↑

Properties

Label 588.3.c.b.197.1
Level $588$
Weight $3$
Character 588.197
Self dual yes
Analytic conductor $16.022$
Analytic rank $0$
Dimension $1$
CM discriminant -3
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] = [588,3,Mod(197,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.197");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.c (of order \(2\), degree \(1\), 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: \(16.0218395444\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 197.1
Character \(\chi\) \(=\) 588.197

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +9.00000 q^{9} +1.00000 q^{13} +37.0000 q^{19} +25.0000 q^{25} +27.0000 q^{27} -59.0000 q^{31} +47.0000 q^{37} +3.00000 q^{39} +83.0000 q^{43} +111.000 q^{57} -74.0000 q^{61} -109.000 q^{67} -143.000 q^{73} +75.0000 q^{75} +131.000 q^{79} +81.0000 q^{81} -177.000 q^{93} -2.00000 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 1.00000 0.0769231 0.0384615 − 0.999260i \(-0.487754\pi\)
0.0384615 + 0.999260i \(0.487754\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 37.0000 1.94737 0.973684 − 0.227901i \(-0.0731864\pi\)
0.973684 + 0.227901i \(0.0731864\pi\)
\(20\) 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
\(27\) 27.0000 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −59.0000 −1.90323 −0.951613 − 0.307299i \(-0.900575\pi\)
−0.951613 + 0.307299i \(0.900575\pi\)
\(32\) 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
\(39\) 3.00000 0.0769231
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 83.0000 1.93023 0.965116 − 0.261822i \(-0.0843232\pi\)
0.965116 + 0.261822i \(0.0843232\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 111.000 1.94737
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −74.0000 −1.21311 −0.606557 − 0.795040i \(-0.707450\pi\)
−0.606557 + 0.795040i \(0.707450\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 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
\(73\) −143.000 −1.95890 −0.979452 − 0.201677i \(-0.935361\pi\)
−0.979452 + 0.201677i \(0.935361\pi\)
\(74\) 0 0
\(75\) 75.0000 1.00000
\(76\) 0 0
\(77\) 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
\(81\) 81.0000 1.00000
\(82\) 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
\(91\) 0 0
\(92\) 0 0
\(93\) −177.000 −1.90323
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.0206186 −0.0103093 − 0.999947i \(-0.503282\pi\)
−0.0103093 + 0.999947i \(0.503282\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 37.0000 0.359223 0.179612 − 0.983738i \(-0.442516\pi\)
0.179612 + 0.983738i \(0.442516\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 143.000 1.31193 0.655963 − 0.754793i \(-0.272263\pi\)
0.655963 + 0.754793i \(0.272263\pi\)
\(110\) 0 0
\(111\) 141.000 1.27027
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.00000 0.0769231
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −253.000 −1.99213 −0.996063 − 0.0886483i \(-0.971745\pi\)
−0.996063 + 0.0886483i \(0.971745\pi\)
\(128\) 0 0
\(129\) 249.000 1.93023
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 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 \(-0.858550\pi\)
−0.902878 + 0.429898i \(0.858550\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −286.000 −1.89404 −0.947020 − 0.321175i \(-0.895922\pi\)
−0.947020 + 0.321175i \(0.895922\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 118.000 0.751592 0.375796 − 0.926702i \(-0.377369\pi\)
0.375796 + 0.926702i \(0.377369\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −262.000 −1.60736 −0.803681 − 0.595060i \(-0.797128\pi\)
−0.803681 + 0.595060i \(0.797128\pi\)
\(164\) 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
\(171\) 333.000 1.94737
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 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
\(181\) 1.00000 0.00552486 0.00276243 − 0.999996i \(-0.499121\pi\)
0.00276243 + 0.999996i \(0.499121\pi\)
\(182\) 0 0
\(183\) −222.000 −1.21311
\(184\) 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
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −386.000 −1.93970 −0.969849 − 0.243706i \(-0.921637\pi\)
−0.969849 + 0.243706i \(0.921637\pi\)
\(200\) 0 0
\(201\) −327.000 −1.62687
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −166.000 −0.786730 −0.393365 − 0.919382i \(-0.628689\pi\)
−0.393365 + 0.919382i \(0.628689\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −429.000 −1.95890
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −338.000 −1.51570 −0.757848 − 0.652432i \(-0.773749\pi\)
