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Properties

Label 43.11.b.a.42.1
Level $43$
Weight $11$
Character 43.42
Self dual yes
Analytic conductor $27.320$
Analytic rank $0$
Dimension $1$
CM discriminant -43
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] = [43,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.42");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 11 \)
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: \(27.3203618650\)
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)
 
\(f(q)\) \(=\) \(q+1024.00 q^{4} +59049.0 q^{9} +O(q^{10})\) \(q+1024.00 q^{4} +59049.0 q^{9} -18501.0 q^{11} +303943. q^{13} +1.04858e6 q^{16} -2.76409e6 q^{17} +4.12644e6 q^{23} +9.76562e6 q^{25} +5.72531e7 q^{31} +6.04662e7 q^{36} +1.42671e8 q^{41} -1.47008e8 q^{43} -1.89450e7 q^{44} +4.51177e8 q^{47} +2.82475e8 q^{49} +3.11238e8 q^{52} -3.39721e7 q^{53} -9.90192e8 q^{59} +1.07374e9 q^{64} -1.50482e9 q^{67} -2.83043e9 q^{68} -2.64142e9 q^{79} +3.48678e9 q^{81} -6.75764e9 q^{83} +4.22548e9 q^{92} -1.50068e10 q^{97} -1.09247e9 q^{99} +O(q^{100})\)

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 \(\pi\)
\(4\) 1024.00 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
\(9\) 59049.0 1.00000
\(10\) 0 0
\(11\) −18501.0 −0.114877 −0.0574383 − 0.998349i \(-0.518293\pi\)
−0.0574383 + 0.998349i \(0.518293\pi\)
\(12\) 0 0
\(13\) 303943. 0.818607 0.409303 − 0.912398i \(-0.365772\pi\)
0.409303 + 0.912398i \(0.365772\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.04858e6 1.00000
\(17\) −2.76409e6 −1.94674 −0.973369 − 0.229245i \(-0.926374\pi\)
−0.973369 + 0.229245i \(0.926374\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.12644e6 0.641116 0.320558 − 0.947229i \(-0.396130\pi\)
0.320558 + 0.947229i \(0.396130\pi\)
\(24\) 0 0
\(25\) 9.76562e6 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 5.72531e7 1.99982 0.999909 − 0.0134807i \(-0.00429117\pi\)
0.999909 + 0.0134807i \(0.00429117\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 6.04662e7 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.42671e8 1.23145 0.615726 − 0.787960i \(-0.288863\pi\)
0.615726 + 0.787960i \(0.288863\pi\)
\(42\) 0 0
\(43\) −1.47008e8 −1.00000
\(44\) −1.89450e7 −0.114877
\(45\) 0 0
\(46\) 0 0
\(47\) 4.51177e8 1.96724 0.983620 − 0.180252i \(-0.0576913\pi\)
0.983620 + 0.180252i \(0.0576913\pi\)
\(48\) 0 0
\(49\) 2.82475e8 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 3.11238e8 0.818607
\(53\) −3.39721e7 −0.0812349 −0.0406174 − 0.999175i \(-0.512932\pi\)
−0.0406174 + 0.999175i \(0.512932\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.90192e8 −1.38503 −0.692515 − 0.721403i \(-0.743497\pi\)
−0.692515 + 0.721403i \(0.743497\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.07374e9 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.50482e9 −1.11458 −0.557289 − 0.830319i \(-0.688158\pi\)
−0.557289 + 0.830319i \(0.688158\pi\)
\(68\) −2.83043e9 −1.94674
\(69\) 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
\(79\) −2.64142e9 −0.858423 −0.429212 − 0.903204i \(-0.641209\pi\)
−0.429212 + 0.903204i \(0.641209\pi\)
\(80\) 0 0
\(81\) 3.48678e9 1.00000
\(82\) 0 0
\(83\) −6.75764e9 −1.71555 −0.857777 − 0.514021i \(-0.828155\pi\)
−0.857777 + 0.514021i \(0.828155\pi\)
\(84\) 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
\(92\) 4.22548e9 0.641116
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.50068e10 −1.74754 −0.873772 − 0.486336i \(-0.838333\pi\)
−0.873772 + 0.486336i \(0.838333\pi\)
\(98\) 0 0
\(99\) −1.09247e9 −0.114877
\(100\) 1.00000e10 1.00000
\(101\) −2.06742e10 −1.96708 −0.983540 − 0.180692i \(-0.942166\pi\)
−0.983540 + 0.180692i \(0.942166\pi\)
\(102\) 0 0
\(103\) 1.84203e10 1.58895 0.794476 − 0.607295i \(-0.207745\pi\)
0.794476 + 0.607295i \(0.207745\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.34162e10 1.66955 0.834773 − 0.550595i \(-0.185599\pi\)
0.834773 + 0.550595i \(0.185599\pi\)
\(108\) 0 0
\(109\) 2.95995e10 1.92376 0.961881 − 0.273470i \(-0.0881713\pi\)
0.961881 + 0.273470i \(0.0881713\pi\)
\(110\) 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
\(117\) 1.79475e10 0.818607
