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Properties

Label 816.3.m.a.305.1
Level $816$
Weight $3$
Character 816.305
Self dual yes
Analytic conductor $22.234$
Analytic rank $0$
Dimension $1$
CM discriminant -51
Inner twists $2$

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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] = [816,3,Mod(305,816)] 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("816.305"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(816, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 816.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3,0,-7,0,0,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.2343895718\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 305.1
Character \(\chi\) \(=\) 816.305

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -7.00000 q^{5} +9.00000 q^{9} -5.00000 q^{11} -25.0000 q^{13} +21.0000 q^{15} -17.0000 q^{17} +13.0000 q^{19} -29.0000 q^{23} +24.0000 q^{25} -27.0000 q^{27} -10.0000 q^{29} +15.0000 q^{33} +75.0000 q^{39} +65.0000 q^{41} -35.0000 q^{43} -63.0000 q^{45} +49.0000 q^{49} +51.0000 q^{51} +35.0000 q^{55} -39.0000 q^{57} +175.000 q^{65} +70.0000 q^{67} +87.0000 q^{69} +130.000 q^{71} -72.0000 q^{75} +81.0000 q^{81} +119.000 q^{85} +30.0000 q^{87} -91.0000 q^{95} -45.0000 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/816\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(511\) \(545\) \(613\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(5\) −7.00000 −1.40000 −0.700000 − 0.714143i \(-0.746817\pi\)
−0.700000 + 0.714143i \(0.746817\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) −5.00000 −0.454545 −0.227273 − 0.973831i \(-0.572981\pi\)
−0.227273 + 0.973831i \(0.572981\pi\)
\(12\) 0 0
\(13\) −25.0000 −1.92308 −0.961538 − 0.274670i \(-0.911431\pi\)
−0.961538 + 0.274670i \(0.911431\pi\)
\(14\) 0 0
\(15\) 21.0000 1.40000
\(16\) 0 0
\(17\) −17.0000 −1.00000
\(18\) 0 0
\(19\) 13.0000 0.684211 0.342105 − 0.939662i \(-0.388860\pi\)
0.342105 + 0.939662i \(0.388860\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −29.0000 −1.26087 −0.630435 − 0.776242i \(-0.717123\pi\)
−0.630435 + 0.776242i \(0.717123\pi\)
\(24\) 0 0
\(25\) 24.0000 0.960000
\(26\) 0 0
\(27\) −27.0000 −1.00000
\(28\) 0 0
\(29\) −10.0000 −0.344828 −0.172414 − 0.985025i \(-0.555157\pi\)
−0.172414 + 0.985025i \(0.555157\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 15.0000 0.454545
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 75.0000 1.92308
\(40\) 0 0
\(41\) 65.0000 1.58537 0.792683 − 0.609634i \(-0.208684\pi\)
0.792683 + 0.609634i \(0.208684\pi\)
\(42\) 0 0
\(43\) −35.0000 −0.813953 −0.406977 − 0.913439i \(-0.633417\pi\)
−0.406977 + 0.913439i \(0.633417\pi\)
\(44\) 0 0
\(45\) −63.0000 −1.40000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 51.0000 1.00000
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 35.0000 0.636364
\(56\) 0 0
\(57\) −39.0000 −0.684211
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 175.000 2.69231
\(66\) 0 0
\(67\) 70.0000 1.04478 0.522388 − 0.852708i \(-0.325041\pi\)
0.522388 + 0.852708i \(0.325041\pi\)
\(68\) 0 0
\(69\) 87.0000 1.26087
\(70\) 0 0
\(71\) 130.000 1.83099 0.915493 − 0.402334i \(-0.131801\pi\)
0.915493 + 0.402334i \(0.131801\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −72.0000 −0.960000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 119.000 1.40000
\(86\) 0 0
\(87\) 30.0000 0.344828
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −91.0000 −0.957895
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) −45.0000 −0.454545
