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Properties

Label 32.13.d.a.15.1
Level $32$
Weight $13$
Character 32.15
Self dual yes
Analytic conductor $29.248$
Analytic rank $0$
Dimension $1$
CM discriminant -8
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] = [32,13,Mod(15,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.15");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 32.d (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: yes
Analytic conductor: \(29.2478021528\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 15.1
Character \(\chi\) \(=\) 32.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-658.000 q^{3} -98477.0 q^{9} -1.92312e6 q^{11} -4.52961e7 q^{17} +8.79314e7 q^{19} +2.44141e8 q^{25} +4.14486e8 q^{27} +1.26541e9 q^{33} +8.62826e9 q^{41} +7.03062e9 q^{43} +1.38413e10 q^{49} +2.98048e10 q^{51} -5.78589e10 q^{57} -8.63831e9 q^{59} -1.75046e11 q^{67} +4.91395e10 q^{73} -1.60645e11 q^{75} -2.20397e11 q^{81} +1.92940e11 q^{83} -8.66326e11 q^{89} +1.65649e12 q^{97} +1.89383e11 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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
\(3\) −658.000 −0.902606 −0.451303 − 0.892371i \(-0.649041\pi\)
−0.451303 + 0.892371i \(0.649041\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −98477.0 −0.185302
\(10\) 0 0
\(11\) −1.92312e6 −1.08555 −0.542776 − 0.839877i \(-0.682627\pi\)
−0.542776 + 0.839877i \(0.682627\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.52961e7 −1.87658 −0.938290 − 0.345850i \(-0.887590\pi\)
−0.938290 + 0.345850i \(0.887590\pi\)
\(18\) 0 0
\(19\) 8.79314e7 1.86906 0.934529 − 0.355888i \(-0.115822\pi\)
0.934529 + 0.355888i \(0.115822\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 2.44141e8 1.00000
\(26\) 0 0
\(27\) 4.14486e8 1.06986
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 1.26541e9 0.979826
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.62826e9 1.81644 0.908218 − 0.418498i \(-0.137443\pi\)
0.908218 + 0.418498i \(0.137443\pi\)
\(42\) 0 0
\(43\) 7.03062e9 1.11220 0.556100 − 0.831115i \(-0.312297\pi\)
0.556100 + 0.831115i \(0.312297\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.38413e10 1.00000
\(50\) 0 0
\(51\) 2.98048e10 1.69381
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.78589e10 −1.68702
\(58\) 0 0
\(59\) −8.63831e9 −0.204794 −0.102397 − 0.994744i \(-0.532651\pi\)
−0.102397 + 0.994744i \(0.532651\pi\)
\(60\) 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
\(67\) −1.75046e11 −1.93510 −0.967549 − 0.252684i \(-0.918687\pi\)
−0.967549 + 0.252684i \(0.918687\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 4.91395e10 0.324708 0.162354 − 0.986733i \(-0.448091\pi\)
0.162354 + 0.986733i \(0.448091\pi\)
\(74\) 0 0
\(75\) −1.60645e11 −0.902606
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −2.20397e11 −0.780361
\(82\) 0 0
\(83\) 1.92940e11 0.590139 0.295069 − 0.955476i \(-0.404657\pi\)
0.295069 + 0.955476i \(0.404657\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.66326e11 −1.74318 −0.871589 − 0.490238i \(-0.836910\pi\)
−0.871589 + 0.490238i \(0.836910\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.65649e12 1.98865 0.994324 − 0.106394i \(-0.0339306\pi\)
0.994324 + 0.106394i \(0.0339306\pi\)
\(98\) 0 0
\(99\) 1.89383e11 0.201155
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.77677e12 1.85028 0.925140 − 0.379626i \(-0.123947\pi\)
0.925140 + 0.379626i \(0.123947\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.74847e12 1.80046 0.900230 − 0.435416i \(-0.143399\pi\)
0.900230 + 0.435416i \(0.143399\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.59970e11 0.178424
\(122\) 0 0
\(123\) −5.67739e12 −1.63953
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) −4.62615e12 −1.00388
