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Properties

Label 784.6.a.g.1.1
Level $784$
Weight $6$
Character 784.1
Self dual yes
Analytic conductor $125.741$
Analytic rank $1$
Dimension $1$
CM discriminant -7
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] = [784,6,Mod(1,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,0,0,-243] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(125.740914733\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(+1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 784.1

$q$-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)
 
\(f(q)\) \(=\) \(q-243.000 q^{9} +76.0000 q^{11} +4952.00 q^{23} -3125.00 q^{25} +7282.00 q^{29} -8886.00 q^{37} -11748.0 q^{43} +24550.0 q^{53} -69364.0 q^{67} +2224.00 q^{71} -80168.0 q^{79} +59049.0 q^{81} -18468.0 q^{99} +O(q^{100})\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −243.000 −1.00000
\(10\) 0 0
\(11\) 76.0000 0.189379 0.0946895 − 0.995507i \(-0.469814\pi\)
0.0946895 + 0.995507i \(0.469814\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4952.00 1.95192 0.975958 − 0.217959i \(-0.0699401\pi\)
0.975958 + 0.217959i \(0.0699401\pi\)
\(24\) 0 0
\(25\) −3125.00 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7282.00 1.60789 0.803944 − 0.594705i \(-0.202731\pi\)
0.803944 + 0.594705i \(0.202731\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8886.00 −1.06709 −0.533546 − 0.845771i \(-0.679141\pi\)
−0.533546 + 0.845771i \(0.679141\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −11748.0 −0.968931 −0.484465 − 0.874810i \(-0.660986\pi\)
−0.484465 + 0.874810i \(0.660986\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 24550.0 1.20050 0.600250 − 0.799813i \(-0.295068\pi\)
0.600250 + 0.799813i \(0.295068\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −69364.0 −1.88776 −0.943881 − 0.330286i \(-0.892855\pi\)
−0.943881 + 0.330286i \(0.892855\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2224.00 0.0523587 0.0261794 − 0.999657i \(-0.491666\pi\)
0.0261794 + 0.999657i \(0.491666\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −80168.0 −1.44522 −0.722609 − 0.691257i \(-0.757057\pi\)
−0.722609 + 0.691257i \(0.757057\pi\)
\(80\) 0 0
\(81\) 59049.0 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 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
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) −18468.0 −0.189379
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 64900.0 0.548006 0.274003 − 0.961729i \(-0.411652\pi\)
0.274003 + 0.961729i \(0.411652\pi\)
\(108\) 0 0
\(109\) −219582. −1.77023 −0.885117 − 0.465369i \(-0.845922\pi\)
−0.885117 + 0.465369i \(0.845922\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 123202. 0.907657 0.453828 − 0.891089i \(-0.350058\pi\)
0.453828 + 0.891089i \(0.350058\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −155275. −0.964136
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 262064. 1.44178 0.720888 − 0.693051i \(-0.243734\pi\)
0.720888 + 0.693051i \(0.243734\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −353450. −1.60889 −0.804445 − 0.594027i \(-0.797537\pi\)
−0.804445 + 0.594027i \(0.797537\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −337018. −1.24362 −0.621810 − 0.783168i \(-0.713602\pi\)
−0.621810 + 0.783168i \(0.713602\pi\)
\(150\) 0 0
\(151\) 261624. 0.933760 0.466880 − 0.884321i \(-0.345378\pi\)
0.466880 + 0.884321i \(0.345378\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −663100. −1.95483 −0.977417 − 0.211318i \(-0.932224\pi\)
−0.977417 + 0.211318i \(0.932224\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −371293. −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
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −584564. −1.36364 −0.681820 − 0.731520i \(-0.738811\pi\)
−0.681820 + 0.731520i \(0.738811\pi\)
\(180\) 0 0
\(181\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.00305e6 −1.98947 −0.994737 − 0.102461i \(-0.967328\pi\)
−0.994737 + 0.102461i \(0.967328\pi\)
\(192\) 0 0
\(193\) −385902. −0.745734 −0.372867 − 0.927885i \(-0.621625\pi\)
