pi\)
−0.0977853 + 0.995208i \(0.531176\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −15525.0 −2.17255
\(372\) 0 0
\(373\) 4840.00 0.671865 0.335933 − 0.941886i \(-0.390949\pi\)
0.335933 + 0.941886i \(0.390949\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −600.000 −0.0819670
\(378\) 0 0
\(379\) −1892.00 −0.256426 −0.128213 − 0.991747i \(-0.540924\pi\)
−0.128213 + 0.991747i \(0.540924\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10704.0 1.42806 0.714032 − 0.700113i \(-0.246867\pi\)
0.714032 + 0.700113i \(0.246867\pi\)
\(384\) 0 0
\(385\) 5625.00 0.744614
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7815.00 1.01860 0.509301 − 0.860588i \(-0.329904\pi\)
0.509301 + 0.860588i \(0.329904\pi\)
\(390\) 0 0
\(391\) 8208.00 1.06163
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7320.00 0.932428
\(396\) 0 0
\(397\) −4700.00 −0.594172 −0.297086 − 0.954851i \(-0.596015\pi\)
−0.297086 + 0.954851i \(0.596015\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2100.00 0.261519 0.130759 − 0.991414i \(-0.458258\pi\)
0.130759 + 0.991414i \(0.458258\pi\)
\(402\) 0 0
\(403\) −2020.00 −0.249686
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6450.00 −0.785540
\(408\) 0 0
\(409\) −10753.0 −1.30000 −0.650002 − 0.759933i \(-0.725232\pi\)
−0.650002 + 0.759933i \(0.725232\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 16500.0 1.96589
\(414\) 0 0
\(415\) 7335.00 0.867617
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2940.00 0.342789 0.171394 − 0.985203i \(-0.445173\pi\)
0.171394 + 0.985203i \(0.445173\pi\)
\(420\) 0 0
\(421\) −8696.00 −1.00669 −0.503346 − 0.864085i \(-0.667898\pi\)
−0.503346 + 0.864085i \(0.667898\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −7200.00 −0.821768
\(426\) 0 0
\(427\) −9400.00 −1.06533
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8370.00 −0.935426 −0.467713 − 0.883880i \(-0.654922\pi\)
−0.467713 + 0.883880i \(0.654922\pi\)
\(432\) 0 0
\(433\) −5155.00 −0.572133 −0.286066 − 0.958210i \(-0.592348\pi\)
−0.286066 + 0.958210i \(0.592348\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 228.000 0.0249582
\(438\) 0 0
\(439\) −10987.0 −1.19449 −0.597245 − 0.802059i \(-0.703738\pi\)
−0.597245 + 0.802059i \(0.703738\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1956.00 0.209780 0.104890 − 0.994484i \(-0.466551\pi\)
0.104890 + 0.994484i \(0.466551\pi\)
\(444\) 0 0
\(445\) 6750.00 0.719058
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8730.00 0.917582 0.458791 − 0.888544i \(-0.348283\pi\)
0.458791 + 0.888544i \(0.348283\pi\)
\(450\) 0 0
\(451\) −450.000 −0.0469838
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 7500.00 0.772759
\(456\) 0 0
\(457\) −8665.00 −0.886940 −0.443470 − 0.896289i \(-0.646253\pi\)
−0.443470 + 0.896289i \(0.646253\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −9825.00 −0.992616 −0.496308 − 0.868147i \(-0.665311\pi\)
−0.496308 + 0.868147i \(0.665311\pi\)
\(462\) 0 0
\(463\) −5245.00 −0.526470 −0.263235 − 0.964732i \(-0.584790\pi\)
−0.263235 + 0.964732i \(0.584790\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −11007.0 −1.09067 −0.545335 − 0.838218i \(-0.683598\pi\)
−0.545335 + 0.838218i \(0.683598\pi\)
\(468\) 0 0
\(469\) −6250.00 −0.615348
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1650.00 0.160396
