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Properties

Label 448.3.c.a.321.1
Level $448$
Weight $3$
Character 448.321
Self dual yes
Analytic conductor $12.207$
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] = [448,3,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, 3); 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, 3, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
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,-7,0,9] 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: \(12.2071158433\)
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)
 
\(f(q)\) \(=\) \(q-7.00000 q^{7} +9.00000 q^{9} +6.00000 q^{11} +18.0000 q^{23} +25.0000 q^{25} +54.0000 q^{29} +38.0000 q^{37} -58.0000 q^{43} +49.0000 q^{49} +6.00000 q^{53} -63.0000 q^{63} +118.000 q^{67} +114.000 q^{71} -42.0000 q^{77} -94.0000 q^{79} +81.0000 q^{81} +54.0000 q^{99} +O(q^{100})\)

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
\(7\) −7.00000 −1.00000
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) 6.00000 0.545455 0.272727 − 0.962091i \(-0.412074\pi\)
0.272727 + 0.962091i \(0.412074\pi\)
\(12\) 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
\(23\) 18.0000 0.782609 0.391304 − 0.920261i \(-0.372024\pi\)
0.391304 + 0.920261i \(0.372024\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 54.0000 1.86207 0.931034 − 0.364931i \(-0.118907\pi\)
0.931034 + 0.364931i \(0.118907\pi\)
\(30\) 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
\(37\) 38.0000 1.02703 0.513514 − 0.858082i \(-0.328344\pi\)
0.513514 + 0.858082i \(0.328344\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −58.0000 −1.34884 −0.674419 − 0.738349i \(-0.735606\pi\)
−0.674419 + 0.738349i \(0.735606\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.113208 0.0566038 − 0.998397i \(-0.481973\pi\)
0.0566038 + 0.998397i \(0.481973\pi\)
\(54\) 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
\(63\) −63.0000 −1.00000
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 118.000 1.76119 0.880597 − 0.473866i \(-0.157142\pi\)
0.880597 + 0.473866i \(0.157142\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 114.000 1.60563 0.802817 − 0.596226i \(-0.203334\pi\)
0.802817 + 0.596226i \(0.203334\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −42.0000 −0.545455
\(78\) 0 0
\(79\) −94.0000 −1.18987 −0.594937 − 0.803773i \(-0.702823\pi\)
−0.594937 + 0.803773i \(0.702823\pi\)
\(80\) 0 0
\(81\) 81.0000 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
\(99\) 54.0000 0.545455
\(100\) 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
\(107\) −186.000 −1.73832 −0.869159 − 0.494533i \(-0.835339\pi\)
−0.869159 + 0.494533i \(0.835339\pi\)
\(108\) 0 0
\(109\) −106.000 −0.972477 −0.486239 − 0.873826i \(-0.661631\pi\)
−0.486239 + 0.873826i \(0.661631\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −222.000 −1.96460 −0.982301 − 0.187310i \(-0.940023\pi\)
−0.982301 + 0.187310i \(0.940023\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −85.0000 −0.702479
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.00000 0.0157480 0.00787402 − 0.999969i \(-0.497494\pi\)
0.00787402 + 0.999969i \(0.497494\pi\)
\(128\) 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
\(137\) −174.000 −1.27007 −0.635036 − 0.772482i \(-0.719015\pi\)
−0.635036 + 0.772482i \(0.719015\pi\)
\(138\) 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
\(149\) −186.000 −1.24832 −0.624161 − 0.781296i \(-0.714559\pi\)
−0.624161 + 0.781296i \(0.714559\pi\)
\(150\) 0 0
\(151\) 274.000 1.81457 0.907285 − 0.420517i \(-0.138151\pi\)
0.907285 + 0.420517i \(0.138151\pi\)
\(152\) 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
