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Properties

Label 27.3.b.a.26.1
Level $27$
Weight $3$
Character 27.26
Self dual yes
Analytic conductor $0.736$
Analytic rank $0$
Dimension $1$
CM discriminant -3
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] = [27,3,Mod(26,27)] 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("27.26"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 27.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.735696713773\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 26.1
Character \(\chi\) \(=\) 27.26

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{4} -13.0000 q^{7} -1.00000 q^{13} +16.0000 q^{16} +11.0000 q^{19} +25.0000 q^{25} -52.0000 q^{28} -46.0000 q^{31} +47.0000 q^{37} -22.0000 q^{43} +120.000 q^{49} -4.00000 q^{52} -121.000 q^{61} +64.0000 q^{64} -109.000 q^{67} -97.0000 q^{73} +44.0000 q^{76} +131.000 q^{79} +13.0000 q^{91} +167.000 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0
\(4\) 4.00000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −13.0000 −1.85714 −0.928571 − 0.371154i \(-0.878962\pi\)
−0.928571 + 0.371154i \(0.878962\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.0769231 −0.0384615 − 0.999260i \(-0.512246\pi\)
−0.0384615 + 0.999260i \(0.512246\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 11.0000 0.578947 0.289474 − 0.957186i \(-0.406520\pi\)
0.289474 + 0.957186i \(0.406520\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −52.0000 −1.85714
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −46.0000 −1.48387 −0.741935 − 0.670471i \(-0.766092\pi\)
−0.741935 + 0.670471i \(0.766092\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 47.0000 1.27027 0.635135 − 0.772401i \(-0.280944\pi\)
0.635135 + 0.772401i \(0.280944\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −22.0000 −0.511628 −0.255814 − 0.966726i \(-0.582343\pi\)
−0.255814 + 0.966726i \(0.582343\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 120.000 2.44898
\(50\) 0 0
\(51\) 0 0
\(52\) −4.00000 −0.0769231
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −121.000 −1.98361 −0.991803 − 0.127774i \(-0.959217\pi\)
−0.991803 + 0.127774i \(0.959217\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −109.000 −1.62687 −0.813433 − 0.581659i \(-0.802404\pi\)
−0.813433 + 0.581659i \(0.802404\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −97.0000 −1.32877 −0.664384 − 0.747392i \(-0.731306\pi\)
−0.664384 + 0.747392i \(0.731306\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 44.0000 0.578947
\(77\) 0 0
\(78\) 0 0
\(79\) 131.000 1.65823 0.829114 − 0.559080i \(-0.188845\pi\)
0.829114 + 0.559080i \(0.188845\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 13.0000 0.142857
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 167.000 1.72165 0.860825 − 0.508902i \(-0.169948\pi\)
0.860825 + 0.508902i \(0.169948\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 100.000 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −37.0000 −0.359223 −0.179612 − 0.983738i \(-0.557484\pi\)
−0.179612 + 0.983738i \(0.557484\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −214.000 −1.96330 −0.981651 − 0.190684i \(-0.938929\pi\)
−0.981651 + 0.190684i \(0.938929\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −208.000 −1.85714
\(113\) 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
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −184.000 −1.48387
\(125\) 0 0
\(126\) 0 0
\(127\) 146.000 1.14961 0.574803 − 0.818292i \(-0.305079\pi\)
0.574803 + 0.818292i \(0.305079\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −143.000 −1.07519
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 251.000 1.80576 0.902878 − 0.429898i \(-0.141450\pi\)
0.902878 + 0.429898i \(0.141450\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 188.000 1.27027
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 59.0000 0.390728 0.195364 − 0.980731i \(-0.437411\pi\)