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.55951e10 −0.986803
\(122\) 0 0
\(123\) 0 0
\(124\) 5.86272e10 1.99982
\(125\) 0 0
\(126\) 0 0
\(127\) −2.48836e10 −0.753172 −0.376586 − 0.926382i \(-0.622902\pi\)
−0.376586 + 0.926382i \(0.622902\pi\)
\(128\) 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
\(139\) −9.37290e10 −1.80634 −0.903171 − 0.429280i \(-0.858767\pi\)
−0.903171 + 0.429280i \(0.858767\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.62325e9 −0.0940388
\(144\) 6.19174e10 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
\(153\) −1.63217e11 −1.94674
\(154\) 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 \(\pi\)
\(164\) 1.46096e11 1.23145
\(165\) 0 0
\(166\) 0 0
\(167\) −2.17810e11 −1.67686 −0.838428 − 0.545012i \(-0.816525\pi\)
−0.838428 + 0.545012i \(0.816525\pi\)
\(168\) 0 0
\(169\) −4.54771e10 −0.329883
\(170\) 0 0
\(171\) 0 0
\(172\) −1.50537e11 −1.00000
\(173\) −2.24429e11 −1.44827 −0.724133 − 0.689661i \(-0.757760\pi\)
−0.724133 + 0.689661i \(0.757760\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.93997e10 −0.114877
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 2.43697e11 1.25446 0.627230 − 0.778834i \(-0.284188\pi\)
0.627230 + 0.778834i \(0.284188\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.11384e10 0.223635
\(188\) 4.62005e11 1.96724
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −3.88554e11 −1.45099 −0.725496 − 0.688226i \(-0.758390\pi\)
−0.725496 + 0.688226i \(0.758390\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2.89255e11 1.00000
\(197\) 5.20556e11 1.75443 0.877216 − 0.480096i \(-0.159398\pi\)
0.877216 + 0.480096i \(0.159398\pi\)
\(198\) 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
\(207\) 2.43662e11 0.641116
\(208\) 3.18707e11 0.818607
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −3.47874e10 −0.0812349
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −8.40126e11 −1.59361
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 5.76650e11 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 8.84229e11 1.40407 0.702033 − 0.712145i \(-0.252276\pi\)
0.702033 + 0.712145i \(0.252276\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.01396e12 −1.38503
\(237\) 0 0
\(238\) 0 0
\(239\) −5.08317e11 −0.651846 −0.325923 − 0.945396i \(-0.605675\pi\)
−0.325923 + 0.945396i \(0.605675\pi\)
\(240\) 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
\(251\) −9.79256e11 −0.982941 −0.491471 − 0.870894i \(-0.663540\pi\)
−0.491471 + 0.870894i \(0.663540\pi\)
\(252\) 0 0
\(253\) −7.63433e10 −0.0736493
\(254\) 0 0
\(255\) 0 0
\(256\) 1.09951e12 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
\(268\) −1.54094e12 −1.11458
\(269\) −2.68710e12 −1.90775 −0.953876 − 0.300200i \(-0.902946\pi\)
−0.953876 + 0.300200i \(0.902946\pi\)
\(270\) 0 0
\(271\) −1.78851e12 −1.22362 −0.611809 − 0.791005i \(-0.709558\pi\)
−0.611809 + 0.791005i \(0.709558\pi\)
\(272\) −2.89836e12 −1.94674
\(273\) 0 0
\(274\) 0 0
\(275\) −1.80674e11 −0.114877
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 3.38074e12 1.99982
\(280\) 0 0
\(281\) 1.77925e12 1.01556 0.507779 − 0.861488i \(-0.330467\pi\)
0.507779 + 0.861488i \(0.330467\pi\)
\(282\) 0 0
\(283\) −1.38279e12 −0.761773 −0.380886 − 0.924622i \(-0.624381\pi\)
−0.380886 + 0.924622i \(0.624381\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5.62419e12 2.78979
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.31425e12 −1.53478 −0.767391 − 0.641179i \(-0.778446\pi\)
−0.767391 + 0.641179i \(0.778446\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.25420e12 0.524822
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5.17261e12 1.89678 0.948392 − 0.317101i \(-0.102709\pi\)
0.948392 + 0.317101i \(0.102709\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.69064e12 1.61224 0.806120 − 0.591752i \(-0.201563\pi\)
0.806120 + 0.591752i \(0.201563\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −2.70481e12 −0.858423
\(317\) −1.77718e12 −0.555183 −0.277591 − 0.960699i \(-0.589536\pi\)
−0.277591 + 0.960699i \(0.589536\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 3.57047e12 1.00000
\(325\) 2.96819e12 0.818607
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −6.91982e12 −1.71555
