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Properties

Label 675.6.a.c.1.1
Level $675$
Weight $6$
Character 675.1
Self dual yes
Analytic conductor $108.259$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,6,Mod(1,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 675.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(108.259078374\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 675.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-32.0000 q^{4} +211.000 q^{7} +775.000 q^{13} +1024.00 q^{16} +3143.00 q^{19} -6752.00 q^{28} -10324.0 q^{31} +9889.00 q^{37} +3352.00 q^{43} +27714.0 q^{49} -24800.0 q^{52} -18301.0 q^{61} -32768.0 q^{64} -73475.0 q^{67} +78127.0 q^{73} -100576. q^{76} +9707.00 q^{79} +163525. q^{91} +43339.0 q^{97} +O(q^{100})\)

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
\(7\) 211.000 1.62756 0.813781 − 0.581172i \(-0.197406\pi\)
0.813781 + 0.581172i \(0.197406\pi\)
\(8\) 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
\(19\) 3143.00 1.99738 0.998689 − 0.0511835i \(-0.0162993\pi\)
0.998689 + 0.0511835i \(0.0162993\pi\)
\(20\) 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
\(28\) −6752.00 −1.62756
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −10324.0 −1.92950 −0.964748 − 0.263176i \(-0.915230\pi\)
−0.964748 + 0.263176i \(0.915230\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9889.00 1.18754 0.593770 − 0.804635i \(-0.297639\pi\)
0.593770 + 0.804635i \(0.297639\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 3352.00 0.276460 0.138230 − 0.990400i \(-0.455859\pi\)
0.138230 + 0.990400i \(0.455859\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 27714.0 1.64896
\(50\) 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
\(61\) −18301.0 −0.629724 −0.314862 − 0.949137i \(-0.601958\pi\)
−0.314862 + 0.949137i \(0.601958\pi\)
\(62\) 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
\(73\) 78127.0 1.71591 0.857954 − 0.513727i \(-0.171735\pi\)
0.857954 + 0.513727i \(0.171735\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −100576. −1.99738
\(77\) 0 0
\(78\) 0 0
\(79\) 9707.00 0.174992 0.0874958 − 0.996165i \(-0.472114\pi\)
0.0874958 + 0.996165i \(0.472114\pi\)
\(80\) 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
\(91\) 163525. 2.07005
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 43339.0 0.467681 0.233840 − 0.972275i \(-0.424871\pi\)
0.233840 + 0.972275i \(0.424871\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −70577.0 −0.655496 −0.327748 − 0.944765i \(-0.606290\pi\)
−0.327748 + 0.944765i \(0.606290\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 114482. 0.922935 0.461467 − 0.887157i \(-0.347323\pi\)
0.461467 + 0.887157i \(0.347323\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 216064. 1.62756
\(113\) 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
\(124\) 330368. 1.92950
\(125\) 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
\(133\) 663173. 3.25086
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 202193. 0.887624 0.443812 − 0.896120i \(-0.353626\pi\)
0.443812 + 0.896120i \(0.353626\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −316448. −1.18754
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 536351. 1.91429 0.957143 − 0.289616i \(-0.0935277\pi\)
0.957143 + 0.289616i \(0.0935277\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −109214. −0.353614 −0.176807 − 0.984246i \(-0.556577\pi\)
−0.176807 + 0.984246i \(0.556577\pi\)
\(158\) 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
\(172\) −107264. −0.276460
\(173\) 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
\(193\) 1.02118e6 1.97337 0.986683 − 0.162653i \(-0.0520053\pi\)
