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Properties

Label 96.3.h.b.17.1
Level $96$
Weight $3$
Character 96.17
Self dual yes
Analytic conductor $2.616$
Analytic rank $0$
Dimension $1$
CM discriminant -24
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [96,3,Mod(17,96)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(96, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 96.h (of order \(2\), degree \(1\), not minimal)

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: \(2.61581053786\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 17.1
Character \(\chi\) \(=\) 96.17

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -2.00000 q^{5} +10.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -2.00000 q^{5} +10.0000 q^{7} +9.00000 q^{9} -10.0000 q^{11} -6.00000 q^{15} +30.0000 q^{21} -21.0000 q^{25} +27.0000 q^{27} -50.0000 q^{29} -38.0000 q^{31} -30.0000 q^{33} -20.0000 q^{35} -18.0000 q^{45} +51.0000 q^{49} +94.0000 q^{53} +20.0000 q^{55} -10.0000 q^{59} +90.0000 q^{63} +50.0000 q^{73} -63.0000 q^{75} -100.000 q^{77} +58.0000 q^{79} +81.0000 q^{81} +134.000 q^{83} -150.000 q^{87} -114.000 q^{93} -190.000 q^{97} -90.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(65\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 1.00000
\(4\) 0 0
\(5\) −2.00000 −0.400000 −0.200000 − 0.979796i \(-0.564094\pi\)
−0.200000 + 0.979796i \(0.564094\pi\)
\(6\) 0 0
\(7\) 10.0000 1.42857 0.714286 − 0.699854i \(-0.246752\pi\)
0.714286 + 0.699854i \(0.246752\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) −10.0000 −0.909091 −0.454545 − 0.890724i \(-0.650198\pi\)
−0.454545 + 0.890724i \(0.650198\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −6.00000 −0.400000
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 30.0000 1.42857
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −21.0000 −0.840000
\(26\) 0 0
\(27\) 27.0000 1.00000
\(28\) 0 0
\(29\) −50.0000 −1.72414 −0.862069 − 0.506791i \(-0.830832\pi\)
−0.862069 + 0.506791i \(0.830832\pi\)
\(30\) 0 0
\(31\) −38.0000 −1.22581 −0.612903 − 0.790158i \(-0.709998\pi\)
−0.612903 + 0.790158i \(0.709998\pi\)
\(32\) 0 0
\(33\) −30.0000 −0.909091
\(34\) 0 0
\(35\) −20.0000 −0.571429
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −18.0000 −0.400000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 51.0000 1.04082
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 94.0000 1.77358 0.886792 − 0.462168i \(-0.152928\pi\)
0.886792 + 0.462168i \(0.152928\pi\)
\(54\) 0 0
\(55\) 20.0000 0.363636
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.0000 −0.169492 −0.0847458 − 0.996403i \(-0.527008\pi\)
−0.0847458 + 0.996403i \(0.527008\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 90.0000 1.42857
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 50.0000 0.684932 0.342466 − 0.939530i \(-0.388738\pi\)
0.342466 + 0.939530i \(0.388738\pi\)
\(74\) 0 0
\(75\) −63.0000 −0.840000
\(76\) 0 0
\(77\) −100.000 −1.29870
\(78\) 0 0
\(79\) 58.0000 0.734177 0.367089 − 0.930186i \(-0.380355\pi\)
0.367089 + 0.930186i \(0.380355\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 134.000 1.61446 0.807229 − 0.590238i \(-0.200966\pi\)
0.807229 + 0.590238i \(0.200966\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −150.000 −1.72414
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −114.000 −1.22581
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −190.000 −1.95876 −0.979381 − 0.202020i \(-0.935249\pi\)
−0.979381 + 0.202020i \(0.935249\pi\)
\(98\) 0 0
\(99\) −90.0000 −0.909091
\(100\) 0 0
\(101\) 190.000 1.88119 0.940594 − 0.339533i \(-0.110269\pi\)
0.940594 + 0.339533i \(0.110269\pi\)
\(102\) 0 0
\(103\) 10.0000 0.0970874 0.0485437 − 0.998821i \(-0.484542\pi\)
0.0485437 + 0.998821i \(0.484542\pi\)
\(104\) 0 0
\(105\) −60.0000 −0.571429
\(106\) 0 0
\(107\) 86.0000 0.803738 0.401869 − 0.915697i \(-0.368361\pi\)
0.401869 + 0.915697i \(0.368361\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −21.0000 −0.173554
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 92.0000 0.736000
\(126\) 0 0
