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Properties

Label 320.9.h.a.319.1
Level $320$
Weight $9$
Character 320.319
Self dual yes
Analytic conductor $130.361$
Analytic rank $0$
Dimension $1$
CM discriminant -20
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [320,9,Mod(319,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 9, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.319"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 320.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-158,0,-625] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(130.361155220\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 319.1
Character \(\chi\) \(=\) 320.319

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-158.000 q^{3} -625.000 q^{5} -1922.00 q^{7} +18403.0 q^{9} +98750.0 q^{15} +303676. q^{21} -211202. q^{23} +390625. q^{25} -1.87104e6 q^{27} -20642.0 q^{29} +1.20125e6 q^{35} -5.41920e6 q^{41} -2.51952e6 q^{43} -1.15019e7 q^{45} -9.61824e6 q^{47} -2.07072e6 q^{49} +1.10616e7 q^{61} -3.53706e7 q^{63} -2.02498e7 q^{67} +3.33699e7 q^{69} -6.17188e7 q^{75} +1.74882e8 q^{81} -3.08846e7 q^{83} +3.26144e6 q^{87} -1.06805e8 q^{89} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −158.000 −1.95062 −0.975309 − 0.220846i \(-0.929118\pi\)
−0.975309 + 0.220846i \(0.929118\pi\)
\(4\) 0 0
\(5\) −625.000 −1.00000
\(6\) 0 0
\(7\) −1922.00 −0.800500 −0.400250 − 0.916406i \(-0.631077\pi\)
−0.400250 + 0.916406i \(0.631077\pi\)
\(8\) 0 0
\(9\) 18403.0 2.80491
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 98750.0 1.95062
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 303676. 1.56147
\(22\) 0 0
\(23\) −211202. −0.754721 −0.377361 − 0.926066i \(-0.623168\pi\)
−0.377361 + 0.926066i \(0.623168\pi\)
\(24\) 0 0
\(25\) 390625. 1.00000
\(26\) 0 0
\(27\) −1.87104e6 −3.52068
\(28\) 0 0
\(29\) −20642.0 −0.0291850 −0.0145925 − 0.999894i \(-0.504645\pi\)
−0.0145925 + 0.999894i \(0.504645\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.20125e6 0.800500
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.41920e6 −1.91778 −0.958892 − 0.283772i \(-0.908414\pi\)
−0.958892 + 0.283772i \(0.908414\pi\)
\(42\) 0 0
\(43\) −2.51952e6 −0.736960 −0.368480 − 0.929636i \(-0.620122\pi\)
−0.368480 + 0.929636i \(0.620122\pi\)
\(44\) 0 0
\(45\) −1.15019e7 −2.80491
\(46\) 0 0
\(47\) −9.61824e6 −1.97108 −0.985540 − 0.169443i \(-0.945803\pi\)
−0.985540 + 0.169443i \(0.945803\pi\)
\(48\) 0 0
\(49\) −2.07072e6 −0.359200
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 1.10616e7 0.798911 0.399456 − 0.916753i \(-0.369199\pi\)
0.399456 + 0.916753i \(0.369199\pi\)
\(62\) 0 0
\(63\) −3.53706e7 −2.24533
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.02498e7 −1.00489 −0.502447 − 0.864608i \(-0.667567\pi\)
−0.502447 + 0.864608i \(0.667567\pi\)
\(68\) 0 0
\(69\) 3.33699e7 1.47217
\(70\) 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
\(75\) −6.17188e7 −1.95062
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.74882e8 4.06260
\(82\) 0 0
\(83\) −3.08846e7 −0.650774 −0.325387 − 0.945581i \(-0.605495\pi\)
−0.325387 + 0.945581i \(0.605495\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.26144e6 0.0569288
\(88\) 0 0
\(89\) −1.06805e8 −1.70228 −0.851139 − 0.524940i \(-0.824088\pi\)
−0.851139 + 0.524940i \(0.824088\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.77672e8 1.70740 0.853699 − 0.520767i \(-0.174354\pi\)
0.853699 + 0.520767i \(0.174354\pi\)
\(102\) 0 0
\(103\) −1.46694e8 −1.30336 −0.651678 − 0.758496i \(-0.725934\pi\)
