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Properties

Label 43.7.b.a.42.1
Level $43$
Weight $7$
Character 43.42
Self dual yes
Analytic conductor $9.892$
Analytic rank $0$
Dimension $1$
CM discriminant -43
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [43,7,Mod(42,43)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(43, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.42");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 43.b (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: \(9.89232559565\)
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 42.1
Character \(\chi\) \(=\) 43.42

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+64.0000 q^{4} +729.000 q^{9} +O(q^{10})\) \(q+64.0000 q^{4} +729.000 q^{9} -1638.00 q^{11} +3706.00 q^{13} +4096.00 q^{16} +7074.00 q^{17} -4734.00 q^{23} +15625.0 q^{25} -47918.0 q^{31} +46656.0 q^{36} -137358. q^{41} -79507.0 q^{43} -104832. q^{44} +42354.0 q^{47} +117649. q^{49} +237184. q^{52} -280854. q^{53} +406458. q^{59} +262144. q^{64} -471926. q^{67} +452736. q^{68} +259378. q^{79} +531441. q^{81} -681174. q^{83} -302976. q^{92} -1.74125e6 q^{97} -1.19410e6 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 64.0000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 729.000 1.00000
\(10\) 0 0
\(11\) −1638.00 −1.23065 −0.615327 − 0.788272i \(-0.710976\pi\)
−0.615327 + 0.788272i \(0.710976\pi\)
\(12\) 0 0
\(13\) 3706.00 1.68685 0.843423 − 0.537250i \(-0.180537\pi\)
0.843423 + 0.537250i \(0.180537\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4096.00 1.00000
\(17\) 7074.00 1.43985 0.719927 − 0.694050i \(-0.244175\pi\)
0.719927 + 0.694050i \(0.244175\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4734.00 −0.389085 −0.194543 − 0.980894i \(-0.562322\pi\)
−0.194543 + 0.980894i \(0.562322\pi\)
\(24\) 0 0
\(25\) 15625.0 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −47918.0 −1.60847 −0.804236 − 0.594310i \(-0.797425\pi\)
−0.804236 + 0.594310i \(0.797425\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 46656.0 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −137358. −1.99298 −0.996489 − 0.0837270i \(-0.973318\pi\)
−0.996489 + 0.0837270i \(0.973318\pi\)
\(42\) 0 0
\(43\) −79507.0 −1.00000
\(44\) −104832. −1.23065
\(45\) 0 0
\(46\) 0 0
\(47\) 42354.0 0.407944 0.203972 − 0.978977i \(-0.434615\pi\)
0.203972 + 0.978977i \(0.434615\pi\)
\(48\) 0 0
\(49\) 117649. 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 237184. 1.68685
\(53\) −280854. −1.88648 −0.943242 − 0.332107i \(-0.892241\pi\)
−0.943242 + 0.332107i \(0.892241\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 406458. 1.97906 0.989532 − 0.144317i \(-0.0460984\pi\)
0.989532 + 0.144317i \(0.0460984\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 262144. 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −471926. −1.56910 −0.784548 − 0.620068i \(-0.787105\pi\)
−0.784548 + 0.620068i \(0.787105\pi\)
\(68\) 452736. 1.43985
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 259378. 0.526080 0.263040 − 0.964785i \(-0.415275\pi\)
0.263040 + 0.964785i \(0.415275\pi\)
\(80\) 0 0
\(81\) 531441. 1.00000
\(82\) 0 0
\(83\) −681174. −1.19131 −0.595654 − 0.803241i \(-0.703107\pi\)
−0.595654 + 0.803241i \(0.703107\pi\)
\(84\) 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
\(92\) −302976. −0.389085
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.74125e6 −1.90785 −0.953927 − 0.300040i \(-0.903000\pi\)
−0.953927 + 0.300040i \(0.903000\pi\)
\(98\) 0 0
\(99\) −1.19410e6 −1.23065
\(100\) 1.00000e6 1.00000
\(101\) −846198. −0.821311 −0.410656 − 0.911790i \(-0.634700\pi\)
−0.410656 + 0.911790i \(0.634700\pi\)
\(102\) 0 0
\(103\) −169054. −0.154708 −0.0773542 − 0.997004i \(-0.524647\pi\)
−0.0773542 + 0.997004i \(0.524647\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.36849e6 −1.11709 −0.558546 − 0.829473i \(-0.688641\pi\)
−0.558546 + 0.829473i \(0.688641\pi\)
\(108\) 0 0
\(109\) 1.19686e6 0.924194 0.462097 − 0.886829i \(-0.347097\pi\)
0.462097 + 0.886829i \(0.347097\pi\)
\(110\) 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
\(117\) 2.70167e6 1.68685
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 911483. 0.514508