\(127\) −230.000 −1.81102 −0.905512 − 0.424321i \(-0.860513\pi\)
−0.905512 + 0.424321i \(0.860513\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −250.000 −1.90840 −0.954198 − 0.299174i \(-0.903289\pi\)
−0.954198 + 0.299174i \(0.903289\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −54.0000 −0.400000
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 100.000 0.689655
\(146\) 0 0
\(147\) 153.000 1.04082
\(148\) 0 0
\(149\) −290.000 −1.94631 −0.973154 − 0.230153i \(-0.926077\pi\)
−0.973154 + 0.230153i \(0.926077\pi\)
\(150\) 0 0
\(151\) 298.000 1.97351 0.986755 − 0.162218i \(-0.0518647\pi\)
0.986755 + 0.162218i \(0.0518647\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 76.0000 0.490323
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 282.000 1.77358
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 60.0000 0.363636
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 46.0000 0.265896 0.132948 − 0.991123i \(-0.457556\pi\)
0.132948 + 0.991123i \(0.457556\pi\)
\(174\) 0 0
\(175\) −210.000 −1.20000
\(176\) 0 0
\(177\) −30.0000 −0.169492
\(178\) 0 0
\(179\) 230.000 1.28492 0.642458 − 0.766321i \(-0.277915\pi\)
0.642458 + 0.766321i \(0.277915\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 270.000 1.42857
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 290.000 1.50259 0.751295 − 0.659966i \(-0.229429\pi\)
0.751295 + 0.659966i \(0.229429\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −194.000 −0.984772 −0.492386 − 0.870377i \(-0.663875\pi\)
−0.492386 + 0.870377i \(0.663875\pi\)
\(198\) 0 0
\(199\) 202.000 1.01508 0.507538 − 0.861630i \(-0.330556\pi\)
0.507538 + 0.861630i \(0.330556\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −500.000 −2.46305
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −380.000 −1.75115
\(218\) 0 0
\(219\) 150.000 0.684932
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −230.000 −1.03139 −0.515695 − 0.856772i \(-0.672466\pi\)
−0.515695 + 0.856772i \(0.672466\pi\)
\(224\) 0 0
\(225\) −189.000 −0.840000
\(226\) 0 0
\(227\) −346.000 −1.52423 −0.762115 − 0.647442i \(-0.775839\pi\)
−0.762115 + 0.647442i \(0.775839\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) −300.000 −1.29870
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 174.000 0.734177
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −382.000 −1.58506 −0.792531 − 0.609831i \(-0.791237\pi\)
−0.792531 + 0.609831i \(0.791237\pi\)
\(242\) 0 0
\(243\) 243.000 1.00000
\(244\) 0 0
\(245\) −102.000 −0.416327
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 402.000 1.61446
\(250\) 0 0
\(251\) 470.000 1.87251 0.936255 − 0.351321i \(-0.114267\pi\)
0.936255 + 0.351321i \(0.114267\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −450.000 −1.72414
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −188.000 −0.709434
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 430.000 1.59851 0.799257 − 0.600990i \(-0.205227\pi\)
0.799257 + 0.600990i \(0.205227\pi\)
\(270\) 0 0
\(271\) 58.0000 0.214022 0.107011 − 0.994258i \(-0.465872\pi\)
0.107011 + 0.994258i \(0.465872\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 210.000 0.763636
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −342.000 −1.22581
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) −570.000 −1.95876
\(292\) 0 0
\(293\) −386.000 −1.31741 −0.658703 − 0.752403i \(-0.728895\pi\)
−0.658703 + 0.752403i \(0.728895\pi\)
\(294\) 0 0
\(295\) 20.0000 0.0677966
\(296\) 0 0
\(297\) −270.000 −0.909091
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 570.000 1.88119
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 30.0000 0.0970874
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 530.000 1.69329 0.846645 − 0.532158i \(-0.178619\pi\)
0.846645 + 0.532158i \(0.178619\pi\)
\(314\) 0 0
\(315\) −180.000 −0.571429
\(316\) 0 0
\(317\) 334.000 1.05363 0.526814 − 0.849981i \(-0.323386\pi\)
0.526814 + 0.849981i \(0.323386\pi\)
\(318\) 0 0
\(319\) 500.000 1.56740
\(320\) 0 0
\(321\) 258.000 0.803738
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −190.000 −0.563798 −0.281899 − 0.959444i \(-0.590964\pi\)