−0.651678 + 0.758496i \(0.725934\pi\)
\(104\) 0 0
\(105\) −1.89798e8 −1.56147
\(106\) 0 0
\(107\) −2.05579e8 −1.56835 −0.784175 − 0.620540i \(-0.786914\pi\)
−0.784175 + 0.620540i \(0.786914\pi\)
\(108\) 0 0
\(109\) −2.15777e8 −1.52862 −0.764309 − 0.644851i \(-0.776920\pi\)
−0.764309 + 0.644851i \(0.776920\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 1.32001e8 0.754721
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.14359e8 1.00000
\(122\) 0 0
\(123\) 8.56233e8 3.74086
\(124\) 0 0
\(125\) −2.44141e8 −1.00000
\(126\) 0 0
\(127\) −2.90506e7 −0.111671 −0.0558354 − 0.998440i \(-0.517782\pi\)
−0.0558354 + 0.998440i \(0.517782\pi\)
\(128\) 0 0
\(129\) 3.98084e8 1.43753
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.16940e9 3.52068
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 1.51968e9 3.84482
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 1.29012e7 0.0291850
\(146\) 0 0
\(147\) 3.27173e8 0.700662
\(148\) 0 0
\(149\) −9.54571e7 −0.193670 −0.0968352 − 0.995300i \(-0.530872\pi\)
−0.0968352 + 0.995300i \(0.530872\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 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
\(161\) 4.05930e8 0.604154
\(162\) 0 0
\(163\) −7.23181e8 −1.02446 −0.512232 − 0.858847i \(-0.671181\pi\)
−0.512232 + 0.858847i \(0.671181\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.54147e9 1.98184 0.990921 − 0.134442i \(-0.0429243\pi\)
0.990921 + 0.134442i \(0.0429243\pi\)
\(168\) 0 0
\(169\) 8.15731e8 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −7.50781e8 −0.800500
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.77745e9 −1.65609 −0.828045 − 0.560661i \(-0.810547\pi\)
−0.828045 + 0.560661i \(0.810547\pi\)
\(182\) 0 0
\(183\) −1.74773e9 −1.55837
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3.59613e9 2.81831
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 3.19946e9 1.96017
\(202\) 0 0
\(203\) 3.96739e7 0.0233626
\(204\) 0 0
\(205\) 3.38700e9 1.91778
\(206\) 0 0
\(207\) −3.88675e9 −2.11692
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.57470e9 0.736960
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −3.26930e9 −1.32201 −0.661006 − 0.750381i \(-0.729870\pi\)
−0.661006 + 0.750381i \(0.729870\pi\)
\(224\) 0 0
\(225\) 7.18867e9 2.80491
\(226\) 0 0
\(227\) −4.74983e9 −1.78885 −0.894426 − 0.447216i \(-0.852416\pi\)
−0.894426 + 0.447216i \(0.852416\pi\)
\(228\) 0 0
\(229\) 4.18692e9 1.52249 0.761243 − 0.648467i \(-0.224590\pi\)
0.761243 + 0.648467i \(0.224590\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 6.01140e9 1.97108
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −6.12390e9 −1.81535 −0.907675 − 0.419675i \(-0.862144\pi\)
−0.907675 + 0.419675i \(0.862144\pi\)
\(242\) 0 0
\(243\) −1.53554e10 −4.40389
\(244\) 0 0
\(245\) 1.29420e9 0.359200
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 4.87977e9 1.26941
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.79875e8 −0.0818612
\(262\) 0 0
\(263\) 6.24149e9 1.30456 0.652282 − 0.757977i \(-0.273812\pi\)
0.652282 + 0.757977i \(0.273812\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.68752e10 3.32049
\(268\) 0 0
\(269\) −1.00564e10 −1.92058 −0.960288 − 0.279010i \(-0.909994\pi\)
−0.960288 + 0.279010i \(0.909994\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 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
\(281\) −1.21874e10 −1.95472 −0.977361 − 0.211581i \(-0.932139\pi\)
−0.977361 + 0.211581i \(0.932139\pi\)
\(282\) 0 0
\(283\) −9.18975e9 −1.43271 −0.716355 − 0.697736i \(-0.754191\pi\)
