Properties

Label 4140.2.n.a.2069.12
Level $4140$
Weight $2$
Character 4140.2069
Analytic conductor $33.058$
Analytic rank $0$
Dimension $16$
CM discriminant -115
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4140,2,Mod(2069,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4140.2069");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.n (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: no
Analytic conductor: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1352 x^{12} - 4836 x^{11} + 18782 x^{10} - 55300 x^{9} + 177369 x^{8} - 421148 x^{7} + 1135954 x^{6} - 2123100 x^{5} + \cdots + 11064600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 2069.12
Root \(1.07874 - 3.36384i\) of defining polynomial
Character \(\chi\) \(=\) 4140.2069
Dual form 4140.2.n.a.2069.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23607i q^{5} -0.918623 q^{7} +O(q^{10})\) \(q+2.23607i q^{5} -0.918623 q^{7} -3.41156i q^{17} +4.79583i q^{23} -5.00000 q^{25} -10.6172i q^{29} -2.76383 q^{31} -2.05410i q^{35} -7.52610 q^{37} +9.53701i q^{41} -0.457775 q^{43} -6.15613 q^{49} +13.5780i q^{53} -7.51139i q^{59} -0.581259 q^{67} -13.7230i q^{71} -18.0501i q^{83} +7.62848 q^{85} +10.4024 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 80 q^{25} + 112 q^{49}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.23607i 1.00000i
\(6\) 0 0
\(7\) −0.918623 −0.347207 −0.173603 0.984816i \(-0.555541\pi\)
−0.173603 + 0.984816i \(0.555541\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.41156i − 0.827425i −0.910408 0.413712i \(-0.864232\pi\)
0.910408 0.413712i \(-0.135768\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.79583i 1.00000i
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 10.6172i − 1.97157i −0.168017 0.985784i \(-0.553736\pi\)
0.168017 0.985784i \(-0.446264\pi\)
\(30\) 0 0
\(31\) −2.76383 −0.496398 −0.248199 0.968709i \(-0.579839\pi\)
−0.248199 + 0.968709i \(0.579839\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 2.05410i − 0.347207i
\(36\) 0 0
\(37\) −7.52610 −1.23728 −0.618642 0.785673i \(-0.712317\pi\)
−0.618642 + 0.785673i \(0.712317\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.53701i 1.48943i 0.667383 + 0.744715i \(0.267415\pi\)
−0.667383 + 0.744715i \(0.732585\pi\)
\(42\) 0 0
\(43\) −0.457775 −0.0698100 −0.0349050 0.999391i \(-0.511113\pi\)
−0.0349050 + 0.999391i \(0.511113\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −6.15613 −0.879447
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.5780i 1.86508i 0.361070 + 0.932539i \(0.382412\pi\)
−0.361070 + 0.932539i \(0.617588\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 7.51139i − 0.977900i −0.872312 0.488950i \(-0.837380\pi\)
0.872312 0.488950i \(-0.162620\pi\)
\(60\) 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
\(67\) −0.581259 −0.0710120 −0.0355060 0.999369i \(-0.511304\pi\)
−0.0355060 + 0.999369i \(0.511304\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 13.7230i − 1.62863i −0.580426 0.814313i \(-0.697114\pi\)
0.580426 0.814313i \(-0.302886\pi\)
\(72\) 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
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 18.0501i − 1.98126i −0.136586 0.990628i \(-0.543613\pi\)
0.136586 0.990628i \(-0.456387\pi\)
\(84\) 0 0
\(85\) 7.62848 0.827425
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.4024 1.05620 0.528101 0.849182i \(-0.322904\pi\)
0.528101 + 0.849182i \(0.322904\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 13.6452i − 1.35775i −0.734254 0.678875i \(-0.762468\pi\)
0.734254 0.678875i \(-0.237532\pi\)
\(102\) 0 0
\(103\) −19.4314 −1.91464 −0.957318 0.289036i \(-0.906665\pi\)
−0.957318 + 0.289036i \(0.906665\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 19.8263i 1.91668i 0.285622 + 0.958342i \(0.407800\pi\)
−0.285622 + 0.958342i \(0.592200\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 4.63370i − 0.435902i −0.975960 0.217951i \(-0.930063\pi\)