−0.757848 + 0.652432i \(0.773749\pi\)
\(224\) 0 0
\(225\) 225.000 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −383.000 −1.67249 −0.836245 − 0.548357i \(-0.815254\pi\)
−0.836245 + 0.548357i \(0.815254\pi\)
\(230\) 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
\(237\) 393.000 1.65823
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 286.000 1.18672 0.593361 − 0.804936i \(-0.297801\pi\)
0.593361 + 0.804936i \(0.297801\pi\)
\(242\) 0 0
\(243\) 243.000 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 37.0000 0.149798
\(248\) 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
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 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
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −242.000 −0.892989 −0.446494 − 0.894786i \(-0.647328\pi\)
−0.446494 + 0.894786i \(0.647328\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −529.000 −1.90975 −0.954874 − 0.297012i \(-0.904010\pi\)
−0.954874 + 0.297012i \(0.904010\pi\)
\(278\) 0 0
\(279\) −531.000 −1.90323
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −59.0000 −0.208481 −0.104240 − 0.994552i \(-0.533241\pi\)
−0.104240 + 0.994552i \(0.533241\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) −6.00000 −0.0206186
\(292\) 0 0
\(293\) 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
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 253.000 0.824104 0.412052 − 0.911160i \(-0.364812\pi\)
0.412052 + 0.911160i \(0.364812\pi\)
\(308\) 0 0
\(309\) 111.000 0.359223
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 457.000 1.46006 0.730032 − 0.683413i \(-0.239505\pi\)
0.730032 + 0.683413i \(0.239505\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 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
\(327\) 429.000 1.31193
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 299.000 0.903323 0.451662 − 0.892189i \(-0.350831\pi\)
0.451662 + 0.892189i \(0.350831\pi\)
\(332\) 0 0
\(333\) 423.000 1.27027
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 167.000 0.495549 0.247774 − 0.968818i \(-0.420301\pi\)
0.247774 + 0.968818i \(0.420301\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 502.000 1.43840 0.719198 − 0.694805i \(-0.244510\pi\)
0.719198 + 0.694805i \(0.244510\pi\)
\(350\) 0 0
\(351\) 27.0000 0.0769231
\(352\) 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
\(361\) 1008.00 2.79224
\(362\) 0 0
\(363\) 363.000 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −227.000 −0.618529 −0.309264 − 0.950976i \(-0.600083\pi\)
−0.309264 + 0.950976i \(0.600083\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −121.000 −0.324397 −0.162198 − 0.986758i \(-0.551858\pi\)
−0.162198 + 0.986758i \(0.551858\pi\)
\(374\) 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
\(381\) −759.000 −1.99213
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 747.000 1.93023
\(388\) 0 0
\(389\) 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
\(397\) 793.000 1.99748 0.998741 − 0.0501728i \(-0.0159772\pi\)
0.998741 + 0.0501728i \(0.0159772\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −59.0000 −0.146402
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −143.000 −0.349633 −0.174817 − 0.984601i \(-0.555933\pi\)
−0.174817 + 0.984601i \(0.555933\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −753.000 −1.80576
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 839.000 1.99287 0.996437 − 0.0843398i \(-0.0268781\pi\)
0.996437 + 0.0843398i \(0.0268781\pi\)
\(422\) 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
\(433\) −503.000 −1.16166 −0.580831 − 0.814024i \(-0.697272\pi\)
−0.580831 + 0.814024i \(0.697272\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 94.0000 0.214123 0.107062 − 0.994252i \(-0.465856\pi\)
0.107062 + 0.994252i \(0.465856\pi\)
\(440\) 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
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −858.000 −1.89404
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 47.0000 0.102845 0.0514223 − 0.998677i \(-0.483625\pi\)
0.0514223 + 0.998677i \(0.483625\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 923.000 1.99352 0.996760 − 0.0804300i \(-0.0256293\pi\)
0.996760 + 0.0804300i \(0.0256293\pi\)
\(464\) 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
\(471\) 354.000 0.751592
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 925.000 1.94737
\(476\) 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