−0.281899 + 0.959444i \(0.590964\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 380.000 1.11437
\(342\) 0 0
\(343\) 20.0000 0.0583090
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −106.000 −0.305476 −0.152738 − 0.988267i \(-0.548809\pi\)
−0.152738 + 0.988267i \(0.548809\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) −63.0000 −0.173554
\(364\) 0 0
\(365\) −100.000 −0.273973
\(366\) 0 0
\(367\) −710.000 −1.93460 −0.967302 − 0.253626i \(-0.918377\pi\)
−0.967302 + 0.253626i \(0.918377\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 940.000 2.53369
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 276.000 0.736000
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −690.000 −1.81102
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 200.000 0.519481
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 190.000 0.488432 0.244216 − 0.969721i \(-0.421469\pi\)
0.244216 + 0.969721i \(0.421469\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −750.000 −1.90840
\(394\) 0 0
\(395\) −116.000 −0.293671
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −162.000 −0.400000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −718.000 −1.75550 −0.877751 − 0.479118i \(-0.840957\pi\)
−0.877751 + 0.479118i \(0.840957\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −100.000 −0.242131
\(414\) 0 0
\(415\) −268.000 −0.645783
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −730.000 −1.74224 −0.871122 − 0.491067i \(-0.836607\pi\)
−0.871122 + 0.491067i \(0.836607\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −670.000 −1.54734 −0.773672 − 0.633586i \(-0.781582\pi\)
−0.773672 + 0.633586i \(0.781582\pi\)
\(434\) 0 0
\(435\) 300.000 0.689655
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −278.000 −0.633257 −0.316629 − 0.948550i \(-0.602551\pi\)
−0.316629 + 0.948550i \(0.602551\pi\)
\(440\) 0 0
\(441\) 459.000 1.04082
\(442\) 0 0
\(443\) 86.0000 0.194131 0.0970655 − 0.995278i \(-0.469054\pi\)
0.0970655 + 0.995278i \(0.469054\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −870.000 −1.94631
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 894.000 1.97351
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 530.000 1.15974 0.579869 − 0.814710i \(-0.303104\pi\)
0.579869 + 0.814710i \(0.303104\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −530.000 −1.14967 −0.574837 − 0.818268i \(-0.694935\pi\)
−0.574837 + 0.818268i \(0.694935\pi\)
\(462\) 0 0
\(463\) 250.000 0.539957 0.269978 − 0.962866i \(-0.412983\pi\)
0.269978 + 0.962866i \(0.412983\pi\)
\(464\) 0 0
\(465\) 228.000 0.490323
\(466\) 0 0
\(467\) −634.000 −1.35760 −0.678801 − 0.734322i \(-0.737500\pi\)
−0.678801 + 0.734322i \(0.737500\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 846.000 1.77358
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 380.000 0.783505
\(486\) 0 0
\(487\) 970.000 1.99179 0.995893 − 0.0905356i \(-0.0288579\pi\)
0.995893 + 0.0905356i \(0.0288579\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 470.000 0.957230 0.478615 − 0.878025i \(-0.341139\pi\)
0.478615 + 0.878025i \(0.341139\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 180.000 0.363636
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −380.000 −0.752475
\(506\) 0 0
\(507\) 507.000 1.00000
\(508\) 0 0
\(509\) −1010.00 −1.98428 −0.992141 − 0.125121i \(-0.960068\pi\)
−0.992141 + 0.125121i \(0.960068\pi\)
\(510\) 0 0
\(511\) 500.000 0.978474
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −20.0000 −0.0388350
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 138.000 0.265896
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) −630.000 −1.20000
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) −90.0000 −0.169492
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −172.000 −0.321495
\(536\) 0 0
\(537\) 690.000 1.28492
\(538\) 0 0
\(539\) −510.000 −0.946197
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 580.000 1.04882
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −914.000 −1.64093 −0.820467 − 0.571694i \(-0.806286\pi\)
−0.820467 + 0.571694i \(0.806286\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 326.000 0.579041 0.289520 − 0.957172i \(-0.406504\pi\)