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7.03191e12 1.61780 0.808898 − 0.587949i \(-0.200065\pi\)
0.808898 + 0.587949i \(0.200065\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.05924e12 −0.229732
\(342\) 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
\(353\) −1.99785e12 −0.364493 −0.182247 − 0.983253i \(-0.558337\pi\)
−0.182247 + 0.983253i \(0.558337\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.12827e12 −1.02770 −0.513850 − 0.857880i \(-0.671781\pi\)
−0.513850 + 0.857880i \(0.671781\pi\)
\(360\) 0 0
\(361\) 6.13107e12 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.24992e13 −1.87737 −0.938686 − 0.344772i \(-0.887956\pi\)
−0.938686 + 0.344772i \(0.887956\pi\)
\(368\) 4.32689e12 0.641116
\(369\) 8.42460e12 1.23145
\(370\) 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
\(379\) −1.30255e13 −1.66571 −0.832855 − 0.553491i \(-0.813295\pi\)
−0.832855 + 0.553491i \(0.813295\pi\)
\(380\) 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
\(387\) −8.68070e12 −1.00000
\(388\) −1.53669e13 −1.74754
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) −1.14059e13 −1.24808
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −1.11868e12 −0.114877
\(397\) 1.22403e13 1.24120 0.620598 − 0.784129i \(-0.286890\pi\)
0.620598 + 0.784129i \(0.286890\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.02400e13 1.00000
\(401\) −2.04525e13 −1.97254 −0.986269 − 0.165150i \(-0.947189\pi\)
−0.986269 + 0.165150i \(0.947189\pi\)
\(402\) 0 0
\(403\) 1.74017e13 1.63706
\(404\) −2.11704e13 −1.96708
\(405\) 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
\(412\) 1.88624e13 1.58895
\(413\) 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
\(423\) 2.66415e13 1.96724
\(424\) 0 0
\(425\) −2.69931e13 −1.94674
\(426\) 0 0
\(427\) 0 0
\(428\) 2.39782e13 1.66955
\(429\) 0 0
\(430\) 0 0
\(431\) −4.50537e11 −0.0302931 −0.0151466 − 0.999885i \(-0.504821\pi\)
−0.0151466 + 0.999885i \(0.504821\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.03098e13 1.92376
\(437\) 0 0
\(438\) 0 0
\(439\) 5.65637e12 0.346909 0.173454 − 0.984842i \(-0.444507\pi\)
0.173454 + 0.984842i \(0.444507\pi\)
\(440\) 0 0
\(441\) 1.66799e13 1.00000
\(442\) 0 0
\(443\) −3.41110e13 −1.99929 −0.999644 − 0.0266768i \(-0.991508\pi\)
−0.999644 + 0.0266768i \(0.991508\pi\)
\(444\) 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
\(451\) −2.63956e12 −0.141465
\(452\) 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
\(461\) 3.99298e13 1.91775 0.958876 − 0.283826i \(-0.0916039\pi\)
0.958876 + 0.283826i \(0.0916039\pi\)
\(462\) 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 \(\pi\)
\(468\) 1.83783e13 0.818607
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.71980e12 0.114877
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2.00602e12 −0.0812349
\(478\) 0 0
\(479\) −2.89808e13 −1.14930 −0.574649 − 0.818400i \(-0.694862\pi\)
−0.574649 + 0.818400i \(0.694862\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −2.62094e13 −0.986803
\(485\) 0 0
\(486\) 0 0
\(487\) 5.20825e11 0.0190128 0.00950641 − 0.999955i \(-0.496974\pi\)
0.00950641 + 0.999955i \(0.496974\pi\)
\(488\) 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
\(496\) 6.00342e13 1.99982
\(497\) 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
\(508\) −2.54808e13 −0.753172
\(509\) −1.88907e13 −0.552917 −0.276459 − 0.961026i \(-0.589161\pi\)
−0.276459 + 0.961026i \(0.589161\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −8.34722e12 −0.225990
\(518\) 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
\(527\) −1.58253e14 −3.89312
\(528\) 0 0
\(529\) −2.43990e13 −0.588970
\(530\) 0 0
\(531\) −5.84698e13 −1.38503
\(532\) 0 0
\(533\) 4.33640e13 1.00808
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.22607e12 −0.114877
\(540\) 0 0
\(541\) 2.99767e12 0.0646842 0.0323421 − 0.999477i \(-0.489703\pi\)
0.0323421 + 0.999477i \(0.489703\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.49911e13 0.714531 0.357265 − 0.934003i \(-0.383709\pi\)
0.357265 + 0.934003i \(0.383709\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −9.59785e13 −1.80634