0.195364 + 0.980731i \(0.437411\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −118.000 −0.751592 −0.375796 − 0.926702i \(-0.622631\pi\)
−0.375796 + 0.926702i \(0.622631\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 299.000 1.83436 0.917178 − 0.398478i \(-0.130461\pi\)
0.917178 + 0.398478i \(0.130461\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −168.000 −0.994083
\(170\) 0 0
\(171\) 0 0
\(172\) −88.0000 −0.511628
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −325.000 −1.85714
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −313.000 −1.72928 −0.864641 − 0.502390i \(-0.832454\pi\)
−0.864641 + 0.502390i \(0.832454\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 143.000 0.740933 0.370466 − 0.928846i \(-0.379198\pi\)
0.370466 + 0.928846i \(0.379198\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 480.000 2.44898
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −277.000 −1.39196 −0.695980 − 0.718061i \(-0.745030\pi\)
−0.695980 + 0.718061i \(0.745030\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −16.0000 −0.0769231
\(209\) 0 0
\(210\) 0 0
\(211\) −253.000 −1.19905 −0.599526 − 0.800355i \(-0.704644\pi\)
−0.599526 + 0.800355i \(0.704644\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 598.000 2.75576
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 338.000 1.51570 0.757848 − 0.652432i \(-0.226251\pi\)
0.757848 + 0.652432i \(0.226251\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 26.0000 0.113537 0.0567686 − 0.998387i \(-0.481920\pi\)
0.0567686 + 0.998387i \(0.481920\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 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
\(241\) −193.000 −0.800830 −0.400415 − 0.916334i \(-0.631134\pi\)
−0.400415 + 0.916334i \(0.631134\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −484.000 −1.98361
\(245\) 0 0
\(246\) 0 0
\(247\) −11.0000 −0.0445344
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −611.000 −2.35907
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −436.000 −1.62687
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 299.000 1.10332 0.551661 − 0.834069i \(-0.313994\pi\)
0.551661 + 0.834069i \(0.313994\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 122.000 0.440433 0.220217 − 0.975451i \(-0.429324\pi\)
0.220217 + 0.975451i \(0.429324\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 458.000 1.61837 0.809187 − 0.587551i \(-0.199908\pi\)
0.809187 + 0.587551i \(0.199908\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −388.000 −1.32877
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 286.000 0.950166
\(302\) 0 0
\(303\) 0 0
\(304\) 176.000 0.578947
\(305\) 0 0
\(306\) 0 0
\(307\) −358.000 −1.16612 −0.583062 − 0.812428i \(-0.698145\pi\)
−0.583062 + 0.812428i \(0.698145\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 599.000 1.91374 0.956869 − 0.290520i \(-0.0938282\pi\)
0.956869 + 0.290520i \(0.0938282\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 524.000 1.65823
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −25.0000 −0.0769231
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −661.000 −1.99698 −0.998489 − 0.0549442i \(-0.982502\pi\)
−0.998489 + 0.0549442i \(0.982502\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −649.000 −1.92582 −0.962908 − 0.269830i \(-0.913033\pi\)
−0.962908 + 0.269830i \(0.913033\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −923.000 −2.69096
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 671.000 1.92264 0.961318 − 0.275441i \(-0.0888238\pi\)
0.961318 + 0.275441i \(0.0888238\pi\)
\(350\) 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
\(361\) −240.000 −0.664820
\(362\) 0 0
\(363\) 0 0
\(364\) 52.0000 0.142857
\(365\) 0 0
\(366\) 0 0
\(367\) 491.000 1.33787 0.668937 − 0.743319i \(-0.266749\pi\)
