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Properties

Label 448.7.c.b.321.1
Level $448$
Weight $7$
Character 448.321
Self dual yes
Analytic conductor $103.064$
Analytic rank $0$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,7,Mod(321,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 7, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.321"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 448.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,343,0,729] 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: \(103.064229462\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 321.1
Character \(\chi\) \(=\) 448.321

$q$-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)
 
\(f(q)\) \(=\) \(q+343.000 q^{7} +729.000 q^{9} +1962.00 q^{11} +22734.0 q^{23} +15625.0 q^{25} +21222.0 q^{29} -101194. q^{37} -126614. q^{43} +117649. q^{49} -50346.0 q^{53} +250047. q^{63} -53926.0 q^{67} +242478. q^{71} +672966. q^{77} -929378. q^{79} +531441. q^{81} +1.43030e6 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 343.000 1.00000
\(8\) 0 0
\(9\) 729.000 1.00000
\(10\) 0 0
\(11\) 1962.00 1.47408 0.737040 − 0.675849i \(-0.236223\pi\)
0.737040 + 0.675849i \(0.236223\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 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
\(21\) 0 0
\(22\) 0 0
\(23\) 22734.0 1.86850 0.934248 − 0.356623i \(-0.116072\pi\)
0.934248 + 0.356623i \(0.116072\pi\)
\(24\) 0 0
\(25\) 15625.0 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 21222.0 0.870146 0.435073 − 0.900395i \(-0.356722\pi\)
0.435073 + 0.900395i \(0.356722\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −101194. −1.99779 −0.998894 − 0.0470096i \(-0.985031\pi\)
−0.998894 + 0.0470096i \(0.985031\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −126614. −1.59249 −0.796244 − 0.604975i \(-0.793183\pi\)
−0.796244 + 0.604975i \(0.793183\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 117649. 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −50346.0 −0.338172 −0.169086 − 0.985601i \(-0.554082\pi\)
−0.169086 + 0.985601i \(0.554082\pi\)
\(54\) 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
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 250047. 1.00000
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −53926.0 −0.179297 −0.0896487 − 0.995973i \(-0.528574\pi\)
−0.0896487 + 0.995973i \(0.528574\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 242478. 0.677481 0.338741 − 0.940880i \(-0.389999\pi\)
0.338741 + 0.940880i \(0.389999\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 672966. 1.47408
\(78\) 0 0
\(79\) −929378. −1.88500 −0.942499 − 0.334208i \(-0.891531\pi\)
−0.942499 + 0.334208i \(0.891531\pi\)
\(80\) 0 0
\(81\) 531441. 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 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
\(99\) 1.43030e6 1.47408
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 46314.0 0.0378060 0.0189030 − 0.999821i \(-0.493983\pi\)
0.0189030 + 0.999821i \(0.493983\pi\)
\(108\) 0 0
\(109\) 2.58714e6 1.99775 0.998874 − 0.0474386i \(-0.0151058\pi\)
0.998874 + 0.0474386i \(0.0151058\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.43689e6 −1.68889 −0.844445 − 0.535642i \(-0.820069\pi\)
−0.844445 + 0.535642i \(0.820069\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.07788e6 1.17291
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 96766.0 0.0472402 0.0236201 − 0.999721i \(-0.492481\pi\)
0.0236201 + 0.999721i \(0.492481\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.52939e6 1.76148 0.880741 − 0.473598i \(-0.157045\pi\)
0.880741 + 0.473598i \(0.157045\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 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
\(149\) 5.95330e6 1.79970 0.899848 − 0.436204i \(-0.143677\pi\)
0.899848 + 0.436204i \(0.143677\pi\)
\(150\) 0 0
\(151\) −1.82840e6 −0.531057 −0.265528 − 0.964103i \(-0.585546\pi\)
−0.265528 + 0.964103i \(0.585546\pi\)
\(152\) 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
\(161\) 7.79776e6 1.86850
\(162\) 0 0
\(163\) −5.49309e6 −1.26839 −0.634197 − 0.773171i \(-0.718669\pi\)
−0.634197 + 0.773171i \(0.718669\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 4.82681e6 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 5.35938e6 1.00000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −7.12762e6 −1.24276 −0.621378 − 0.783511i \(-0.713427\pi\)
−0.621378 + 0.783511i \(0.713427\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.64544e6 −0.379663 −0.189831 − 0.981817i \(-0.560794\pi\)
−0.189831 + 0.981817i \(0.560794\pi\)
