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Properties

Label 1472.2.c.c.1471.1
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 1471.1
Root \(1.33454 + 0.467979i\) of defining polynomial
Character \(\chi\) \(=\) 1472.1471
Dual form 1472.2.c.c.1471.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.43410i q^{3} -8.79306 q^{9} -4.88325 q^{13} +4.79583i q^{23} +5.00000 q^{25} +19.8940i q^{27} -6.70287 q^{29} -0.309728i q^{31} +16.7696i q^{39} -3.97345 q^{41} -6.55848i q^{47} -7.00000 q^{49} +9.59166i q^{59} +16.4694 q^{69} -14.0461i q^{71} -7.61268 q^{73} -17.1705i q^{75} +41.9388 q^{81} +23.0183i q^{87} -1.06364 q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 18 q^{9} + 30 q^{25} - 42 q^{49} + 54 q^{81} + 12 q^{93}+O(q^{100}) \) 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
\(3\) − 3.43410i − 1.98268i −0.131319 − 0.991340i \(-0.541921\pi\)
0.131319 − 0.991340i \(-0.458079\pi\)
\(4\) 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
\(9\) −8.79306 −2.93102
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −4.88325 −1.35437 −0.677185 − 0.735812i \(-0.736801\pi\)
−0.677185 + 0.735812i \(0.736801\pi\)
\(14\) 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
\(27\) 19.8940i 3.82860i
\(28\) 0 0
\(29\) −6.70287 −1.24469 −0.622346 − 0.782742i \(-0.713820\pi\)
−0.622346 + 0.782742i \(0.713820\pi\)
\(30\) 0 0
\(31\) − 0.309728i − 0.0556288i −0.999613 − 0.0278144i \(-0.991145\pi\)
0.999613 − 0.0278144i \(-0.00885474\pi\)
\(32\) 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
\(39\) 16.7696i 2.68528i
\(40\) 0 0
\(41\) −3.97345 −0.620548 −0.310274 − 0.950647i \(-0.600421\pi\)
−0.310274 + 0.950647i \(0.600421\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.55848i − 0.956652i −0.878182 − 0.478326i \(-0.841244\pi\)
0.878182 − 0.478326i \(-0.158756\pi\)
\(48\) 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 \(0.214642\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 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
\(69\) 16.4694 1.98268
\(70\) 0 0
\(71\) − 14.0461i − 1.66697i −0.552542 − 0.833485i \(-0.686342\pi\)
0.552542 − 0.833485i \(-0.313658\pi\)
\(72\) 0 0
\(73\) −7.61268 −0.890997 −0.445498 − 0.895283i \(-0.646973\pi\)
−0.445498 + 0.895283i \(0.646973\pi\)
\(74\) 0 0
\(75\) − 17.1705i − 1.98268i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 41.9388 4.65986
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 23.0183i 2.46783i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.06364 −0.110294
\(94\) 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
\(117\) 42.9388 3.96969
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 13.6452i 1.23035i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 20.9143i 1.85585i 0.372769 + 0.927924i \(0.378408\pi\)
−0.372769 + 0.927924i \(0.621592\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.81465i 0.245917i 0.992412 + 0.122958i \(0.0392382\pi\)
−0.992412 + 0.122958i \(0.960762\pi\)
\(132\) 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
\(139\) − 10.9218i − 0.926372i −0.886261 − 0.463186i \(-0.846706\pi\)
0.886261 − 0.463186i \(-0.153294\pi\)
\(140\) 0 0
\(141\) −22.5225 −1.89674
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 24.0387i 1.98268i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) − 20.2949i − 1.65157i −0.563982 − 0.825787i \(-0.690731\pi\)
0.563982 − 0.825787i \(-0.309269\pi\)
\(152\) 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
\(163\) − 9.68285i − 0.758420i −0.925311 − 0.379210i \(-0.876196\pi\)
0.925311 − 0.379210i \(-0.123804\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.59166i 0.742225i 0.928588 + 0.371113i \(0.121024\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) 10.8462 0.834321
\(170\) 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
\(177\) 32.9388 2.47583
\(178\) 0 0
\(179\) 17.7900i 1.32968i 0.746984 + 0.664842i \(0.231501\pi\)
−0.746984 + 0.664842i \(0.768499\pi\)
\(180\) 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
\(193\) −27.1457 −1.95399 −0.976995 − 0.213262i \(-0.931591\pi\)
−0.976995 + 0.213262i \(0.931591\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −28.0555 −1.99887 −0.999436 − 0.0335834i \(-0.989308\pi\)
