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Properties

Label 448.5.c.b.321.1
Level $448$
Weight $5$
Character 448.321
Self dual yes
Analytic conductor $46.310$
Analytic rank $0$
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] = [448,5,Mod(321,448)] 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("448.321"); S:= CuspForms(chi, 5); N := Newforms(S);
 
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, 5, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 448.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,49,0,81] 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: \(46.3097434616\)
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)
 
\(f(q)\) \(=\) \(q+49.0000 q^{7} +81.0000 q^{9} +206.000 q^{11} -734.000 q^{23} +625.000 q^{25} -1234.00 q^{29} +1294.00 q^{37} +334.000 q^{43} +2401.00 q^{49} +5582.00 q^{53} +3969.00 q^{63} -4946.00 q^{67} +2914.00 q^{71} +10094.0 q^{77} -3646.00 q^{79} +6561.00 q^{81} +16686.0 q^{99} +O(q^{100})\)

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
\(7\) 49.0000 1.00000
\(8\) 0 0
\(9\) 81.0000 1.00000
\(10\) 0 0
\(11\) 206.000 1.70248 0.851240 − 0.524777i \(-0.175851\pi\)
0.851240 + 0.524777i \(0.175851\pi\)
\(12\) 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
\(23\) −734.000 −1.38752 −0.693762 − 0.720205i \(-0.744048\pi\)
−0.693762 + 0.720205i \(0.744048\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1234.00 −1.46730 −0.733650 − 0.679527i \(-0.762185\pi\)
−0.733650 + 0.679527i \(0.762185\pi\)
\(30\) 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
\(37\) 1294.00 0.945215 0.472608 − 0.881273i \(-0.343313\pi\)
0.472608 + 0.881273i \(0.343313\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 334.000 0.180638 0.0903191 − 0.995913i \(-0.471211\pi\)
0.0903191 + 0.995913i \(0.471211\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 2401.00 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5582.00 1.98718 0.993592 − 0.113026i \(-0.0360544\pi\)
0.993592 + 0.113026i \(0.0360544\pi\)
\(54\) 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
\(63\) 3969.00 1.00000
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4946.00 −1.10180 −0.550902 − 0.834570i \(-0.685716\pi\)
−0.550902 + 0.834570i \(0.685716\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2914.00 0.578060 0.289030 − 0.957320i \(-0.406667\pi\)
0.289030 + 0.957320i \(0.406667\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10094.0 1.70248
\(78\) 0 0
\(79\) −3646.00 −0.584201 −0.292101 − 0.956388i \(-0.594354\pi\)
−0.292101 + 0.956388i \(0.594354\pi\)
\(80\) 0 0
\(81\) 6561.00 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
\(99\) 16686.0 1.70248
\(100\) 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
\(107\) −11698.0 −1.02175 −0.510874 − 0.859655i \(-0.670678\pi\)
−0.510874 + 0.859655i \(0.670678\pi\)
\(108\) 0 0
\(109\) 12526.0 1.05429 0.527144 − 0.849776i \(-0.323263\pi\)
0.527144 + 0.849776i \(0.323263\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 23746.0 1.85966 0.929830 − 0.367989i \(-0.119954\pi\)
0.929830 + 0.367989i \(0.119954\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 27795.0 1.89844
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −32254.0 −1.99975 −0.999876 − 0.0157475i \(-0.994987\pi\)
−0.999876 + 0.0157475i \(0.994987\pi\)
\(128\) 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
\(137\) −7262.00 −0.386915 −0.193457 − 0.981109i \(-0.561970\pi\)
−0.193457 + 0.981109i \(0.561970\pi\)
\(138\) 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
\(149\) 9806.00 0.441692 0.220846 − 0.975309i \(-0.429118\pi\)
0.220846 + 0.975309i \(0.429118\pi\)