\(161\) −126.000 −0.782609
\(162\) 0 0
\(163\) −74.0000 −0.453988 −0.226994 − 0.973896i \(-0.572890\pi\)
−0.226994 + 0.973896i \(0.572890\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −175.000 −1.00000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 342.000 1.91061 0.955307 − 0.295615i \(-0.0955244\pi\)
0.955307 + 0.295615i \(0.0955244\pi\)
\(180\) 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
\(191\) −318.000 −1.66492 −0.832461 − 0.554084i \(-0.813069\pi\)
−0.832461 + 0.554084i \(0.813069\pi\)
\(192\) 0 0
\(193\) −62.0000 −0.321244 −0.160622 − 0.987016i \(-0.551350\pi\)
−0.160622 + 0.987016i \(0.551350\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −282.000 −1.43147 −0.715736 − 0.698371i \(-0.753909\pi\)
−0.715736 + 0.698371i \(0.753909\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −378.000 −1.86207
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 162.000 0.782609
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 278.000 1.31754 0.658768 − 0.752346i \(-0.271078\pi\)
0.658768 + 0.752346i \(0.271078\pi\)
\(212\) 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
\(225\) 225.000 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
\(233\) 18.0000 0.0772532 0.0386266 − 0.999254i \(-0.487702\pi\)
0.0386266 + 0.999254i \(0.487702\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −222.000 −0.928870 −0.464435 − 0.885607i \(-0.653743\pi\)
−0.464435 + 0.885607i \(0.653743\pi\)
\(240\) 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
\(253\) 108.000 0.426877
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −266.000 −1.02703
\(260\) 0 0
\(261\) 486.000 1.86207
\(262\) 0 0
\(263\) 498.000 1.89354 0.946768 − 0.321917i \(-0.104327\pi\)
0.946768 + 0.321917i \(0.104327\pi\)
\(264\) 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
\(275\) 150.000 0.545455
\(276\) 0 0
\(277\) 454.000 1.63899 0.819495 − 0.573087i \(-0.194254\pi\)
0.819495 + 0.573087i \(0.194254\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 114.000 0.405694 0.202847 − 0.979210i \(-0.434981\pi\)
0.202847 + 0.979210i \(0.434981\pi\)
\(282\) 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
\(289\) 289.000 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
\(301\) 406.000 1.34884
\(302\) 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
\(317\) −522.000 −1.64669 −0.823344 − 0.567543i \(-0.807894\pi\)
−0.823344 + 0.567543i \(0.807894\pi\)
\(318\) 0 0
\(319\) 324.000 1.01567
\(320\) 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
\(331\) −634.000 −1.91541 −0.957704 − 0.287755i \(-0.907091\pi\)
−0.957704 + 0.287755i \(0.907091\pi\)
\(332\) 0 0
\(333\) 342.000 1.02703
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 226.000 0.670623 0.335312 − 0.942107i \(-0.391158\pi\)
0.335312 + 0.942107i \(0.391158\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −343.000 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 678.000 1.95389 0.976945 − 0.213490i \(-0.0684831\pi\)
0.976945 + 0.213490i \(0.0684831\pi\)
\(348\) 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
\(359\) −654.000 −1.82173 −0.910864 − 0.412708i \(-0.864583\pi\)
−0.910864 + 0.412708i \(0.864583\pi\)
\(360\) 0 0
\(361\) 361.000 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
\(371\) −42.0000 −0.113208
\(372\) 0 0
\(373\) 262.000 0.702413 0.351206 − 0.936298i \(-0.385772\pi\)
0.351206 + 0.936298i \(0.385772\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 614.000 1.62005 0.810026 − 0.586393i \(-0.199453\pi\)
0.810026 + 0.586393i \(0.199453\pi\)
\(380\) 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
\(387\) −522.000 −1.34884
\(388\) 0 0
\(389\) −666.000 −1.71208 −0.856041 − 0.516908i \(-0.827083\pi\)