\(192\) 0 0
\(193\) 6.68999e6 0.930579 0.465290 − 0.885159i \(-0.345950\pi\)
0.465290 + 0.885159i \(0.345950\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.04066e7 1.36117 0.680585 − 0.732670i \(-0.261726\pi\)
0.680585 + 0.732670i \(0.261726\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7.27915e6 0.870146
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.65731e7 1.86850
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 1.56456e7 1.66550 0.832748 − 0.553652i \(-0.186766\pi\)
0.832748 + 0.553652i \(0.186766\pi\)
\(212\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 1.13906e7 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.92577e6 −0.231299 −0.115649 − 0.993290i \(-0.536895\pi\)
−0.115649 + 0.993290i \(0.536895\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.71015e7 −1.98518 −0.992591 − 0.121505i \(-0.961228\pi\)
−0.992591 + 0.121505i \(0.961228\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 4.46041e7 2.75431
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −3.47095e7 −1.99779
\(260\) 0 0
\(261\) 1.54708e7 0.870146
\(262\) 0 0
\(263\) −2.01675e7 −1.10863 −0.554313 − 0.832308i \(-0.687019\pi\)
−0.554313 + 0.832308i \(0.687019\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.06562e7 1.47408
\(276\) 0 0
\(277\) −1.09282e7 −0.514175 −0.257087 − 0.966388i \(-0.582763\pi\)
−0.257087 + 0.966388i \(0.582763\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.55231e7 −1.15031 −0.575155 − 0.818045i \(-0.695058\pi\)
−0.575155 + 0.818045i \(0.695058\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.41376e7 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
\(301\) −4.34286e7 −1.59249
\(302\) 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
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.51291e7 0.474937 0.237469 − 0.971395i \(-0.423682\pi\)
0.237469 + 0.971395i \(0.423682\pi\)
\(318\) 0 0
\(319\) 4.16376e7 1.28267
\(320\) 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
\(331\) 4.64551e7 1.28100 0.640500 − 0.767958i \(-0.278727\pi\)
0.640500 + 0.767958i \(0.278727\pi\)
\(332\) 0 0
\(333\) −7.37704e7 −1.99779
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −6.54566e7 −1.71027 −0.855133 − 0.518408i \(-0.826525\pi\)
−0.855133 + 0.518408i \(0.826525\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 4.03536e7 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.67538e7 −1.59767 −0.798836 − 0.601548i \(-0.794551\pi\)
−0.798836 + 0.601548i \(0.794551\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 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
\(359\) 2.68617e7 0.580565 0.290282 − 0.956941i \(-0.406251\pi\)
0.290282 + 0.956941i \(0.406251\pi\)
\(360\) 0 0
\(361\) 4.70459e7 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.72687e7 −0.338172
\(372\) 0 0
\(373\) −9.13707e7 −1.76068 −0.880340 − 0.474344i \(-0.842685\pi\)
−0.880340 + 0.474344i \(0.842685\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 3.31112e7 0.608215 0.304107 − 0.952638i \(-0.401642\pi\)
0.304107 + 0.952638i \(0.401642\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −9.23016e7 −1.59249
\(388\) 0 0
\(389\) 6.93106e6 0.117747 0.0588737 − 0.998265i \(-0.481249\pi\)
0.0588737 + 0.998265i \(0.481249\pi\)
\(390\) 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
\(401\) −1.26409e8 −1.96040 −0.980199 − 0.198016i \(-0.936550\pi\)
−0.980199 + 0.198016i \(0.936550\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.98543e8 −2.94490
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −8.36917e7 −1.12160 −0.560798 − 0.827953i \(-0.689505\pi\)
−0.560798 + 0.827953i \(0.689505\pi\)
\(422\) 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
\(431\) 8.60283e7 1.07451 0.537254 − 0.843421i \(-0.319462\pi\)
0.537254 + 0.843421i \(0.319462\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 8.57661e7 1.00000
\(442\) 0 0
\(443\) 1.71340e8 1.97082 0.985410 − 0.170196i \(-0.0544402\pi\)
0.985410 + 0.170196i \(0.0544402\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.73823e8 −1.92030 −0.960149 − 0.279490i \(-0.909835\pi\)
−0.960149 + 0.279490i \(0.909835\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.26728e8 −1.32777 −0.663886 − 0.747834i \(-0.731094\pi\)
−0.663886 + 0.747834i \(0.731094\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 1.27272e8 1.28230 0.641151 − 0.767415i \(-0.278457\pi\)
0.641151 + 0.767415i \(0.278457\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −1.84966e7 −0.179297
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.48417e8 −2.34746
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −3.67022e7 −0.338172