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −613.000 −1.25873 −0.629363 − 0.777111i \(-0.716684\pi\)
−0.629363 + 0.777111i \(0.716684\pi\)
\(488\) 0 0
\(489\) −786.000 −1.60736
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −877.000 −1.75752 −0.878758 − 0.477269i \(-0.841627\pi\)
−0.878758 + 0.477269i \(0.841627\pi\)
\(500\) 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
\(507\) −504.000 −0.994083
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 999.000 1.94737
\(514\) 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 \(-0.778592\pi\)
−0.767686 + 0.640826i \(0.778592\pi\)
\(524\) 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
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −793.000 −1.46580 −0.732902 − 0.680334i \(-0.761835\pi\)
−0.732902 + 0.680334i \(0.761835\pi\)
\(542\) 0 0
\(543\) 3.00000 0.00552486
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 506.000 0.925046 0.462523 − 0.886607i \(-0.346944\pi\)
0.462523 + 0.886607i \(0.346944\pi\)
\(548\) 0 0
\(549\) −666.000 −1.21311
\(550\) 0 0
\(551\) 0 0
\(552\) 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
\(559\) 83.0000 0.148479
\(560\) 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
\(571\) 1067.00 1.86865 0.934326 − 0.356420i \(-0.116003\pi\)
0.934326 + 0.356420i \(0.116003\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1033.00 1.79029 0.895147 − 0.445770i \(-0.147070\pi\)
0.895147 + 0.445770i \(0.147070\pi\)
\(578\) 0 0
\(579\) 429.000 0.740933
\(580\) 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
\(589\) −2183.00 −3.70628
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1158.00 −1.93970
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 673.000 1.11980 0.559900 − 0.828560i \(-0.310839\pi\)
0.559900 + 0.828560i \(0.310839\pi\)
\(602\) 0 0
\(603\) −981.000 −1.62687
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 373.000 0.614498 0.307249 − 0.951629i \(-0.400592\pi\)
0.307249 + 0.951629i \(0.400592\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1126.00 −1.83687 −0.918434 − 0.395574i \(-0.870546\pi\)
−0.918434 + 0.395574i \(0.870546\pi\)
\(614\) 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 \(-0.221967\pi\)
0.766559 + 0.642174i \(0.221967\pi\)
\(620\) 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
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 674.000 1.06815 0.534073 − 0.845438i \(-0.320661\pi\)
0.534073 + 0.845438i \(0.320661\pi\)
\(632\) 0 0
\(633\) −498.000 −0.786730
\(634\) 0 0
\(635\) 0 0
\(636\) 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
\(643\) −923.000 −1.43546 −0.717729 − 0.696322i \(-0.754819\pi\)
−0.717729 + 0.696322i \(0.754819\pi\)
\(644\) 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
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1287.00 −1.95890
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1201.00 1.81694 0.908472 − 0.417946i \(-0.137250\pi\)
0.908472 + 0.417946i \(0.137250\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1014.00 −1.51570
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1177.00 −1.74889 −0.874443 − 0.485129i \(-0.838773\pi\)
−0.874443 + 0.485129i \(0.838773\pi\)
\(674\) 0 0
\(675\) 675.000 1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 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
\(687\) −1149.00 −1.67249
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −299.000 −0.432706 −0.216353 − 0.976315i \(-0.569416\pi\)
−0.216353 + 0.976315i \(0.569416\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 1739.00 2.47368
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −934.000 −1.31735 −0.658674 − 0.752428i \(-0.728882\pi\)
−0.658674 + 0.752428i \(0.728882\pi\)
\(710\) 0 0
\(711\) 1179.00 1.65823
\(712\) 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
\(721\) 0 0
\(722\) 0 0
\(723\) 858.000 1.18672
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1429.00 1.96561 0.982806 − 0.184641i \(-0.0591122\pi\)
0.982806 + 0.184641i \(0.0591122\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −383.000 −0.522510 −0.261255 − 0.965270i \(-0.584136\pi\)
−0.261255 + 0.965270i \(0.584136\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −109.000 −0.147497 −0.0737483 − 0.997277i \(-0.523496\pi\)
−0.0737483 + 0.997277i \(0.523496\pi\)
\(740\) 0 0
\(741\) 111.000 0.149798
\(742\) 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
\(751\) −1381.00 −1.83888 −0.919441 − 0.393229i \(-0.871358\pi\)
−0.919441 + 0.393229i \(0.871358\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −838.000 −1.10700 −0.553501 − 0.832849i \(-0.686708\pi\)
