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Properties

Label 225.6.a.e.1.1
Level $225$
Weight $6$
Character 225.1
Self dual yes
Analytic conductor $36.086$
Analytic rank $1$
Dimension $1$
CM discriminant -3
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] = [225,6,Mod(1,225)] 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("225.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,-32,0,0,25] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(36.0863594579\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 225.1

$q$-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)
 
\(f(q)\) \(=\) \(q-32.0000 q^{4} +25.0000 q^{7} +775.000 q^{13} +1024.00 q^{16} -1711.00 q^{19} -800.000 q^{28} +2723.00 q^{31} -16550.0 q^{37} -22475.0 q^{43} -16182.0 q^{49} -24800.0 q^{52} +56927.0 q^{61} -32768.0 q^{64} -73475.0 q^{67} +1450.00 q^{73} +54752.0 q^{76} -100564. q^{79} +19375.0 q^{91} -177725. q^{97} +O(q^{100})\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −32.0000 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 25.0000 0.192839 0.0964195 − 0.995341i \(-0.469261\pi\)
0.0964195 + 0.995341i \(0.469261\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 775.000 1.27187 0.635936 − 0.771742i \(-0.280614\pi\)
0.635936 + 0.771742i \(0.280614\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1024.00 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −1711.00 −1.08734 −0.543671 − 0.839299i \(-0.682966\pi\)
−0.543671 + 0.839299i \(0.682966\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) −800.000 −0.192839
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2723.00 0.508913 0.254456 − 0.967084i \(-0.418103\pi\)
0.254456 + 0.967084i \(0.418103\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −16550.0 −1.98744 −0.993719 − 0.111902i \(-0.964306\pi\)
−0.993719 + 0.111902i \(0.964306\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −22475.0 −1.85365 −0.926827 − 0.375489i \(-0.877475\pi\)
−0.926827 + 0.375489i \(0.877475\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −16182.0 −0.962813
\(50\) 0 0
\(51\) 0 0
\(52\) −24800.0 −1.27187
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 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
\(61\) 56927.0 1.95882 0.979408 − 0.201890i \(-0.0647084\pi\)
0.979408 + 0.201890i \(0.0647084\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −32768.0 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −73475.0 −1.99964 −0.999822 − 0.0188789i \(-0.993990\pi\)
−0.999822 + 0.0188789i \(0.993990\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 1450.00 0.0318464 0.0159232 − 0.999873i \(-0.494931\pi\)
0.0159232 + 0.999873i \(0.494931\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 54752.0 1.08734
\(77\) 0 0
\(78\) 0 0
\(79\) −100564. −1.81290 −0.906452 − 0.422309i \(-0.861220\pi\)
−0.906452 + 0.422309i \(0.861220\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 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
\(91\) 19375.0 0.245266
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −177725. −1.91787 −0.958935 − 0.283626i \(-0.908463\pi\)
−0.958935 + 0.283626i \(0.908463\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −140900. −1.30863 −0.654317 − 0.756221i \(-0.727044\pi\)
−0.654317 + 0.756221i \(0.727044\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 133361. 1.07513 0.537567 − 0.843221i \(-0.319344\pi\)
0.537567 + 0.843221i \(0.319344\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 25600.0 0.192839
\(113\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −161051. −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −87136.0 −0.508913
\(125\) 0 0
\(126\) 0 0
\(127\) 267100. 1.46948 0.734742 − 0.678347i \(-0.237303\pi\)
0.734742 + 0.678347i \(0.237303\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −42775.0 −0.209682
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 252464. 1.10831 0.554157 − 0.832412i \(-0.313041\pi\)
0.554157 + 0.832412i \(0.313041\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 529600. 1.98744
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −127627. −0.455512 −0.227756 − 0.973718i \(-0.573139\pi\)
−0.227756 + 0.973718i \(0.573139\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 581125. 1.88157 0.940785 − 0.339004i \(-0.110090\pi\)
0.940785 + 0.339004i \(0.110090\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 352375. 1.03881 0.519405 − 0.854528i \(-0.326154\pi\)
0.519405 + 0.854528i \(0.326154\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 229332. 0.617658
\(170\) 0 0
\(171\) 0 0
\(172\) 719200. 1.85365
\(173\) 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
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −853027. −1.93538 −0.967690 − 0.252142i \(-0.918865\pi\)
−0.967690 + 0.252142i \(0.918865\pi\)