0.986683 + 0.162653i \(0.0520053\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −886848. −1.64896
\(197\) 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
\(217\) −2.17836e6 −3.14037
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −304052. −0.409436 −0.204718 − 0.978821i \(-0.565628\pi\)
−0.204718 + 0.978821i \(0.565628\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 1.26982e6 1.60012 0.800060 − 0.599919i \(-0.204801\pi\)
0.800060 + 0.599919i \(0.204801\pi\)
\(230\) 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
\(244\) 585632. 0.629724
\(245\) 0 0
\(246\) 0 0
\(247\) 2.43582e6 2.54041
\(248\) 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
\(259\) 2.08658e6 1.93279
\(260\) 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
\(271\) −1.88700e6 −1.56081 −0.780403 − 0.625277i \(-0.784986\pi\)
−0.780403 + 0.625277i \(0.784986\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.48661e6 1.94719 0.973596 − 0.228276i \(-0.0733089\pi\)
0.973596 + 0.228276i \(0.0733089\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −1952.00 −0.00144882 −0.000724409 − 1.00000i \(-0.500231\pi\)
−0.000724409 1.00000i \(0.500231\pi\)
\(284\) 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
\(292\) −2.50006e6 −1.71591
\(293\) 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
\(301\) 707272. 0.449956
\(302\) 0 0
\(303\) 0 0
\(304\) 3.21843e6 1.99738
\(305\) 0 0
\(306\) 0 0
\(307\) 2.30114e6 1.39347 0.696733 − 0.717331i \(-0.254636\pi\)
0.696733 + 0.717331i \(0.254636\pi\)
\(308\) 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
\(316\) −310624. −0.174992
\(317\) 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
\(331\) −546151. −0.273995 −0.136998 − 0.990571i \(-0.543745\pi\)
−0.136998 + 0.990571i \(0.543745\pi\)
\(332\) 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
\(343\) 2.30138e6 1.05622
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −3.48766e6 −1.53275 −0.766373 − 0.642396i \(-0.777941\pi\)
−0.766373 + 0.642396i \(0.777941\pi\)
\(350\) 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
\(361\) 7.40235e6 2.98952
\(362\) 0 0
\(363\) 0 0
\(364\) −5.23280e6 −2.07005
\(365\) 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
\(373\) 5.31727e6 1.97887 0.989434 − 0.144983i \(-0.0463128\pi\)
0.989434 + 0.144983i \(0.0463128\pi\)
\(374\) 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
\(388\) −1.38685e6 −0.467681
\(389\) 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
\(397\) −5.78949e6 −1.84359 −0.921794 − 0.387681i \(-0.873276\pi\)
−0.921794 + 0.387681i \(0.873276\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −8.00110e6 −2.45407
\(404\) 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
\(412\) 2.25846e6 0.655496
\(413\) 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
\(421\) −4.87580e6 −1.34073 −0.670364 − 0.742033i \(-0.733862\pi\)
−0.670364 + 0.742033i \(0.733862\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −3.86151e6 −1.02491
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −1.85760e6 −0.476138 −0.238069 − 0.971248i \(-0.576514\pi\)
−0.238069 + 0.971248i \(0.576514\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.66342e6 −0.922935
\(437\) 0 0
\(438\) 0 0
\(439\) 3.99333e6 0.988950 0.494475 − 0.869192i \(-0.335360\pi\)
0.494475 + 0.869192i \(0.335360\pi\)
\(440\) 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
\(448\) −6.91405e6 −1.62756
\(449\) 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
\(463\) −2.92217e6 −0.633510 −0.316755 − 0.948507i \(-0.602593\pi\)
−0.316755 + 0.948507i \(0.602593\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −1.55032e7 −3.25454