0.289520 + 0.957172i \(0.406504\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 810.000 1.42857
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 290.000 0.502600 0.251300 − 0.967909i \(-0.419142\pi\)
0.251300 + 0.967909i \(0.419142\pi\)
\(578\) 0 0
\(579\) 870.000 1.50259
\(580\) 0 0
\(581\) 1340.00 2.30637
\(582\) 0 0
\(583\) −940.000 −1.61235
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −874.000 −1.48893 −0.744463 − 0.667663i \(-0.767295\pi\)
−0.744463 + 0.667663i \(0.767295\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −582.000 −0.984772
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 606.000 1.01508
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1198.00 −1.99334 −0.996672 − 0.0815138i \(-0.974025\pi\)
−0.996672 + 0.0815138i \(0.974025\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 42.0000 0.0694215
\(606\) 0 0
\(607\) 730.000 1.20264 0.601318 − 0.799010i \(-0.294643\pi\)
0.601318 + 0.799010i \(0.294643\pi\)
\(608\) 0 0
\(609\) −1500.00 −2.46305
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 341.000 0.545600
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1238.00 −1.96197 −0.980983 − 0.194096i \(-0.937823\pi\)
−0.980983 + 0.194096i \(0.937823\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 460.000 0.724409
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 100.000 0.154083
\(650\) 0 0
\(651\) −1140.00 −1.75115
\(652\) 0 0
\(653\) 1006.00 1.54058 0.770291 − 0.637693i \(-0.220111\pi\)
0.770291 + 0.637693i \(0.220111\pi\)
\(654\) 0 0
\(655\) 500.000 0.763359
\(656\) 0 0
\(657\) 450.000 0.684932
\(658\) 0 0
\(659\) −730.000 −1.10774 −0.553869 − 0.832603i \(-0.686849\pi\)
−0.553869 + 0.832603i \(0.686849\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −690.000 −1.03139
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −190.000 −0.282318 −0.141159 − 0.989987i \(-0.545083\pi\)
−0.141159 + 0.989987i \(0.545083\pi\)
\(674\) 0 0
\(675\) −567.000 −0.840000
\(676\) 0 0
\(677\) −1346.00 −1.98818 −0.994092 − 0.108545i \(-0.965381\pi\)
−0.994092 + 0.108545i \(0.965381\pi\)
\(678\) 0 0
\(679\) −1900.00 −2.79823
\(680\) 0 0
\(681\) −1038.00 −1.52423
\(682\) 0 0
\(683\) 1334.00 1.95315 0.976574 − 0.215182i \(-0.0690345\pi\)
0.976574 + 0.215182i \(0.0690345\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) −900.000 −1.29870
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −50.0000 −0.0713267 −0.0356633 − 0.999364i \(-0.511354\pi\)
−0.0356633 + 0.999364i \(0.511354\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1900.00 2.68741
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 522.000 0.734177
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 100.000 0.138696
\(722\) 0 0
\(723\) −1146.00 −1.58506
\(724\) 0 0
\(725\) 1050.00 1.44828
\(726\) 0 0
\(727\) 1450.00 1.99450 0.997249 − 0.0741249i \(-0.0236163\pi\)
0.997249 + 0.0741249i \(0.0236163\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −306.000 −0.416327
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 580.000 0.778523
\(746\) 0 0
\(747\) 1206.00 1.61446
\(748\) 0 0
\(749\) 860.000 1.14820
\(750\) 0 0
\(751\) 1402.00 1.86684 0.933422 − 0.358780i \(-0.116807\pi\)
0.933422 + 0.358780i \(0.116807\pi\)
\(752\) 0 0
\(753\) 1410.00 1.87251
\(754\) 0 0
\(755\) −596.000 −0.789404
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −862.000 −1.12094 −0.560468 − 0.828176i \(-0.689379\pi\)
−0.560468 + 0.828176i \(0.689379\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1154.00 −1.49288 −0.746442 − 0.665450i \(-0.768240\pi\)
−0.746442 + 0.665450i \(0.768240\pi\)
\(774\) 0 0
\(775\) 798.000 1.02968
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1350.00 −1.72414
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −564.000 −0.709434
\(796\) 0 0
\(797\) 1294.00 1.62359 0.811794 − 0.583944i \(-0.198491\pi\)
0.811794 + 0.583944i \(0.198491\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −500.000 −0.622665
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1290.00 1.59851