0.975960 0.217951i \(-0.0699372\pi\)
\(114\) 0 0
\(115\) −10.7238 −1.00000
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.13394i 0.287288i
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 11.1803i − 1.00000i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 9.50890i − 0.830796i −0.909640 0.415398i \(-0.863642\pi\)
0.909640 0.415398i \(-0.136358\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 19.1833i − 1.63894i −0.573121 0.819471i \(-0.694267\pi\)
0.573121 0.819471i \(-0.305733\pi\)
\(138\) 0 0
\(139\) −23.5484 −1.99735 −0.998676 0.0514389i \(-0.983619\pi\)
−0.998676 + 0.0514389i \(0.983619\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 23.7408 1.97157
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −21.4476 −1.74538 −0.872691 0.488273i \(-0.837627\pi\)
−0.872691 + 0.488273i \(0.837627\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 6.18010i − 0.496398i
\(156\) 0 0
\(157\) 24.9999 1.99521 0.997604 0.0691817i \(-0.0220388\pi\)
0.997604 + 0.0691817i \(0.0220388\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 4.40556i − 0.347207i
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 4.59312 0.347207
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 26.4795i − 1.97917i −0.143958 0.989584i \(-0.545983\pi\)
0.143958 0.989584i \(-0.454017\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 16.8289i − 1.23728i
\(186\) 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
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.75322i 0.684542i
\(204\) 0 0
\(205\) −21.3254 −1.48943
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 18.0208 1.24060 0.620301 0.784364i \(-0.287010\pi\)
0.620301 + 0.784364i \(0.287010\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 1.02362i − 0.0698100i
\(216\) 0 0
\(217\) 2.53892 0.172353
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 8.94427i − 0.593652i −0.954932 0.296826i \(-0.904072\pi\)
0.954932 0.296826i \(-0.0959282\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 25.2363i − 1.63240i −0.577768 0.816201i \(-0.696076\pi\)
0.577768 0.816201i \(-0.303924\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 13.7655i − 0.879447i
\(246\) 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
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 6.91365 0.429593
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 26.9944i 1.66454i 0.554367 + 0.832272i \(0.312960\pi\)
−0.554367 + 0.832272i \(0.687040\pi\)
\(264\) 0 0
\(265\) −30.3613 −1.86508
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 32.7192i 1.99493i 0.0711840 + 0.997463i \(0.477322\pi\)
−0.0711840 + 0.997463i \(0.522678\pi\)
\(270\) 0 0
\(271\) −24.3613 −1.47984 −0.739921 0.672694i \(-0.765137\pi\)
−0.739921 + 0.672694i \(0.765137\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −31.8482 −1.89318 −0.946589 0.322442i \(-0.895496\pi\)
−0.946589 + 0.322442i \(0.895496\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 8.76092i − 0.517140i
\(288\) 0 0
\(289\) 5.36126 0.315368
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 22.5949i 1.32001i 0.751262 + 0.660004i \(0.229445\pi\)
−0.751262 + 0.660004i \(0.770555\pi\)
\(294\) 0 0
\(295\) 16.7960 0.977900
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0.420522 0.0242385
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 34.9647i − 1.98267i −0.131363 0.991334i \(-0.541935\pi\)
0.131363 0.991334i \(-0.458065\pi\)
\(312\) 0 0
\(313\) 19.4881 1.10154 0.550768 0.834659i \(-0.314335\pi\)
0.550768 + 0.834659i \(0.314335\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −36.3613 −1.99860 −0.999298 0.0374662i \(-0.988071\pi\)
−0.999298 + 0.0374662i \(0.988071\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 1.29973i − 0.0710120i
\(336\) 0 0
\(337\) −8.57128 −0.466908 −0.233454 0.972368i \(-0.575003\pi\)