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −155.000 −1.50485 −0.752427 − 0.658675i \(-0.771117\pi\)
−0.752427 + 0.658675i \(0.771117\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 211.000 1.97196 0.985981 − 0.166856i \(-0.0533614\pi\)
0.985981 + 0.166856i \(0.0533614\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −199.000 −1.76106 −0.880531 − 0.473989i \(-0.842814\pi\)
−0.880531 + 0.473989i \(0.842814\pi\)
\(114\) 0 0
\(115\) 203.000 1.76522
\(116\) 0 0
\(117\) −225.000 −1.92308
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −96.0000 −0.793388
\(122\) 0 0
\(123\) −195.000 −1.58537
\(124\) 0 0
\(125\) 7.00000 0.0560000
\(126\) 0 0
\(127\) 205.000 1.61417 0.807087 − 0.590433i \(-0.201043\pi\)
0.807087 + 0.590433i \(0.201043\pi\)
\(128\) 0 0
\(129\) 105.000 0.813953
\(130\) 0 0
\(131\) −245.000 −1.87023 −0.935115 − 0.354346i \(-0.884704\pi\)
−0.935115 + 0.354346i \(0.884704\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 189.000 1.40000
\(136\) 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
\(143\) 125.000 0.874126
\(144\) 0 0
\(145\) 70.0000 0.482759
\(146\) 0 0
\(147\) −147.000 −1.00000
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −98.0000 −0.649007 −0.324503 − 0.945885i \(-0.605197\pi\)
−0.324503 + 0.945885i \(0.605197\pi\)
\(152\) 0 0
\(153\) −153.000 −1.00000
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −145.000 −0.923567 −0.461783 − 0.886993i \(-0.652790\pi\)
−0.461783 + 0.886993i \(0.652790\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) −105.000 −0.636364
\(166\) 0 0
\(167\) 91.0000 0.544910 0.272455 − 0.962169i \(-0.412164\pi\)
0.272455 + 0.962169i \(0.412164\pi\)
\(168\) 0 0
\(169\) 456.000 2.69822
\(170\) 0 0
\(171\) 117.000 0.684211
\(172\) 0 0
\(173\) 329.000 1.90173 0.950867 − 0.309599i \(-0.100195\pi\)
0.950867 + 0.309599i \(0.100195\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 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
\(187\) 85.0000 0.454545
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) −525.000 −2.69231
\(196\) 0 0
\(197\) −31.0000 −0.157360 −0.0786802 − 0.996900i \(-0.525071\pi\)
−0.0786802 + 0.996900i \(0.525071\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) −210.000 −1.04478
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −455.000 −2.21951
\(206\) 0 0
\(207\) −261.000 −1.26087
\(208\) 0 0
\(209\) −65.0000 −0.311005
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) −390.000 −1.83099
\(214\) 0 0
\(215\) 245.000 1.13953
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 425.000 1.92308
\(222\) 0 0
\(223\) −395.000 −1.77130 −0.885650 − 0.464353i \(-0.846287\pi\)
−0.885650 + 0.464353i \(0.846287\pi\)
\(224\) 0 0
\(225\) 216.000 0.960000
\(226\) 0 0
\(227\) 379.000 1.66960 0.834802 − 0.550551i \(-0.185582\pi\)
0.834802 + 0.550551i \(0.185582\pi\)
\(228\) 0 0
\(229\) −358.000 −1.56332 −0.781659 − 0.623706i \(-0.785626\pi\)
−0.781659 + 0.623706i \(0.785626\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 41.0000 0.175966 0.0879828 − 0.996122i \(-0.471958\pi\)
0.0879828 + 0.996122i \(0.471958\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −243.000 −1.00000
\(244\) 0 0
\(245\) −343.000 −1.40000
\(246\) 0 0
\(247\) −325.000 −1.31579
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 145.000 0.573123
\(254\) 0 0
\(255\) −357.000 −1.40000
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −90.0000 −0.344828
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −295.000 −1.09665 −0.548327 − 0.836264i \(-0.684735\pi\)
−0.548327 + 0.836264i \(0.684735\pi\)
\(270\) 0 0