\(130\) 0 0
\(131\) 1.38397e12 0.273841 0.136920 − 0.990582i \(-0.456280\pi\)
0.136920 + 0.990582i \(0.456280\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.32173e13 −1.99903 −0.999514 − 0.0311883i \(-0.990071\pi\)
−0.999514 + 0.0311883i \(0.990071\pi\)
\(138\) 0 0
\(139\) 4.19430e12 0.581528 0.290764 − 0.956795i \(-0.406091\pi\)
0.290764 + 0.956795i \(0.406091\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −9.10757e12 −0.902606
\(148\) 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
\(153\) 4.46062e12 0.347734
\(154\) 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
\(163\) −2.20968e13 −1.17816 −0.589080 − 0.808075i \(-0.700510\pi\)
−0.589080 + 0.808075i \(0.700510\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 2.32981e13 1.00000
\(170\) 0 0
\(171\) −8.65922e12 −0.346340
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.68401e12 0.184848
\(178\) 0 0
\(179\) 5.53626e13 1.68305 0.841527 − 0.540215i \(-0.181657\pi\)
0.841527 + 0.540215i \(0.181657\pi\)
\(180\) 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
\(187\) 8.71099e13 2.03712
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −3.16470e12 −0.0612335 −0.0306167 − 0.999531i \(-0.509747\pi\)
−0.0306167 + 0.999531i \(0.509747\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 1.15180e14 1.74663
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.69103e14 −2.02896
\(210\) 0 0
\(211\) −1.71035e14 −1.93816 −0.969079 − 0.246750i \(-0.920637\pi\)
−0.969079 + 0.246750i \(0.920637\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3.23338e13 −0.293084
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −2.40422e13 −0.185302
\(226\) 0 0
\(227\) −1.17238e14 −0.856870 −0.428435 − 0.903573i \(-0.640935\pi\)
−0.428435 + 0.903573i \(0.640935\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.97639e14 −1.23520 −0.617599 − 0.786494i \(-0.711894\pi\)
−0.617599 + 0.786494i \(0.711894\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 3.10482e14 1.58465 0.792326 − 0.610097i \(-0.208870\pi\)
0.792326 + 0.610097i \(0.208870\pi\)
\(242\) 0 0
\(243\) −7.52536e13 −0.365502
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −1.26955e14 −0.532663
\(250\) 0 0
\(251\) 3.28002e14 1.31170 0.655849 − 0.754892i \(-0.272311\pi\)
0.655849 + 0.754892i \(0.272311\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.16326e14 −0.750775 −0.375388 − 0.926868i \(-0.622490\pi\)
−0.375388 + 0.926868i \(0.622490\pi\)
\(258\) 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
\(267\) 5.70043e14 1.57340
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.69512e14 −1.08555
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.55524e14 1.73778 0.868889 − 0.495007i \(-0.164834\pi\)
0.868889 + 0.495007i \(0.164834\pi\)
\(282\) 0 0
\(283\) 6.60679e14 1.28609 0.643046 − 0.765828i \(-0.277671\pi\)
0.643046 + 0.765828i \(0.277671\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.46911e15 2.52155
\(290\) 0 0
\(291\) −1.08997e15 −1.79497
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −7.97107e14 −1.16139
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.63875e15 1.95741 0.978706 − 0.205269i \(-0.0658069\pi\)
0.978706 + 0.205269i \(0.0658069\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −1.79845e15 −1.91264 −0.956319 − 0.292326i \(-0.905571\pi\)
−0.956319 + 0.292326i \(0.905571\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1.82712e15 −1.67007
\(322\) 0 0
\(323\) −3.98295e15 −3.50743
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.60911e15 1.98392 0.991959 − 0.126557i \(-0.0403928\pi\)
0.991959 + 0.126557i \(0.0403928\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −5.20794e13 −0.0355538 −0.0177769 − 0.999842i \(-0.505659\pi\)
−0.0177769 + 0.999842i \(0.505659\pi\)
\(338\) 0 0
