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Properties

Label 1472.2.c.c.1471.2
Level $1472$
Weight $2$
Character 1472.1471
Analytic conductor $11.754$
Analytic rank $0$
Dimension $6$
CM discriminant -23
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1472,2,Mod(1471,1472)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1472, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1472.1471");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1472.c (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: no
Analytic conductor: \(11.7539791775\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8869743.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{3} + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 92)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1471.2
Root \(-1.07255 - 0.921756i\) of defining polynomial
Character \(\chi\) \(=\) 1472.1471
Dual form 1472.2.c.c.1471.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.11101i q^{3} -1.45636 q^{9} +7.03677 q^{13} -4.79583i q^{23} +5.00000 q^{25} -3.25863i q^{27} -3.94950 q^{29} +9.48506i q^{31} -14.8547i q^{39} +12.5299 q^{41} -13.7071i q^{47} -7.00000 q^{49} -9.59166i q^{59} -10.1240 q^{69} +1.04102i q^{71} -9.44264 q^{73} -10.5550i q^{75} -11.2481 q^{81} +8.33743i q^{87} +20.0230 q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 18 q^{9} + 30 q^{25} - 42 q^{49} + 54 q^{81} + 12 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1472\mathbb{Z}\right)^\times\).

\(n\) \(645\) \(833\) \(1151\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) − 2.11101i − 1.21879i −0.792866 − 0.609396i \(-0.791412\pi\)
0.792866 − 0.609396i \(-0.208588\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −1.45636 −0.485454
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 7.03677 1.95165 0.975825 − 0.218554i \(-0.0701339\pi\)
0.975825 + 0.218554i \(0.0701339\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.79583i − 1.00000i
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) − 3.25863i − 0.627124i
\(28\) 0 0
\(29\) −3.94950 −0.733404 −0.366702 − 0.930339i \(-0.619513\pi\)
−0.366702 + 0.930339i \(0.619513\pi\)
\(30\) 0 0
\(31\) 9.48506i 1.70357i 0.523895 + 0.851783i \(0.324479\pi\)
−0.523895 + 0.851783i \(0.675521\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) − 14.8547i − 2.37866i
\(40\) 0 0
\(41\) 12.5299 1.95684 0.978422 − 0.206618i \(-0.0662459\pi\)
0.978422 + 0.206618i \(0.0662459\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 13.7071i − 1.99938i −0.0248485 − 0.999691i \(-0.507910\pi\)
0.0248485 − 0.999691i \(-0.492090\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 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
\(57\) 0 0
\(58\) 0 0
\(59\) − 9.59166i − 1.24873i −0.781133 − 0.624364i \(-0.785358\pi\)
0.781133 − 0.624364i \(-0.214642\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −10.1240 −1.21879
\(70\) 0 0
\(71\) 1.04102i 0.123546i 0.998090 + 0.0617729i \(0.0196755\pi\)
−0.998090 + 0.0617729i \(0.980325\pi\)
\(72\) 0 0
\(73\) −9.44264 −1.10518 −0.552588 − 0.833454i \(-0.686360\pi\)
−0.552588 + 0.833454i \(0.686360\pi\)
\(74\) 0 0
\(75\) − 10.5550i − 1.21879i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −11.2481 −1.24979
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.33743i 0.893867i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 20.0230 2.07629
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 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
\(117\) −10.2481 −0.947437
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) − 26.4508i − 2.38499i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.18100i 0.282268i 0.989990 + 0.141134i \(0.0450749\pi\)
−0.989990 + 0.141134i \(0.954925\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 21.0811i 1.84187i 0.389721 + 0.920933i \(0.372572\pi\)
−0.389721 + 0.920933i \(0.627428\pi\)
\(132\) 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
\(139\) 12.6371i 1.07186i 0.844261 + 0.535932i \(0.180040\pi\)
−0.844261 + 0.535932i \(0.819960\pi\)
\(140\) 0 0
\(141\) −28.9358 −2.43683
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 14.7771i 1.21879i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) − 22.1511i − 1.80263i −0.433162 − 0.901316i \(-0.642602\pi\)
0.433162 − 0.901316i \(-0.357398\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 25.3031i − 1.98189i −0.134250 − 0.990947i \(-0.542863\pi\)
0.134250 − 0.990947i \(-0.457137\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 9.59166i − 0.742225i −0.928588 − 0.371113i \(-0.878976\pi\)