−0.553501 + 0.832849i \(0.686708\pi\)
\(758\) 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
\(769\) −671.000 −0.872562 −0.436281 − 0.899811i \(-0.643705\pi\)
−0.436281 + 0.899811i \(0.643705\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −1475.00 −1.90323
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1562.00 −1.98475 −0.992376 − 0.123246i \(-0.960669\pi\)
−0.992376 + 0.123246i \(0.960669\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −74.0000 −0.0933165
\(794\) 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
\(807\) 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 \(-0.883187\pi\)
−0.933416 + 0.358797i \(0.883187\pi\)
\(812\) 0 0
\(813\) −726.000 −0.892989
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3071.00 3.75887
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1058.00 1.28554 0.642770 − 0.766059i \(-0.277785\pi\)
0.642770 + 0.766059i \(0.277785\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1151.00 −1.38842 −0.694210 − 0.719773i \(-0.744246\pi\)
−0.694210 + 0.719773i \(0.744246\pi\)
\(830\) 0 0
\(831\) −1587.00 −1.90975
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1593.00 −1.90323
\(838\) 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
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −177.000 −0.208481
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1177.00 1.37984 0.689918 − 0.723888i \(-0.257647\pi\)
0.689918 + 0.723888i \(0.257647\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1418.00 −1.65076 −0.825378 − 0.564580i \(-0.809038\pi\)
−0.825378 + 0.564580i \(0.809038\pi\)
\(860\) 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
\(867\) 867.000 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −109.000 −0.125144
\(872\) 0 0
\(873\) −18.0000 −0.0206186
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −598.000 −0.681870 −0.340935 − 0.940087i \(-0.610744\pi\)
−0.340935 + 0.940087i \(0.610744\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1259.00 1.42582 0.712911 − 0.701255i \(-0.247377\pi\)
0.712911 + 0.701255i \(0.247377\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 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
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 971.000 1.05658 0.528292 − 0.849063i \(-0.322833\pi\)
0.528292 + 0.849063i \(0.322833\pi\)
\(920\) 0 0
\(921\) 759.000 0.824104
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1175.00 1.27027
\(926\) 0 0
\(927\) 333.000 0.359223
\(928\) 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
\(937\) −1847.00 −1.97118 −0.985592 − 0.169138i \(-0.945902\pi\)
−0.985592 + 0.169138i \(0.945902\pi\)
\(938\) 0 0
\(939\) 1371.00 1.46006
\(940\) 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
\(949\) −143.000 −0.150685
\(950\) 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
\(961\) 2520.00 2.62227
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 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
\(973\) 0 0
\(974\) 0 0
\(975\) 75.0000 0.0769231
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1287.00 1.31193
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 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
\(993\) 897.000 0.903323
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1487.00 −1.49147 −0.745737 − 0.666240i \(-0.767903\pi\)
−0.745737 + 0.666240i \(0.767903\pi\)
\(998\) 0 0
\(999\) 1269.00 1.27027
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 588.3.c.b.197.1 1
3.2 odd 2 CM 588.3.c.b.197.1 1
7.2 even 3 588.3.p.a.557.1 2
7.3 odd 6 84.3.p.a.65.1 yes 2
7.4 even 3 588.3.p.a.569.1 2
7.5 odd 6 84.3.p.a.53.1 ✓ 2
7.6 odd 2 588.3.c.a.197.1 1
21.2 odd 6 588.3.p.a.557.1 2
21.5 even 6 84.3.p.a.53.1 ✓ 2
21.11 odd 6 588.3.p.a.569.1 2
21.17 even 6 84.3.p.a.65.1 yes 2
21.20 even 2 588.3.c.a.197.1 1
28.3 even 6 336.3.bn.a.65.1 2
28.19 even 6 336.3.bn.a.305.1 2
84.47 odd 6 336.3.bn.a.305.1 2
84.59 odd 6 336.3.bn.a.65.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.3.p.a.53.1 ✓ 2 7.5 odd 6
84.3.p.a.53.1 ✓ 2 21.5 even 6
84.3.p.a.65.1 yes 2 7.3 odd 6
84.3.p.a.65.1 yes 2 21.17 even 6
336.3.bn.a.65.1 2 28.3 even 6
336.3.bn.a.65.1 2 84.59 odd 6
336.3.bn.a.305.1 2 28.19 even 6
336.3.bn.a.305.1 2 84.47 odd 6
588.3.c.a.197.1 1 7.6 odd 2
588.3.c.a.197.1 1 21.20 even 2
588.3.c.b.197.1 1 1.1 even 1 trivial
588.3.c.b.197.1 1 3.2 odd 2 CM
588.3.p.a.557.1 2 7.2 even 3
588.3.p.a.557.1 2 21.2 odd 6
588.3.p.a.569.1 2 7.4 even 3
588.3.p.a.569.1 2 21.11 odd 6