−0.233454 + 0.972368i \(0.575003\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 12.0855 0.652557
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −18.3613 −0.982856 −0.491428 0.870918i \(-0.663525\pi\)
−0.491428 + 0.870918i \(0.663525\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 30.6857 1.62863
\(356\) 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
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −14.4709 −0.755377 −0.377689 0.925933i \(-0.623281\pi\)
−0.377689 + 0.925933i \(0.623281\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 12.4730i − 0.647568i
\(372\) 0 0
\(373\) −30.2916 −1.56844 −0.784220 0.620483i \(-0.786937\pi\)
−0.784220 + 0.620483i \(0.786937\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 0.643015i − 0.0328565i −0.999865 0.0164283i \(-0.994770\pi\)
0.999865 0.0164283i \(-0.00522951\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 16.3613 0.827425
\(392\) 0 0
\(393\) 0 0
\(394\) 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
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 28.4131 1.40494 0.702468 0.711715i \(-0.252081\pi\)
0.702468 + 0.711715i \(0.252081\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.90014i 0.339534i
\(414\) 0 0
\(415\) 40.3613 1.98126
\(416\) 0 0
\(417\) 0 0
\(418\) 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
\(423\) 0 0
\(424\) 0 0
\(425\) 17.0578i 0.827425i
\(426\) 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
\(433\) −38.7930 −1.86427 −0.932137 0.362106i \(-0.882058\pi\)
−0.932137 + 0.362106i \(0.882058\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −21.4476 −1.02364 −0.511819 0.859093i \(-0.671028\pi\)
−0.511819 + 0.859093i \(0.671028\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 36.8274i − 1.73799i −0.494817 0.868997i \(-0.664765\pi\)
0.494817 0.868997i \(-0.335235\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.36359 0.297676 0.148838 0.988862i \(-0.452447\pi\)
0.148838 + 0.988862i \(0.452447\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 31.7457i 1.47855i 0.673406 + 0.739273i \(0.264831\pi\)
−0.673406 + 0.739273i \(0.735169\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 36.2618i − 1.67799i −0.544135 0.838997i \(-0.683142\pi\)
0.544135 0.838997i \(-0.316858\pi\)
\(468\) 0 0
\(469\) 0.533958 0.0246559
\(470\) 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
\(485\) 23.2604i 1.05620i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 44.3103i 1.99970i 0.0173933 + 0.999849i \(0.494463\pi\)
−0.0173933 + 0.999849i \(0.505537\pi\)
\(492\) 0 0
\(493\) −36.2213 −1.63132
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.6063i 0.565470i
\(498\) 0 0
\(499\) −12.3613 −0.553366 −0.276683 0.960961i \(-0.589235\pi\)
−0.276683 + 0.960961i \(0.589235\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 8.78271i 0.391602i 0.980644 + 0.195801i \(0.0627306\pi\)
−0.980644 + 0.195801i \(0.937269\pi\)
\(504\) 0 0
\(505\) 30.5116 1.35775
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 40.2310i 1.78321i 0.452816 + 0.891604i \(0.350419\pi\)
−0.452816 + 0.891604i \(0.649581\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 43.4500i − 1.91464i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −38.4051 −1.67934 −0.839669 0.543098i \(-0.817251\pi\)
−0.839669 + 0.543098i \(0.817251\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.42896i 0.410732i
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −44.3330 −1.91668
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 42.8952 1.84421 0.922105 0.386940i \(-0.126468\pi\)
0.922105 + 0.386940i \(0.126468\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 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
\(557\) 43.0642i 1.82469i 0.409422 + 0.912345i \(0.365730\pi\)
−0.409422 + 0.912345i \(0.634270\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 45.8328i 1.93162i 0.259248 + 0.965811i \(0.416525\pi\)