−0.372867 + 0.927885i \(0.621625\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 385814. 0.708292 0.354146 − 0.935190i \(-0.384772\pi\)
0.354146 + 0.935190i \(0.384772\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.20334e6 −1.95192
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 1.09705e6 1.69637 0.848186 − 0.529699i \(-0.177695\pi\)
0.848186 + 0.529699i \(0.177695\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 759375. 1.00000
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.27950e6 −1.54401 −0.772004 − 0.635617i \(-0.780746\pi\)
−0.772004 + 0.635617i \(0.780746\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −723536. −0.819342 −0.409671 − 0.912233i \(-0.634357\pi\)
−0.409671 + 0.912233i \(0.634357\pi\)
\(240\) 0 0
\(241\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 376352. 0.369652
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.76953e6 −1.60789
\(262\) 0 0
\(263\) −1.53155e6 −1.36534 −0.682672 − 0.730725i \(-0.739182\pi\)
−0.682672 + 0.730725i \(0.739182\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −237500. −0.189379
\(276\) 0 0
\(277\) −2.55145e6 −1.99796 −0.998982 − 0.0451116i \(-0.985636\pi\)
−0.998982 + 0.0451116i \(0.985636\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.54797e6 1.92499 0.962497 − 0.271294i \(-0.0874515\pi\)
0.962497 + 0.271294i \(0.0874515\pi\)
\(282\) 0 0
\(283\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.41986e6 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 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
\(307\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −221714. −0.123921 −0.0619605 − 0.998079i \(-0.519735\pi\)
−0.0619605 + 0.998079i \(0.519735\pi\)
\(318\) 0 0
\(319\) 553432. 0.304500
\(320\) 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
\(331\) −2.97148e6 −1.49074 −0.745371 − 0.666650i \(-0.767727\pi\)
−0.745371 + 0.666650i \(0.767727\pi\)
\(332\) 0 0
\(333\) 2.15930e6 1.06709
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.15965e6 −1.99518 −0.997590 − 0.0693859i \(-0.977896\pi\)
−0.997590 + 0.0693859i \(0.977896\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.29816e6 −1.02461 −0.512304 − 0.858804i \(-0.671208\pi\)
−0.512304 + 0.858804i \(0.671208\pi\)
\(348\) 0 0
\(349\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4.26897e6 −1.74818 −0.874091 − 0.485762i \(-0.838542\pi\)
−0.874091 + 0.485762i \(0.838542\pi\)
\(360\) 0 0
\(361\) −2.47610e6 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 599302. 0.223035 0.111518 − 0.993762i \(-0.464429\pi\)
0.111518 + 0.993762i \(0.464429\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 5.59273e6 1.99998 0.999991 − 0.00429827i \(-0.00136819\pi\)
0.999991 + 0.00429827i \(0.00136819\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.85476e6 0.968931
\(388\) 0 0
\(389\) −1.26012e6 −0.422218 −0.211109 − 0.977462i \(-0.567708\pi\)
−0.211109 + 0.977462i \(0.567708\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.03293e6 1.87356 0.936779 − 0.349922i \(-0.113792\pi\)
0.936779 + 0.349922i \(0.113792\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −675336. −0.202085
\(408\) 0 0
\(409\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 2.07477e6 0.570513 0.285257 − 0.958451i \(-0.407921\pi\)
0.285257 + 0.958451i \(0.407921\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.33885e6 1.90298 0.951491 − 0.307676i \(-0.0995514\pi\)
0.951491 + 0.307676i \(0.0995514\pi\)
\(432\) 0 0
\(433\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5.10490e6 1.23588 0.617942 − 0.786223i \(-0.287967\pi\)
0.617942 + 0.786223i \(0.287967\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.99808e6 0.467732 0.233866 − 0.972269i \(-0.424862\pi\)
0.233866 + 0.972269i \(0.424862\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −5.77969e6 −1.29453 −0.647267 − 0.762263i \(-0.724088\pi\)
−0.647267 + 0.762263i \(0.724088\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −2.88620e6 −0.625711 −0.312856 − 0.949801i \(-0.601286\pi\)
−0.312856 + 0.949801i \(0.601286\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −892848. −0.183495