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 174.000 0.214022
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 670.000 0.816078 0.408039 − 0.912965i \(-0.366213\pi\)
0.408039 + 0.912965i \(0.366213\pi\)
\(822\) 0 0
\(823\) −470.000 −0.571081 −0.285541 − 0.958367i \(-0.592173\pi\)
−0.285541 + 0.958367i \(0.592173\pi\)
\(824\) 0 0
\(825\) 630.000 0.763636
\(826\) 0 0
\(827\) −1546.00 −1.86941 −0.934704 − 0.355428i \(-0.884335\pi\)
−0.934704 + 0.355428i \(0.884335\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1026.00 −1.22581
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1659.00 1.97265
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −338.000 −0.400000
\(846\) 0 0
\(847\) −210.000 −0.247934
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −92.0000 −0.106358
\(866\) 0 0
\(867\) 867.000 1.00000
\(868\) 0 0
\(869\) −580.000 −0.667434
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −1710.00 −1.95876
\(874\) 0 0
\(875\) 920.000 1.05143
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) −1158.00 −1.31741
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 60.0000 0.0677966
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −2300.00 −2.58718
\(890\) 0 0
\(891\) −810.000 −0.909091
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −460.000 −0.513966
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1900.00 2.11346
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 1710.00 1.88119
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −1340.00 −1.46769
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2500.00 −2.72628
\(918\) 0 0
\(919\) −662.000 −0.720348 −0.360174 − 0.932885i \(-0.617283\pi\)
−0.360174 + 0.932885i \(0.617283\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 90.0000 0.0970874
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1490.00 1.59018 0.795091 − 0.606491i \(-0.207423\pi\)
0.795091 + 0.606491i \(0.207423\pi\)
\(938\) 0 0
\(939\) 1590.00 1.69329
\(940\) 0 0
\(941\) 430.000 0.456961 0.228480 − 0.973549i \(-0.426624\pi\)
0.228480 + 0.973549i \(0.426624\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −540.000 −0.571429
\(946\) 0 0
\(947\) −1306.00 −1.37909 −0.689546 − 0.724242i \(-0.742190\pi\)
−0.689546 + 0.724242i \(0.742190\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 1002.00 1.05363
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1500.00 1.56740
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 483.000 0.502601
\(962\) 0 0
\(963\) 774.000 0.803738
\(964\) 0 0
\(965\) −580.000 −0.601036
\(966\) 0 0
\(967\) −1910.00 −1.97518 −0.987590 − 0.157051i \(-0.949801\pi\)
−0.987590 + 0.157051i \(0.949801\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1930.00 −1.98764 −0.993821 − 0.110996i \(-0.964596\pi\)
−0.993821 + 0.110996i \(0.964596\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 388.000 0.393909
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1382.00 −1.39455 −0.697275 − 0.716803i \(-0.745605\pi\)
−0.697275 + 0.716803i \(0.745605\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −404.000 −0.406030
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 96.3.h.b.17.1 1
3.2 odd 2 96.3.h.a.17.1 1
4.3 odd 2 24.3.h.b.5.1 yes 1
8.3 odd 2 24.3.h.a.5.1 ✓ 1
8.5 even 2 96.3.h.a.17.1 1
12.11 even 2 24.3.h.a.5.1 ✓ 1
16.3 odd 4 768.3.e.d.257.1 2
16.5 even 4 768.3.e.c.257.1 2
16.11 odd 4 768.3.e.d.257.2 2
16.13 even 4 768.3.e.c.257.2 2
24.5 odd 2 CM 96.3.h.b.17.1 1
24.11 even 2 24.3.h.b.5.1 yes 1
48.5 odd 4 768.3.e.c.257.2 2
48.11 even 4 768.3.e.d.257.1 2
48.29 odd 4 768.3.e.c.257.1 2
48.35 even 4 768.3.e.d.257.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.3.h.a.5.1 ✓ 1 8.3 odd 2
24.3.h.a.5.1 ✓ 1 12.11 even 2
24.3.h.b.5.1 yes 1 4.3 odd 2
24.3.h.b.5.1 yes 1 24.11 even 2
96.3.h.a.17.1 1 3.2 odd 2
96.3.h.a.17.1 1 8.5 even 2
96.3.h.b.17.1 1 1.1 even 1 trivial
96.3.h.b.17.1 1 24.5 odd 2 CM
768.3.e.c.257.1 2 16.5 even 4
768.3.e.c.257.1 2 48.29 odd 4
768.3.e.c.257.2 2 16.13 even 4
768.3.e.c.257.2 2 48.5 odd 4
768.3.e.d.257.1 2 16.3 odd 4
768.3.e.d.257.1 2 48.11 even 4
768.3.e.d.257.2 2 16.11 odd 4
768.3.e.d.257.2 2 48.35 even 4