\(474\) 0 0
\(475\) −200.000 −0.0193192
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −16950.0 −1.61684 −0.808419 − 0.588608i \(-0.799676\pi\)
−0.808419 + 0.588608i \(0.799676\pi\)
\(480\) 0 0
\(481\) −8600.00 −0.815231
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −16575.0 −1.55182
\(486\) 0 0
\(487\) 10640.0 0.990030 0.495015 − 0.868885i \(-0.335163\pi\)
0.495015 + 0.868885i \(0.335163\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1635.00 0.150278 0.0751390 − 0.997173i \(-0.476060\pi\)
0.0751390 + 0.997173i \(0.476060\pi\)
\(492\) 0 0
\(493\) −2160.00 −0.197326
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9000.00 −0.812284
\(498\) 0 0
\(499\) 15802.0 1.41762 0.708812 − 0.705397i \(-0.249231\pi\)
0.708812 + 0.705397i \(0.249231\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 7866.00 0.697272 0.348636 − 0.937258i \(-0.386645\pi\)
0.348636 + 0.937258i \(0.386645\pi\)
\(504\) 0 0
\(505\) 21375.0 1.88351
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −11955.0 −1.04105 −0.520527 − 0.853845i \(-0.674264\pi\)
−0.520527 + 0.853845i \(0.674264\pi\)
\(510\) 0 0
\(511\) −19625.0 −1.69894
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −15900.0 −1.36046
\(516\) 0 0
\(517\) −4950.00 −0.421085
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −19260.0 −1.61957 −0.809785 − 0.586727i \(-0.800416\pi\)
−0.809785 + 0.586727i \(0.800416\pi\)
\(522\) 0 0
\(523\) 18520.0 1.54842 0.774209 − 0.632930i \(-0.218148\pi\)
0.774209 + 0.632930i \(0.218148\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7272.00 −0.601088
\(528\) 0 0
\(529\) 829.000 0.0681351
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −600.000 −0.0487596
\(534\) 0 0
\(535\) 22275.0 1.80006
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4230.00 −0.338032
\(540\) 0 0
\(541\) −8372.00 −0.665324 −0.332662 − 0.943046i \(-0.607947\pi\)
−0.332662 + 0.943046i \(0.607947\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12930.0 1.01626
\(546\) 0 0
\(547\) −17120.0 −1.33821 −0.669103 − 0.743170i \(-0.733321\pi\)
−0.669103 + 0.743170i \(0.733321\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −60.0000 −0.00463899
\(552\) 0 0
\(553\) −12200.0 −0.938150
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10575.0 0.804447 0.402224 − 0.915541i \(-0.368237\pi\)
0.402224 + 0.915541i \(0.368237\pi\)
\(558\) 0 0
\(559\) 2200.00 0.166458
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10455.0 0.782639 0.391319 − 0.920255i \(-0.372019\pi\)
0.391319 + 0.920255i \(0.372019\pi\)
\(564\) 0 0
\(565\) −10350.0 −0.770669
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24540.0 1.80803 0.904016 − 0.427498i \(-0.140605\pi\)
0.904016 + 0.427498i \(0.140605\pi\)
\(570\) 0 0
\(571\) −24644.0 −1.80616 −0.903082 − 0.429469i \(-0.858701\pi\)
−0.903082 + 0.429469i \(0.858701\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −11400.0 −0.826805
\(576\) 0 0
\(577\) −9610.00 −0.693361 −0.346681 − 0.937983i \(-0.612691\pi\)
−0.346681 + 0.937983i \(0.612691\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12225.0 −0.872941
\(582\) 0 0
\(583\) −9315.00 −0.661729
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4017.00 0.282452 0.141226 − 0.989977i \(-0.454896\pi\)
0.141226 + 0.989977i \(0.454896\pi\)
\(588\) 0 0
\(589\) −202.000 −0.0141312