\(182\) 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
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −656375. −1.26841 −0.634204 − 0.773166i \(-0.718672\pi\)
−0.634204 + 0.773166i \(0.718672\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 517824. 0.962813
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 1.01476e6 1.81648 0.908241 − 0.418448i \(-0.137426\pi\)
0.908241 + 0.418448i \(0.137426\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 793600. 1.27187
\(209\) 0 0
\(210\) 0 0
\(211\) −947323. −1.46485 −0.732423 − 0.680850i \(-0.761611\pi\)
−0.732423 + 0.680850i \(0.761611\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 68075.0 0.0981383
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.10698e6 −1.49065 −0.745325 − 0.666701i \(-0.767706\pi\)
−0.745325 + 0.666701i \(0.767706\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 189689. 0.239031 0.119515 − 0.992832i \(-0.461866\pi\)
0.119515 + 0.992832i \(0.461866\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 436577. 0.484193 0.242096 − 0.970252i \(-0.422165\pi\)
0.242096 + 0.970252i \(0.422165\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −1.82166e6 −1.95882
\(245\) 0 0
\(246\) 0 0
\(247\) −1.32602e6 −1.38296
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.04858e6 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −413750. −0.383256
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 2.35120e6 1.99964
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 2.25285e6 1.86341 0.931707 − 0.363210i \(-0.118319\pi\)
0.931707 + 0.363210i \(0.118319\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.74822e6 −1.36898 −0.684491 − 0.729021i \(-0.739976\pi\)
−0.684491 + 0.729021i \(0.739976\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −2.33262e6 −1.73133 −0.865663 − 0.500627i \(-0.833103\pi\)
−0.865663 + 0.500627i \(0.833103\pi\)
\(284\) 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
\(292\) −46400.0 −0.0318464
\(293\) 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
\(301\) −561875. −0.357457
\(302\) 0 0
\(303\) 0 0
\(304\) −1.75206e6 −1.08734
\(305\) 0 0
\(306\) 0 0
\(307\) −3.20232e6 −1.93919 −0.969593 − 0.244723i \(-0.921303\pi\)
−0.969593 + 0.244723i \(0.921303\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −2.56708e6 −1.48108 −0.740539 − 0.672014i \(-0.765430\pi\)
−0.740539 + 0.672014i \(0.765430\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 3.21805e6 1.81290
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 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
\(331\) −3.14685e6 −1.57872 −0.789361 − 0.613929i \(-0.789588\pi\)
−0.789361 + 0.613929i \(0.789588\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.63172e6 −1.26231 −0.631155 − 0.775657i \(-0.717419\pi\)
−0.631155 + 0.775657i \(0.717419\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −824725. −0.378507
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 4.27561e6 1.87904 0.939518 − 0.342501i \(-0.111274\pi\)
0.939518 + 0.342501i \(0.111274\pi\)
\(350\) 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
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 451422. 0.182312
\(362\) 0 0
\(363\) 0 0
\(364\) −620000. −0.245266
\(365\) 0 0
\(366\) 0 0
\(367\) 2.58318e6 1.00113 0.500563 − 0.865700i \(-0.333126\pi\)
0.500563 + 0.865700i \(0.333126\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.98388e6 −0.738316 −0.369158 − 0.929367i \(-0.620354\pi\)
−0.369158 + 0.929367i \(0.620354\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −4.87806e6 −1.74441 −0.872206 − 0.489138i \(-0.837311\pi\)
−0.872206 + 0.489138i \(0.837311\pi\)
\(380\) 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
\(387\) 0 0
\(388\) 5.68720e6 1.91787
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.00342e6 1.59328 0.796638 − 0.604456i \(-0.206610\pi\)
0.796638 + 0.604456i \(0.206610\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 2.11033e6 0.647272
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.36489e6 1.88141 0.940703 − 0.339231i \(-0.110167\pi\)
0.940703 + 0.339231i \(0.110167\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.50880e6 1.30863
\(413\) 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
\(421\) −2.23610e6 −0.614874 −0.307437 − 0.951568i \(-0.599471\pi\)
−0.307437 + 0.951568i \(0.599471\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.42318e6 0.377736
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 7.49192e6 1.92032 0.960160 − 0.279450i \(-0.0901522\pi\)
0.960160 + 0.279450i \(0.0901522\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.26755e6 −1.07513
\(437\) 0 0
\(438\) 0 0
\(439\) 4.08241e6 1.01101 0.505505 − 0.862824i \(-0.331306\pi\)
0.505505 + 0.862824i \(0.331306\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −819200. −0.192839