\(470\) 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
\(481\) 7.66398e6 1.51040
\(482\) 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
\(496\) −1.05718e7 −1.92950
\(497\) 0 0
\(498\) 0 0
\(499\) −8.36157e6 −1.50327 −0.751634 − 0.659581i \(-0.770734\pi\)
−0.751634 + 0.659581i \(0.770734\pi\)
\(500\) 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
\(511\) 1.64848e7 2.79275
\(512\) 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
\(523\) −2.09363e6 −0.334692 −0.167346 − 0.985898i \(-0.553520\pi\)
−0.167346 + 0.985898i \(0.553520\pi\)
\(524\) 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
\(532\) −2.12215e7 −3.25086
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −3.02235e6 −0.443968 −0.221984 − 0.975050i \(-0.571253\pi\)
−0.221984 + 0.975050i \(0.571253\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.49331e6 0.213394 0.106697 − 0.994292i \(-0.465973\pi\)
0.106697 + 0.994292i \(0.465973\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 2.04818e6 0.284810
\(554\) 0 0
\(555\) 0 0
\(556\) −6.47018e6 −0.887624
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 2.59780e6 0.351622
\(560\) 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
\(571\) 5.88780e6 0.755723 0.377862 − 0.925862i \(-0.376660\pi\)
0.377862 + 0.925862i \(0.376660\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.46322e7 −1.82966 −0.914830 − 0.403839i \(-0.867676\pi\)
−0.914830 + 0.403839i \(0.867676\pi\)
\(578\) 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
\(589\) −3.24483e7 −3.85393
\(590\) 0 0
\(591\) 0 0
\(592\) 1.01263e7 1.18754
\(593\) 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
\(601\) 6.02163e6 0.680029 0.340015 − 0.940420i \(-0.389568\pi\)
0.340015 + 0.940420i \(0.389568\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.71632e7 −1.91429
\(605\) 0 0
\(606\) 0 0
\(607\) 1.56805e7 1.72739 0.863693 − 0.504019i \(-0.168146\pi\)
0.863693 + 0.504019i \(0.168146\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 566773. 0.0609197 0.0304599 − 0.999536i \(-0.490303\pi\)
0.0304599 + 0.999536i \(0.490303\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 1.87828e7 1.97031 0.985153 − 0.171676i \(-0.0549183\pi\)
0.985153 + 0.171676i \(0.0549183\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 3.49485e6 0.353614
\(629\) 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
\(637\) 2.14784e7 2.09726
\(638\) 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
\(661\) −690949. −0.0615095 −0.0307548 − 0.999527i \(-0.509791\pi\)
−0.0307548 + 0.999527i \(0.509791\pi\)
\(662\) 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
\(673\) −1.73115e7 −1.47332 −0.736661 − 0.676262i \(-0.763599\pi\)
−0.736661 + 0.676262i \(0.763599\pi\)
\(674\) 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
\(679\) 9.14453e6 0.761179
\(680\) 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
\(688\) 3.43245e6 0.276460
\(689\) 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
\(703\) 3.10811e7 2.37197
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.36370e7 1.76594 0.882971 − 0.469427i \(-0.155539\pi\)
0.882971 + 0.469427i \(0.155539\pi\)
\(710\) 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
\(721\) −1.48917e7 −1.06686
\(722\) 0 0
\(723\) 0 0
\(724\) 2.72969e7 1.93538
\(725\) 0 0
\(726\) 0 0
\(727\) 2.84512e7 1.99648 0.998240 − 0.0592978i \(-0.0188862\pi\)
0.998240 + 0.0592978i \(0.0188862\pi\)
\(728\) 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
\(751\) −2.73219e7 −1.76771 −0.883857 − 0.467758i \(-0.845062\pi\)
−0.883857 + 0.467758i \(0.845062\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.12909e7 −0.716124 −0.358062 − 0.933698i \(-0.616562\pi\)