−0.716355 + 0.697736i \(0.754191\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.04157e10 1.53519
\(288\) 0 0
\(289\) 6.97576e9 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 4.84251e9 0.589936
\(302\) 0 0
\(303\) −2.80723e10 −3.33048
\(304\) 0 0
\(305\) −6.91350e9 −0.798911
\(306\) 0 0
\(307\) 1.00291e10 1.12904 0.564521 − 0.825418i \(-0.309061\pi\)
0.564521 + 0.825418i \(0.309061\pi\)
\(308\) 0 0
\(309\) 2.31776e10 2.54235
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 2.21066e10 2.24533
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 3.24814e10 3.05925
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3.40927e10 2.98175
\(328\) 0 0
\(329\) 1.84863e10 1.57785
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.26561e10 1.00489
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.50599e10 1.08804
\(344\) 0 0
\(345\) −2.08562e10 −1.47217
\(346\) 0 0
\(347\) 2.26968e10 1.56548 0.782739 − 0.622350i \(-0.213822\pi\)
0.782739 + 0.622350i \(0.213822\pi\)
\(348\) 0 0
\(349\) −2.94354e10 −1.98412 −0.992061 − 0.125761i \(-0.959863\pi\)
−0.992061 + 0.125761i \(0.959863\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 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
\(361\) 1.69836e10 1.00000
\(362\) 0 0
\(363\) −3.38687e10 −1.95062
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2.86398e10 −1.57872 −0.789360 − 0.613930i \(-0.789588\pi\)
−0.789360 + 0.613930i \(0.789588\pi\)
\(368\) 0 0
\(369\) −9.97295e10 −5.37921
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 3.85742e10 1.95062
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 4.58999e9 0.217827
\(382\) 0 0
\(383\) −4.19031e10 −1.94738 −0.973691 − 0.227872i \(-0.926823\pi\)
−0.973691 + 0.227872i \(0.926823\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.63667e10 −2.06710
\(388\) 0 0
\(389\) −2.27630e10 −0.994104 −0.497052 − 0.867721i \(-0.665584\pi\)
−0.497052 + 0.867721i \(0.665584\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 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
\(401\) −4.30430e10 −1.66466 −0.832328 − 0.554283i \(-0.812993\pi\)
−0.832328 + 0.554283i \(0.812993\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.09301e11 −4.06260
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 3.92940e10 1.40421 0.702107 − 0.712072i \(-0.252243\pi\)
0.702107 + 0.712072i \(0.252243\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1.93029e10 0.650774
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −7.28990e7 −0.00232056 −0.00116028 − 0.999999i \(-0.500369\pi\)
−0.00116028 + 0.999999i \(0.500369\pi\)
\(422\) 0 0
\(423\) −1.77005e11 −5.52870
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.12604e10 −0.639528
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) −2.03840e9 −0.0569288
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −3.81074e10 −1.00752
\(442\) 0 0
\(443\) −1.88895e10 −0.490461 −0.245230 − 0.969465i \(-0.578864\pi\)
−0.245230 + 0.969465i \(0.578864\pi\)
\(444\) 0 0
\(445\) 6.67530e10 1.70228
\(446\) 0 0
\(447\) 1.50822e10 0.377777
\(448\) 0 0
\(449\) −2.13599e10 −0.525549 −0.262774 − 0.964857i \(-0.584638\pi\)
−0.262774 + 0.964857i \(0.584638\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 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
\(461\) 9.71865e9 0.215180 0.107590 − 0.994195i \(-0.465687\pi\)
0.107590 + 0.994195i \(0.465687\pi\)
\(462\) 0 0
\(463\) 6.78962e10 1.47748 0.738739 − 0.673991i \(-0.235421\pi\)
0.738739 + 0.673991i \(0.235421\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.19487e10 1.72296 0.861479 − 0.507794i \(-0.169539\pi\)