\(122\) 0 0
\(123\) 0 0
\(124\) −3.06675e6 −1.60847
\(125\) 0 0
\(126\) 0 0
\(127\) 4.08283e6 1.99320 0.996599 − 0.0824008i \(-0.0262588\pi\)
0.996599 + 0.0824008i \(0.0262588\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 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
\(139\) 5.02294e6 1.87031 0.935155 − 0.354240i \(-0.115260\pi\)
0.935155 + 0.354240i \(0.115260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −6.07043e6 −2.07592
\(144\) 2.98598e6 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 5.15695e6 1.43985
\(154\) 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
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −8.79091e6 −1.99298
\(165\) 0 0
\(166\) 0 0
\(167\) 295074. 0.0633551 0.0316775 − 0.999498i \(-0.489915\pi\)
0.0316775 + 0.999498i \(0.489915\pi\)
\(168\) 0 0
\(169\) 8.90763e6 1.84545
\(170\) 0 0
\(171\) 0 0
\(172\) −5.08845e6 −1.00000
\(173\) −9.29453e6 −1.79510 −0.897551 − 0.440910i \(-0.854656\pi\)
−0.897551 + 0.440910i \(0.854656\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6.70925e6 −1.23065
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −2.60572e6 −0.439432 −0.219716 − 0.975564i \(-0.570513\pi\)
−0.219716 + 0.975564i \(0.570513\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.15872e7 −1.77196
\(188\) 2.71066e6 0.407944
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 2.02439e6 0.281593 0.140796 − 0.990039i \(-0.455034\pi\)
0.140796 + 0.990039i \(0.455034\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 7.52954e6 1.00000
\(197\) 8.81735e6 1.15329 0.576646 − 0.816994i \(-0.304361\pi\)
0.576646 + 0.816994i \(0.304361\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.45109e6 −0.389085
\(208\) 1.51798e7 1.68685
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −1.79747e7 −1.88648
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.62162e7 2.42881
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 1.13906e7 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −1.08120e7 −0.900327 −0.450164 − 0.892946i \(-0.648634\pi\)
−0.450164 + 0.892946i \(0.648634\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 2.60133e7 1.97906
\(237\) 0 0
\(238\) 0 0
\(239\) −2.37845e7 −1.74221 −0.871103 − 0.491100i \(-0.836595\pi\)
−0.871103 + 0.491100i \(0.836595\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 9.95020e6 0.629232 0.314616 − 0.949219i \(-0.398124\pi\)
0.314616 + 0.949219i \(0.398124\pi\)
\(252\) 0 0
\(253\) 7.75429e6 0.478829
\(254\) 0 0
\(255\) 0 0
\(256\) 1.67772e7 1.00000
\(257\) 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
\(268\) −3.02033e7 −1.56910
\(269\) −3.82806e7 −1.96663 −0.983313 − 0.181923i \(-0.941768\pi\)
−0.983313 + 0.181923i \(0.941768\pi\)
\(270\) 0 0
\(271\) −3.39873e7 −1.70769 −0.853844 − 0.520529i \(-0.825735\pi\)
−0.853844 + 0.520529i \(0.825735\pi\)
\(272\) 2.89751e7 1.43985
\(273\) 0 0
\(274\) 0 0
\(275\) −2.55938e7 −1.23065
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −3.49322e7 −1.60847
\(280\) 0 0
\(281\) −1.34847e7 −0.607747 −0.303874 − 0.952712i \(-0.598280\pi\)
−0.303874 + 0.952712i \(0.598280\pi\)
\(282\) 0 0
\(283\) 1.73960e7 0.767522 0.383761 − 0.923432i \(-0.374629\pi\)
0.383761 + 0.923432i \(0.374629\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.59039e7 1.07318
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.52086e7 1.00218 0.501090 − 0.865395i \(-0.332933\pi\)
0.501090 + 0.865395i \(0.332933\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.75442e7 −0.656327
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −5.24865e7 −1.81398 −0.906990 − 0.421153i \(-0.861626\pi\)
−0.906990 + 0.421153i \(0.861626\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.84842e7 1.27938 0.639692 − 0.768631i \(-0.279062\pi\)
0.639692 + 0.768631i \(0.279062\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.66002e7 0.526080
\(317\) 2.82581e7 0.887084 0.443542 − 0.896254i \(-0.353722\pi\)
0.443542 + 0.896254i \(0.353722\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 3.40122e7 1.00000
\(325\) 5.79062e7 1.68685
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −4.35951e7 −1.19131