\(150\) 0 0
\(151\) 29474.0 1.29266 0.646331 − 0.763057i \(-0.276302\pi\)
0.646331 + 0.763057i \(0.276302\pi\)
\(152\) 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
\(161\) −35966.0 −1.38752
\(162\) 0 0
\(163\) 47662.0 1.79390 0.896948 − 0.442137i \(-0.145779\pi\)
0.896948 + 0.442137i \(0.145779\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 28561.0 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 30625.0 1.00000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −52882.0 −1.65045 −0.825224 − 0.564806i \(-0.808951\pi\)
−0.825224 + 0.564806i \(0.808951\pi\)
\(180\) 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
\(191\) 28162.0 0.771963 0.385982 − 0.922506i \(-0.373863\pi\)
0.385982 + 0.922506i \(0.373863\pi\)
\(192\) 0 0
\(193\) −70654.0 −1.89680 −0.948401 − 0.317073i \(-0.897300\pi\)
−0.948401 + 0.317073i \(0.897300\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1906.00 −0.0491123 −0.0245562 − 0.999698i \(-0.507817\pi\)
−0.0245562 + 0.999698i \(0.507817\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −60466.0 −1.46730
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −59454.0 −1.38752
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 11758.0 0.264100 0.132050 − 0.991243i \(-0.457844\pi\)
0.132050 + 0.991243i \(0.457844\pi\)
\(212\) 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
\(225\) 50625.0 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
\(233\) −108254. −1.99403 −0.997016 − 0.0771956i \(-0.975403\pi\)
−0.997016 + 0.0771956i \(0.975403\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −64958.0 −1.13720 −0.568600 − 0.822614i \(-0.692515\pi\)
−0.568600 + 0.822614i \(0.692515\pi\)
\(240\) 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
\(253\) −151204. −2.36223
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 63406.0 0.945215
\(260\) 0 0
\(261\) −99954.0 −1.46730
\(262\) 0 0
\(263\) 109666. 1.58548 0.792740 − 0.609561i \(-0.208654\pi\)
0.792740 + 0.609561i \(0.208654\pi\)
\(264\) 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
\(275\) 128750. 1.70248
\(276\) 0 0
\(277\) −52658.0 −0.686285 −0.343143 − 0.939283i \(-0.611491\pi\)
−0.343143 + 0.939283i \(0.611491\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −144926. −1.83541 −0.917706 − 0.397260i \(-0.869961\pi\)
−0.917706 + 0.397260i \(0.869961\pi\)
\(282\) 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
\(289\) 83521.0 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
\(301\) 16366.0 0.180638
\(302\) 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
\(317\) −71506.0 −0.711580 −0.355790 − 0.934566i \(-0.615788\pi\)
−0.355790 + 0.934566i \(0.615788\pi\)
\(318\) 0 0
\(319\) −254204. −2.49805
\(320\) 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
\(331\) −182834. −1.66879 −0.834394 − 0.551169i \(-0.814182\pi\)
−0.834394 + 0.551169i \(0.814182\pi\)
\(332\) 0 0
\(333\) 104814. 0.945215
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −176062. −1.55026 −0.775132 − 0.631799i \(-0.782317\pi\)
−0.775132 + 0.631799i \(0.782317\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 117649. 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −218866. −1.81769 −0.908844 − 0.417136i \(-0.863034\pi\)
−0.908844 + 0.417136i \(0.863034\pi\)
\(348\) 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
\(359\) 169954. 1.31869 0.659345 − 0.751841i \(-0.270834\pi\)
0.659345 + 0.751841i \(0.270834\pi\)
\(360\) 0 0
\(361\) 130321. 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
\(371\) 273518. 1.98718
\(372\) 0 0
\(373\) 209614. 1.50662 0.753308 − 0.657668i \(-0.228457\pi\)