−0.358062 + 0.933698i \(0.616562\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 2.41557e7 1.50213
\(764\) 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
\(772\) −3.26777e7 −1.97337
\(773\) 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
\(784\) 2.83791e7 1.64896
\(785\) 0 0
\(786\) 0 0
\(787\) 7.40376e6 0.426104 0.213052 − 0.977041i \(-0.431660\pi\)
0.213052 + 0.977041i \(0.431660\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.41833e7 −0.800928
\(794\) 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
\(811\) 2.27708e7 1.21570 0.607849 − 0.794053i \(-0.292033\pi\)
0.607849 + 0.794053i \(0.292033\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.05353e7 0.552196
\(818\) 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
\(829\) −2.26485e7 −1.14460 −0.572299 − 0.820045i \(-0.693948\pi\)
−0.572299 + 0.820045i \(0.693948\pi\)
\(830\) 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
\(847\) −3.39818e7 −1.62756
\(848\) 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
\(859\) −3.88974e7 −1.79861 −0.899307 − 0.437317i \(-0.855929\pi\)
−0.899307 + 0.437317i \(0.855929\pi\)
\(860\) 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
\(868\) 6.97076e7 3.14037
\(869\) 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
\(877\) −4.12304e7 −1.81017 −0.905084 − 0.425232i \(-0.860192\pi\)
−0.905084 + 0.425232i \(0.860192\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 4.58066e7 1.97709 0.988545 − 0.150925i \(-0.0482253\pi\)
0.988545 + 0.150925i \(0.0482253\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 5.63581e7 2.39167
\(890\) 0 0
\(891\) 0 0
\(892\) 9.72966e6 0.409436
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.95222e7 1.99886 0.999428 − 0.0338109i \(-0.0107644\pi\)
0.999428 + 0.0338109i \(0.0107644\pi\)
\(908\) 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
\(916\) −4.06342e7 −1.60012
\(917\) 0 0
\(918\) 0 0
\(919\) 4.63039e7 1.80854 0.904271 − 0.426959i \(-0.140415\pi\)
0.904271 + 0.426959i \(0.140415\pi\)
\(920\) 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
\(931\) 8.71051e7 3.29359
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.89709e7 1.82217 0.911085 − 0.412219i \(-0.135246\pi\)
0.911085 + 0.412219i \(0.135246\pi\)
\(938\) 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
\(949\) 6.05484e7 2.18241
\(950\) 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
\(961\) 7.79558e7 2.72295
\(962\) 0 0
\(963\) 0 0
\(964\) −1.39705e7 −0.484193
\(965\) 0 0
\(966\) 0 0
\(967\) 2.56836e7 0.883262 0.441631 − 0.897197i \(-0.354400\pi\)
0.441631 + 0.897197i \(0.354400\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 4.26627e7 1.44466
\(974\) 0 0
\(975\) 0 0
\(976\) −1.87402e7 −0.629724
\(977\) 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
\(988\) −7.79464e7 −2.54041
\(989\) 0 0
\(990\) 0 0
\(991\) −1.94402e7 −0.628804 −0.314402 − 0.949290i \(-0.601804\pi\)
−0.314402 + 0.949290i \(0.601804\pi\)
\(992\) 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 675.6.a.c.1.1 1
3.2 odd 2 CM 675.6.a.c.1.1 1
5.4 even 2 27.6.a.a.1.1 ✓ 1
15.14 odd 2 27.6.a.a.1.1 ✓ 1
20.19 odd 2 432.6.a.f.1.1 1
45.4 even 6 81.6.c.b.55.1 2
45.14 odd 6 81.6.c.b.55.1 2
45.29 odd 6 81.6.c.b.28.1 2
45.34 even 6 81.6.c.b.28.1 2
60.59 even 2 432.6.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.6.a.a.1.1 ✓ 1 5.4 even 2
27.6.a.a.1.1 ✓ 1 15.14 odd 2
81.6.c.b.28.1 2 45.29 odd 6
81.6.c.b.28.1 2 45.34 even 6
81.6.c.b.55.1 2 45.4 even 6
81.6.c.b.55.1 2 45.14 odd 6
432.6.a.f.1.1 1 20.19 odd 2
432.6.a.f.1.1 1 60.59 even 2
675.6.a.c.1.1 1 1.1 even 1 trivial
675.6.a.c.1.1 1 3.2 odd 2 CM