\(449\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.25745e6 1.84950 0.924752 − 0.380569i \(-0.124272\pi\)
0.924752 + 0.380569i \(0.124272\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −6.11690e6 −1.32611 −0.663054 − 0.748572i \(-0.730740\pi\)
−0.663054 + 0.748572i \(0.730740\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −1.83688e6 −0.385609
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −1.28262e7 −2.52777
\(482\) 0 0
\(483\) 0 0
\(484\) 5.15363e6 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 1.36487e6 0.260778 0.130389 − 0.991463i \(-0.458377\pi\)
0.130389 + 0.991463i \(0.458377\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 2.78835e6 0.508913
\(497\) 0 0
\(498\) 0 0
\(499\) 1.05353e7 1.89406 0.947030 − 0.321144i \(-0.104067\pi\)
0.947030 + 0.321144i \(0.104067\pi\)
\(500\) 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
\(508\) −8.54720e6 −1.46948
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 36250.0 0.00614124
\(512\) 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
\(523\) 1.17287e7 1.87497 0.937486 − 0.348023i \(-0.113147\pi\)
0.937486 + 0.348023i \(0.113147\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6.43634e6 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 1.36880e6 0.209682
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.30081e7 1.91082 0.955410 − 0.295281i \(-0.0954134\pi\)
0.955410 + 0.295281i \(0.0954134\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.27982e7 −1.82886 −0.914430 − 0.404744i \(-0.867361\pi\)
−0.914430 + 0.404744i \(0.867361\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −2.51410e6 −0.349599
\(554\) 0 0
\(555\) 0 0
\(556\) −8.07885e6 −1.10831
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −1.74181e7 −2.35761
\(560\) 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
\(569\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 9.54998e6 1.22578 0.612889 − 0.790169i \(-0.290007\pi\)
0.612889 + 0.790169i \(0.290007\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.29099e7 1.61430 0.807150 − 0.590347i \(-0.201009\pi\)
0.807150 + 0.590347i \(0.201009\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −4.65905e6 −0.553362
\(590\) 0 0
\(591\) 0 0
\(592\) −1.69472e7 −1.98744
\(593\) 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
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 1.14126e7 1.28884 0.644420 − 0.764671i \(-0.277099\pi\)
0.644420 + 0.764671i \(0.277099\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 4.08406e6 0.455512
\(605\) 0 0
\(606\) 0 0
\(607\) −1.57649e7 −1.73668 −0.868339 − 0.495970i \(-0.834812\pi\)
−0.868339 + 0.495970i \(0.834812\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.58234e7 1.70079 0.850394 − 0.526147i \(-0.176364\pi\)
0.850394 + 0.526147i \(0.176364\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −1.22260e7 −1.28251 −0.641253 − 0.767330i \(-0.721585\pi\)
−0.641253 + 0.767330i \(0.721585\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) −1.85960e7 −1.88157
\(629\) 0 0
\(630\) 0 0
\(631\) 1.82315e7 1.82284 0.911422 − 0.411472i \(-0.134985\pi\)
0.911422 + 0.411472i \(0.134985\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.25411e7 −1.22457
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −2.06702e7 −1.97159 −0.985796 − 0.167944i \(-0.946287\pi\)
−0.985796 + 0.167944i \(0.946287\pi\)
\(644\) 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
\(652\) −1.12760e7 −1.03881
\(653\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −1.91018e7 −1.70048 −0.850238 − 0.526398i \(-0.823542\pi\)
−0.850238 + 0.526398i \(0.823542\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −5.10725e6 −0.434660 −0.217330 − 0.976098i \(-0.569735\pi\)
−0.217330 + 0.976098i \(0.569735\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −7.33862e6 −0.617658
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) −4.44312e6 −0.369840
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −2.30144e7 −1.85365
\(689\) 0 0
\(690\) 0 0
\(691\) 1.73630e7 1.38335 0.691673 − 0.722211i \(-0.256874\pi\)
0.691673 + 0.722211i \(0.256874\pi\)
\(692\) 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
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 2.83171e7 2.16102
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −935611. −0.0699004 −0.0349502 − 0.999389i \(-0.511127\pi\)
−0.0349502 + 0.999389i \(0.511127\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −3.52250e6 −0.252356
\(722\) 0 0
\(723\) 0 0
\(724\) 2.72969e7 1.93538
\(725\) 0 0
\(726\) 0 0
\(727\) −1.27620e7 −0.895534 −0.447767 − 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.13028e7 0.777006 0.388503 − 0.921448i \(-0.372992\pi\)
0.388503 + 0.921448i \(0.372992\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −2.96035e7 −1.99403 −0.997017 − 0.0771842i \(-0.975407\pi\)