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −5.96565e6 −1.20050
\(478\) 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
\(485\) 0 0
\(486\) 0 0
\(487\) 2.76146e6 0.527615 0.263807 − 0.964575i \(-0.415022\pi\)
0.263807 + 0.964575i \(0.415022\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8.82732e6 −1.65244 −0.826219 − 0.563349i \(-0.809513\pi\)
−0.826219 + 0.563349i \(0.809513\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.11204e7 1.99925 0.999626 − 0.0273386i \(-0.00870324\pi\)
0.999626 + 0.0273386i \(0.00870324\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 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
\(523\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.80860e7 2.80997
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.11261e7 1.63437 0.817186 − 0.576374i \(-0.195533\pi\)
0.817186 + 0.576374i \(0.195533\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.23604e6 0.319529 0.159765 − 0.987155i \(-0.448926\pi\)
0.159765 + 0.987155i \(0.448926\pi\)
\(548\) 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
\(557\) 6.23949e6 0.852140 0.426070 − 0.904690i \(-0.359898\pi\)
0.426070 + 0.904690i \(0.359898\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.04738e7 1.35620 0.678099 − 0.734971i \(-0.262804\pi\)
0.678099 + 0.734971i \(0.262804\pi\)
\(570\) 0 0
\(571\) −6.33912e6 −0.813653 −0.406826 − 0.913506i \(-0.633365\pi\)
−0.406826 + 0.913506i \(0.633365\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.54750e7 −1.95192
\(576\) 0 0
\(577\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1.86580e6 0.227349
\(584\) 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
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 7.17757e6 0.817354 0.408677 − 0.912679i \(-0.365990\pi\)
0.408677 + 0.912679i \(0.365990\pi\)
\(600\) 0 0
\(601\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 1.68555e7 1.88776
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.75514e7 1.88652 0.943258 − 0.332060i \(-0.107744\pi\)
0.943258 + 0.332060i \(0.107744\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.14899e6 −0.756017 −0.378008 − 0.925802i \(-0.623391\pi\)
−0.378008 + 0.925802i \(0.623391\pi\)
\(618\) 0 0
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.76562e6 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.99786e7 −1.99752 −0.998760 − 0.0497844i \(-0.984147\pi\)
−0.998760 + 0.0497844i \(0.984147\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −540432. −0.0523587
\(640\) 0 0
\(641\) −1.14963e7 −1.10513 −0.552563 − 0.833471i \(-0.686350\pi\)
−0.552563 + 0.833471i \(0.686350\pi\)
\(642\) 0 0
\(643\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 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
\(653\) 1.09772e7 1.00742 0.503710 − 0.863873i \(-0.331968\pi\)
0.503710 + 0.863873i \(0.331968\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.02232e7 1.81400 0.907000 − 0.421131i \(-0.138367\pi\)
0.907000 + 0.421131i \(0.138367\pi\)
\(660\) 0 0
\(661\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.60605e7 3.13846
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.38435e6 0.117817 0.0589085 − 0.998263i \(-0.481238\pi\)
0.0589085 + 0.998263i \(0.481238\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.13149e7 1.74836 0.874179 − 0.485603i \(-0.161400\pi\)
0.874179 + 0.485603i \(0.161400\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 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
\(701\) 4.88600e6 0.375542 0.187771 − 0.982213i \(-0.439874\pi\)
0.187771 + 0.982213i \(0.439874\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.43225e7 −1.07005 −0.535023 − 0.844837i \(-0.679697\pi\)
−0.535023 + 0.844837i \(0.679697\pi\)
\(710\) 0 0
\(711\) 1.94808e7 1.44522
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.27562e7 −1.60789
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −1.43489e7 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.27166e6 −0.357502
\(738\) 0 0
\(739\) 2.64893e6 0.178427 0.0892133 − 0.996013i \(-0.471565\pi\)
0.0892133 + 0.996013i \(0.471565\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −2.49502e7 −1.65807 −0.829033 − 0.559199i \(-0.811109\pi\)