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7.41430e7 −1.93723 −0.968613 − 0.248572i \(-0.920039\pi\)
−0.968613 + 0.248572i \(0.920039\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 7.84897e7 1.97947
\(342\) 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
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.35869e7 0.990906 0.495453 − 0.868635i \(-0.335002\pi\)
0.495453 + 0.868635i \(0.335002\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9.25323e7 1.99991 0.999954 − 0.00964023i \(-0.00306863\pi\)
0.999954 + 0.00964023i \(0.00306863\pi\)
\(360\) 0 0
\(361\) 4.70459e7 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4.95813e7 −1.00304 −0.501522 − 0.865145i \(-0.667226\pi\)
−0.501522 + 0.865145i \(0.667226\pi\)
\(368\) −1.93905e7 −0.389085
\(369\) −1.00134e8 −1.99298
\(370\) 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
\(377\) 0 0
\(378\) 0 0
\(379\) 1.04748e8 1.92409 0.962047 − 0.272883i \(-0.0879773\pi\)
0.962047 + 0.272883i \(0.0879773\pi\)
\(380\) 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
\(387\) −5.79606e7 −1.00000
\(388\) −1.11440e8 −1.90785
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) −3.34883e7 −0.560226
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −7.64225e7 −1.23065
\(397\) −4.89286e7 −0.781973 −0.390986 − 0.920396i \(-0.627866\pi\)
−0.390986 + 0.920396i \(0.627866\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 6.40000e7 1.00000
\(401\) 1.11350e8 1.72685 0.863427 − 0.504474i \(-0.168314\pi\)
0.863427 + 0.504474i \(0.168314\pi\)
\(402\) 0 0
\(403\) −1.77584e8 −2.71324
\(404\) −5.41567e7 −0.821311
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.08195e7 −0.154708
\(413\) 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
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 3.08761e7 0.407944
\(424\) 0 0
\(425\) 1.10531e8 1.43985
\(426\) 0 0
\(427\) 0 0
\(428\) −8.75831e7 −1.11709
\(429\) 0 0
\(430\) 0 0
\(431\) 9.29385e7 1.16082 0.580409 − 0.814325i \(-0.302893\pi\)
0.580409 + 0.814325i \(0.302893\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7.65989e7 0.924194
\(437\) 0 0
\(438\) 0 0
\(439\) −1.65507e8 −1.95625 −0.978123 − 0.208027i \(-0.933296\pi\)
−0.978123 + 0.208027i \(0.933296\pi\)
\(440\) 0 0
\(441\) 8.57661e7 1.00000
\(442\) 0 0
\(443\) −5.63710e7 −0.648402 −0.324201 − 0.945988i \(-0.605095\pi\)
−0.324201 + 0.945988i \(0.605095\pi\)
\(444\) 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
\(451\) 2.24992e8 2.45266
\(452\) 0 0
\(453\) 0 0
\(454\) 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
\(461\) −1.36377e8 −1.39200 −0.695998 − 0.718044i \(-0.745038\pi\)
−0.695998 + 0.718044i \(0.745038\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 1.72907e8 1.68685
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.30232e8 1.23065
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2.04743e8 −1.88648
\(478\) 0 0
\(479\) 7.89322e7 0.718204 0.359102 − 0.933298i \(-0.383083\pi\)
0.359102 + 0.933298i \(0.383083\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 5.83349e7 0.514508
\(485\) 0 0
\(486\) 0 0
\(487\) 2.19286e8 1.89856 0.949278 − 0.314437i \(-0.101816\pi\)
0.949278 + 0.314437i \(0.101816\pi\)
\(488\) 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
\(496\) −1.96272e8 −1.60847
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 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
\(508\) 2.61301e8 1.99320
\(509\) 4.41177e7 0.334549 0.167274 − 0.985910i \(-0.446503\pi\)
0.167274 + 0.985910i \(0.446503\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −6.93759e7 −0.502038
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.38972e8 −2.31596
\(528\) 0 0
\(529\) −1.25625e8 −0.848613
\(530\) 0 0
\(531\) 2.96308e8 1.97906
\(532\) 0 0
\(533\) −5.09049e8 −3.36185
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.92709e8 −1.23065
\(540\) 0 0
\(541\) −6.14596e6 −0.0388148 −0.0194074 − 0.999812i \(-0.506178\pi\)
−0.0194074 + 0.999812i \(0.506178\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.30213e8 −0.795596 −0.397798 − 0.917473i \(-0.630225\pi\)
−0.397798 + 0.917473i \(0.630225\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 3.21468e8 1.87031
\(557\) −5.49149e7 −0.317778 −0.158889 − 0.987296i \(-0.550791\pi\)