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.20135e8 −1.90591 −0.952955 − 0.303113i \(-0.901974\pi\)
−0.952955 + 0.303113i \(0.901974\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.78277e8 1.50609 0.753044 − 0.657970i \(-0.228585\pi\)
0.753044 + 0.657970i \(0.228585\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.31700e7 0.677481
\(498\) 0 0
\(499\) −1.96143e8 −1.57860 −0.789300 − 0.614008i \(-0.789556\pi\)
−0.789300 + 0.614008i \(0.789556\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.68799e8 2.49128
\(530\) 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
\(539\) 2.30827e8 1.47408
\(540\) 0 0
\(541\) 6.45700e7 0.407792 0.203896 − 0.978993i \(-0.434640\pi\)
0.203896 + 0.978993i \(0.434640\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.58854e8 −0.970592 −0.485296 − 0.874350i \(-0.661288\pi\)
−0.485296 + 0.874350i \(0.661288\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −3.18777e8 −1.88500
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.34035e7 −0.424767 −0.212384 − 0.977186i \(-0.568123\pi\)
−0.212384 + 0.977186i \(0.568123\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.82284e8 1.00000
\(568\) 0 0
\(569\) 3.55493e8 1.92972 0.964859 − 0.262767i \(-0.0846349\pi\)
0.964859 + 0.262767i \(0.0846349\pi\)
\(570\) 0 0
\(571\) −3.26262e8 −1.75250 −0.876250 − 0.481857i \(-0.839963\pi\)
−0.876250 + 0.481857i \(0.839963\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.55219e8 1.86850
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −9.87789e7 −0.498492
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 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
\(599\) −1.82026e8 −0.846941 −0.423471 − 0.905910i \(-0.639188\pi\)
−0.423471 + 0.905910i \(0.639188\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −3.93121e7 −0.179297
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 2.35393e8 1.02191 0.510954 − 0.859608i \(-0.329292\pi\)
0.510954 + 0.859608i \(0.329292\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.63532e8 1.97344 0.986721 − 0.162423i \(-0.0519308\pi\)
0.986721 + 0.162423i \(0.0519308\pi\)
\(618\) 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
\(625\) 2.44141e8 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.83542e8 −0.730544 −0.365272 − 0.930901i \(-0.619024\pi\)
−0.365272 + 0.930901i \(0.619024\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 1.76766e8 0.677481
\(640\) 0 0
\(641\) −4.47931e8 −1.70073 −0.850367 − 0.526189i \(-0.823620\pi\)
−0.850367 + 0.526189i \(0.823620\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.74812e8 −0.627816 −0.313908 − 0.949453i \(-0.601638\pi\)
−0.313908 + 0.949453i \(0.601638\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −5.69128e8 −1.98863 −0.994314 − 0.106485i \(-0.966040\pi\)
−0.994314 + 0.106485i \(0.966040\pi\)
\(660\) 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
\(667\) 4.82461e8 1.62587
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 5.17302e8 1.69707 0.848534 − 0.529141i \(-0.177486\pi\)
0.848534 + 0.529141i \(0.177486\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5.22860e8 1.64105 0.820527 − 0.571607i \(-0.193680\pi\)
0.820527 + 0.571607i \(0.193680\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 4.90592e8 1.47408
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.71321e8 1.94884 0.974420 − 0.224735i \(-0.0721516\pi\)
0.974420 + 0.224735i \(0.0721516\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 5.55400e8 1.55836 0.779178 − 0.626802i \(-0.215637\pi\)
0.779178 + 0.626802i \(0.215637\pi\)
\(710\) 0 0
\(711\) −6.77517e8 −1.88500
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.31594e8 0.870146
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 3.87420e8 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.05803e8 −0.264299
\(738\) 0 0
\(739\) −1.65862e8 −0.410973 −0.205486 − 0.978660i \(-0.565878\pi\)
−0.205486 + 0.978660i \(0.565878\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.87319e8 0.456684 0.228342 − 0.973581i \(-0.426670\pi\)
0.228342 + 0.973581i \(0.426670\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.58857e7 0.0378060
\(750\) 0 0
\(751\) 8.41137e8 1.98585 0.992926 − 0.118736i \(-0.0378842\pi\)
0.992926 + 0.118736i \(0.0378842\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −3.45533e8 −0.796529 −0.398264 − 0.917271i \(-0.630387\pi\)
−0.398264 + 0.917271i \(0.630387\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 8.87390e8 1.99775
\(764\) 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
\(773\) 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