\(557\) 7.55065e13 1.40834 0.704171 − 0.710030i \(-0.251319\pi\)
0.704171 + 0.710030i \(0.251319\pi\)
\(558\) 0 0
\(559\) −4.46822e13 −0.818607
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.07590e13 0.366999 0.183500 − 0.983020i \(-0.441257\pi\)
0.183500 + 0.983020i \(0.441257\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.10700e14 1.85603 0.928014 − 0.372544i \(-0.121515\pi\)
0.928014 + 0.372544i \(0.121515\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) −5.75821e12 −0.0940388
\(573\) 0 0
\(574\) 0 0
\(575\) 4.02973e13 0.641116
\(576\) 6.34034e13 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
\(583\) 6.28517e11 0.00933199
\(584\) 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
\(599\) 1.41732e14 1.83795 0.918977 − 0.394310i \(-0.129016\pi\)
0.918977 + 0.394310i \(0.129016\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −8.88581e13 −1.11458
\(604\) 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
\(611\) 1.37132e14 1.61040
\(612\) −1.67134e14 −1.94674
\(613\) 1.32331e14 1.52883 0.764417 − 0.644722i \(-0.223027\pi\)
0.764417 + 0.644722i \(0.223027\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.26768e14 1.41770 0.708848 − 0.705361i \(-0.249215\pi\)
0.708848 + 0.705361i \(0.249215\pi\)
\(618\) 0 0
\(619\) −1.61680e14 −1.77911 −0.889555 − 0.456827i \(-0.848986\pi\)
−0.889555 + 0.456827i \(0.848986\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.53674e13 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
\(637\) 8.58564e13 0.818607
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −1.87987e14 −1.71030 −0.855152 − 0.518377i \(-0.826536\pi\)
−0.855152 + 0.518377i \(0.826536\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 1.83195e13 0.159108
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.49602e14 1.23145
\(657\) 0 0
\(658\) 0 0
\(659\) 1.67370e13 0.134663 0.0673317 − 0.997731i \(-0.478551\pi\)
0.0673317 + 0.997731i \(0.478551\pi\)
\(660\) 0 0
\(661\) 7.26589e13 0.575813 0.287907 − 0.957659i \(-0.407041\pi\)
0.287907 + 0.957659i \(0.407041\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −2.23038e14 −1.67686
\(669\) 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
\(676\) −4.65686e13 −0.329883
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −8.06568e13 −0.542672 −0.271336 − 0.962485i \(-0.587466\pi\)
−0.271336 + 0.962485i \(0.587466\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −1.54150e14 −1.00000
\(689\) −1.03256e13 −0.0664994
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) −2.29815e14 −1.44827
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −3.94356e14 −2.39731
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.53292e14 0.905588 0.452794 − 0.891615i \(-0.350427\pi\)
0.452794 + 0.891615i \(0.350427\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.98653e13 −0.114877
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.03692e14 −1.13696 −0.568478 − 0.822698i \(-0.692468\pi\)
−0.568478 + 0.822698i \(0.692468\pi\)
\(710\) 0 0
\(711\) −1.55973e14 −0.858423
\(712\) 0 0
\(713\) 2.36252e14 1.28212
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −3.00981e14 −1.56637 −0.783185 − 0.621789i \(-0.786406\pi\)
−0.783185 + 0.621789i \(0.786406\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 2.49546e14 1.25446
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2.05891e14 1.00000
\(730\) 0 0
\(731\) 4.06344e14 1.94674
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.78407e13 0.128039
\(738\) 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
\(747\) −3.99032e14 −1.71555
\(748\) 5.23657e13 0.223635
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 4.73093e14 1.96724
\(753\) 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
\(767\) −3.00962e14 −1.13380
\(768\) 0 0
\(769\) 5.30656e14 1.97325 0.986623 − 0.163018i \(-0.0521228\pi\)
0.986623 + 0.163018i \(0.0521228\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −3.97879e14 −1.45099
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 5.59112e14 1.99982
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 2.96197e14 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 4.98159e13 0.165004 0.0825020 − 0.996591i \(-0.473709\pi\)
