Properties

Label 4140.2.n.a.2069.1
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 2069.1
Root \(2.65988 - 2.13910i\) of defining polynomial
Character \(\chi\) \(=\) 4140.2069
Dual form 4140.2.n.a.2069.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23607i q^{5} -5.21115 q^{7} +O(q^{10})\) \(q-2.23607i q^{5} -5.21115 q^{7} -3.41156i q^{17} +4.79583i q^{23} -5.00000 q^{25} -1.80960i q^{29} +2.76383 q^{31} +11.6525i q^{35} -9.55813 q^{37} +8.54667i q^{41} -13.1069 q^{43} +20.1561 q^{49} -13.5780i q^{53} -13.4007i q^{59} +16.3604 q^{67} +9.78151i q^{71} +18.0501i q^{83} -7.62848 q^{85} +16.7269 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) 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
\(7\) −5.21115 −1.96963 −0.984816 0.173603i \(-0.944459\pi\)
−0.984816 + 0.173603i \(0.944459\pi\)
\(8\) 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
\(29\) − 1.80960i − 0.336034i −0.985784 0.168017i \(-0.946264\pi\)
0.985784 0.168017i \(-0.0537364\pi\)
\(30\) 0 0
\(31\) 2.76383 0.496398 0.248199 0.968709i \(-0.420161\pi\)
0.248199 + 0.968709i \(0.420161\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 11.6525i 1.96963i
\(36\) 0 0
\(37\) −9.55813 −1.57135 −0.785673 0.618642i \(-0.787683\pi\)
−0.785673 + 0.618642i \(0.787683\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.54667i 1.33477i 0.744715 + 0.667383i \(0.232585\pi\)
−0.744715 + 0.667383i \(0.767415\pi\)
\(42\) 0 0
\(43\) −13.1069 −1.99878 −0.999391 0.0349050i \(-0.988887\pi\)
−0.999391 + 0.0349050i \(0.988887\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 20.1561 2.87945
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 13.5780i − 1.86508i −0.361070 0.932539i \(-0.617588\pi\)
0.361070 0.932539i \(-0.382412\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 13.4007i − 1.74462i −0.488950 0.872312i \(-0.662620\pi\)
0.488950 0.872312i \(-0.337380\pi\)
\(60\) 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
\(67\) 16.3604 1.99874 0.999369 0.0355060i \(-0.0113043\pi\)
0.999369 + 0.0355060i \(0.0113043\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.78151i 1.16085i 0.814313 + 0.580426i \(0.197114\pi\)
−0.814313 + 0.580426i \(0.802886\pi\)
\(72\) 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 \(0.456387\pi\)
−0.136586 + 0.990628i \(0.543613\pi\)
\(84\) 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
\(97\) 16.7269 1.69836 0.849182 0.528101i \(-0.177096\pi\)
0.849182 + 0.528101i \(0.177096\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.7583i 1.46851i 0.678875 + 0.734254i \(0.262468\pi\)
−0.678875 + 0.734254i \(0.737532\pi\)
\(102\) 0 0
\(103\) 5.86678 0.578071 0.289036 0.957318i \(-0.406665\pi\)
0.289036 + 0.957318i \(0.406665\pi\)
\(104\) 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 \(0.0699372\pi\)
−0.975960 + 0.217951i \(0.930063\pi\)
\(114\) 0 0
\(115\) 10.7238 1.00000
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 17.7782i 1.62972i
\(120\) 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
\(131\) 20.8226i 1.81928i 0.415398 + 0.909640i \(0.363642\pi\)
−0.415398 + 0.909640i \(0.636358\pi\)
\(132\) 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 \(-0.0163808\pi\)
0.998676 + 0.0514389i \(0.0163808\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −4.04639 −0.336034
\(146\) 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 \(-0.162373\pi\)
0.872691 + 0.488273i \(0.162373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 6.18010i − 0.496398i
\(156\) 0 0
\(157\) 1.73369 0.138363 0.0691817 0.997604i \(-0.477961\pi\)
0.0691817 + 0.997604i \(0.477961\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 24.9918i − 1.96963i
\(162\) 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
\(175\) 26.0558 1.96963
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.85204i 0.287915i 0.989584 + 0.143958i \(0.0459829\pi\)
−0.989584 + 0.143958i \(0.954017\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 21.3726i 1.57135i
\(186\) 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
\(203\) 9.43010i 0.661863i
\(204\) 0 0
\(205\) 19.1109 1.33477
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −18.0208 −1.24060 −0.620301 0.784364i \(-0.712990\pi\)
−0.620301 + 0.784364i \(0.712990\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 29.3079i 1.99878i
\(216\) 0 0
\(217\) −14.4027 −0.977721
\(218\) 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 \(0.0959282\pi\)
−0.954932 + 0.296826i \(0.904072\pi\)
\(228\) 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
\(239\) 17.8642i 1.15554i 0.816201 + 0.577768i \(0.196076\pi\)
−0.816201 + 0.577768i \(0.803924\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 45.0705i − 2.87945i
\(246\) 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
\(259\) 49.8089 3.09497
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 26.9944i − 1.66454i −0.554367 0.832272i \(-0.687040\pi\)