0.668937 + 0.743319i \(0.266749\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −577.000 −1.54692 −0.773458 − 0.633847i \(-0.781475\pi\)
−0.773458 + 0.633847i \(0.781475\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 83.0000 0.218997 0.109499 − 0.993987i \(-0.465075\pi\)
0.109499 + 0.993987i \(0.465075\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 668.000 1.72165
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 362.000 0.911839 0.455919 − 0.890021i \(-0.349311\pi\)
0.455919 + 0.890021i \(0.349311\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 400.000 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 46.0000 0.114144
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 143.000 0.349633 0.174817 − 0.984601i \(-0.444067\pi\)
0.174817 + 0.984601i \(0.444067\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −148.000 −0.359223
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −481.000 −1.14252 −0.571259 − 0.820770i \(-0.693545\pi\)
−0.571259 + 0.820770i \(0.693545\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1573.00 3.68384
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −862.000 −1.99076 −0.995381 − 0.0960028i \(-0.969394\pi\)
−0.995381 + 0.0960028i \(0.969394\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −856.000 −1.96330
\(437\) 0 0
\(438\) 0 0
\(439\) −94.0000 −0.214123 −0.107062 − 0.994252i \(-0.534144\pi\)
−0.107062 + 0.994252i \(0.534144\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −832.000 −1.85714
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −814.000 −1.78118 −0.890591 − 0.454805i \(-0.849709\pi\)
−0.890591 + 0.454805i \(0.849709\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −397.000 −0.857451 −0.428726 − 0.903435i \(-0.641037\pi\)
−0.428726 + 0.903435i \(0.641037\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 1417.00 3.02132
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 275.000 0.578947
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −47.0000 −0.0977131
\(482\) 0 0
\(483\) 0 0
\(484\) 484.000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −349.000 −0.716632 −0.358316 − 0.933600i \(-0.616649\pi\)
−0.358316 + 0.933600i \(0.616649\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −736.000 −1.48387
\(497\) 0 0
\(498\) 0 0
\(499\) 26.0000 0.0521042 0.0260521 − 0.999661i \(-0.491706\pi\)
0.0260521 + 0.999661i \(0.491706\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 584.000 1.14961
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 1261.00 2.46771
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 803.000 1.53537 0.767686 − 0.640826i \(-0.221408\pi\)
0.767686 + 0.640826i \(0.221408\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −572.000 −1.07519
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −241.000 −0.445471 −0.222736 − 0.974879i \(-0.571499\pi\)
−0.222736 + 0.974879i \(0.571499\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1093.00 −1.99817 −0.999086 − 0.0427471i \(-0.986389\pi\)
−0.999086 + 0.0427471i \(0.986389\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1703.00 −3.07957
\(554\) 0 0
\(555\) 0 0
\(556\) 1004.00 1.80576
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 22.0000 0.0393560
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −181.000 −0.316988 −0.158494 − 0.987360i \(-0.550664\pi\)
−0.158494 + 0.987360i \(0.550664\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1033.00 −1.79029 −0.895147 − 0.445770i \(-0.852930\pi\)
−0.895147 + 0.445770i \(0.852930\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) −506.000 −0.859083
\(590\) 0 0
\(591\) 0 0
\(592\) 752.000 1.27027
\(593\) 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
\(601\) −526.000 −0.875208 −0.437604 − 0.899168i \(-0.644173\pi\)
−0.437604 + 0.899168i \(0.644173\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 236.000 0.390728
\(605\) 0 0
\(606\) 0 0
\(607\) 1187.00 1.95552 0.977759 − 0.209729i \(-0.0672583\pi\)