0.861479 + 0.507794i \(0.169539\pi\)
\(468\) 0 0
\(469\) 3.89200e10 0.804418
\(470\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −6.41370e10 −1.17847
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 5.48589e10 0.975284 0.487642 − 0.873044i \(-0.337857\pi\)
0.487642 + 0.873044i \(0.337857\pi\)
\(488\) 0 0
\(489\) 1.14263e11 1.99834
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) −2.43552e11 −3.86582
\(502\) 0 0
\(503\) 1.71128e10 0.267331 0.133665 − 0.991027i \(-0.457325\pi\)
0.133665 + 0.991027i \(0.457325\pi\)
\(504\) 0 0
\(505\) −1.11045e11 −1.70740
\(506\) 0 0
\(507\) −1.28885e11 −1.95062
\(508\) 0 0
\(509\) −6.48146e10 −0.965610 −0.482805 − 0.875728i \(-0.660382\pi\)
−0.482805 + 0.875728i \(0.660382\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9.16836e10 1.30336
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.46894e11 −1.99367 −0.996834 − 0.0795049i \(-0.974666\pi\)
−0.996834 + 0.0795049i \(0.974666\pi\)
\(522\) 0 0
\(523\) 1.20932e11 1.61635 0.808175 − 0.588942i \(-0.200455\pi\)
0.808175 + 0.588942i \(0.200455\pi\)
\(524\) 0 0
\(525\) 1.18623e11 1.56147
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.37047e10 −0.430396
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.28487e11 1.56835
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −3.50805e10 −0.409522 −0.204761 − 0.978812i \(-0.565642\pi\)
−0.204761 + 0.978812i \(0.565642\pi\)
\(542\) 0 0
\(543\) 2.80838e11 3.23040
\(544\) 0 0
\(545\) 1.34861e11 1.52862
\(546\) 0 0
\(547\) 1.53459e11 1.71413 0.857065 − 0.515208i \(-0.172285\pi\)
0.857065 + 0.515208i \(0.172285\pi\)
\(548\) 0 0
\(549\) 2.03567e11 2.24087
\(550\) 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
\(561\) 0 0
\(562\) 0 0
\(563\) 1.95254e11 1.94341 0.971707 − 0.236190i \(-0.0758989\pi\)
0.971707 + 0.236190i \(0.0758989\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −3.36122e11 −3.25211
\(568\) 0 0
\(569\) 1.77936e11 1.69752 0.848760 − 0.528778i \(-0.177350\pi\)
0.848760 + 0.528778i \(0.177350\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.25008e10 −0.754721
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.93603e10 0.520944
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.78340e10 −0.150209 −0.0751046 − 0.997176i \(-0.523929\pi\)
−0.0751046 + 0.997176i \(0.523929\pi\)
\(588\) 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
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 3.90189e10 0.299073 0.149536 − 0.988756i \(-0.452222\pi\)
0.149536 + 0.988756i \(0.452222\pi\)
\(602\) 0 0
\(603\) −3.72656e11 −2.81864
\(604\) 0 0
\(605\) −1.33974e11 −1.00000
\(606\) 0 0
\(607\) 2.34492e11 1.72732 0.863662 − 0.504071i \(-0.168165\pi\)
0.863662 + 0.504071i \(0.168165\pi\)
\(608\) 0 0
\(609\) −6.26848e9 −0.0455715
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) −5.35146e11 −3.74086
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 3.95167e11 2.65714
\(622\) 0 0
\(623\) 2.05279e11 1.36267
\(624\) 0 0
\(625\) 1.52588e11 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
\(635\) 1.81566e10 0.111671
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.13651e11 −0.673193 −0.336596 − 0.941649i \(-0.609276\pi\)
−0.336596 + 0.941649i \(0.609276\pi\)
\(642\) 0 0
\(643\) 9.85936e10 0.576773 0.288387 − 0.957514i \(-0.406881\pi\)
0.288387 + 0.957514i \(0.406881\pi\)
\(644\) 0 0
\(645\) −2.48802e11 −1.43753
\(646\) 0 0
\(647\) 3.44581e11 1.96641 0.983206 − 0.182498i \(-0.0584182\pi\)