\(271\) −83.0000 −0.306273 −0.153137 − 0.988205i \(-0.548937\pi\)
−0.153137 + 0.988205i \(0.548937\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −120.000 −0.436364
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 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
\(285\) 273.000 0.957895
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 135.000 0.454545
\(298\) 0 0
\(299\) 725.000 2.42475
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −410.000 −1.33550 −0.667752 − 0.744383i \(-0.732744\pi\)
−0.667752 + 0.744383i \(0.732744\pi\)
\(308\) 0 0
\(309\) 465.000 1.50485
\(310\) 0 0
\(311\) −350.000 −1.12540 −0.562701 − 0.826661i \(-0.690238\pi\)
−0.562701 + 0.826661i \(0.690238\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 566.000 1.78549 0.892744 − 0.450563i \(-0.148777\pi\)
0.892744 + 0.450563i \(0.148777\pi\)
\(318\) 0 0
\(319\) 50.0000 0.156740
\(320\) 0 0
\(321\) −633.000 −1.97196
\(322\) 0 0
\(323\) −221.000 −0.684211
\(324\) 0 0
\(325\) −600.000 −1.84615
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 613.000 1.85196 0.925982 − 0.377568i \(-0.123240\pi\)
0.925982 + 0.377568i \(0.123240\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −490.000 −1.46269
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 597.000 1.76106
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −609.000 −1.76522
\(346\) 0 0
\(347\) 394.000 1.13545 0.567723 − 0.823219i \(-0.307824\pi\)
0.567723 + 0.823219i \(0.307824\pi\)
\(348\) 0 0
\(349\) −577.000 −1.65330 −0.826648 − 0.562720i \(-0.809755\pi\)
−0.826648 + 0.562720i \(0.809755\pi\)
\(350\) 0 0
\(351\) 675.000 1.92308
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −910.000 −2.56338
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −192.000 −0.531856
\(362\) 0 0
\(363\) 288.000 0.793388
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 585.000 1.58537
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −70.0000 −0.187668 −0.0938338 − 0.995588i \(-0.529912\pi\)
−0.0938338 + 0.995588i \(0.529912\pi\)
\(374\) 0 0
\(375\) −21.0000 −0.0560000
\(376\) 0 0
\(377\) 250.000 0.663130
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −615.000 −1.61417
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −315.000 −0.813953
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 493.000 1.26087
\(392\) 0 0
\(393\) 735.000 1.87023
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 785.000 1.95761 0.978803 − 0.204804i \(-0.0656557\pi\)
0.978803 + 0.204804i \(0.0656557\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −567.000 −1.40000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −457.000 −1.11736 −0.558680 − 0.829383i \(-0.688692\pi\)
−0.558680 + 0.829383i \(0.688692\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 250.000 0.596659 0.298329 − 0.954463i \(-0.403571\pi\)
0.298329 + 0.954463i \(0.403571\pi\)
\(420\) 0 0
\(421\) 383.000 0.909739 0.454869 − 0.890558i \(-0.349686\pi\)
0.454869 + 0.890558i \(0.349686\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −408.000 −0.960000
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −375.000 −0.874126
\(430\) 0 0
\(431\) −590.000 −1.36891 −0.684455 − 0.729055i \(-0.739960\pi\)
−0.684455 + 0.729055i \(0.739960\pi\)
\(432\) 0 0
\(433\) 815.000 1.88222 0.941109 − 0.338105i \(-0.109786\pi\)
0.941109 + 0.338105i \(0.109786\pi\)
\(434\) 0 0
\(435\) −210.000 −0.482759
\(436\) 0 0
\(437\) −377.000 −0.862700
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 830.000 1.84855 0.924276 − 0.381724i \(-0.124670\pi\)
0.924276 + 0.381724i \(0.124670\pi\)
\(450\) 0 0
\(451\) −325.000 −0.720621