\(339\) −2.46649e15 −1.62511
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.26334e15 0.723675 0.361837 − 0.932241i \(-0.382150\pi\)
0.361837 + 0.932241i \(0.382150\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.84078e14 0.198505 0.0992523 − 0.995062i \(-0.468355\pi\)
0.0992523 + 0.995062i \(0.468355\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 5.51862e15 2.49337
\(362\) 0 0
\(363\) −3.68460e14 −0.161046
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −8.49685e14 −0.336589
\(370\) 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
\(379\) 5.18672e15 1.75008 0.875039 − 0.484052i \(-0.160835\pi\)
0.875039 + 0.484052i \(0.160835\pi\)
\(380\) 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
\(387\) −6.92354e14 −0.206093
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −9.10650e14 −0.247170
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.92614e15 −0.463257 −0.231629 − 0.972804i \(-0.574405\pi\)
−0.231629 + 0.972804i \(0.574405\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 7.63062e15 1.63012 0.815061 − 0.579375i \(-0.196703\pi\)
0.815061 + 0.579375i \(0.196703\pi\)
\(410\) 0 0
\(411\) 8.69697e15 1.80433
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2.75985e15 −0.524891
\(418\) 0 0
\(419\) −7.38079e15 −1.36401 −0.682007 − 0.731346i \(-0.738892\pi\)
−0.682007 + 0.731346i \(0.738892\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.10586e16 −1.87658
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 4.24415e15 0.643966 0.321983 − 0.946745i \(-0.395651\pi\)
0.321983 + 0.946745i \(0.395651\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −1.36305e15 −0.185302
\(442\) 0 0
\(443\) −1.04567e16 −1.38348 −0.691738 − 0.722149i \(-0.743155\pi\)
−0.691738 + 0.722149i \(0.743155\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −5.86206e14 −0.0715438 −0.0357719 − 0.999360i \(-0.511389\pi\)
−0.0357719 + 0.999360i \(0.511389\pi\)
\(450\) 0 0
\(451\) −1.65932e16 −1.97184
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.78261e14 0.0195686 0.00978428 − 0.999952i \(-0.496886\pi\)
0.00978428 + 0.999952i \(0.496886\pi\)
\(458\) 0 0
\(459\) −1.87746e16 −2.00768
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.02528e16 1.95246 0.976232 − 0.216730i \(-0.0695392\pi\)
0.976232 + 0.216730i \(0.0695392\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.35207e16 −1.20735
\(474\) 0 0
\(475\) 2.14676e16 1.86906
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 1.45397e16 1.06341
\(490\) 0 0
\(491\) 2.03908e16 1.45527 0.727637 − 0.685963i \(-0.240619\pi\)
0.727637 + 0.685963i \(0.240619\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.39468e16 1.55112 0.775559 − 0.631275i \(-0.217468\pi\)
0.775559 + 0.631275i \(0.217468\pi\)
\(500\) 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
\(507\) −1.53301e16 −0.902606
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.64464e16 1.99963
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −4.38740e14 −0.0219372 −0.0109686 − 0.999940i \(-0.503491\pi\)
−0.0109686 + 0.999940i \(0.503491\pi\)
\(522\) 0 0
\(523\) −2.85281e16 −1.39400 −0.697001 − 0.717070i \(-0.745483\pi\)
−0.697001 + 0.717070i \(0.745483\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 2.19146e16 1.00000
\(530\) 0 0
\(531\) 8.50675e14 0.0379487
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.64286e16 −1.51914
\(538\) 0 0
\(539\) −2.66185e16 −1.08555
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.10476e16 1.15906 0.579528 − 0.814952i \(-0.303237\pi\)
0.579528 + 0.814952i \(0.303237\pi\)
\(548\) 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
\(561\) −5.73183e16 −1.83872
\(562\) 0 0
\(563\) 2.23353e16 0.701361 0.350681 − 0.936495i \(-0.385950\pi\)