0.983206 + 0.182498i \(0.0584182\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 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
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −2.34485e11 −1.22831 −0.614156 − 0.789185i \(-0.710503\pi\)
−0.614156 + 0.789185i \(0.710503\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.35963e9 0.0220265
\(668\) 0 0
\(669\) 5.16549e11 2.57874
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −7.30873e11 −3.52068
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 7.50473e11 3.48937
\(682\) 0 0
\(683\) −4.60451e10 −0.211593 −0.105796 − 0.994388i \(-0.533739\pi\)
−0.105796 + 0.994388i \(0.533739\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −6.61534e11 −2.96979
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.54322e11 1.88145 0.940723 − 0.339177i \(-0.110148\pi\)
0.940723 + 0.339177i \(0.110148\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −9.49801e11 −3.84482
\(706\) 0 0
\(707\) −3.41487e11 −1.36677
\(708\) 0 0
\(709\) 2.36889e11 0.937474 0.468737 − 0.883338i \(-0.344709\pi\)
0.468737 + 0.883338i \(0.344709\pi\)
\(710\) 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
\(721\) 2.81945e11 1.04334
\(722\) 0 0
\(723\) 9.67577e11 3.54105
\(724\) 0 0
\(725\) −8.06328e9 −0.0291850
\(726\) 0 0
\(727\) −4.51416e11 −1.61599 −0.807995 − 0.589189i \(-0.799447\pi\)
−0.807995 + 0.589189i \(0.799447\pi\)
\(728\) 0 0
\(729\) 1.27876e12 4.52771
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −2.04483e11 −0.700662
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.38698e11 1.11137 0.555683 − 0.831394i \(-0.312457\pi\)
0.555683 + 0.831394i \(0.312457\pi\)
\(744\) 0 0
\(745\) 5.96607e10 0.193670
\(746\) 0 0
\(747\) −5.68370e11 −1.82536
\(748\) 0 0
\(749\) 3.95122e11 1.25546
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 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
\(761\) 5.38529e11 1.60572 0.802862 − 0.596165i \(-0.203310\pi\)
0.802862 + 0.596165i \(0.203310\pi\)
\(762\) 0 0
\(763\) 4.14723e11 1.22366
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −3.17192e11 −0.907021 −0.453511 − 0.891251i \(-0.649829\pi\)
−0.453511 + 0.891251i \(0.649829\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 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
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 3.86219e10 0.102751
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 7.34080e11 1.91357 0.956785 − 0.290795i \(-0.0939198\pi\)
0.956785 + 0.290795i \(0.0939198\pi\)
\(788\) 0 0
\(789\) −9.86155e11 −2.54470
\(790\) 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
\(801\) −1.96553e12 −4.77474
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −2.53706e11 −0.604154
\(806\) 0 0
\(807\) 1.58890e12 3.74631
\(808\) 0 0
\(809\) −7.15434e11 −1.67023 −0.835113 − 0.550078i \(-0.814598\pi\)
−0.835113 + 0.550078i \(0.814598\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.51988e11 1.02446
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.08574e10 0.177970 0.0889850 − 0.996033i \(-0.471638\pi\)
0.0889850 + 0.996033i \(0.471638\pi\)
\(822\) 0 0
\(823\) 5.64525e11 1.23051 0.615254 − 0.788329i \(-0.289054\pi\)
0.615254 + 0.788329i \(0.289054\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8.99261e10 0.192249 0.0961244 − 0.995369i \(-0.469355\pi\)
0.0961244 + 0.995369i \(0.469355\pi\)
\(828\) 0 0
\(829\) 2.88497e11 0.610834 0.305417 − 0.952219i \(-0.401204\pi\)
0.305417 + 0.952219i \(0.401204\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −9.63419e11 −1.98184