0.977759 + 0.209729i \(0.0672583\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 983.000 1.60359 0.801794 − 0.597600i \(-0.203879\pi\)
0.801794 + 0.597600i \(0.203879\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −949.000 −1.53312 −0.766559 − 0.642174i \(-0.778033\pi\)
−0.766559 + 0.642174i \(0.778033\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −472.000 −0.751592
\(629\) 0 0
\(630\) 0 0
\(631\) 587.000 0.930269 0.465135 − 0.885240i \(-0.346006\pi\)
0.465135 + 0.885240i \(0.346006\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −120.000 −0.188383
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 314.000 0.488336 0.244168 − 0.969733i \(-0.421485\pi\)
0.244168 + 0.969733i \(0.421485\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 1196.00 1.83436
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1079.00 1.63238 0.816188 − 0.577787i \(-0.196084\pi\)
0.816188 + 0.577787i \(0.196084\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 23.0000 0.0341753 0.0170877 − 0.999854i \(-0.494561\pi\)
0.0170877 + 0.999854i \(0.494561\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −672.000 −0.994083
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −2171.00 −3.19735
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −352.000 −0.511628
\(689\) 0 0
\(690\) 0 0
\(691\) −1318.00 −1.90738 −0.953690 − 0.300790i \(-0.902750\pi\)
−0.953690 + 0.300790i \(0.902750\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1300.00 −1.85714
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 517.000 0.735420
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1391.00 1.96192 0.980959 − 0.194214i \(-0.0622158\pi\)
0.980959 + 0.194214i \(0.0622158\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 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
\(721\) 481.000 0.667129
\(722\) 0 0
\(723\) 0 0
\(724\) −1252.00 −1.72928
\(725\) 0 0
\(726\) 0 0
\(727\) 482.000 0.662999 0.331499 − 0.943455i \(-0.392446\pi\)
0.331499 + 0.943455i \(0.392446\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1034.00 1.41064 0.705321 − 0.708888i \(-0.250803\pi\)
0.705321 + 0.708888i \(0.250803\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1222.00 −1.65359 −0.826793 − 0.562506i \(-0.809837\pi\)
−0.826793 + 0.562506i \(0.809837\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 179.000 0.238349 0.119174 − 0.992873i \(-0.461975\pi\)
0.119174 + 0.992873i \(0.461975\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −673.000 −0.889036 −0.444518 − 0.895770i \(-0.646625\pi\)
−0.444518 + 0.895770i \(0.646625\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 2782.00 3.64613
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 863.000 1.12224 0.561118 − 0.827736i \(-0.310371\pi\)
0.561118 + 0.827736i \(0.310371\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 572.000 0.740933
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −1150.00 −1.48387
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1920.00 2.44898
\(785\) 0 0
\(786\) 0 0
\(787\) −613.000 −0.778907 −0.389454 − 0.921046i \(-0.627336\pi\)
−0.389454 + 0.921046i \(0.627336\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 121.000 0.152585
\(794\) 0 0
\(795\) 0 0
\(796\) −1108.00 −1.39196
\(797\) 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
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 1514.00 1.86683 0.933416 − 0.358797i \(-0.116813\pi\)
0.933416 + 0.358797i \(0.116813\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −242.000 −0.296206
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −1621.00 −1.96962 −0.984812 − 0.173626i \(-0.944452\pi\)
−0.984812 + 0.173626i \(0.944452\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1609.00 −1.94089 −0.970446 − 0.241317i \(-0.922421\pi\)
−0.970446 + 0.241317i \(0.922421\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −64.0000 −0.0769231