−0.856041 + 0.516908i \(0.827083\pi\)
\(390\) 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
\(401\) 354.000 0.882793 0.441397 − 0.897312i \(-0.354483\pi\)
0.441397 + 0.897312i \(0.354483\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 228.000 0.560197
\(408\) 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
\(421\) 166.000 0.394299 0.197150 − 0.980373i \(-0.436832\pi\)
0.197150 + 0.980373i \(0.436832\pi\)
\(422\) 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
\(431\) 162.000 0.375870 0.187935 − 0.982181i \(-0.439821\pi\)
0.187935 + 0.982181i \(0.439821\pi\)
\(432\) 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
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) 486.000 1.09707 0.548533 − 0.836129i \(-0.315187\pi\)
0.548533 + 0.836129i \(0.315187\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −894.000 −1.99109 −0.995546 − 0.0942807i \(-0.969945\pi\)
−0.995546 + 0.0942807i \(0.969945\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −878.000 −1.92123 −0.960613 − 0.277891i \(-0.910365\pi\)
−0.960613 + 0.277891i \(0.910365\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 674.000 1.45572 0.727862 − 0.685724i \(-0.240514\pi\)
0.727862 + 0.685724i \(0.240514\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −826.000 −1.76119
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −348.000 −0.735729
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 54.0000 0.113208
\(478\) 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
\(487\) −398.000 −0.817248 −0.408624 − 0.912703i \(-0.633991\pi\)
−0.408624 + 0.912703i \(0.633991\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −954.000 −1.94297 −0.971487 − 0.237094i \(-0.923805\pi\)
−0.971487 + 0.237094i \(0.923805\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −798.000 −1.60563
\(498\) 0 0
\(499\) −298.000 −0.597194 −0.298597 − 0.954379i \(-0.596519\pi\)
−0.298597 + 0.954379i \(0.596519\pi\)
\(500\) 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
\(529\) −205.000 −0.387524
\(530\) 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
\(539\) 294.000 0.545455
\(540\) 0 0
\(541\) −74.0000 −0.136784 −0.0683919 − 0.997659i \(-0.521787\pi\)
−0.0683919 + 0.997659i \(0.521787\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −842.000 −1.53931 −0.769653 − 0.638463i \(-0.779571\pi\)
−0.769653 + 0.638463i \(0.779571\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 658.000 1.18987
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1002.00 −1.79892 −0.899461 − 0.437000i \(-0.856041\pi\)
−0.899461 + 0.437000i \(0.856041\pi\)
\(558\) 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
\(567\) −567.000 −1.00000
\(568\) 0 0
\(569\) −654.000 −1.14938 −0.574692 − 0.818369i \(-0.694878\pi\)
−0.574692 + 0.818369i \(0.694878\pi\)
\(570\) 0 0
\(571\) 1126.00 1.97198 0.985989 − 0.166807i \(-0.0533458\pi\)
0.985989 + 0.166807i \(0.0533458\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 450.000 0.782609
\(576\) 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
\(583\) 36.0000 0.0617496
\(584\) 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
\(599\) −174.000 −0.290484 −0.145242 − 0.989396i \(-0.546396\pi\)
−0.145242 + 0.989396i \(0.546396\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 1062.00 1.76119
\(604\) 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
\(613\) −218.000 −0.355628 −0.177814 − 0.984064i \(-0.556903\pi\)
−0.177814 + 0.984064i \(0.556903\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −558.000 −0.904376 −0.452188 − 0.891923i \(-0.649356\pi\)
−0.452188 + 0.891923i \(0.649356\pi\)
\(618\) 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