−0.999436 + 0.0335834i \(0.989308\pi\)
\(198\) 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
\(207\) − 42.1700i − 2.93102i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) − 28.7750i − 1.98095i −0.137686 − 0.990476i \(-0.543966\pi\)
0.137686 − 0.990476i \(-0.456034\pi\)
\(212\) 0 0
\(213\) −48.2359 −3.30507
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 26.1427i 1.76656i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 28.7750i − 1.92692i −0.267860 − 0.963458i \(-0.586316\pi\)
0.267860 − 0.963458i \(-0.413684\pi\)
\(224\) 0 0
\(225\) −43.9653 −2.93102
\(226\) 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
\(233\) −9.43229 −0.617930 −0.308965 − 0.951073i \(-0.599983\pi\)
−0.308965 + 0.951073i \(0.599983\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 27.1631i 1.75703i 0.477711 + 0.878517i \(0.341467\pi\)
−0.477711 + 0.878517i \(0.658533\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) − 84.3401i − 5.41042i
\(244\) 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
\(257\) −25.3261 −1.57980 −0.789899 − 0.613237i \(-0.789867\pi\)
−0.789899 + 0.613237i \(0.789867\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 58.9388 3.64822
\(262\) 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
\(269\) −1.24402 −0.0758493 −0.0379247 − 0.999281i \(-0.512075\pi\)
−0.0379247 + 0.999281i \(0.512075\pi\)
\(270\) 0 0
\(271\) − 28.7750i − 1.74796i −0.485965 − 0.873978i \(-0.661532\pi\)
0.485965 − 0.873978i \(-0.338468\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −29.8751 −1.79502 −0.897511 − 0.440992i \(-0.854627\pi\)
−0.897511 + 0.440992i \(0.854627\pi\)
\(278\) 0 0
\(279\) 2.72346i 0.163049i
\(280\) 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
\(299\) − 23.4193i − 1.35437i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 20.6046i 1.18370i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 28.7750i − 1.64228i −0.570730 − 0.821138i \(-0.693340\pi\)
0.570730 − 0.821138i \(-0.306660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 34.6508i 1.96486i 0.186621 + 0.982432i \(0.440246\pi\)
−0.186621 + 0.982432i \(0.559754\pi\)
\(312\) 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
\(325\) −24.4163 −1.35437
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 31.5264i 1.73285i 0.499310 + 0.866423i \(0.333587\pi\)
−0.499310 + 0.866423i \(0.666413\pi\)
\(332\) 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 \(0.0828835\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 0 0
\(349\) −10.3421 −0.553600 −0.276800 − 0.960928i \(-0.589274\pi\)
−0.276800 + 0.960928i \(0.589274\pi\)
\(350\) 0 0
\(351\) − 97.1473i − 5.18534i
\(352\) 0 0
\(353\) −30.7849 −1.63852 −0.819258 − 0.573425i \(-0.805614\pi\)
−0.819258 + 0.573425i \(0.805614\pi\)
\(354\) 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
\(363\) 37.7751i 1.98268i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 34.9388 1.81884
\(370\) 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
\(377\) 32.7318 1.68577
\(378\) 0 0
\(379\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 71.8220 3.67955
\(382\) 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
\(393\) 9.66579 0.487574
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 39.6416 1.98956 0.994778 − 0.102061i \(-0.0325437\pi\)
0.994778 + 0.102061i \(0.0325437\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 1.51248i 0.0753420i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 36.9122 1.82519 0.912595 − 0.408864i \(-0.134075\pi\)
0.912595 + 0.408864i \(0.134075\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −37.5065 −1.83670
\(418\) 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
\(423\) 57.6691i 2.80397i
\(424\) 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
\(439\) − 41.5190i − 1.98159i −0.135364 − 0.990796i \(-0.543220\pi\)
0.135364 − 0.990796i \(-0.456780\pi\)
\(440\) 0 0
\(441\) 61.5514 2.93102
\(442\) 0 0
\(443\) − 38.3946i − 1.82418i −0.409988 − 0.912091i \(-0.634467\pi\)
0.409988 − 0.912091i \(-0.365533\pi\)
\(444\) 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
\(453\) −69.6947 −3.27454
\(454\) 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