\(781\) 4.75742e8 0.998661
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −8.35855e8 −1.68889
\(792\) 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
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.36371e8 0.635292 0.317646 − 0.948209i \(-0.397108\pi\)
0.317646 + 0.948209i \(0.397108\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 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
\(821\) −7.88782e8 −1.42537 −0.712685 − 0.701484i \(-0.752521\pi\)
−0.712685 + 0.701484i \(0.752521\pi\)
\(822\) 0 0
\(823\) −1.02325e9 −1.83563 −0.917813 − 0.397014i \(-0.870046\pi\)
−0.917813 + 0.397014i \(0.870046\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −5.56178e8 −0.983326 −0.491663 − 0.870786i \(-0.663611\pi\)
−0.491663 + 0.870786i \(0.663611\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −1.44450e8 −0.242845
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 7.12714e8 1.17291
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.30054e9 −3.73286
\(852\) 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
\(861\) 0 0
\(862\) 0 0
\(863\) 8.77431e8 1.36515 0.682576 − 0.730815i \(-0.260860\pi\)
0.682576 + 0.730815i \(0.260860\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.82344e9 −2.77864
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.30615e9 1.93640 0.968198 − 0.250185i \(-0.0804915\pi\)
0.968198 + 0.250185i \(0.0804915\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −4.73327e8 −0.687510 −0.343755 − 0.939059i \(-0.611699\pi\)
−0.343755 + 0.939059i \(0.611699\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 3.31907e7 0.0472402
\(890\) 0 0
\(891\) 1.04269e9 1.47408
\(892\) 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
\(907\) 1.28922e9 1.72785 0.863925 − 0.503621i \(-0.167999\pi\)
0.863925 + 0.503621i \(0.167999\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.85794e7 −0.0774800 −0.0387400 − 0.999249i \(-0.512334\pi\)
−0.0387400 + 0.999249i \(0.512334\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.07950e9 1.39084 0.695419 − 0.718604i \(-0.255219\pi\)
0.695419 + 0.718604i \(0.255219\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.58116e9 −1.99779
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 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
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.84836e8 0.806374 0.403187 − 0.915118i \(-0.367902\pi\)
0.403187 + 0.915118i \(0.367902\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −8.73142e8 −1.00880 −0.504401 − 0.863469i \(-0.668287\pi\)
−0.504401 + 0.863469i \(0.668287\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.55358e9 1.76148
\(960\) 0 0
\(961\) 8.87504e8 1.00000
\(962\) 0 0
\(963\) 3.37629e7 0.0378060
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −8.99699e8 −0.994988 −0.497494 − 0.867467i \(-0.665746\pi\)
−0.497494 + 0.867467i \(0.665746\pi\)
\(968\) 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
\(977\) −4.59650e8 −0.492882 −0.246441 − 0.969158i \(-0.579261\pi\)
−0.246441 + 0.969158i \(0.579261\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.88603e9 1.99775
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.87844e9 −2.97556
\(990\) 0 0
\(991\) −1.36299e9 −1.40046 −0.700230 − 0.713918i \(-0.746919\pi\)
−0.700230 + 0.713918i \(0.746919\pi\)
\(992\) 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 448.7.c.b.321.1 1
4.3 odd 2 448.7.c.a.321.1 1
7.6 odd 2 CM 448.7.c.b.321.1 1
8.3 odd 2 7.7.b.a.6.1 ✓ 1
8.5 even 2 112.7.c.a.97.1 1
24.11 even 2 63.7.d.a.55.1 1
28.27 even 2 448.7.c.a.321.1 1
40.3 even 4 175.7.c.a.174.1 2
40.19 odd 2 175.7.d.a.76.1 1
40.27 even 4 175.7.c.a.174.2 2
56.3 even 6 49.7.d.a.19.1 2
56.11 odd 6 49.7.d.a.19.1 2
56.13 odd 2 112.7.c.a.97.1 1
56.19 even 6 49.7.d.a.31.1 2
56.27 even 2 7.7.b.a.6.1 ✓ 1
56.51 odd 6 49.7.d.a.31.1 2
168.83 odd 2 63.7.d.a.55.1 1
280.27 odd 4 175.7.c.a.174.2 2
280.83 odd 4 175.7.c.a.174.1 2
280.139 even 2 175.7.d.a.76.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.b.a.6.1 ✓ 1 8.3 odd 2
7.7.b.a.6.1 ✓ 1 56.27 even 2
49.7.d.a.19.1 2 56.3 even 6
49.7.d.a.19.1 2 56.11 odd 6
49.7.d.a.31.1 2 56.19 even 6
49.7.d.a.31.1 2 56.51 odd 6
63.7.d.a.55.1 1 24.11 even 2
63.7.d.a.55.1 1 168.83 odd 2
112.7.c.a.97.1 1 8.5 even 2
112.7.c.a.97.1 1 56.13 odd 2
175.7.c.a.174.1 2 40.3 even 4
175.7.c.a.174.1 2 280.83 odd 4
175.7.c.a.174.2 2 40.27 even 4
175.7.c.a.174.2 2 280.27 odd 4
175.7.d.a.76.1 1 40.19 odd 2
175.7.d.a.76.1 1 280.139 even 2
448.7.c.a.321.1 1 4.3 odd 2
448.7.c.a.321.1 1 28.27 even 2
448.7.c.b.321.1 1 1.1 even 1 trivial
448.7.c.b.321.1 1 7.6 odd 2 CM