0.928588 − 0.371113i \(-0.121024\pi\)
\(168\) 0 0
\(169\) 36.5162 2.80894
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −20.2481 −1.52194
\(178\) 0 0
\(179\) − 8.41506i − 0.628971i −0.949262 − 0.314486i \(-0.898168\pi\)
0.949262 − 0.314486i \(-0.101832\pi\)
\(180\) 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
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 18.7045 1.34638 0.673188 − 0.739471i \(-0.264924\pi\)
0.673188 + 0.739471i \(0.264924\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.2113 0.941268 0.470634 − 0.882329i \(-0.344025\pi\)
0.470634 + 0.882329i \(0.344025\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.98447i 0.485454i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 28.7750i 1.98095i 0.137686 + 0.990476i \(0.456034\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 2.19760 0.150577
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 19.9335i 1.34698i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 28.7750i 1.92692i 0.267860 + 0.963458i \(0.413684\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) −7.28181 −0.485454
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −20.4289 −1.33834 −0.669171 − 0.743108i \(-0.733351\pi\)
−0.669171 + 0.743108i \(0.733351\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.3731i 1.70594i 0.521963 + 0.852968i \(0.325200\pi\)
−0.521963 + 0.852968i \(0.674800\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 13.9689i 0.896108i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 29.6907 1.85206 0.926028 − 0.377454i \(-0.123200\pi\)
0.926028 + 0.377454i \(0.123200\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 5.75190 0.356034
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 29.0093 1.76873 0.884365 − 0.466797i \(-0.154592\pi\)
0.884365 + 0.466797i \(0.154592\pi\)
\(270\) 0 0
\(271\) 28.7750i 1.74796i 0.485965 + 0.873978i \(0.338468\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.22505 0.133690 0.0668451 − 0.997763i \(-0.478707\pi\)
0.0668451 + 0.997763i \(0.478707\pi\)
\(278\) 0 0
\(279\) − 13.8137i − 0.827004i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000 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
\(297\) 0 0
\(298\) 0 0
\(299\) − 33.7472i − 1.95165i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 12.6661i 0.727646i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 28.7750i 1.64228i 0.570730 + 0.821138i \(0.306660\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 11.6250i 0.659196i 0.944121 + 0.329598i \(0.106913\pi\)
−0.944121 + 0.329598i \(0.893087\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −30.0000 −1.68497 −0.842484 − 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 35.1839 1.95165
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.0289779i 0.00159277i 1.00000 0.000796384i \(0.000253497\pi\)
−1.00000 0.000796384i \(0.999747\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 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
\(347\) − 9.59166i − 0.514907i −0.966291 − 0.257454i \(-0.917117\pi\)
0.966291 − 0.257454i \(-0.0828835\pi\)
\(348\) 0 0
\(349\) −25.9220 −1.38758 −0.693788 − 0.720180i \(-0.744059\pi\)
−0.693788 + 0.720180i \(0.744059\pi\)
\(350\) 0 0
\(351\) − 22.9303i − 1.22393i
\(352\) 0 0
\(353\) −3.26809 −0.173943 −0.0869714 − 0.996211i \(-0.527719\pi\)
−0.0869714 + 0.996211i \(0.527719\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 23.2211i 1.21879i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −18.2481 −0.949958
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −27.7917 −1.43135
\(378\) 0 0
\(379\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 6.71513 0.344027
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 44.5025 2.24485
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −16.2986 −0.818003 −0.409002 − 0.912534i \(-0.634123\pi\)
−0.409002 + 0.912534i \(0.634123\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 66.7442i 3.32476i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −32.7780 −1.62077 −0.810384 − 0.585899i \(-0.800742\pi\)
−0.810384 + 0.585899i \(0.800742\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 26.6770 1.30638
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 19.9625i 0.970609i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) − 15.8471i − 0.756339i −0.925736 − 0.378170i \(-0.876554\pi\)
0.925736 − 0.378170i \(-0.123446\pi\)
\(440\) 0 0
\(441\) 10.1945 0.485454
\(442\) 0 0
\(443\) − 4.25100i − 0.201971i −0.994888 − 0.100985i \(-0.967800\pi\)