\(461\) 36.0024 1.67680 0.838399 − 0.545056i \(-0.183492\pi\)
0.838399 + 0.545056i \(0.183492\pi\)
\(462\) 0 0
\(463\) − 28.7750i − 1.33729i −0.743583 − 0.668644i \(-0.766875\pi\)
0.743583 − 0.668644i \(-0.233125\pi\)
\(464\) 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
\(487\) 40.8995i 1.85333i 0.375884 + 0.926667i \(0.377339\pi\)
−0.375884 + 0.926667i \(0.622661\pi\)
\(488\) 0 0
\(489\) −33.2519 −1.50370
\(490\) 0 0
\(491\) − 4.67301i − 0.210890i −0.994425 − 0.105445i \(-0.966373\pi\)
0.994425 − 0.105445i \(-0.0336267\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 11.5412i 0.516656i 0.966057 + 0.258328i \(0.0831715\pi\)
−0.966057 + 0.258328i \(0.916828\pi\)
\(500\) 0 0
\(501\) 32.9388 1.47160
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 37.2469i − 1.65419i
\(508\) 0 0
\(509\) 41.4612 1.83774 0.918869 − 0.394564i \(-0.129104\pi\)
0.918869 + 0.394564i \(0.129104\pi\)
\(510\) 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
\(519\) − 61.8139i − 2.71333i
\(520\) 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
\(531\) − 84.3401i − 3.66005i
\(532\) 0 0
\(533\) 19.4033 0.840452
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 61.0926 2.63634
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.575595 0.0247468 0.0123734 − 0.999923i \(-0.496061\pi\)
0.0123734 + 0.999923i \(0.496061\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 9.06340i 0.387523i 0.981049 + 0.193761i \(0.0620688\pi\)
−0.981049 + 0.193761i \(0.937931\pi\)
\(548\) 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
\(577\) −32.6045 −1.35734 −0.678672 − 0.734441i \(-0.737444\pi\)
−0.678672 + 0.734441i \(0.737444\pi\)
\(578\) 0 0
\(579\) 93.2211i 3.87414i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 25.2776i 1.04332i 0.853154 + 0.521660i \(0.174687\pi\)
−0.853154 + 0.521660i \(0.825313\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 96.3455i 3.96312i
\(592\) 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 \(0.0627798\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) 0 0
\(601\) 42.3711 1.72835 0.864176 − 0.503190i \(-0.167841\pi\)
0.864176 + 0.503190i \(0.167841\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 28.7750i − 1.16794i −0.811775 − 0.583970i \(-0.801498\pi\)
0.811775 − 0.583970i \(-0.198502\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 32.0267i 1.29566i
\(612\) 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
\(621\) −95.4081 −3.82860
\(622\) 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
\(633\) −98.8163 −3.92759
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 34.1828 1.35437
\(638\) 0 0
\(639\) 123.509i 4.88592i
\(640\) 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
\(647\) − 47.7677i − 1.87794i −0.343996 − 0.938971i \(-0.611781\pi\)
0.343996 − 0.938971i \(-0.388219\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −12.1617 −0.475925 −0.237962 − 0.971274i \(-0.576480\pi\)
−0.237962 + 0.971274i \(0.576480\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 66.9388 2.61153
\(658\) 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
\(667\) − 32.1458i − 1.24469i
\(668\) 0 0
\(669\) −98.8163 −3.82046
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −21.6868 −0.835966 −0.417983 − 0.908455i \(-0.637263\pi\)
−0.417983 + 0.908455i \(0.637263\pi\)
\(674\) 0 0
\(675\) 99.4699i 3.82860i
\(676\) 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
\(683\) 44.0239i 1.68453i 0.539066 + 0.842263i \(0.318777\pi\)
−0.539066 + 0.842263i \(0.681223\pi\)
\(684\) 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 \(-0.815646\pi\)
0.836919 − 0.547326i \(-0.184354\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 32.3915i 1.22516i
\(700\) 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
\(713\) 1.48540 0.0556288
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 93.2809 3.48364
\(718\) 0 0
\(719\) 47.9583i 1.78854i 0.447524 + 0.894272i \(0.352306\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −33.5144 −1.24469
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −163.816 −6.06727
\(730\) 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
\(739\) − 52.1310i − 1.91767i −0.283964 − 0.958835i \(-0.591650\pi\)