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −594.000 −0.0411343 −0.0205672 − 0.999788i \(-0.506547\pi\)
−0.0205672 + 0.999788i \(0.506547\pi\)
\(594\) 0 0
\(595\) 27000.0 1.86032
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8790.00 −0.599582 −0.299791 − 0.954005i \(-0.596917\pi\)
−0.299791 + 0.954005i \(0.596917\pi\)
\(600\) 0 0
\(601\) 9371.00 0.636025 0.318013 − 0.948087i \(-0.396985\pi\)
0.318013 + 0.948087i \(0.396985\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −16590.0 −1.11484
\(606\) 0 0
\(607\) −14560.0 −0.973595 −0.486798 − 0.873515i \(-0.661835\pi\)
−0.486798 + 0.873515i \(0.661835\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −6600.00 −0.437001
\(612\) 0 0
\(613\) 18250.0 1.20246 0.601232 − 0.799074i \(-0.294677\pi\)
0.601232 + 0.799074i \(0.294677\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −19662.0 −1.28292 −0.641461 − 0.767156i \(-0.721671\pi\)
−0.641461 + 0.767156i \(0.721671\pi\)
\(618\) 0 0
\(619\) −12044.0 −0.782050 −0.391025 − 0.920380i \(-0.627879\pi\)
−0.391025 + 0.920380i \(0.627879\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −11250.0 −0.723470
\(624\) 0 0
\(625\) −18125.0 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −30960.0 −1.96257
\(630\) 0 0
\(631\) 14879.0 0.938706 0.469353 − 0.883011i \(-0.344487\pi\)
0.469353 + 0.883011i \(0.344487\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 27975.0 1.74827
\(636\) 0 0
\(637\) −5640.00 −0.350808
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8850.00 −0.545326 −0.272663 − 0.962110i \(-0.587904\pi\)
−0.272663 + 0.962110i \(0.587904\pi\)
\(642\) 0 0
\(643\) −18380.0 −1.12727 −0.563636 − 0.826023i \(-0.690598\pi\)
−0.563636 + 0.826023i \(0.690598\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −3888.00 −0.236249 −0.118124 − 0.992999i \(-0.537688\pi\)
−0.118124 + 0.992999i \(0.537688\pi\)
\(648\) 0 0
\(649\) 9900.00 0.598781
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6789.00 −0.406852 −0.203426 − 0.979090i \(-0.565208\pi\)
−0.203426 + 0.979090i \(0.565208\pi\)
\(654\) 0 0
\(655\) −17325.0 −1.03350
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 28335.0 1.67492 0.837462 − 0.546496i \(-0.184038\pi\)
0.837462 + 0.546496i \(0.184038\pi\)
\(660\) 0 0
\(661\) 6082.00 0.357886 0.178943 − 0.983859i \(-0.442732\pi\)
0.178943 + 0.983859i \(0.442732\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 750.000 0.0437350
\(666\) 0 0
\(667\) −3420.00 −0.198535
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5640.00 −0.324486
\(672\) 0 0
\(673\) 9965.00 0.570762 0.285381 − 0.958414i \(-0.407880\pi\)
0.285381 + 0.958414i \(0.407880\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8130.00 0.461538 0.230769 − 0.973009i \(-0.425876\pi\)
0.230769 + 0.973009i \(0.425876\pi\)
\(678\) 0 0
\(679\) 27625.0 1.56134
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 33516.0 1.87768 0.938839 − 0.344356i \(-0.111903\pi\)
0.938839 + 0.344356i \(0.111903\pi\)
\(684\) 0 0
\(685\) 41670.0 2.32428
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12420.0 −0.686741
\(690\) 0 0
\(691\) 22084.0 1.21580 0.607898 − 0.794015i \(-0.292013\pi\)
0.607898 + 0.794015i \(0.292013\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 28860.0 1.57514
\(696\) 0 0
\(697\) −2160.00 −0.117383