\(452\) 0 0
\(453\) 294.000 0.649007
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 455.000 0.995624 0.497812 − 0.867285i \(-0.334137\pi\)
0.497812 + 0.867285i \(0.334137\pi\)
\(458\) 0 0
\(459\) 459.000 1.00000
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 910.000 1.96544 0.982721 − 0.185091i \(-0.0592580\pi\)
0.982721 + 0.185091i \(0.0592580\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 435.000 0.923567
\(472\) 0 0
\(473\) 175.000 0.369979
\(474\) 0 0
\(475\) 312.000 0.656842
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −125.000 −0.260960 −0.130480 − 0.991451i \(-0.541652\pi\)
−0.130480 + 0.991451i \(0.541652\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 170.000 0.344828
\(494\) 0 0
\(495\) 315.000 0.636364
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) −273.000 −0.544910
\(502\) 0 0
\(503\) −581.000 −1.15507 −0.577535 − 0.816366i \(-0.695985\pi\)
−0.577535 + 0.816366i \(0.695985\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1368.00 −2.69822
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −351.000 −0.684211
\(514\) 0 0
\(515\) 1085.00 2.10680
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −987.000 −1.90173
\(520\) 0 0
\(521\) −1015.00 −1.94818 −0.974088 − 0.226168i \(-0.927380\pi\)
−0.974088 + 0.226168i \(0.927380\pi\)
\(522\) 0 0
\(523\) 790.000 1.51052 0.755258 − 0.655427i \(-0.227512\pi\)
0.755258 + 0.655427i \(0.227512\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 312.000 0.589792
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1625.00 −3.04878
\(534\) 0 0
\(535\) −1477.00 −2.76075
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −245.000 −0.454545
\(540\) 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
\(551\) −130.000 −0.235935
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 875.000 1.56530
\(560\) 0 0
\(561\) −255.000 −0.454545
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 1393.00 2.46549
\(566\) 0 0
\(567\) 0 0
\(568\) 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
\(575\) −696.000 −1.21043
\(576\) 0 0
\(577\) 695.000 1.20451 0.602253 − 0.798305i \(-0.294270\pi\)
0.602253 + 0.798305i \(0.294270\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 1575.00 2.69231
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 93.0000 0.157360
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 630.000 1.04478
\(604\) 0 0
\(605\) 672.000 1.11074
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1175.00 1.91680 0.958401 − 0.285424i \(-0.0921345\pi\)
0.958401 + 0.285424i \(0.0921345\pi\)
\(614\) 0 0
\(615\) 1365.00 2.21951
\(616\) 0 0
\(617\) −466.000 −0.755267 −0.377634 − 0.925955i \(-0.623262\pi\)
−0.377634 + 0.925955i \(0.623262\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 783.000 1.26087
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −649.000 −1.03840
\(626\) 0 0
\(627\) 195.000 0.311005
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1237.00 1.96038 0.980190 − 0.198059i \(-0.0634636\pi\)
0.980190 + 0.198059i \(0.0634636\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1435.00 −2.25984
\(636\) 0 0
\(637\) −1225.00 −1.92308
\(638\) 0 0
\(639\) 1170.00 1.83099
\(640\) 0 0
\(641\) −775.000 −1.20905 −0.604524 − 0.796587i \(-0.706637\pi\)
−0.604524 + 0.796587i \(0.706637\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) −735.000 −1.13953
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 881.000 1.34916 0.674579 − 0.738203i \(-0.264325\pi\)
0.674579 + 0.738203i \(0.264325\pi\)
\(654\) 0 0
\(655\) 1715.00 2.61832
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 47.0000 0.0711044 0.0355522 − 0.999368i \(-0.488681\pi\)