0.283964 − 0.958835i \(-0.408350\pi\)
\(740\) 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
\(761\) 33.2730 1.20614 0.603072 − 0.797687i \(-0.293943\pi\)
0.603072 + 0.797687i \(0.293943\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 46.8385i − 1.69124i
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 86.9724i 3.13223i
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) − 1.54864i − 0.0556288i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 133.347i − 4.76542i
\(784\) 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
\(807\) 4.27210i 0.150385i
\(808\) 0 0
\(809\) 42.0000 1.47664 0.738321 − 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) − 29.6680i − 1.04178i −0.853622 − 0.520892i \(-0.825599\pi\)
0.853622 − 0.520892i \(-0.174401\pi\)
\(812\) 0 0
\(813\) −98.8163 −3.46564
\(814\) 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
\(823\) 22.1533i 0.772214i 0.922454 + 0.386107i \(0.126180\pi\)
−0.922454 + 0.386107i \(0.873820\pi\)
\(824\) 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
\(831\) 102.594i 3.55895i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 6.16172 0.212980
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 15.9285 0.549258
\(842\) 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
\(857\) 44.1907 1.50952 0.754762 − 0.655998i \(-0.227752\pi\)
0.754762 + 0.655998i \(0.227752\pi\)
\(858\) 0 0
\(859\) − 50.8921i − 1.73642i −0.496201 − 0.868208i \(-0.665272\pi\)
0.496201 − 0.868208i \(-0.334728\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 12.1878i 0.414877i 0.978248 + 0.207438i \(0.0665126\pi\)
−0.978248 + 0.207438i \(0.933487\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 58.3797i − 1.98268i
\(868\) 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 \(-0.839119\pi\)
0.874970 − 0.484178i \(-0.160881\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 29.0215i − 0.974445i −0.873278 − 0.487223i \(-0.838010\pi\)
0.873278 − 0.487223i \(-0.161990\pi\)
\(888\) 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
\(897\) −80.4242 −2.68528
\(898\) 0 0
\(899\) 2.07607i 0.0692407i
\(900\) 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
\(909\) 52.7584 1.74988
\(910\) 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
\(921\) −98.8163 −3.25611
\(922\) 0 0
\(923\) 68.5909i 2.25770i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 60.0845 1.97131 0.985653 − 0.168782i \(-0.0539833\pi\)
0.985653 + 0.168782i \(0.0539833\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 118.994 3.89570
\(934\) 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
\(943\) − 19.0560i − 0.620548i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36.5362i 1.18727i 0.804735 + 0.593634i \(0.202307\pi\)
−0.804735 + 0.593634i \(0.797693\pi\)
\(948\) 0 0
\(949\) 37.1746 1.20674
\(950\) 0 0
\(951\) 103.023i 3.34075i
\(952\) 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
\(961\) 30.9041 0.996905
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 62.1236i 1.99776i 0.0473194 + 0.998880i \(0.484932\pi\)
−0.0473194 + 0.998880i \(0.515068\pi\)
\(968\) 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
\(975\) 83.8480i 2.68528i
\(976\) 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 \(-0.848912\pi\)
0.889449 − 0.457034i \(-0.151088\pi\)
\(992\) 0 0
\(993\) 108.265 3.43568
\(994\) 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.1471.1 6
4.3 odd 2 inner 1472.2.c.c.1471.6 6
8.3 odd 2 92.2.b.b.91.1 ✓ 6
8.5 even 2 92.2.b.b.91.2 yes 6
23.22 odd 2 CM 1472.2.c.c.1471.1 6
24.5 odd 2 828.2.e.b.91.5 6
24.11 even 2 828.2.e.b.91.6 6
92.91 even 2 inner 1472.2.c.c.1471.6 6
184.45 odd 2 92.2.b.b.91.2 yes 6
184.91 even 2 92.2.b.b.91.1 ✓ 6
552.275 odd 2 828.2.e.b.91.6 6
552.413 even 2 828.2.e.b.91.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
92.2.b.b.91.1 ✓ 6 8.3 odd 2
92.2.b.b.91.1 ✓ 6 184.91 even 2
92.2.b.b.91.2 yes 6 8.5 even 2
92.2.b.b.91.2 yes 6 184.45 odd 2
828.2.e.b.91.5 6 24.5 odd 2
828.2.e.b.91.5 6 552.413 even 2
828.2.e.b.91.6 6 24.11 even 2
828.2.e.b.91.6 6 552.275 odd 2
1472.2.c.c.1471.1 6 1.1 even 1 trivial
1472.2.c.c.1471.1 6 23.22 odd 2 CM
1472.2.c.c.1471.6 6 4.3 odd 2 inner
1472.2.c.c.1471.6 6 92.91 even 2 inner