0.350681 + 0.936495i \(0.385950\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.36836e16 1.87652 0.938262 − 0.345925i \(-0.112435\pi\)
0.938262 + 0.345925i \(0.112435\pi\)
\(570\) 0 0
\(571\) −5.93581e16 −1.71263 −0.856314 − 0.516455i \(-0.827251\pi\)
−0.856314 + 0.516455i \(0.827251\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −7.38009e16 −1.99989 −0.999946 − 0.0103984i \(-0.996690\pi\)
−0.999946 + 0.0103984i \(0.996690\pi\)
\(578\) 0 0
\(579\) 2.08237e15 0.0552697
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −6.62782e15 −0.162010 −0.0810051 − 0.996714i \(-0.525813\pi\)
−0.0810051 + 0.996714i \(0.525813\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.46818e16 −0.337638 −0.168819 − 0.985647i \(-0.553995\pi\)
−0.168819 + 0.985647i \(0.553995\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −4.27830e16 −0.907872 −0.453936 − 0.891034i \(-0.649980\pi\)
−0.453936 + 0.891034i \(0.649980\pi\)
\(602\) 0 0
\(603\) 1.72380e16 0.358577
\(604\) 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
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.12519e15 0.147273 0.0736364 − 0.997285i \(-0.476540\pi\)
0.0736364 + 0.997285i \(0.476540\pi\)
\(618\) 0 0
\(619\) −1.06992e17 −1.90198 −0.950989 − 0.309225i \(-0.899930\pi\)
−0.950989 + 0.309225i \(0.899930\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 5.96046e16 1.00000
\(626\) 0 0
\(627\) 1.11270e17 1.83135
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 1.12541e17 1.74939
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.37398e16 1.35137 0.675687 − 0.737188i \(-0.263847\pi\)
0.675687 + 0.737188i \(0.263847\pi\)
\(642\) 0 0
\(643\) 6.10304e16 0.863536 0.431768 − 0.901985i \(-0.357890\pi\)
0.431768 + 0.901985i \(0.357890\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 1.66125e16 0.222314
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4.83911e15 −0.0601691
\(658\) 0 0
\(659\) 6.58930e16 0.804501 0.402251 − 0.915530i \(-0.368228\pi\)
0.402251 + 0.915530i \(0.368228\pi\)
\(660\) 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
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.27565e17 −1.37290 −0.686451 − 0.727176i \(-0.740833\pi\)
−0.686451 + 0.727176i \(0.740833\pi\)
\(674\) 0 0
\(675\) 1.01193e17 1.06986
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 7.71429e16 0.773416
\(682\) 0 0
\(683\) −4.09526e16 −0.403419 −0.201710 − 0.979445i \(-0.564650\pi\)
−0.201710 + 0.979445i \(0.564650\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −2.12572e17 −1.95271 −0.976354 − 0.216178i \(-0.930641\pi\)
−0.976354 + 0.216178i \(0.930641\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −3.90826e17 −3.40869
\(698\) 0 0
\(699\) 1.30046e17 1.11490
\(700\) 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
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −2.04297e17 −1.43032
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.66645e17 1.11027
\(730\) 0 0
\(731\) −3.18459e17 −2.08713
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.36634e17 2.10065
\(738\) 0 0
\(739\) 8.15585e16 0.500729 0.250364 − 0.968152i \(-0.419450\pi\)
0.250364 + 0.968152i \(0.419450\pi\)
\(740\) 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
\(747\) −1.90002e16 −0.109354
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −2.15825e17 −1.18395
\(754\) 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
\(761\) −3.06091e17 −1.57595 −0.787975 − 0.615707i \(-0.788871\pi\)
−0.787975 + 0.615707i \(0.788871\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 7.49631e16 0.362485 0.181242 − 0.983438i \(-0.441988\pi\)
0.181242 + 0.983438i \(0.441988\pi\)
\(770\) 0 0
\(771\) 1.42342e17 0.677654
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.58695e17 3.39502
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −3.83356e17 −1.61345 −0.806723 − 0.590930i \(-0.798761\pi\)