−0.158889 + 0.987296i \(0.550791\pi\)
\(558\) 0 0
\(559\) −2.94653e8 −1.68685
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.40407e8 1.34717 0.673583 − 0.739112i \(-0.264754\pi\)
0.673583 + 0.739112i \(0.264754\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.13200e7 0.170014 0.0850071 − 0.996380i \(-0.472909\pi\)
0.0850071 + 0.996380i \(0.472909\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) −3.88507e8 −2.07592
\(573\) 0 0
\(574\) 0 0
\(575\) −7.39688e7 −0.389085
\(576\) 1.91103e8 1.00000
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 4.60039e8 2.32161
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 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
\(597\) 0 0
\(598\) 0 0
\(599\) 2.27361e8 1.05788 0.528939 − 0.848660i \(-0.322590\pi\)
0.528939 + 0.848660i \(0.322590\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −3.44034e8 −1.56910
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.56964e8 0.688139
\(612\) 3.30045e8 1.43985
\(613\) −4.50770e8 −1.95692 −0.978462 − 0.206428i \(-0.933816\pi\)
−0.978462 + 0.206428i \(0.933816\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.31686e8 1.41212 0.706060 − 0.708152i \(-0.250471\pi\)
0.706060 + 0.708152i \(0.250471\pi\)
\(618\) 0 0
\(619\) −1.39934e7 −0.0589998 −0.0294999 − 0.999565i \(-0.509391\pi\)
−0.0294999 + 0.999565i \(0.509391\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.44141e8 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4.36007e8 1.68685
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 3.06986e8 1.15474 0.577372 − 0.816481i \(-0.304078\pi\)
0.577372 + 0.816481i \(0.304078\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −6.65778e8 −2.43554
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −5.62618e8 −1.99298
\(657\) 0 0
\(658\) 0 0
\(659\) 5.36774e8 1.87558 0.937789 − 0.347205i \(-0.112869\pi\)
0.937789 + 0.347205i \(0.112869\pi\)
\(660\) 0 0
\(661\) 4.15775e8 1.43964 0.719820 − 0.694161i \(-0.244224\pi\)
0.719820 + 0.694161i \(0.244224\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 1.88847e7 0.0633551
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 5.70088e8 1.84545
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.30136e8 1.97775 0.988877 − 0.148737i \(-0.0475207\pi\)
0.988877 + 0.148737i \(0.0475207\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −3.25661e8 −1.00000
\(689\) −1.04084e9 −3.18221
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) −5.94850e8 −1.79510
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −9.71670e8 −2.86960
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −6.88587e8 −1.99896 −0.999481 − 0.0322157i \(-0.989744\pi\)
−0.999481 + 0.0322157i \(0.989744\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −4.29392e8 −1.23065
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 7.11959e8 1.99764 0.998818 − 0.0486144i \(-0.0154805\pi\)
0.998818 + 0.0486144i \(0.0154805\pi\)
\(710\) 0 0
\(711\) 1.89087e8 0.526080
\(712\) 0 0
\(713\) 2.26844e8 0.625833
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.56712e7 0.176680 0.0883402 − 0.996090i \(-0.471844\pi\)
0.0883402 + 0.996090i \(0.471844\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −1.66766e8 −0.439432
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 3.87420e8 1.00000
\(730\) 0 0
\(731\) −5.62433e8 −1.43985
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.73015e8 1.93101
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 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
\(747\) −4.96576e8 −1.19131
\(748\) −7.41582e8 −1.77196
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 1.73482e8 0.407944
\(753\) 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
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.50633e9 3.33837
\(768\) 0 0
\(769\) 3.64548e8 0.801634 0.400817 − 0.916158i \(-0.368726\pi\)
0.400817 + 0.916158i \(0.368726\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.29561e8 0.281593
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −7.48719e8 −1.60847
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 4.81890e8 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −4.82932e7 −0.0990745 −0.0495372 − 0.998772i \(-0.515775\pi\)