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10395.0 −0.560077 −0.280038 − 0.959989i \(-0.590347\pi\)
−0.280038 + 0.959989i \(0.590347\pi\)
\(702\) 0 0
\(703\) −860.000 −0.0461387
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −35625.0 −1.89507
\(708\) 0 0
\(709\) 4804.00 0.254468 0.127234 − 0.991873i \(-0.459390\pi\)
0.127234 + 0.991873i \(0.459390\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −11514.0 −0.604772
\(714\) 0 0
\(715\) 4500.00 0.235371
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −10980.0 −0.569520 −0.284760 − 0.958599i \(-0.591914\pi\)
−0.284760 + 0.958599i \(0.591914\pi\)
\(720\) 0 0
\(721\) 26500.0 1.36881
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3000.00 0.153679
\(726\) 0 0
\(727\) −25945.0 −1.32359 −0.661793 − 0.749687i \(-0.730204\pi\)
−0.661793 + 0.749687i \(0.730204\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7920.00 0.400727
\(732\) 0 0
\(733\) −18650.0 −0.939773 −0.469886 − 0.882727i \(-0.655705\pi\)
−0.469886 + 0.882727i \(0.655705\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3750.00 −0.187426
\(738\) 0 0
\(739\) 5128.00 0.255259 0.127630 − 0.991822i \(-0.459263\pi\)
0.127630 + 0.991822i \(0.459263\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 32700.0 1.61460 0.807299 − 0.590142i \(-0.200928\pi\)
0.807299 + 0.590142i \(0.200928\pi\)
\(744\) 0 0
\(745\) 21825.0 1.07330
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −37125.0 −1.81111
\(750\) 0 0
\(751\) 21161.0 1.02820 0.514098 − 0.857731i \(-0.328127\pi\)
0.514098 + 0.857731i \(0.328127\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −10905.0 −0.525660
\(756\) 0 0
\(757\) −7130.00 −0.342331 −0.171165 − 0.985242i \(-0.554753\pi\)
−0.171165 + 0.985242i \(0.554753\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3360.00 0.160052 0.0800262 − 0.996793i \(-0.474500\pi\)
0.0800262 + 0.996793i \(0.474500\pi\)
\(762\) 0 0
\(763\) −21550.0 −1.02249
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 13200.0 0.621414
\(768\) 0 0
\(769\) 33473.0 1.56966 0.784829 − 0.619712i \(-0.212751\pi\)
0.784829 + 0.619712i \(0.212751\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −3546.00 −0.164995 −0.0824973 − 0.996591i \(-0.526290\pi\)
−0.0824973 + 0.996591i \(0.526290\pi\)
\(774\) 0 0
\(775\) 10100.0 0.468133
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −60.0000 −0.00275959
\(780\) 0 0
\(781\) −5400.00 −0.247410
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −48900.0 −2.22333
\(786\) 0 0
\(787\) 31840.0 1.44215 0.721076 − 0.692856i \(-0.243648\pi\)
0.721076 + 0.692856i \(0.243648\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 17250.0 0.775397
\(792\) 0 0
\(793\) −7520.00 −0.336750
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −15717.0 −0.698525 −0.349263 − 0.937025i \(-0.613568\pi\)
−0.349263 + 0.937025i \(0.613568\pi\)
\(798\) 0 0
\(799\) −23760.0 −1.05203
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11775.0 −0.517473
\(804\) 0 0
\(805\) 42750.0 1.87173
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −10530.0 −0.457621 −0.228810 − 0.973471i \(-0.573484\pi\)
−0.228810 + 0.973471i \(0.573484\pi\)
\(810\) 0 0
\(811\) 26782.0 1.15961 0.579805 − 0.814755i \(-0.303129\pi\)
0.579805 + 0.814755i \(0.303129\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −38100.0 −1.63753
\(816\) 0 0