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1006.00 −1.59429 −0.797147 − 0.603785i \(-0.793659\pi\)
−0.797147 + 0.603785i \(0.793659\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 1026.00 1.60563
\(640\) 0 0
\(641\) 834.000 1.30109 0.650546 − 0.759467i \(-0.274540\pi\)
0.650546 + 0.759467i \(0.274540\pi\)
\(642\) 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
\(653\) −1194.00 −1.82848 −0.914242 − 0.405169i \(-0.867213\pi\)
−0.914242 + 0.405169i \(0.867213\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −618.000 −0.937785 −0.468892 − 0.883255i \(-0.655347\pi\)
−0.468892 + 0.883255i \(0.655347\pi\)
\(660\) 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
\(667\) 972.000 1.45727
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −446.000 −0.662704 −0.331352 − 0.943507i \(-0.607505\pi\)
−0.331352 + 0.943507i \(0.607505\pi\)
\(674\) 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
\(683\) −1338.00 −1.95900 −0.979502 − 0.201433i \(-0.935440\pi\)
−0.979502 + 0.201433i \(0.935440\pi\)
\(684\) 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
\(693\) −378.000 −0.545455
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1398.00 1.99429 0.997147 − 0.0754851i \(-0.0240505\pi\)
0.997147 + 0.0754851i \(0.0240505\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1382.00 1.94922 0.974612 − 0.223900i \(-0.0718789\pi\)
0.974612 + 0.223900i \(0.0718789\pi\)
\(710\) 0 0
\(711\) −846.000 −1.18987
\(712\) 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
\(725\) 1350.00 1.86207
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 729.000 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
\(737\) 708.000 0.960651
\(738\) 0 0
\(739\) −1226.00 −1.65900 −0.829499 − 0.558508i \(-0.811374\pi\)
−0.829499 + 0.558508i \(0.811374\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 114.000 0.153432 0.0767160 − 0.997053i \(-0.475557\pi\)
0.0767160 + 0.997053i \(0.475557\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1302.00 1.73832
\(750\) 0 0
\(751\) 802.000 1.06791 0.533955 − 0.845513i \(-0.320705\pi\)
0.533955 + 0.845513i \(0.320705\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1402.00 −1.85205 −0.926024 − 0.377465i \(-0.876796\pi\)
−0.926024 + 0.377465i \(0.876796\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 742.000 0.972477
\(764\) 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
\(781\) 684.000 0.875800
\(782\) 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
\(791\) 1554.00 1.96460
\(792\) 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
\(809\) −174.000 −0.215080 −0.107540 − 0.994201i \(-0.534297\pi\)
−0.107540 + 0.994201i \(0.534297\pi\)
\(810\) 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
\(821\) 1158.00 1.41048 0.705238 − 0.708971i \(-0.250840\pi\)
0.705238 + 0.708971i \(0.250840\pi\)
\(822\) 0 0
\(823\) −622.000 −0.755772 −0.377886 − 0.925852i \(-0.623349\pi\)
−0.377886 + 0.925852i \(0.623349\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −282.000 −0.340992 −0.170496 − 0.985358i \(-0.554537\pi\)
−0.170496 + 0.985358i \(0.554537\pi\)
\(828\) 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
\(841\) 2075.00 2.46730
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 595.000 0.702479
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 684.000 0.803760
\(852\) 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
\(863\) −1662.00 −1.92584 −0.962920 − 0.269787i \(-0.913047\pi\)
−0.962920 + 0.269787i \(0.913047\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −564.000 −0.649022
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −746.000 −0.850627 −0.425314 − 0.905046i \(-0.639836\pi\)
