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Properties

Label 675.6.a.d.1.1
Level $675$
Weight $6$
Character 675.1
Self dual yes
Analytic conductor $108.259$
Analytic rank $1$
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: \(1\)
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 \(\chi\) \(=\) 675.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-32.0000 q^{4} +236.000 q^{7} +O(q^{10})\) \(q-32.0000 q^{4} +236.000 q^{7} -775.000 q^{13} +1024.00 q^{16} -1432.00 q^{19} -7552.00 q^{28} +7601.00 q^{31} -6661.00 q^{37} -19123.0 q^{43} +38889.0 q^{49} +24800.0 q^{52} -38626.0 q^{61} -32768.0 q^{64} +73475.0 q^{67} +79577.0 q^{73} +45824.0 q^{76} +90857.0 q^{79} -182900. q^{91} -134386. q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(7\) 236.000 1.82040 0.910200 − 0.414169i \(-0.135928\pi\)
0.910200 + 0.414169i \(0.135928\pi\)
\(8\) 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 \(-0.719386\pi\)
−0.635936 + 0.771742i \(0.719386\pi\)
\(14\) 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
\(19\) −1432.00 −0.910037 −0.455018 − 0.890482i \(-0.650367\pi\)
−0.455018 + 0.890482i \(0.650367\pi\)
\(20\) 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
\(28\) −7552.00 −1.82040
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 7601.00 1.42058 0.710291 − 0.703908i \(-0.248563\pi\)
0.710291 + 0.703908i \(0.248563\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6661.00 −0.799899 −0.399949 − 0.916537i \(-0.630972\pi\)
−0.399949 + 0.916537i \(0.630972\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −19123.0 −1.57719 −0.788597 − 0.614911i \(-0.789192\pi\)
−0.788597 + 0.614911i \(0.789192\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 38889.0 2.31386
\(50\) 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
\(61\) −38626.0 −1.32909 −0.664546 − 0.747247i \(-0.731375\pi\)
−0.664546 + 0.747247i \(0.731375\pi\)
\(62\) 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 \(-0.00600969\pi\)
0.999822 + 0.0188789i \(0.00600969\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 79577.0 1.74775 0.873877 − 0.486147i \(-0.161598\pi\)
0.873877 + 0.486147i \(0.161598\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 45824.0 0.910037
\(77\) 0 0
\(78\) 0 0
\(79\) 90857.0 1.63791 0.818956 − 0.573856i \(-0.194553\pi\)
0.818956 + 0.573856i \(0.194553\pi\)
\(80\) 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
\(91\) −182900. −2.31532
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −134386. −1.45019 −0.725095 − 0.688649i \(-0.758204\pi\)
−0.725095 + 0.688649i \(0.758204\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −211477. −1.96413 −0.982065 − 0.188544i \(-0.939623\pi\)
−0.982065 + 0.188544i \(0.939623\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −247843. −1.99807 −0.999034 − 0.0439362i \(-0.986010\pi\)
−0.999034 + 0.0439362i \(0.986010\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 241664. 1.82040
\(113\) 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
\(124\) −243232. −1.42058
\(125\) 0 0
\(126\) 0 0
\(127\) −267100. −1.46948 −0.734742 − 0.678347i \(-0.762697\pi\)
−0.734742 + 0.678347i \(0.762697\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −337952. −1.65663
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −454657. −1.99594 −0.997969 − 0.0637074i \(-0.979708\pi\)
−0.997969 + 0.0637074i \(0.979708\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 213152. 0.799899
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −408724. −1.45877 −0.729387 − 0.684102i \(-0.760194\pi\)
−0.729387 + 0.684102i \(0.760194\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 471911. 1.52796 0.763978 − 0.645242i \(-0.223243\pi\)
0.763978 + 0.645242i \(0.223243\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −352375. −1.03881 −0.519405 − 0.854528i \(-0.673846\pi\)
−0.519405 + 0.854528i \(0.673846\pi\)
\(164\) 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
\(172\) 611936. 1.57719
\(173\) 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
\(193\) 364802. 0.704959 0.352480 − 0.935820i \(-0.385339\pi\)
0.352480 + 0.935820i \(0.385339\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.24445e6 −2.31386
\(197\) 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
\(217\) 1.79384e6 2.58603
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.41103e6 −1.90009 −0.950043 − 0.312120i \(-0.898961\pi\)
−0.950043 + 0.312120i \(0.898961\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −1.45951e6 −1.83915 −0.919576 − 0.392913i \(-0.871467\pi\)
−0.919576 + 0.392913i \(0.871467\pi\)
\(230\) 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
\(244\) 1.23603e6 1.32909
\(245\) 0 0
\(246\) 0 0
\(247\) 1.10980e6 1.15745
\(248\) 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
\(259\) −1.57200e6 −1.45614
\(260\) 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
\(271\) −365851. −0.302608 −0.151304 − 0.988487i \(-0.548347\pi\)
−0.151304 + 0.988487i \(0.548347\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 738389. 0.578210 0.289105 − 0.957297i \(-0.406642\pi\)
0.289105 + 0.957297i \(0.406642\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −2.33458e6 −1.73277 −0.866387 − 0.499373i \(-0.833564\pi\)
−0.866387 + 0.499373i \(0.833564\pi\)