−0.259248 + 0.965811i \(0.583475\pi\)
\(564\) 0 0
\(565\) 10.3613 0.435902
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 23.9792i − 1.00000i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 16.5812i 0.687906i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) −7.00770 −0.287288
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 15.9469i 0.651575i 0.945443 + 0.325787i \(0.105629\pi\)
−0.945443 + 0.325787i \(0.894371\pi\)
\(600\) 0 0
\(601\) 46.3613 1.89112 0.945558 0.325455i \(-0.105517\pi\)
0.945558 + 0.325455i \(0.105517\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 24.5967i − 1.00000i
\(606\) 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
\(613\) −49.2653 −1.98981 −0.994903 0.100840i \(-0.967847\pi\)
−0.994903 + 0.100840i \(0.967847\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 49.6782i − 1.99997i −0.00563284 0.999984i \(-0.501793\pi\)
0.00563284 0.999984i \(-0.498207\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 25.6757i 1.02376i
\(630\) 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
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 45.4067 1.79066 0.895332 0.445400i \(-0.146938\pi\)
0.895332 + 0.445400i \(0.146938\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 21.2625 0.830796
\(656\) 0 0
\(657\) 0 0
\(658\) 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
\(667\) 50.9184 1.97157
\(668\) 0 0
\(669\) 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
\(676\) 0 0
\(677\) − 22.8454i − 0.878019i −0.898482 0.439009i \(-0.855329\pi\)
0.898482 0.439009i \(-0.144671\pi\)
\(678\) 0 0
\(679\) −9.55587 −0.366721
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 42.8952 1.63894
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −21.4476 −0.815906 −0.407953 0.913003i \(-0.633757\pi\)
−0.407953 + 0.913003i \(0.633757\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 52.6559i − 1.99735i
\(696\) 0 0
\(697\) 32.5361 1.23239
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.5348i 0.471420i
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 13.2548i − 0.496398i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 12.9117i − 0.481525i −0.970584 0.240763i \(-0.922602\pi\)
0.970584 0.240763i \(-0.0773975\pi\)
\(720\) 0 0
\(721\) 17.8502 0.664775
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 53.0861i 1.97157i
\(726\) 0 0
\(727\) 48.0629 1.78255 0.891277 0.453459i \(-0.149810\pi\)
0.891277 + 0.453459i \(0.149810\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.56173i 0.0577625i
\(732\) 0 0
\(733\) −32.3489 −1.19483 −0.597416 0.801931i \(-0.703806\pi\)
−0.597416 + 0.801931i \(0.703806\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 34.6037 1.27292 0.636460 0.771310i \(-0.280398\pi\)
0.636460 + 0.771310i \(0.280398\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.59166i 0.351884i 0.984401 + 0.175942i \(0.0562971\pi\)
−0.984401 + 0.175942i \(0.943703\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 18.2129i − 0.665486i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 47.9583i − 1.74538i
\(756\) 0 0
\(757\) 55.0078 1.99929 0.999645 0.0266296i \(-0.00847746\pi\)
0.999645 + 0.0266296i \(0.00847746\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 1.32059i − 0.0478713i −0.999714 0.0239357i \(-0.992380\pi\)
0.999714 0.0239357i \(-0.00761969\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 38.3667i 1.37995i 0.723832 + 0.689976i \(0.242379\pi\)
−0.723832 + 0.689976i \(0.757621\pi\)
\(774\) 0 0
\(775\) 13.8191 0.496398
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 55.9014i 1.99521i
\(786\) 0 0
\(787\) −52.7556 −1.88054 −0.940268 0.340436i \(-0.889425\pi\)
−0.940268 + 0.340436i \(0.889425\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.25662i 0.151348i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 8.94865i − 0.316977i −0.987361 0.158489i \(-0.949338\pi\)