0.554367 0.832272i \(-0.312960\pi\)
\(264\) 0 0
\(265\) −30.3613 −1.86508
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.33501i 0.142368i 0.997463 + 0.0711840i \(0.0226778\pi\)
−0.997463 + 0.0711840i \(0.977322\pi\)
\(270\) 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
\(283\) 10.8486 0.644885 0.322442 0.946589i \(-0.395496\pi\)
0.322442 + 0.946589i \(0.395496\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 44.5380i − 2.62900i
\(288\) 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
\(295\) −29.9649 −1.74462
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 68.3020 3.93686
\(302\) 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
\(311\) − 4.63324i − 0.262727i −0.991334 0.131363i \(-0.958065\pi\)
0.991334 0.131363i \(-0.0419355\pi\)
\(312\) 0 0
\(313\) −29.5332 −1.66932 −0.834659 0.550768i \(-0.814335\pi\)
−0.834659 + 0.550768i \(0.814335\pi\)
\(314\) 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
\(335\) − 36.5829i − 1.99874i
\(336\) 0 0
\(337\) 35.7006 1.94474 0.972368 0.233454i \(-0.0750028\pi\)
0.972368 + 0.233454i \(0.0750028\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −68.5586 −3.70182
\(344\) 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
\(355\) 21.8721 1.16085
\(356\) 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
\(367\) −35.4766 −1.85187 −0.925933 0.377689i \(-0.876719\pi\)
−0.925933 + 0.377689i \(0.876719\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 70.7569i 3.67352i
\(372\) 0 0
\(373\) −23.9670 −1.24097 −0.620483 0.784220i \(-0.713063\pi\)
−0.620483 + 0.784220i \(0.713063\pi\)
\(374\) 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 \(-0.747919\pi\)
−0.702468 + 0.711715i \(0.747919\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 69.8332i 3.43627i
\(414\) 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
\(433\) −15.0699 −0.724212 −0.362106 0.932137i \(-0.617942\pi\)
−0.362106 + 0.932137i \(0.617942\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 21.4476 1.02364 0.511819 0.859093i \(-0.328972\pi\)
0.511819 + 0.859093i \(0.328972\pi\)
\(440\) 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
\(449\) 20.9700i 0.989635i 0.868997 + 0.494817i \(0.164765\pi\)
−0.868997 + 0.494817i \(0.835235\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 42.2789 1.97772 0.988862 0.148838i \(-0.0475533\pi\)
0.988862 + 0.148838i \(0.0475533\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 28.9173i − 1.34681i −0.739273 0.673406i \(-0.764831\pi\)
0.739273 0.673406i \(-0.235169\pi\)
\(462\) 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 \(0.316858\pi\)
−0.544135 + 0.838997i \(0.683142\pi\)
\(468\) 0 0
\(469\) −85.2565 −3.93678
\(470\) 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
\(485\) − 37.4026i − 1.69836i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 0.770818i − 0.0347865i −0.999849 0.0173933i \(-0.994463\pi\)
0.999849 0.0173933i \(-0.00553673\pi\)
\(492\) 0 0
\(493\) −6.17355 −0.278043
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 50.9730i − 2.28645i
\(498\) 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 \(-0.937269\pi\)
0.980644 0.195801i \(-0.0627306\pi\)
\(504\) 0 0
\(505\) 33.0006 1.46851
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 20.4320i − 0.905633i −0.891604 0.452816i \(-0.850419\pi\)
0.891604 0.452816i \(-0.149581\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 13.1185i − 0.578071i
\(516\) 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
\(523\) 24.8404 1.08620 0.543098 0.839669i \(-0.317251\pi\)
0.543098 + 0.839669i \(0.317251\pi\)
\(524\) 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 \(-0.873532\pi\)
−0.922105 + 0.386940i \(0.873532\pi\)
\(542\) 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
\(581\) − 94.0619i − 3.90234i
\(582\) 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
\(595\) 39.7532 1.62972
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 46.2784i 1.89089i 0.325787 + 0.945443i \(0.394371\pi\)
−0.325787 + 0.945443i \(0.605629\pi\)
\(600\) 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
\(613\) −4.99338 −0.201681 −0.100840 0.994903i \(-0.532153\pi\)
−0.100840 + 0.994903i \(0.532153\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 49.6782i 1.99997i 0.00563284 + 0.999984i \(0.498207\pi\)
−0.00563284 + 0.999984i \(0.501793\pi\)
\(618\) 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
\(629\) 32.6081i 1.30017i
\(630\) 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
\(643\) −22.5884 −0.890799 −0.445400 0.895332i \(-0.646938\pi\)
−0.445400 + 0.895332i \(0.646938\pi\)
\(644\) 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
\(655\) 46.5608 1.81928
\(656\) 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
\(667\) 8.67853 0.336034
\(668\) 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 \(0.144671\pi\)
−0.898482 + 0.439009i \(0.855329\pi\)
\(678\) 0 0
\(679\) −87.1667 −3.34515