0.753308 + 0.657668i \(0.228457\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −89714.0 −0.624571 −0.312285 − 0.949988i \(-0.601095\pi\)
−0.312285 + 0.949988i \(0.601095\pi\)
\(380\) 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
\(387\) 27054.0 0.180638
\(388\) 0 0
\(389\) −140914. −0.931226 −0.465613 − 0.884989i \(-0.654166\pi\)
−0.465613 + 0.884989i \(0.654166\pi\)
\(390\) 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
\(401\) −196286. −1.22068 −0.610338 − 0.792141i \(-0.708966\pi\)
−0.610338 + 0.792141i \(0.708966\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 266564. 1.60921
\(408\) 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
\(421\) 326926. 1.84453 0.922264 − 0.386561i \(-0.126337\pi\)
0.922264 + 0.386561i \(0.126337\pi\)
\(422\) 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
\(431\) −345278. −1.85872 −0.929361 − 0.369173i \(-0.879641\pi\)
−0.929361 + 0.369173i \(0.879641\pi\)
\(432\) 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
\(441\) 194481. 1.00000
\(442\) 0 0
\(443\) 156302. 0.796447 0.398224 − 0.917288i \(-0.369627\pi\)
0.398224 + 0.917288i \(0.369627\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 396034. 1.96444 0.982222 − 0.187721i \(-0.0601102\pi\)
0.982222 + 0.187721i \(0.0601102\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 353186. 1.69111 0.845553 − 0.533891i \(-0.179271\pi\)
0.845553 + 0.533891i \(0.179271\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 25538.0 0.119131 0.0595655 − 0.998224i \(-0.481028\pi\)
0.0595655 + 0.998224i \(0.481028\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −242354. −1.10180
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 68804.0 0.307533
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 452142. 1.98718
\(478\) 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
\(487\) −315934. −1.33210 −0.666052 − 0.745905i \(-0.732017\pi\)
−0.666052 + 0.745905i \(0.732017\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −427954. −1.77515 −0.887573 − 0.460667i \(-0.847610\pi\)
−0.887573 + 0.460667i \(0.847610\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 142786. 0.578060
\(498\) 0 0
\(499\) 409198. 1.64336 0.821679 − 0.569950i \(-0.193037\pi\)
0.821679 + 0.569950i \(0.193037\pi\)
\(500\) 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
\(529\) 258915. 0.925222
\(530\) 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
\(539\) 494606. 1.70248
\(540\) 0 0
\(541\) 579886. 1.98129 0.990645 − 0.136463i \(-0.0435736\pi\)
0.990645 + 0.136463i \(0.0435736\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −110546. −0.369461 −0.184730 − 0.982789i \(-0.559141\pi\)
−0.184730 + 0.982789i \(0.559141\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −178654. −0.584201
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −383506. −1.23612 −0.618062 − 0.786130i \(-0.712082\pi\)
−0.618062 + 0.786130i \(0.712082\pi\)
\(558\) 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
\(567\) 321489. 1.00000
\(568\) 0 0
\(569\) −219806. −0.678914 −0.339457 − 0.940622i \(-0.610243\pi\)
−0.339457 + 0.940622i \(0.610243\pi\)
\(570\) 0 0
\(571\) −615794. −1.88870 −0.944351 − 0.328941i \(-0.893308\pi\)
−0.944351 + 0.328941i \(0.893308\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −458750. −1.38752
\(576\) 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
\(583\) 1.14989e6 3.38314
\(584\) 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
\(599\) −687326. −1.91562 −0.957809 − 0.287404i \(-0.907208\pi\)
−0.957809 + 0.287404i \(0.907208\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −400626. −1.10180
\(604\) 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