−0.0495372 + 0.998772i \(0.515775\pi\)
\(788\) 5.64311e8 1.15329
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −9.82821e8 −1.94133 −0.970665 − 0.240437i \(-0.922709\pi\)
−0.970665 + 0.240437i \(0.922709\pi\)
\(798\) 0 0
\(799\) 2.99612e8 0.587380
\(800\) 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
\(809\) 7.68270e8 1.45100 0.725502 − 0.688220i \(-0.241608\pi\)
0.725502 + 0.688220i \(0.241608\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9.64259e8 −1.74247 −0.871233 − 0.490869i \(-0.836679\pi\)
−0.871233 + 0.490869i \(0.836679\pi\)
\(822\) 0 0
\(823\) 8.61808e8 1.54601 0.773003 − 0.634403i \(-0.218754\pi\)
0.773003 + 0.634403i \(0.218754\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.14204e8 0.378713 0.189357 − 0.981908i \(-0.439360\pi\)
0.189357 + 0.981908i \(0.439360\pi\)
\(828\) −2.20870e8 −0.389085
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 9.71506e8 1.68685
\(833\) 8.32249e8 1.43985
\(834\) 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
\(841\) 5.94823e8 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −1.15038e9 −1.88648
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −5.84404e8 −0.941599 −0.470800 − 0.882240i \(-0.656035\pi\)
−0.470800 + 0.882240i \(0.656035\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.16869e9 1.85676 0.928380 − 0.371632i \(-0.121202\pi\)
0.928380 + 0.371632i \(0.121202\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 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
\(868\) 0 0
\(869\) −4.24861e8 −0.647422
\(870\) 0 0
\(871\) −1.74896e9 −2.64682
\(872\) 0 0
\(873\) −1.26937e9 −1.90785
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.17927e9 −1.74830 −0.874148 − 0.485660i \(-0.838579\pi\)
−0.874148 + 0.485660i \(0.838579\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.07692e9 1.57490 0.787452 − 0.616376i \(-0.211400\pi\)
0.787452 + 0.616376i \(0.211400\pi\)
\(882\) 0 0
\(883\) −1.04569e9 −1.51887 −0.759436 − 0.650582i \(-0.774525\pi\)
−0.759436 + 0.650582i \(0.774525\pi\)
\(884\) 1.67784e9 2.42881
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −8.70500e8 −1.23065
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 7.29000e8 1.00000
\(901\) −1.98676e9 −2.71626
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 6.97955e8 0.935417 0.467709 − 0.883883i \(-0.345080\pi\)
0.467709 + 0.883883i \(0.345080\pi\)
\(908\) 0 0
\(909\) −6.16878e8 −0.821311
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 1.11576e9 1.46609
\(914\) 0 0
\(915\) 0 0
\(916\) −6.91969e8 −0.900327
\(917\) 0 0
\(918\) 0 0
\(919\) −1.48901e9 −1.91845 −0.959223 − 0.282650i \(-0.908787\pi\)
−0.959223 + 0.282650i \(0.908787\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1.23240e8 −0.154708
\(928\) 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
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.33415e9 1.60117 0.800584 − 0.599220i \(-0.204523\pi\)
0.800584 + 0.599220i \(0.204523\pi\)
\(942\) 0 0
\(943\) 6.50253e8 0.775438
\(944\) 1.66485e9 1.97906
\(945\) 0 0
\(946\) 0 0
\(947\) 5.63397e8 0.663384 0.331692 − 0.943388i \(-0.392381\pi\)
0.331692 + 0.943388i \(0.392381\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.52221e9 −1.74221
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.40863e9 1.58718
\(962\) 0 0
\(963\) −9.97626e8 −1.11709
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −6.37454e8 −0.704968 −0.352484 − 0.935818i \(-0.614663\pi\)
−0.352484 + 0.935818i \(0.614663\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1.80062e9 −1.96681 −0.983407 − 0.181414i \(-0.941933\pi\)
−0.983407 + 0.181414i \(0.941933\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.77697e9 1.90545 0.952724 − 0.303837i \(-0.0982679\pi\)
0.952724 + 0.303837i \(0.0982679\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 8.72509e8 0.924194
\(982\) 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
\(989\) 3.76386e8 0.389085
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 43.7.b.a.42.1 ✓ 1
43.42 odd 2 CM 43.7.b.a.42.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.7.b.a.42.1 ✓ 1 1.1 even 1 trivial
43.7.b.a.42.1 ✓ 1 43.42 odd 2 CM