\(680\) 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 \(-0.366243\pi\)
0.407953 + 0.913003i \(0.366243\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 52.6559i − 1.99735i
\(696\) 0 0
\(697\) 29.1575 1.10442
\(698\) 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
\(707\) − 76.9079i − 2.89242i
\(708\) 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
\(719\) − 52.0508i − 1.94117i −0.240763 0.970584i \(-0.577398\pi\)
0.240763 0.970584i \(-0.422602\pi\)
\(720\) 0 0
\(721\) −30.5727 −1.13859
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9.04800i 0.336034i
\(726\) 0 0
\(727\) −24.4532 −0.906917 −0.453459 0.891277i \(-0.649810\pi\)
−0.453459 + 0.891277i \(0.649810\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 44.7149i 1.65384i
\(732\) 0 0
\(733\) −43.4229 −1.60386 −0.801931 0.597416i \(-0.796194\pi\)
−0.801931 + 0.597416i \(0.796194\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −34.6037 −1.27292 −0.636460 0.771310i \(-0.719602\pi\)
−0.636460 + 0.771310i \(0.719602\pi\)
\(740\) 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
\(749\) − 103.318i − 3.77516i
\(750\) 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
\(757\) 1.46535 0.0532592 0.0266296 0.999645i \(-0.491523\pi\)
0.0266296 + 0.999645i \(0.491523\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 55.1567i − 1.99943i −0.0239357 0.999714i \(-0.507620\pi\)
0.0239357 0.999714i \(-0.492380\pi\)
\(762\) 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
\(785\) − 3.87665i − 0.138363i
\(786\) 0 0
\(787\) −19.1008 −0.680871 −0.340436 0.940268i \(-0.610575\pi\)
−0.340436 + 0.940268i \(0.610575\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 24.1469i − 0.858565i
\(792\) 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
\(805\) −55.8834 −1.96963
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 32.2302i 1.13315i 0.824009 + 0.566577i \(0.191733\pi\)
−0.824009 + 0.566577i \(0.808267\pi\)
\(810\) 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
\(821\) − 3.46144i − 0.120805i −0.998174 0.0604026i \(-0.980762\pi\)
0.998174 0.0604026i \(-0.0192385\pi\)
\(822\) 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 \(-0.896502\pi\)
0.947603 0.319450i \(-0.103498\pi\)
\(828\) 0 0
\(829\) −44.9960 −1.56278 −0.781389 0.624045i \(-0.785488\pi\)
−0.781389 + 0.624045i \(0.785488\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 68.7638i − 2.38253i
\(834\) 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
\(841\) 25.7254 0.887081
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 29.0689i − 1.00000i
\(846\) 0 0
\(847\) 57.3227 1.96963
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 45.8392i − 1.57135i
\(852\) 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
\(875\) − 58.2625i − 1.96963i
\(876\) 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
\(895\) 8.61343 0.287915
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 5.00142i − 0.166807i
\(900\) 0 0
\(901\) −46.3220 −1.54321
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 43.4330 1.44217 0.721085 0.692847i \(-0.243644\pi\)
0.721085 + 0.692847i \(0.243644\pi\)
\(908\) 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
\(917\) − 108.510i − 3.58331i
\(918\) 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
\(925\) 47.7906 1.57135
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 14.1342i − 0.463729i −0.972748 0.231864i \(-0.925517\pi\)
0.972748 0.231864i \(-0.0744826\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 54.6743 1.78613 0.893065 0.449927i \(-0.148550\pi\)
0.893065 + 0.449927i \(0.148550\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) −40.9884 −1.33477
\(944\) 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 \(0.117963\pi\)
−0.932113 + 0.362167i \(0.882037\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 99.9673i 3.22811i
\(960\) 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
\(973\) −122.715 −3.93405
\(974\) 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
\(989\) − 62.8584i − 1.99878i
\(990\) 0 0
\(991\) −55.3884 −1.75947 −0.879734 0.475465i \(-0.842280\pi\)
−0.879734 + 0.475465i \(0.842280\pi\)
\(992\) 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.2069.1 16
3.2 odd 2 inner 4140.2.n.a.2069.9 yes 16
5.4 even 2 inner 4140.2.n.a.2069.16 yes 16
15.14 odd 2 inner 4140.2.n.a.2069.8 yes 16
23.22 odd 2 inner 4140.2.n.a.2069.16 yes 16
69.68 even 2 inner 4140.2.n.a.2069.8 yes 16
115.114 odd 2 CM 4140.2.n.a.2069.1 16
345.344 even 2 inner 4140.2.n.a.2069.9 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.2.n.a.2069.1 16 1.1 even 1 trivial
4140.2.n.a.2069.1 16 115.114 odd 2 CM
4140.2.n.a.2069.8 yes 16 15.14 odd 2 inner
4140.2.n.a.2069.8 yes 16 69.68 even 2 inner
4140.2.n.a.2069.9 yes 16 3.2 odd 2 inner
4140.2.n.a.2069.9 yes 16 345.344 even 2 inner
4140.2.n.a.2069.16 yes 16 5.4 even 2 inner
4140.2.n.a.2069.16 yes 16 23.22 odd 2 inner