\(817\) 220.000 0.00942084
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10110.0 0.429770 0.214885 − 0.976639i \(-0.431062\pi\)
0.214885 + 0.976639i \(0.431062\pi\)
\(822\) 0 0
\(823\) −12535.0 −0.530914 −0.265457 − 0.964123i \(-0.585523\pi\)
−0.265457 + 0.964123i \(0.585523\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9792.00 0.411731 0.205865 − 0.978580i \(-0.433999\pi\)
0.205865 + 0.978580i \(0.433999\pi\)
\(828\) 0 0
\(829\) 4534.00 0.189955 0.0949773 − 0.995479i \(-0.469722\pi\)
0.0949773 + 0.995479i \(0.469722\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −20304.0 −0.844528
\(834\) 0 0
\(835\) −52470.0 −2.17461
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 8880.00 0.365401 0.182701 − 0.983169i \(-0.441516\pi\)
0.182701 + 0.983169i \(0.441516\pi\)
\(840\) 0 0
\(841\) −23489.0 −0.963098
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −26955.0 −1.09737
\(846\) 0 0
\(847\) 27650.0 1.12168
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −49020.0 −1.97460
\(852\) 0 0
\(853\) −2270.00 −0.0911176 −0.0455588 − 0.998962i \(-0.514507\pi\)
−0.0455588 + 0.998962i \(0.514507\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19608.0 0.781560 0.390780 − 0.920484i \(-0.372205\pi\)
0.390780 + 0.920484i \(0.372205\pi\)
\(858\) 0 0
\(859\) 952.000 0.0378135 0.0189068 − 0.999821i \(-0.493981\pi\)
0.0189068 + 0.999821i \(0.493981\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 17604.0 0.694377 0.347188 − 0.937795i \(-0.387136\pi\)
0.347188 + 0.937795i \(0.387136\pi\)
\(864\) 0 0
\(865\) −17235.0 −0.677465
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −7320.00 −0.285747
\(870\) 0 0
\(871\) −5000.00 −0.194510
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9375.00 0.362209
\(876\) 0 0
\(877\) −21890.0 −0.842842 −0.421421 − 0.906865i \(-0.638469\pi\)
−0.421421 + 0.906865i \(0.638469\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −23940.0 −0.915504 −0.457752 − 0.889080i \(-0.651345\pi\)
−0.457752 + 0.889080i \(0.651345\pi\)
\(882\) 0 0
\(883\) 34990.0 1.33353 0.666765 − 0.745268i \(-0.267678\pi\)
0.666765 + 0.745268i \(0.267678\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 22188.0 0.839910 0.419955 − 0.907545i \(-0.362046\pi\)
0.419955 + 0.907545i \(0.362046\pi\)
\(888\) 0 0
\(889\) −46625.0 −1.75900
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −660.000 −0.0247324
\(894\) 0 0
\(895\) 4725.00 0.176469
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3030.00 0.112410
\(900\) 0 0
\(901\) −44712.0 −1.65324
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −17040.0 −0.625888
\(906\) 0 0
\(907\) −37370.0 −1.36808 −0.684041 − 0.729444i \(-0.739779\pi\)
−0.684041 + 0.729444i \(0.739779\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −40710.0 −1.48055 −0.740276 − 0.672303i \(-0.765305\pi\)
−0.740276 + 0.672303i \(0.765305\pi\)
\(912\) 0 0
\(913\) −7335.00 −0.265885
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 28875.0 1.03984
\(918\) 0 0
\(919\) 20981.0 0.753100 0.376550 − 0.926396i \(-0.377110\pi\)
0.376550 + 0.926396i \(0.377110\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −7200.00 −0.256762
\(924\) 0 0
\(925\) 43000.0 1.52847
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −20100.0 −0.709860 −0.354930 − 0.934893i \(-0.615495\pi\)