0.994888 − 0.100985i \(-0.0321996\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 − 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −46.7612 −2.19703
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −38.2711 −1.78246 −0.891232 − 0.453547i \(-0.850158\pi\)
−0.891232 + 0.453547i \(0.850158\pi\)
\(462\) 0 0
\(463\) 28.7750i 1.33729i 0.743583 + 0.668644i \(0.233125\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 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
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 34.8172i 1.57772i 0.614575 + 0.788858i \(0.289328\pi\)
−0.614575 + 0.788858i \(0.710672\pi\)
\(488\) 0 0
\(489\) −53.4152 −2.41552
\(490\) 0 0
\(491\) 35.8292i 1.61695i 0.588531 + 0.808475i \(0.299707\pi\)
−0.588531 + 0.808475i \(0.700293\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 31.6072i − 1.41493i −0.706747 − 0.707466i \(-0.749838\pi\)
0.706747 − 0.707466i \(-0.250162\pi\)
\(500\) 0 0
\(501\) −20.2481 −0.904618
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 77.0860i − 3.42351i
\(508\) 0 0
\(509\) −5.31232 −0.235465 −0.117732 − 0.993045i \(-0.537562\pi\)
−0.117732 + 0.993045i \(0.537562\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) − 37.9982i − 1.66793i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 13.9689i 0.606200i
\(532\) 0 0
\(533\) 88.1701 3.81907
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −17.7643 −0.766585
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 39.9956 1.71954 0.859772 − 0.510677i \(-0.170605\pi\)
0.859772 + 0.510677i \(0.170605\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 44.2733i 1.89299i 0.322722 + 0.946494i \(0.395402\pi\)
−0.322722 + 0.946494i \(0.604598\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 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
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 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
\(571\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 23.9792i − 1.00000i
\(576\) 0 0
\(577\) −14.2544 −0.593417 −0.296708 − 0.954968i \(-0.595889\pi\)
−0.296708 + 0.954968i \(0.595889\pi\)
\(578\) 0 0
\(579\) − 39.4853i − 1.64095i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 23.1632i − 0.956046i −0.878347 − 0.478023i \(-0.841354\pi\)
0.878347 − 0.478023i \(-0.158646\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) − 27.8892i − 1.14721i
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 9.59166i − 0.391905i −0.980613 − 0.195952i \(-0.937220\pi\)
0.980613 − 0.195952i \(-0.0627798\pi\)
\(600\) 0 0
\(601\) 0.180813 0.00737552 0.00368776 − 0.999993i \(-0.498826\pi\)
0.00368776 + 0.999993i \(0.498826\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 28.7750i 1.16794i 0.811775 + 0.583970i \(0.198502\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 96.4536i − 3.90209i
\(612\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) −15.6279 −0.627124
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 60.7443 2.41437
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −49.2574 −1.95165
\(638\) 0 0
\(639\) − 1.51610i − 0.0599759i
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 39.0392i − 1.53479i −0.641175 − 0.767395i \(-0.721553\pi\)
0.641175 − 0.767395i \(-0.278447\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −36.9083 −1.44433 −0.722167 − 0.691719i \(-0.756854\pi\)
−0.722167 + 0.691719i \(0.756854\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 13.7519 0.536513
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 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
\(667\) 18.9411i 0.733404i
\(668\) 0 0
\(669\) 60.7443 2.34851
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 51.6633 1.99147 0.995737 − 0.0922433i \(-0.0294037\pi\)
0.995737 + 0.0922433i \(0.0294037\pi\)
\(674\) 0 0
\(675\) − 16.2932i − 0.627124i
\(676\) 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
\(683\) 46.4132i 1.77595i 0.459889 + 0.887977i \(0.347889\pi\)
−0.459889 + 0.887977i \(0.652111\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 28.7750i 1.09465i 0.836919 + 0.547326i \(0.184354\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 43.1256i 1.63116i
\(700\) 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
\(713\) 45.4887 1.70357
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 55.6739 2.07918
\(718\) 0 0
\(719\) − 47.9583i − 1.78854i −0.447524 − 0.894272i \(-0.647694\pi\)
0.447524 − 0.894272i \(-0.352306\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −19.7475 −0.733404
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −4.25571 −0.157619
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 12.6950i − 0.466994i −0.972357 − 0.233497i \(-0.924983\pi\)