−0.997017 + 0.0771842i \(0.975407\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.13875e6 0.0736763 0.0368381 − 0.999321i \(-0.488271\pi\)
0.0368381 + 0.999321i \(0.488271\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3.11435e7 1.97527 0.987637 − 0.156758i \(-0.0501042\pi\)
0.987637 + 0.156758i \(0.0501042\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 3.33402e6 0.207328
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.50017e7 1.52459 0.762296 − 0.647228i \(-0.224072\pi\)
0.762296 + 0.647228i \(0.224072\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.10040e7 1.26841
\(773\) 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
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −1.65704e7 −0.962813
\(785\) 0 0
\(786\) 0 0
\(787\) 2.57024e7 1.47923 0.739616 − 0.673029i \(-0.235007\pi\)
0.739616 + 0.673029i \(0.235007\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 4.41184e7 2.49136
\(794\) 0 0
\(795\) 0 0
\(796\) −3.24724e7 −1.81648
\(797\) 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
\(809\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 1.43756e7 0.767491 0.383745 − 0.923439i \(-0.374634\pi\)
0.383745 + 0.923439i \(0.374634\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3.84547e7 2.01555
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −1.64220e7 −0.845134 −0.422567 − 0.906332i \(-0.638871\pi\)
−0.422567 + 0.906332i \(0.638871\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 3.94293e7 1.99266 0.996329 − 0.0856034i \(-0.0272818\pi\)
0.996329 + 0.0856034i \(0.0272818\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.53952e7 −1.27187
\(833\) 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
\(841\) −2.05111e7 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 3.03143e7 1.46485
\(845\) 0 0
\(846\) 0 0
\(847\) −4.02628e6 −0.192839
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 3.00842e6 0.141568 0.0707842 − 0.997492i \(-0.477450\pi\)
0.0707842 + 0.997492i \(0.477450\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 3.06774e6 0.141852 0.0709259 − 0.997482i \(-0.477405\pi\)
0.0709259 + 0.997482i \(0.477405\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) −2.17840e6 −0.0981383
\(869\) 0 0
\(870\) 0 0
\(871\) −5.69431e7 −2.54329
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.73911e7 1.64161 0.820804 − 0.571210i \(-0.193526\pi\)
0.820804 + 0.571210i \(0.193526\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −1.68468e7 −0.727135 −0.363567 − 0.931568i \(-0.618441\pi\)
−0.363567 + 0.931568i \(0.618441\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 6.67750e6 0.283374
\(890\) 0 0
\(891\) 0 0
\(892\) 3.54232e7 1.49065
\(893\) 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
\(907\) −2.33102e7 −0.940866 −0.470433 − 0.882436i \(-0.655902\pi\)
−0.470433 + 0.882436i \(0.655902\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −6.07005e6 −0.239031
\(917\) 0 0
\(918\) 0 0
\(919\) −4.21821e6 −0.164755 −0.0823777 − 0.996601i \(-0.526251\pi\)
−0.0823777 + 0.996601i \(0.526251\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 2.76874e7 1.04691
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −5.29708e6 −0.197100 −0.0985501 − 0.995132i \(-0.531420\pi\)
−0.0985501 + 0.995132i \(0.531420\pi\)
\(938\) 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
\(947\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 1.12375e6 0.0405046
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.12144e7 −0.741008
\(962\) 0 0
\(963\) 0 0
\(964\) −1.39705e7 −0.484193
\(965\) 0 0
\(966\) 0 0
\(967\) −5.80289e7 −1.99562 −0.997811 − 0.0661347i \(-0.978933\pi\)
−0.997811 + 0.0661347i \(0.978933\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 6.31160e6 0.213726
\(974\) 0 0
\(975\) 0 0
\(976\) 5.82932e7 1.95882
\(977\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 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
\(988\) 4.24328e7 1.38296
\(989\) 0 0
\(990\) 0 0
\(991\) 6.05528e7 1.95862 0.979310 − 0.202365i \(-0.0648626\pi\)
0.979310 + 0.202365i \(0.0648626\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 4.48154e7 1.42787 0.713937 − 0.700210i \(-0.246910\pi\)
0.713937 + 0.700210i \(0.246910\pi\)
\(998\) 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 225.6.a.e.1.1 yes 1
3.2 odd 2 CM 225.6.a.e.1.1 yes 1
5.2 odd 4 225.6.b.f.199.2 2
5.3 odd 4 225.6.b.f.199.1 2
5.4 even 2 225.6.a.d.1.1 ✓ 1
15.2 even 4 225.6.b.f.199.2 2
15.8 even 4 225.6.b.f.199.1 2
15.14 odd 2 225.6.a.d.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.6.a.d.1.1 ✓ 1 5.4 even 2
225.6.a.d.1.1 ✓ 1 15.14 odd 2
225.6.a.e.1.1 yes 1 1.1 even 1 trivial
225.6.a.e.1.1 yes 1 3.2 odd 2 CM
225.6.b.f.199.1 2 5.3 odd 4
225.6.b.f.199.1 2 15.8 even 4
225.6.b.f.199.2 2 5.2 odd 4
225.6.b.f.199.2 2 15.2 even 4