−0.354930 + 0.934893i \(0.615495\pi\)
\(930\) 0 0
\(931\) −564.000 −0.0198543
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 16200.0 0.566627
\(936\) 0 0
\(937\) 15635.0 0.545115 0.272558 − 0.962139i \(-0.412130\pi\)
0.272558 + 0.962139i \(0.412130\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 23955.0 0.829873 0.414937 − 0.909850i \(-0.363804\pi\)
0.414937 + 0.909850i \(0.363804\pi\)
\(942\) 0 0
\(943\) −3420.00 −0.118102
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −36393.0 −1.24880 −0.624400 − 0.781105i \(-0.714656\pi\)
−0.624400 + 0.781105i \(0.714656\pi\)
\(948\) 0 0
\(949\) −15700.0 −0.537032
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −43020.0 −1.46228 −0.731141 − 0.682227i \(-0.761012\pi\)
−0.731141 + 0.682227i \(0.761012\pi\)
\(954\) 0 0
\(955\) −36900.0 −1.25032
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −69450.0 −2.33854
\(960\) 0 0
\(961\) −19590.0 −0.657581
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 14475.0 0.482867
\(966\) 0 0
\(967\) −43585.0 −1.44943 −0.724715 − 0.689049i \(-0.758029\pi\)
−0.724715 + 0.689049i \(0.758029\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −43335.0 −1.43222 −0.716110 − 0.697987i \(-0.754079\pi\)
−0.716110 + 0.697987i \(0.754079\pi\)
\(972\) 0 0
\(973\) −48100.0 −1.58480
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −30390.0 −0.995151 −0.497575 − 0.867421i \(-0.665776\pi\)
−0.497575 + 0.867421i \(0.665776\pi\)
\(978\) 0 0
\(979\) −6750.00 −0.220358
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 59226.0 1.92168 0.960842 − 0.277096i \(-0.0893719\pi\)
0.960842 + 0.277096i \(0.0893719\pi\)
\(984\) 0 0
\(985\) 37395.0 1.20965
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12540.0 0.403184
\(990\) 0 0
\(991\) 8399.00 0.269226 0.134613 − 0.990898i \(-0.457021\pi\)
0.134613 + 0.990898i \(0.457021\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −7665.00 −0.244218
\(996\) 0 0
\(997\) −13340.0 −0.423753 −0.211877 − 0.977296i \(-0.567958\pi\)
−0.211877 + 0.977296i \(0.567958\pi\)
\(998\) 0 0
\(999\) 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 1728.4.a.bc.1.1 1
3.2 odd 2 1728.4.a.c.1.1 1
4.3 odd 2 1728.4.a.bd.1.1 1
8.3 odd 2 432.4.a.a.1.1 1
8.5 even 2 27.4.a.a.1.1 ✓ 1
12.11 even 2 1728.4.a.d.1.1 1
24.5 odd 2 27.4.a.b.1.1 yes 1
24.11 even 2 432.4.a.n.1.1 1
40.13 odd 4 675.4.b.b.649.2 2
40.29 even 2 675.4.a.j.1.1 1
40.37 odd 4 675.4.b.b.649.1 2
56.13 odd 2 1323.4.a.d.1.1 1
72.5 odd 6 81.4.c.a.55.1 2
72.13 even 6 81.4.c.c.55.1 2
72.29 odd 6 81.4.c.a.28.1 2
72.61 even 6 81.4.c.c.28.1 2
120.29 odd 2 675.4.a.a.1.1 1
120.53 even 4 675.4.b.a.649.1 2
120.77 even 4 675.4.b.a.649.2 2
168.125 even 2 1323.4.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.4.a.a.1.1 ✓ 1 8.5 even 2
27.4.a.b.1.1 yes 1 24.5 odd 2
81.4.c.a.28.1 2 72.29 odd 6
81.4.c.a.55.1 2 72.5 odd 6
81.4.c.c.28.1 2 72.61 even 6
81.4.c.c.55.1 2 72.13 even 6
432.4.a.a.1.1 1 8.3 odd 2
432.4.a.n.1.1 1 24.11 even 2
675.4.a.a.1.1 1 120.29 odd 2
675.4.a.j.1.1 1 40.29 even 2
675.4.b.a.649.1 2 120.53 even 4
675.4.b.a.649.2 2 120.77 even 4
675.4.b.b.649.1 2 40.37 odd 4
675.4.b.b.649.2 2 40.13 odd 4
1323.4.a.d.1.1 1 56.13 odd 2
1323.4.a.k.1.1 1 168.125 even 2
1728.4.a.c.1.1 1 3.2 odd 2
1728.4.a.d.1.1 1 12.11 even 2
1728.4.a.bc.1.1 1 1.1 even 1 trivial
1728.4.a.bd.1.1 1 4.3 odd 2