−0.829033 + 0.559199i \(0.811109\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.51088e7 −1.62452 −0.812260 − 0.583295i \(-0.801763\pi\)
−0.812260 + 0.583295i \(0.801763\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.78870e7 −1.13448 −0.567242 − 0.823551i \(-0.691989\pi\)
−0.567242 + 0.823551i \(0.691989\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 169024. 0.00991564
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.83707e7 0.986856 0.493428 − 0.869787i \(-0.335744\pi\)
0.493428 + 0.869787i \(0.335744\pi\)
\(810\) 0 0
\(811\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 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
\(821\) 3.55881e7 1.84267 0.921334 − 0.388773i \(-0.127101\pi\)
0.921334 + 0.388773i \(0.127101\pi\)
\(822\) 0 0
\(823\) −7.08675e6 −0.364710 −0.182355 − 0.983233i \(-0.558372\pi\)
−0.182355 + 0.983233i \(0.558372\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.68552e7 1.87385 0.936926 − 0.349527i \(-0.113658\pi\)
0.936926 + 0.349527i \(0.113658\pi\)
\(828\) 0 0
\(829\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 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
\(841\) 3.25164e7 1.58530
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.40035e7 −2.08287
\(852\) 0 0
\(853\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.76142e7 −1.26214 −0.631068 − 0.775727i \(-0.717383\pi\)
−0.631068 + 0.775727i \(0.717383\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6.09277e6 −0.273694
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.33428e7 −1.90291 −0.951453 − 0.307793i \(-0.900410\pi\)
−0.951453 + 0.307793i \(0.900410\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 3.94011e7 1.70062 0.850308 − 0.526286i \(-0.176416\pi\)
0.850308 + 0.526286i \(0.176416\pi\)
\(884\) 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
\(891\) 4.48772e6 0.189379
\(892\) 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
\(907\) −4.48347e7 −1.80966 −0.904828 − 0.425777i \(-0.860001\pi\)
−0.904828 + 0.425777i \(0.860001\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −4.87850e7 −1.94756 −0.973778 − 0.227498i \(-0.926945\pi\)
−0.973778 + 0.227498i \(0.926945\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −5.06716e7 −1.97914 −0.989569 − 0.144059i \(-0.953984\pi\)
−0.989569 + 0.144059i \(0.953984\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.77688e7 1.06709
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 5.50009e7 1.99294 0.996471 − 0.0839326i \(-0.0267480\pi\)
0.996471 + 0.0839326i \(0.0267480\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −9.97130e6 −0.355647 −0.177824 − 0.984062i \(-0.556906\pi\)
−0.177824 + 0.984062i \(0.556906\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.86292e7 −1.00000
\(962\) 0 0
\(963\) −1.57707e7 −0.548006
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −2.00222e7 −0.688566 −0.344283 − 0.938866i \(-0.611878\pi\)
−0.344283 + 0.938866i \(0.611878\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.99817e7 −1.67523 −0.837616 − 0.546260i \(-0.816051\pi\)
−0.837616 + 0.546260i \(0.816051\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 5.33584e7 1.77023
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −5.81761e7 −1.89127
\(990\) 0 0
\(991\) 5.73144e7 1.85387 0.926936 − 0.375219i \(-0.122433\pi\)
0.926936 + 0.375219i \(0.122433\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 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 784.6.a.g.1.1 1
4.3 odd 2 49.6.a.b.1.1 ✓ 1
7.6 odd 2 CM 784.6.a.g.1.1 1
12.11 even 2 441.6.a.a.1.1 1
28.3 even 6 49.6.c.a.30.1 2
28.11 odd 6 49.6.c.a.30.1 2
28.19 even 6 49.6.c.a.18.1 2
28.23 odd 6 49.6.c.a.18.1 2
28.27 even 2 49.6.a.b.1.1 ✓ 1
84.83 odd 2 441.6.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.b.1.1 ✓ 1 4.3 odd 2
49.6.a.b.1.1 ✓ 1 28.27 even 2
49.6.c.a.18.1 2 28.19 even 6
49.6.c.a.18.1 2 28.23 odd 6
49.6.c.a.30.1 2 28.3 even 6
49.6.c.a.30.1 2 28.11 odd 6
441.6.a.a.1.1 1 12.11 even 2
441.6.a.a.1.1 1 84.83 odd 2
784.6.a.g.1.1 1 1.1 even 1 trivial
784.6.a.g.1.1 1 7.6 odd 2 CM