−0.806723 + 0.590930i \(0.798761\pi\)
\(788\) 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
\(801\) 8.53132e16 0.323014
\(802\) 0 0
\(803\) −9.45012e16 −0.352488
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.67164e17 0.596283 0.298142 − 0.954522i \(-0.403633\pi\)
0.298142 + 0.954522i \(0.403633\pi\)
\(810\) 0 0
\(811\) 4.49628e17 1.58026 0.790130 − 0.612939i \(-0.210013\pi\)
0.790130 + 0.612939i \(0.210013\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 6.18212e17 2.07876
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 3.08939e17 0.979826
\(826\) 0 0
\(827\) 3.04079e17 0.950502 0.475251 − 0.879850i \(-0.342357\pi\)
0.475251 + 0.879850i \(0.342357\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.26956e17 −1.87658
\(834\) 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
\(841\) 3.53815e17 1.00000
\(842\) 0 0
\(843\) −5.62935e17 −1.56853
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −4.34727e17 −1.16083
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.89760e16 0.0983813 0.0491907 − 0.998789i \(-0.484336\pi\)
0.0491907 + 0.998789i \(0.484336\pi\)
\(858\) 0 0
\(859\) 1.37849e17 0.343119 0.171560 − 0.985174i \(-0.445119\pi\)
0.171560 + 0.985174i \(0.445119\pi\)
\(860\) 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
\(867\) −9.66675e17 −2.27597
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −1.63126e17 −0.368500
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −7.94685e17 −1.69957 −0.849786 − 0.527128i \(-0.823269\pi\)
−0.849786 + 0.527128i \(0.823269\pi\)
\(882\) 0 0
\(883\) −8.71688e17 −1.83906 −0.919532 − 0.393015i \(-0.871432\pi\)
−0.919532 + 0.393015i \(0.871432\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4.23851e17 0.847123
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.64915e17 0.835084 0.417542 − 0.908658i \(-0.362892\pi\)
0.417542 + 0.908658i \(0.362892\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −3.71048e17 −0.640627
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −1.07830e18 −1.76677
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.13239e18 1.76157 0.880786 − 0.473515i \(-0.157015\pi\)
0.880786 + 0.473515i \(0.157015\pi\)
\(930\) 0 0
\(931\) 1.21708e18 1.86906
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 9.59909e17 1.41838 0.709190 − 0.705018i \(-0.249061\pi\)
0.709190 + 0.705018i \(0.249061\pi\)
\(938\) 0 0
\(939\) 1.18338e18 1.72636
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.42092e18 −1.97001 −0.985005 − 0.172528i \(-0.944807\pi\)
−0.985005 + 0.172528i \(0.944807\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.35078e18 −1.80313 −0.901565 − 0.432645i \(-0.857581\pi\)
−0.901565 + 0.432645i \(0.857581\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.87663e17 1.00000
\(962\) 0 0
\(963\) −2.73448e17 −0.342860
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 2.62078e18 3.16583
\(970\) 0 0
\(971\) −1.67618e18 −1.99989 −0.999943 − 0.0107080i \(-0.996591\pi\)
−0.999943 + 0.0107080i \(0.996591\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 7.04930e17 0.810548 0.405274 − 0.914195i \(-0.367176\pi\)
0.405274 + 0.914195i \(0.367176\pi\)
\(978\) 0 0
\(979\) 1.66605e18 1.89231
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −1.71679e18 −1.79070
\(994\) 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 32.13.d.a.15.1 1
4.3 odd 2 8.13.d.a.3.1 ✓ 1
8.3 odd 2 CM 32.13.d.a.15.1 1
8.5 even 2 8.13.d.a.3.1 ✓ 1
12.11 even 2 72.13.b.a.19.1 1
24.5 odd 2 72.13.b.a.19.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.13.d.a.3.1 ✓ 1 4.3 odd 2
8.13.d.a.3.1 ✓ 1 8.5 even 2
32.13.d.a.15.1 1 1.1 even 1 trivial
32.13.d.a.15.1 1 8.3 odd 2 CM
72.13.b.a.19.1 1 12.11 even 2
72.13.b.a.19.1 1 24.5 odd 2