0.987361 0.158489i \(-0.0506621\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 9.85114 0.347207
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 46.8744i 1.64802i 0.566577 + 0.824009i \(0.308267\pi\)
−0.566577 + 0.824009i \(0.691733\pi\)
\(810\) 0 0
\(811\) 4.36126 0.153145 0.0765723 0.997064i \(-0.475602\pi\)
0.0765723 + 0.997064i \(0.475602\pi\)
\(812\) 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
\(821\) 57.2016i 1.99635i 0.0604026 + 0.998174i \(0.480762\pi\)
−0.0604026 + 0.998174i \(0.519238\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 18.3732i 0.638900i 0.947603 + 0.319450i \(0.103498\pi\)
−0.947603 + 0.319450i \(0.896502\pi\)
\(828\) 0 0
\(829\) 44.9960 1.56278 0.781389 0.624045i \(-0.214512\pi\)
0.781389 + 0.624045i \(0.214512\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 21.0020i 0.727676i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −83.7254 −2.88708
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 29.0689i 1.00000i
\(846\) 0 0
\(847\) 10.1049 0.347207
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 36.0939i − 1.23728i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 52.3613 1.78654 0.893272 0.449517i \(-0.148404\pi\)
0.893272 + 0.449517i \(0.148404\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 10.2705i 0.347207i
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 59.2099 1.97917
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 29.3442i 0.978682i
\(900\) 0 0
\(901\) 46.3220 1.54321
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −41.7322 −1.38569 −0.692847 0.721085i \(-0.743644\pi\)
−0.692847 + 0.721085i \(0.743644\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 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
\(917\) 8.73509i 0.288458i
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 37.6305 1.23728
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 59.2978i 1.94550i 0.231864 + 0.972748i \(0.425517\pi\)
−0.231864 + 0.972748i \(0.574483\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −27.5449 −0.899854 −0.449927 0.893065i \(-0.648550\pi\)
−0.449927 + 0.893065i \(0.648550\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) −45.7379 −1.48943
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 22.3607i − 0.724333i −0.932113 0.362167i \(-0.882037\pi\)
0.932113 0.362167i \(-0.117963\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 17.6222i 0.569052i
\(960\) 0 0
\(961\) −23.3613 −0.753589
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 21.6321 0.693495
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 55.4244i − 1.77319i −0.462551 0.886593i \(-0.653066\pi\)
0.462551 0.886593i \(-0.346934\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 25.3634i 0.808968i 0.914545 + 0.404484i \(0.132549\pi\)
−0.914545 + 0.404484i \(0.867451\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 2.19541i − 0.0698100i
\(990\) 0 0
\(991\) 55.3884 1.75947 0.879734 0.475465i \(-0.157720\pi\)
0.879734 + 0.475465i \(0.157720\pi\)
\(992\) 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 4140.2.n.a.2069.12 yes 16
3.2 odd 2 inner 4140.2.n.a.2069.4 16
5.4 even 2 inner 4140.2.n.a.2069.5 yes 16
15.14 odd 2 inner 4140.2.n.a.2069.13 yes 16
23.22 odd 2 inner 4140.2.n.a.2069.5 yes 16
69.68 even 2 inner 4140.2.n.a.2069.13 yes 16
115.114 odd 2 CM 4140.2.n.a.2069.12 yes 16
345.344 even 2 inner 4140.2.n.a.2069.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.2.n.a.2069.4 16 3.2 odd 2 inner
4140.2.n.a.2069.4 16 345.344 even 2 inner
4140.2.n.a.2069.5 yes 16 5.4 even 2 inner
4140.2.n.a.2069.5 yes 16 23.22 odd 2 inner
4140.2.n.a.2069.12 yes 16 1.1 even 1 trivial
4140.2.n.a.2069.12 yes 16 115.114 odd 2 CM
4140.2.n.a.2069.13 yes 16 15.14 odd 2 inner
4140.2.n.a.2069.13 yes 16 69.68 even 2 inner