\(613\) 704014. 1.87353 0.936764 − 0.349961i \(-0.113805\pi\)
0.936764 + 0.349961i \(0.113805\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −450014. −1.18210 −0.591052 − 0.806633i \(-0.701287\pi\)
−0.591052 + 0.806633i \(0.701287\pi\)
\(618\) 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
\(625\) 390625. 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 215714. 0.541776 0.270888 − 0.962611i \(-0.412683\pi\)
0.270888 + 0.962611i \(0.412683\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 236034. 0.578060
\(640\) 0 0
\(641\) −126206. −0.307159 −0.153580 − 0.988136i \(-0.549080\pi\)
−0.153580 + 0.988136i \(0.549080\pi\)
\(642\) 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
\(653\) −572818. −1.34335 −0.671677 − 0.740844i \(-0.734426\pi\)
−0.671677 + 0.740844i \(0.734426\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 486638. 1.12056 0.560280 − 0.828303i \(-0.310693\pi\)
0.560280 + 0.828303i \(0.310693\pi\)
\(660\) 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
\(667\) 905756. 2.03591
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −706942. −1.56082 −0.780411 − 0.625266i \(-0.784990\pi\)
−0.780411 + 0.625266i \(0.784990\pi\)
\(674\) 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
\(683\) −857266. −1.83770 −0.918849 − 0.394609i \(-0.870880\pi\)
−0.918849 + 0.394609i \(0.870880\pi\)
\(684\) 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
\(693\) 817614. 1.70248
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −971602. −1.97721 −0.988604 − 0.150539i \(-0.951899\pi\)
−0.988604 + 0.150539i \(0.951899\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −904562. −1.79948 −0.899738 − 0.436431i \(-0.856242\pi\)
−0.899738 + 0.436431i \(0.856242\pi\)
\(710\) 0 0
\(711\) −295326. −0.584201
\(712\) 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
\(725\) −771250. −1.46730
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 531441. 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
\(737\) −1.01888e6 −1.87580
\(738\) 0 0
\(739\) −410834. −0.752277 −0.376138 − 0.926564i \(-0.622748\pi\)
−0.376138 + 0.926564i \(0.622748\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.09110e6 −1.97646 −0.988229 − 0.152980i \(-0.951113\pi\)
−0.988229 + 0.152980i \(0.951113\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −573202. −1.02175
\(750\) 0 0
\(751\) −484798. −0.859569 −0.429785 − 0.902931i \(-0.641411\pi\)
−0.429785 + 0.902931i \(0.641411\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −819506. −1.43008 −0.715040 − 0.699083i \(-0.753592\pi\)
−0.715040 + 0.699083i \(0.753592\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 613774. 1.05429
\(764\) 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
\(781\) 600284. 0.984135
\(782\) 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
\(791\) 1.16355e6 1.85966
\(792\) 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
\(809\) −1.27869e6 −1.95374 −0.976870 − 0.213833i \(-0.931405\pi\)
−0.976870 + 0.213833i \(0.931405\pi\)
\(810\) 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
\(821\) 7118.00 0.0105602 0.00528009 − 0.999986i \(-0.498319\pi\)
0.00528009 + 0.999986i \(0.498319\pi\)
\(822\) 0 0
\(823\) −967774. −1.42881 −0.714405 − 0.699733i \(-0.753302\pi\)
−0.714405 + 0.699733i \(0.753302\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.28833e6 1.88372 0.941862 − 0.335999i \(-0.109074\pi\)
0.941862 + 0.335999i \(0.109074\pi\)
\(828\) 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
\(841\) 815475. 1.15297
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.36196e6 1.89844
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −949796. −1.31151