0.0355522 + 0.999368i \(0.488681\pi\)
\(662\) 0 0
\(663\) −1275.00 −1.92308
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 290.000 0.434783
\(668\) 0 0
\(669\) 1185.00 1.77130
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −648.000 −0.960000
\(676\) 0 0
\(677\) 521.000 0.769572 0.384786 − 0.923006i \(-0.374275\pi\)
0.384786 + 0.923006i \(0.374275\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −1137.00 −1.66960
\(682\) 0 0
\(683\) 691.000 1.01171 0.505857 − 0.862618i \(-0.331177\pi\)
0.505857 + 0.862618i \(0.331177\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1074.00 1.56332
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1105.00 −1.58537
\(698\) 0 0
\(699\) −123.000 −0.175966
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −875.000 −1.22378
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1435.00 1.99583 0.997914 − 0.0645609i \(-0.0205647\pi\)
0.997914 + 0.0645609i \(0.0205647\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −240.000 −0.331034
\(726\) 0 0
\(727\) −1250.00 −1.71939 −0.859697 − 0.510804i \(-0.829348\pi\)
−0.859697 + 0.510804i \(0.829348\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 595.000 0.813953
\(732\) 0 0
\(733\) 650.000 0.886767 0.443383 − 0.896332i \(-0.353778\pi\)
0.443383 + 0.896332i \(0.353778\pi\)
\(734\) 0 0
\(735\) 1029.00 1.40000
\(736\) 0 0
\(737\) −350.000 −0.474898
\(738\) 0 0
\(739\) −203.000 −0.274696 −0.137348 − 0.990523i \(-0.543858\pi\)
−0.137348 + 0.990523i \(0.543858\pi\)
\(740\) 0 0
\(741\) 975.000 1.31579
\(742\) 0 0
\(743\) −1214.00 −1.63392 −0.816958 − 0.576697i \(-0.804341\pi\)
−0.816958 + 0.576697i \(0.804341\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 686.000 0.908609
\(756\) 0 0
\(757\) −985.000 −1.30119 −0.650594 − 0.759425i \(-0.725480\pi\)
−0.650594 + 0.759425i \(0.725480\pi\)
\(758\) 0 0
\(759\) −435.000 −0.573123
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1071.00 1.40000
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1487.00 1.93368 0.966840 − 0.255383i \(-0.0822015\pi\)
0.966840 + 0.255383i \(0.0822015\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 845.000 1.08472
\(780\) 0 0
\(781\) −650.000 −0.832266
\(782\) 0 0
\(783\) 270.000 0.344828
\(784\) 0 0
\(785\) 1015.00 1.29299
\(786\) 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
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 885.000 1.09665
\(808\) 0 0
\(809\) −1255.00 −1.55130 −0.775649 − 0.631165i \(-0.782577\pi\)
−0.775649 + 0.631165i \(0.782577\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 249.000 0.306273
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −455.000 −0.556916
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1625.00 1.97929 0.989647 − 0.143524i \(-0.0458436\pi\)
0.989647 + 0.143524i \(0.0458436\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 360.000 0.436364
\(826\) 0 0
\(827\) −1229.00 −1.48609 −0.743047 − 0.669239i \(-0.766620\pi\)
−0.743047 + 0.669239i \(0.766620\pi\)
\(828\) 0 0
\(829\) 842.000 1.01568 0.507841 − 0.861451i \(-0.330444\pi\)
0.507841 + 0.861451i \(0.330444\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −833.000 −1.00000
\(834\) 0 0
\(835\) −637.000 −0.762874
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −845.000 −1.00715 −0.503576 − 0.863951i \(-0.667982\pi\)
−0.503576 + 0.863951i \(0.667982\pi\)
\(840\) 0 0
\(841\) −741.000 −0.881094
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3192.00 −3.77751
\(846\) 0 0
\(847\) 0 0
\(848\) 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