0.972357 − 0.233497i \(-0.0750170\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −54.7506 −1.98471 −0.992353 − 0.123432i \(-0.960610\pi\)
−0.992353 + 0.123432i \(0.960610\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 67.4944i − 2.43708i
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) − 62.6774i − 2.25727i
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 47.4253i 1.70357i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 12.8700i 0.459935i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 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
\(807\) − 61.2390i − 2.15571i
\(808\) 0 0
\(809\) 42.0000 1.47664 0.738321 − 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) − 56.9393i − 1.99941i −0.0242949 − 0.999705i \(-0.507734\pi\)
0.0242949 − 0.999705i \(-0.492266\pi\)
\(812\) 0 0
\(813\) 60.7443 2.13040
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −54.0000 −1.88461 −0.942306 − 0.334751i \(-0.891348\pi\)
−0.942306 + 0.334751i \(0.891348\pi\)
\(822\) 0 0
\(823\) − 34.7592i − 1.21163i −0.795605 − 0.605815i \(-0.792847\pi\)
0.795605 − 0.605815i \(-0.207153\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) − 4.69710i − 0.162941i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 30.9083 1.06835
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −13.4015 −0.462119
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 − 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.1671 0.381460 0.190730 − 0.981642i \(-0.438914\pi\)
0.190730 + 0.981642i \(0.438914\pi\)
\(858\) 0 0
\(859\) − 50.6353i − 1.72765i −0.503790 − 0.863826i \(-0.668061\pi\)
0.503790 − 0.863826i \(-0.331939\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 55.8693i 1.90181i 0.309477 + 0.950907i \(0.399846\pi\)
−0.309477 + 0.950907i \(0.600154\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 35.8872i − 1.21879i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −14.0000 −0.472746 −0.236373 − 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 28.7750i 0.968355i 0.874970 + 0.484178i \(0.160881\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 30.5372i 1.02534i 0.858586 + 0.512669i \(0.171343\pi\)
−0.858586 + 0.512669i \(0.828657\pi\)
\(888\) 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
\(897\) −71.2406 −2.37866
\(898\) 0 0
\(899\) − 37.4612i − 1.24940i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 8.73818 0.289827
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 60.7443 2.00159
\(922\) 0 0
\(923\) 7.32539i 0.241118i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −38.9526 −1.27799 −0.638996 − 0.769210i \(-0.720650\pi\)
−0.638996 + 0.769210i \(0.720650\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 24.5406 0.803423
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) − 60.0913i − 1.95684i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 61.1613i 1.98748i 0.111734 + 0.993738i \(0.464359\pi\)
−0.111734 + 0.993738i \(0.535641\pi\)
\(948\) 0 0
\(949\) −66.4457 −2.15692
\(950\) 0 0
\(951\) 63.3303i 2.05363i
\(952\) 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
\(961\) −58.9663 −1.90214
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 28.5131i 0.916920i 0.888715 + 0.458460i \(0.151599\pi\)
−0.888715 + 0.458460i \(0.848401\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) − 74.2735i − 2.37866i
\(976\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 28.7750i 0.914068i 0.889449 + 0.457034i \(0.151088\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) 0 0
\(993\) 0.0611725 0.00194125
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 − 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 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 1472.2.c.c.1471.2 6
4.3 odd 2 inner 1472.2.c.c.1471.5 6
8.3 odd 2 92.2.b.b.91.6 yes 6
8.5 even 2 92.2.b.b.91.5 ✓ 6
23.22 odd 2 CM 1472.2.c.c.1471.2 6
24.5 odd 2 828.2.e.b.91.2 6
24.11 even 2 828.2.e.b.91.1 6
92.91 even 2 inner 1472.2.c.c.1471.5 6
184.45 odd 2 92.2.b.b.91.5 ✓ 6
184.91 even 2 92.2.b.b.91.6 yes 6
552.275 odd 2 828.2.e.b.91.1 6
552.413 even 2 828.2.e.b.91.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
92.2.b.b.91.5 ✓ 6 8.5 even 2
92.2.b.b.91.5 ✓ 6 184.45 odd 2
92.2.b.b.91.6 yes 6 8.3 odd 2
92.2.b.b.91.6 yes 6 184.91 even 2
828.2.e.b.91.1 6 24.11 even 2
828.2.e.b.91.1 6 552.275 odd 2
828.2.e.b.91.2 6 24.5 odd 2
828.2.e.b.91.2 6 552.413 even 2
1472.2.c.c.1471.2 6 1.1 even 1 trivial
1472.2.c.c.1471.2 6 23.22 odd 2 CM
1472.2.c.c.1471.5 6 4.3 odd 2 inner
1472.2.c.c.1471.5 6 92.91 even 2 inner