\(852\) 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
\(863\) 1.27271e6 1.70886 0.854430 − 0.519566i \(-0.173907\pi\)
0.854430 + 0.519566i \(0.173907\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −751076. −0.994591
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 981742. 1.27643 0.638217 − 0.769857i \(-0.279672\pi\)
0.638217 + 0.769857i \(0.279672\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1.07151e6 −1.37427 −0.687137 − 0.726528i \(-0.741133\pi\)
−0.687137 + 0.726528i \(0.741133\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −1.58045e6 −1.99975
\(890\) 0 0
\(891\) 1.35157e6 1.70248
\(892\) 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
\(907\) −1.54450e6 −1.87747 −0.938735 − 0.344641i \(-0.888001\pi\)
−0.938735 + 0.344641i \(0.888001\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 792514. 0.954927 0.477464 − 0.878652i \(-0.341556\pi\)
0.477464 + 0.878652i \(0.341556\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −1.47197e6 −1.74288 −0.871439 − 0.490505i \(-0.836812\pi\)
−0.871439 + 0.490505i \(0.836812\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 808750. 0.945215
\(926\) 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
\(947\) −438418. −0.488864 −0.244432 − 0.969666i \(-0.578602\pi\)
−0.244432 + 0.969666i \(0.578602\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 309346. 0.340611 0.170306 − 0.985391i \(-0.445525\pi\)
0.170306 + 0.985391i \(0.445525\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −355838. −0.386915
\(960\) 0 0
\(961\) 923521. 1.00000
\(962\) 0 0
\(963\) −947538. −1.02175
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.75862e6 −1.88070 −0.940350 − 0.340208i \(-0.889502\pi\)
−0.940350 + 0.340208i \(0.889502\pi\)
\(968\) 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
\(977\) −1.88281e6 −1.97251 −0.986253 − 0.165243i \(-0.947159\pi\)
−0.986253 + 0.165243i \(0.947159\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.01461e6 1.05429
\(982\) 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
\(989\) −245156. −0.250640
\(990\) 0 0
\(991\) 12674.0 0.0129052 0.00645262 − 0.999979i \(-0.497946\pi\)
0.00645262 + 0.999979i \(0.497946\pi\)
\(992\) 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 448.5.c.b.321.1 1
4.3 odd 2 448.5.c.a.321.1 1
7.6 odd 2 CM 448.5.c.b.321.1 1
8.3 odd 2 112.5.c.a.97.1 1
8.5 even 2 7.5.b.a.6.1 ✓ 1
24.5 odd 2 63.5.d.a.55.1 1
24.11 even 2 1008.5.f.a.433.1 1
28.27 even 2 448.5.c.a.321.1 1
40.13 odd 4 175.5.c.a.174.1 2
40.29 even 2 175.5.d.a.76.1 1
40.37 odd 4 175.5.c.a.174.2 2
56.5 odd 6 49.5.d.a.31.1 2
56.13 odd 2 7.5.b.a.6.1 ✓ 1
56.27 even 2 112.5.c.a.97.1 1
56.37 even 6 49.5.d.a.31.1 2
56.45 odd 6 49.5.d.a.19.1 2
56.53 even 6 49.5.d.a.19.1 2
168.83 odd 2 1008.5.f.a.433.1 1
168.125 even 2 63.5.d.a.55.1 1
280.13 even 4 175.5.c.a.174.1 2
280.69 odd 2 175.5.d.a.76.1 1
280.237 even 4 175.5.c.a.174.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.5.b.a.6.1 ✓ 1 8.5 even 2
7.5.b.a.6.1 ✓ 1 56.13 odd 2
49.5.d.a.19.1 2 56.45 odd 6
49.5.d.a.19.1 2 56.53 even 6
49.5.d.a.31.1 2 56.5 odd 6
49.5.d.a.31.1 2 56.37 even 6
63.5.d.a.55.1 1 24.5 odd 2
63.5.d.a.55.1 1 168.125 even 2
112.5.c.a.97.1 1 8.3 odd 2
112.5.c.a.97.1 1 56.27 even 2
175.5.c.a.174.1 2 40.13 odd 4
175.5.c.a.174.1 2 280.13 even 4
175.5.c.a.174.2 2 40.37 odd 4
175.5.c.a.174.2 2 280.237 even 4
175.5.d.a.76.1 1 40.29 even 2
175.5.d.a.76.1 1 280.69 odd 2
448.5.c.a.321.1 1 4.3 odd 2
448.5.c.a.321.1 1 28.27 even 2
448.5.c.b.321.1 1 1.1 even 1 trivial
448.5.c.b.321.1 1 7.6 odd 2 CM
1008.5.f.a.433.1 1 24.11 even 2
1008.5.f.a.433.1 1 168.83 odd 2