0.0825020 + 0.996591i \(0.473709\pi\)
\(788\) 5.33049e14 1.75443
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.62961e13 0.237252 0.118626 − 0.992939i \(-0.462151\pi\)
0.118626 + 0.992939i \(0.462151\pi\)
\(798\) 0 0
\(799\) −1.24709e15 −3.82970
\(800\) 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
\(809\) 4.68069e14 1.35073 0.675363 − 0.737486i \(-0.263987\pi\)
0.675363 + 0.737486i \(0.263987\pi\)
\(810\) 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
\(821\) 7.32296e14 1.96323 0.981615 − 0.190874i \(-0.0611321\pi\)
0.981615 + 0.190874i \(0.0611321\pi\)
\(822\) 0 0
\(823\) 3.11639e14 0.825377 0.412689 − 0.910872i \(-0.364590\pi\)
0.412689 + 0.910872i \(0.364590\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −5.15758e14 −1.33327 −0.666636 − 0.745384i \(-0.732266\pi\)
−0.666636 + 0.745384i \(0.732266\pi\)
\(828\) 2.49510e14 0.641116
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.26356e14 0.818607
\(833\) −7.80787e14 −1.94674
\(834\) 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
\(841\) 4.20707e14 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −3.56223e13 −0.0812349
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −8.64586e14 −1.91453 −0.957267 − 0.289205i \(-0.906609\pi\)
−0.957267 + 0.289205i \(0.906609\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.02426e14 0.221567 0.110784 − 0.993845i \(-0.464664\pi\)
0.110784 + 0.993845i \(0.464664\pi\)
\(858\) 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
\(869\) 4.88689e13 0.0986128
\(870\) 0 0
\(871\) −4.57379e14 −0.912401
\(872\) 0 0
\(873\) −8.86134e14 −1.74754
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6.88532e14 −1.32717 −0.663585 − 0.748101i \(-0.730966\pi\)
−0.663585 + 0.748101i \(0.730966\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 4.74954e14 0.894895 0.447447 − 0.894310i \(-0.352333\pi\)
0.447447 + 0.894310i \(0.352333\pi\)
\(882\) 0 0
\(883\) −4.08372e14 −0.760769 −0.380384 − 0.924828i \(-0.624208\pi\)
−0.380384 + 0.924828i \(0.624208\pi\)
\(884\) −8.60289e14 −1.59361
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −6.45090e13 −0.114877
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 5.90490e14 1.00000
\(901\) 9.39018e13 0.158143
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −8.90137e14 −1.45017 −0.725087 − 0.688657i \(-0.758201\pi\)
−0.725087 + 0.688657i \(0.758201\pi\)
\(908\) 0 0
\(909\) −1.22079e15 −1.96708
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 1.25023e14 0.197077
\(914\) 0 0
\(915\) 0 0
\(916\) 9.05450e14 1.40407
\(917\) 0 0
\(918\) 0 0
\(919\) 1.10402e15 1.68423 0.842114 − 0.539300i \(-0.181311\pi\)
0.842114 + 0.539300i \(0.181311\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.08770e15 1.58895
\(928\) 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
\(941\) 7.07351e14 0.958709 0.479354 − 0.877621i \(-0.340871\pi\)
0.479354 + 0.877621i \(0.340871\pi\)
\(942\) 0 0
\(943\) 5.88725e14 0.789504
\(944\) −1.03829e15 −1.38503
\(945\) 0 0
\(946\) 0 0
\(947\) −8.13650e14 −1.06829 −0.534143 − 0.845394i \(-0.679366\pi\)
−0.534143 + 0.845394i \(0.679366\pi\)
\(948\) 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
\(956\) −5.20516e14 −0.651846
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.45829e15 2.99927
\(962\) 0 0
\(963\) 1.38271e15 1.66955
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.68609e15 −1.99411 −0.997054 − 0.0766992i \(-0.975562\pi\)
−0.997054 + 0.0766992i \(0.975562\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.27117e15 1.47267 0.736336 − 0.676616i \(-0.236554\pi\)
0.736336 + 0.676616i \(0.236554\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.53368e15 −1.72291 −0.861455 − 0.507834i \(-0.830446\pi\)
−0.861455 + 0.507834i \(0.830446\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.74782e15 1.92376
\(982\) 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
\(989\) −6.06622e14 −0.641116
\(990\) 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 43.11.b.a.42.1 ✓ 1
43.42 odd 2 CM 43.11.b.a.42.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.11.b.a.42.1 ✓ 1 1.1 even 1 trivial
43.11.b.a.42.1 ✓ 1 43.42 odd 2 CM