\(855\) −819.000 −0.957895
\(856\) 0 0
\(857\) 14.0000 0.0163361 0.00816803 − 0.999967i \(-0.497400\pi\)
0.00816803 + 0.999967i \(0.497400\pi\)
\(858\) 0 0
\(859\) 118.000 0.137369 0.0686845 − 0.997638i \(-0.478120\pi\)
0.0686845 + 0.997638i \(0.478120\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −2303.00 −2.66243
\(866\) 0 0
\(867\) −867.000 −1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −1750.00 −2.00918
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1570.00 −1.78207 −0.891033 − 0.453939i \(-0.850019\pi\)
−0.891033 + 0.453939i \(0.850019\pi\)
\(882\) 0 0
\(883\) −1715.00 −1.94224 −0.971121 − 0.238587i \(-0.923316\pi\)
−0.971121 + 0.238587i \(0.923316\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1099.00 1.23901 0.619504 − 0.784994i \(-0.287334\pi\)
0.619504 + 0.784994i \(0.287334\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −405.000 −0.454545
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2175.00 −2.42475
\(898\) 0 0
\(899\) 0 0
\(900\) 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
\(909\) 0 0
\(910\) 0 0
\(911\) 235.000 0.257958 0.128979 − 0.991647i \(-0.458830\pi\)
0.128979 + 0.991647i \(0.458830\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −563.000 −0.612622 −0.306311 − 0.951931i \(-0.599095\pi\)
−0.306311 + 0.951931i \(0.599095\pi\)
\(920\) 0 0
\(921\) 1230.00 1.33550
\(922\) 0 0
\(923\) −3250.00 −3.52113
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1395.00 −1.50485
\(928\) 0 0
\(929\) 1025.00 1.10334 0.551668 − 0.834063i \(-0.313991\pi\)
0.551668 + 0.834063i \(0.313991\pi\)
\(930\) 0 0
\(931\) 637.000 0.684211
\(932\) 0 0
\(933\) 1050.00 1.12540
\(934\) 0 0
\(935\) −595.000 −0.636364
\(936\) 0 0
\(937\) −1390.00 −1.48346 −0.741729 − 0.670700i \(-0.765994\pi\)
−0.741729 + 0.670700i \(0.765994\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1450.00 −1.54091 −0.770457 − 0.637492i \(-0.779972\pi\)
−0.770457 + 0.637492i \(0.779972\pi\)
\(942\) 0 0
\(943\) −1885.00 −1.99894
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −806.000 −0.851109 −0.425554 − 0.904933i \(-0.639921\pi\)
−0.425554 + 0.904933i \(0.639921\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −1698.00 −1.78549
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −150.000 −0.156740
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 0 0
\(963\) 1899.00 1.97196
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 565.000 0.584281 0.292141 − 0.956375i \(-0.405632\pi\)
0.292141 + 0.956375i \(0.405632\pi\)
\(968\) 0 0
\(969\) 663.000 0.684211
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 1800.00 1.84615
\(976\) 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
\(983\) 91.0000 0.0925738 0.0462869 − 0.998928i \(-0.485261\pi\)
0.0462869 + 0.998928i \(0.485261\pi\)
\(984\) 0 0
\(985\) 217.000 0.220305
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1015.00 1.02629
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −1839.00 −1.85196
\(994\) 0 0
\(995\) 0 0
\(996\) 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 816.3.m.a.305.1 1
3.2 odd 2 816.3.m.b.305.1 1
4.3 odd 2 51.3.c.b.50.1 yes 1
12.11 even 2 51.3.c.a.50.1 ✓ 1
17.16 even 2 816.3.m.b.305.1 1
51.50 odd 2 CM 816.3.m.a.305.1 1
68.67 odd 2 51.3.c.a.50.1 ✓ 1
204.203 even 2 51.3.c.b.50.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
51.3.c.a.50.1 ✓ 1 12.11 even 2
51.3.c.a.50.1 ✓ 1 68.67 odd 2
51.3.c.b.50.1 yes 1 4.3 odd 2
51.3.c.b.50.1 yes 1 204.203 even 2
816.3.m.a.305.1 1 1.1 even 1 trivial
816.3.m.a.305.1 1 51.50 odd 2 CM
816.3.m.b.305.1 1 3.2 odd 2
816.3.m.b.305.1 1 17.16 even 2