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −4.99820e11 −0.999148
\(842\) 0 0
\(843\) 1.92560e12 3.81291
\(844\) 0 0
\(845\) −5.09832e11 −1.00000
\(846\) 0 0
\(847\) −4.11998e11 −0.800500
\(848\) 0 0
\(849\) 1.45198e12 2.79467
\(850\) 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
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) −1.64568e12 −2.99456
\(862\) 0 0
\(863\) −2.99508e11 −0.539964 −0.269982 − 0.962865i \(-0.587018\pi\)
−0.269982 + 0.962865i \(0.587018\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.10217e12 −1.95062
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.69238e11 0.800500
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −6.93442e10 −0.115108 −0.0575541 − 0.998342i \(-0.518330\pi\)
−0.0575541 + 0.998342i \(0.518330\pi\)
\(882\) 0 0
\(883\) −1.21110e12 −1.99221 −0.996107 − 0.0881518i \(-0.971904\pi\)
−0.996107 + 0.0881518i \(0.971904\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.22963e11 1.32949 0.664746 − 0.747069i \(-0.268540\pi\)
0.664746 + 0.747069i \(0.268540\pi\)
\(888\) 0 0
\(889\) 5.58352e10 0.0893925
\(890\) 0 0
\(891\) 0 0
\(892\) 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
\(903\) −7.65117e11 −1.15074
\(904\) 0 0
\(905\) 1.11091e12 1.65609
\(906\) 0 0
\(907\) 1.14390e12 1.69028 0.845142 − 0.534542i \(-0.179516\pi\)
0.845142 + 0.534542i \(0.179516\pi\)
\(908\) 0 0
\(909\) 3.26971e12 4.78909
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 1.09233e12 1.55837
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −1.58460e12 −2.20233
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −2.69961e12 −3.65579
\(928\) 0 0
\(929\) 4.99092e11 0.670066 0.335033 − 0.942206i \(-0.391253\pi\)
0.335033 + 0.942206i \(0.391253\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.51903e12 −1.93735 −0.968674 − 0.248336i \(-0.920116\pi\)
−0.968674 + 0.248336i \(0.920116\pi\)
\(942\) 0 0
\(943\) 1.14455e12 1.44739
\(944\) 0 0
\(945\) −2.24758e12 −2.81831
\(946\) 0 0
\(947\) 5.25398e11 0.653264 0.326632 − 0.945152i \(-0.394086\pi\)
0.326632 + 0.945152i \(0.394086\pi\)
\(948\) 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
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 8.52891e11 1.00000
\(962\) 0 0
\(963\) −3.78327e12 −4.39908
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.52964e12 −1.74938 −0.874689 − 0.484684i \(-0.838935\pi\)
−0.874689 + 0.484684i \(0.838935\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −3.97094e12 −4.28763
\(982\) 0 0
\(983\) −1.56219e12 −1.67309 −0.836544 − 0.547900i \(-0.815427\pi\)
−0.836544 + 0.547900i \(0.815427\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −2.92083e12 −3.07778
\(988\) 0 0
\(989\) 5.32127e11 0.556199
\(990\) 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 320.9.h.a.319.1 1
4.3 odd 2 320.9.h.b.319.1 1
5.4 even 2 320.9.h.b.319.1 1
8.3 odd 2 20.9.d.b.19.1 yes 1
8.5 even 2 20.9.d.a.19.1 ✓ 1
20.19 odd 2 CM 320.9.h.a.319.1 1
40.3 even 4 100.9.b.b.51.1 2
40.13 odd 4 100.9.b.b.51.2 2
40.19 odd 2 20.9.d.a.19.1 ✓ 1
40.27 even 4 100.9.b.b.51.2 2
40.29 even 2 20.9.d.b.19.1 yes 1
40.37 odd 4 100.9.b.b.51.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.9.d.a.19.1 ✓ 1 8.5 even 2
20.9.d.a.19.1 ✓ 1 40.19 odd 2
20.9.d.b.19.1 yes 1 8.3 odd 2
20.9.d.b.19.1 yes 1 40.29 even 2
100.9.b.b.51.1 2 40.3 even 4
100.9.b.b.51.1 2 40.37 odd 4
100.9.b.b.51.2 2 40.13 odd 4
100.9.b.b.51.2 2 40.27 even 4
320.9.h.a.319.1 1 1.1 even 1 trivial
320.9.h.a.319.1 1 20.19 odd 2 CM
320.9.h.b.319.1 1 4.3 odd 2
320.9.h.b.319.1 1 5.4 even 2