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −1012.00 −1.19905
\(845\) 0 0
\(846\) 0 0
\(847\) −1573.00 −1.85714
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −481.000 −0.563892 −0.281946 − 0.959430i \(-0.590980\pi\)
−0.281946 + 0.959430i \(0.590980\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1549.00 −1.80326 −0.901630 − 0.432509i \(-0.857629\pi\)
−0.901630 + 0.432509i \(0.857629\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 2392.00 2.75576
\(869\) 0 0
\(870\) 0 0
\(871\) 109.000 0.125144
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1727.00 1.96921 0.984607 − 0.174785i \(-0.0559231\pi\)
0.984607 + 0.174785i \(0.0559231\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 443.000 0.501699 0.250849 − 0.968026i \(-0.419290\pi\)
0.250849 + 0.968026i \(0.419290\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −1898.00 −2.13498
\(890\) 0 0
\(891\) 0 0
\(892\) 1352.00 1.51570
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 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
\(907\) −1453.00 −1.60198 −0.800992 − 0.598675i \(-0.795694\pi\)
−0.800992 + 0.598675i \(0.795694\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 104.000 0.113537
\(917\) 0 0
\(918\) 0 0
\(919\) 866.000 0.942329 0.471164 − 0.882045i \(-0.343834\pi\)
0.471164 + 0.882045i \(0.343834\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1175.00 1.27027
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1320.00 1.41783
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1847.00 1.97118 0.985592 − 0.169138i \(-0.0540985\pi\)
0.985592 + 0.169138i \(0.0540985\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 97.0000 0.102213
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1155.00 1.20187
\(962\) 0 0
\(963\) 0 0
\(964\) −772.000 −0.800830
\(965\) 0 0
\(966\) 0 0
\(967\) −253.000 −0.261634 −0.130817 − 0.991407i \(-0.541760\pi\)
−0.130817 + 0.991407i \(0.541760\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) −3263.00 −3.35355
\(974\) 0 0
\(975\) 0 0
\(976\) −1936.00 −1.98361
\(977\) 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
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −44.0000 −0.0445344
\(989\) 0 0
\(990\) 0 0
\(991\) 1739.00 1.75479 0.877397 − 0.479766i \(-0.159278\pi\)
0.877397 + 0.479766i \(0.159278\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1894.00 −1.89970 −0.949850 − 0.312707i \(-0.898764\pi\)
−0.949850 + 0.312707i \(0.898764\pi\)
\(998\) 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 27.3.b.a.26.1 ✓ 1
3.2 odd 2 CM 27.3.b.a.26.1 ✓ 1
4.3 odd 2 432.3.e.b.161.1 1
5.2 odd 4 675.3.d.c.674.1 2
5.3 odd 4 675.3.d.c.674.2 2
5.4 even 2 675.3.c.c.26.1 1
8.3 odd 2 1728.3.e.d.1025.1 1
8.5 even 2 1728.3.e.a.1025.1 1
9.2 odd 6 81.3.d.a.53.1 2
9.4 even 3 81.3.d.a.26.1 2
9.5 odd 6 81.3.d.a.26.1 2
9.7 even 3 81.3.d.a.53.1 2
12.11 even 2 432.3.e.b.161.1 1
15.2 even 4 675.3.d.c.674.1 2
15.8 even 4 675.3.d.c.674.2 2
15.14 odd 2 675.3.c.c.26.1 1
24.5 odd 2 1728.3.e.a.1025.1 1
24.11 even 2 1728.3.e.d.1025.1 1
36.7 odd 6 1296.3.q.a.1025.1 2
36.11 even 6 1296.3.q.a.1025.1 2
36.23 even 6 1296.3.q.a.593.1 2
36.31 odd 6 1296.3.q.a.593.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.3.b.a.26.1 ✓ 1 1.1 even 1 trivial
27.3.b.a.26.1 ✓ 1 3.2 odd 2 CM
81.3.d.a.26.1 2 9.4 even 3
81.3.d.a.26.1 2 9.5 odd 6
81.3.d.a.53.1 2 9.2 odd 6
81.3.d.a.53.1 2 9.7 even 3
432.3.e.b.161.1 1 4.3 odd 2
432.3.e.b.161.1 1 12.11 even 2
675.3.c.c.26.1 1 5.4 even 2
675.3.c.c.26.1 1 15.14 odd 2
675.3.d.c.674.1 2 5.2 odd 4
675.3.d.c.674.1 2 15.2 even 4
675.3.d.c.674.2 2 5.3 odd 4
675.3.d.c.674.2 2 15.8 even 4
1296.3.q.a.593.1 2 36.23 even 6
1296.3.q.a.593.1 2 36.31 odd 6
1296.3.q.a.1025.1 2 36.7 odd 6
1296.3.q.a.1025.1 2 36.11 even 6
1728.3.e.a.1025.1 1 8.5 even 2
1728.3.e.a.1025.1 1 24.5 odd 2
1728.3.e.d.1025.1 1 8.3 odd 2
1728.3.e.d.1025.1 1 24.11 even 2