−0.425314 + 0.905046i \(0.639836\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1622.00 1.83692 0.918460 − 0.395514i \(-0.129434\pi\)
0.918460 + 0.395514i \(0.129434\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.0157480
\(890\) 0 0
\(891\) 486.000 0.545455
\(892\) 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
\(907\) −1786.00 −1.96913 −0.984564 − 0.175022i \(-0.944000\pi\)
−0.984564 + 0.175022i \(0.944000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1566.00 −1.71899 −0.859495 − 0.511144i \(-0.829222\pi\)
−0.859495 + 0.511144i \(0.829222\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 466.000 0.507073 0.253536 − 0.967326i \(-0.418406\pi\)
0.253536 + 0.967326i \(0.418406\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 950.000 1.02703
\(926\) 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
\(947\) 1494.00 1.57761 0.788807 − 0.614641i \(-0.210699\pi\)
0.788807 + 0.614641i \(0.210699\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1458.00 1.52991 0.764953 − 0.644086i \(-0.222762\pi\)
0.764953 + 0.644086i \(0.222762\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1218.00 1.27007
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 0 0
\(963\) −1674.00 −1.73832
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −334.000 −0.345398 −0.172699 − 0.984975i \(-0.555249\pi\)
−0.172699 + 0.984975i \(0.555249\pi\)
\(968\) 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
\(977\) 162.000 0.165814 0.0829069 − 0.996557i \(-0.473580\pi\)
0.0829069 + 0.996557i \(0.473580\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −954.000 −0.972477
\(982\) 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
\(989\) −1044.00 −1.05561
\(990\) 0 0
\(991\) −1406.00 −1.41877 −0.709384 − 0.704822i \(-0.751027\pi\)
−0.709384 + 0.704822i \(0.751027\pi\)
\(992\) 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 448.3.c.a.321.1 1
4.3 odd 2 448.3.c.b.321.1 1
7.6 odd 2 CM 448.3.c.a.321.1 1
8.3 odd 2 112.3.c.a.97.1 1
8.5 even 2 7.3.b.a.6.1 ✓ 1
24.5 odd 2 63.3.d.a.55.1 1
24.11 even 2 1008.3.f.a.433.1 1
28.27 even 2 448.3.c.b.321.1 1
40.13 odd 4 175.3.c.a.174.2 2
40.29 even 2 175.3.d.a.76.1 1
40.37 odd 4 175.3.c.a.174.1 2
56.3 even 6 784.3.s.a.705.1 2
56.5 odd 6 49.3.d.a.31.1 2
56.11 odd 6 784.3.s.a.705.1 2
56.13 odd 2 7.3.b.a.6.1 ✓ 1
56.19 even 6 784.3.s.a.129.1 2
56.27 even 2 112.3.c.a.97.1 1
56.37 even 6 49.3.d.a.31.1 2
56.45 odd 6 49.3.d.a.19.1 2
56.51 odd 6 784.3.s.a.129.1 2
56.53 even 6 49.3.d.a.19.1 2
168.5 even 6 441.3.m.a.325.1 2
168.53 odd 6 441.3.m.a.19.1 2
168.83 odd 2 1008.3.f.a.433.1 1
168.101 even 6 441.3.m.a.19.1 2
168.125 even 2 63.3.d.a.55.1 1
168.149 odd 6 441.3.m.a.325.1 2
280.13 even 4 175.3.c.a.174.2 2
280.69 odd 2 175.3.d.a.76.1 1
280.237 even 4 175.3.c.a.174.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.3.b.a.6.1 ✓ 1 8.5 even 2
7.3.b.a.6.1 ✓ 1 56.13 odd 2
49.3.d.a.19.1 2 56.45 odd 6
49.3.d.a.19.1 2 56.53 even 6
49.3.d.a.31.1 2 56.5 odd 6
49.3.d.a.31.1 2 56.37 even 6
63.3.d.a.55.1 1 24.5 odd 2
63.3.d.a.55.1 1 168.125 even 2
112.3.c.a.97.1 1 8.3 odd 2
112.3.c.a.97.1 1 56.27 even 2
175.3.c.a.174.1 2 40.37 odd 4
175.3.c.a.174.1 2 280.237 even 4
175.3.c.a.174.2 2 40.13 odd 4
175.3.c.a.174.2 2 280.13 even 4
175.3.d.a.76.1 1 40.29 even 2
175.3.d.a.76.1 1 280.69 odd 2
441.3.m.a.19.1 2 168.53 odd 6
441.3.m.a.19.1 2 168.101 even 6
441.3.m.a.325.1 2 168.5 even 6
441.3.m.a.325.1 2 168.149 odd 6
448.3.c.a.321.1 1 1.1 even 1 trivial
448.3.c.a.321.1 1 7.6 odd 2 CM
448.3.c.b.321.1 1 4.3 odd 2
448.3.c.b.321.1 1 28.27 even 2
784.3.s.a.129.1 2 56.19 even 6
784.3.s.a.129.1 2 56.51 odd 6
784.3.s.a.705.1 2 56.3 even 6
784.3.s.a.705.1 2 56.11 odd 6
1008.3.f.a.433.1 1 24.11 even 2
1008.3.f.a.433.1 1 168.83 odd 2