\(284\) 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
\(292\) −2.54646e6 −1.74775
\(293\) 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
\(301\) −4.51303e6 −2.87112
\(302\) 0 0
\(303\) 0 0
\(304\) −1.46637e6 −0.910037
\(305\) 0 0
\(306\) 0 0
\(307\) −901189. −0.545720 −0.272860 − 0.962054i \(-0.587970\pi\)
−0.272860 + 0.962054i \(0.587970\pi\)
\(308\) 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 \(-0.234570\pi\)
0.740539 + 0.672014i \(0.234570\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −2.90742e6 −1.63791
\(317\) 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
\(331\) 3.69300e6 1.85272 0.926359 − 0.376642i \(-0.122921\pi\)
0.926359 + 0.376642i \(0.122921\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.63172e6 1.26231 0.631155 − 0.775657i \(-0.282581\pi\)
0.631155 + 0.775657i \(0.282581\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 5.21135e6 2.39175
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −787957. −0.346289 −0.173145 − 0.984896i \(-0.555393\pi\)
−0.173145 + 0.984896i \(0.555393\pi\)
\(350\) 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
\(361\) −425475. −0.171833
\(362\) 0 0
\(363\) 0 0
\(364\) 5.85280e6 2.31532
\(365\) 0 0
\(366\) 0 0
\(367\) −2.58318e6 −1.00113 −0.500563 − 0.865700i \(-0.666874\pi\)
−0.500563 + 0.865700i \(0.666874\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 3.33340e6 1.24055 0.620276 − 0.784384i \(-0.287021\pi\)
0.620276 + 0.784384i \(0.287021\pi\)
\(374\) 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
\(388\) 4.30035e6 1.45019
\(389\) 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
\(397\) −786061. −0.250311 −0.125156 − 0.992137i \(-0.539943\pi\)
−0.125156 + 0.992137i \(0.539943\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −5.89078e6 −1.80680
\(404\) 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
\(412\) 6.76726e6 1.96413
\(413\) 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
\(421\) 7.11190e6 1.95560 0.977801 − 0.209536i \(-0.0671953\pi\)
0.977801 + 0.209536i \(0.0671953\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −9.11574e6 −2.41948
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 5.63432e6 1.44418 0.722091 − 0.691798i \(-0.243181\pi\)
0.722091 + 0.691798i \(0.243181\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7.93098e6 1.99807
\(437\) 0 0
\(438\) 0 0
\(439\) −8.07574e6 −1.99996 −0.999980 − 0.00636830i \(-0.997973\pi\)
−0.999980 + 0.00636830i \(0.997973\pi\)
\(440\) 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
\(448\) −7.73325e6 −1.82040
\(449\) 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 \(-0.875728\pi\)
−0.924752 + 0.380569i \(0.875728\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −9.03907e6 −1.95962 −0.979809 − 0.199936i \(-0.935927\pi\)
−0.979809 + 0.199936i \(0.935927\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 1.73401e7 3.64015
\(470\) 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
\(481\) 5.16228e6 1.01737
\(482\) 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 \(-0.541623\pi\)
−0.130389 + 0.991463i \(0.541623\pi\)
\(488\) 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
\(496\) 7.78342e6 1.42058
\(497\) 0 0
\(498\) 0 0
\(499\) −2.17369e6 −0.390793 −0.195397 − 0.980724i \(-0.562599\pi\)
−0.195397 + 0.980724i \(0.562599\pi\)
\(500\) 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
\(511\) 1.87802e7 3.18161
\(512\) 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
\(523\) 9.63505e6 1.54028 0.770140 − 0.637875i \(-0.220186\pi\)
0.770140 + 0.637875i \(0.220186\pi\)
\(524\) 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
\(532\) 1.08145e7 1.65663
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −9.98573e6 −1.46685 −0.733426 − 0.679769i \(-0.762080\pi\)
−0.733426 + 0.679769i \(0.762080\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.13049e7 −1.61547 −0.807733 − 0.589548i \(-0.799306\pi\)
−0.807733 + 0.589548i \(0.799306\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 2.14423e7 2.98166
\(554\) 0 0
\(555\) 0 0
\(556\) 1.45490e7 1.99594
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 1.48203e7 2.00599
\(560\) 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
\(571\) −1.54378e7 −1.98150 −0.990751 − 0.135693i \(-0.956674\pi\)
−0.990751 + 0.135693i \(0.956674\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.72229e6 −0.215360 −0.107680 − 0.994186i \(-0.534342\pi\)
−0.107680 + 0.994186i \(0.534342\pi\)
\(578\) 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
\(589\) −1.08846e7 −1.29278
\(590\) 0 0
\(591\) 0 0
\(592\) −6.82086e6 −0.799899
\(593\) 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
\(601\) −1.74342e7 −1.96887 −0.984435 − 0.175749i \(-0.943765\pi\)
−0.984435 + 0.175749i \(0.943765\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.30792e7 1.45877
\(605\) 0 0
\(606\) 0 0
\(607\) −84361.0 −0.00929330 −0.00464665 − 0.999989i \(-0.501479\pi\)
−0.00464665 + 0.999989i \(0.501479\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.63902e7 1.76171 0.880853 − 0.473389i \(-0.156969\pi\)
0.880853 + 0.473389i \(0.156969\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −6.55677e6 −0.687802 −0.343901 − 0.939006i \(-0.611748\pi\)
−0.343901 + 0.939006i \(0.611748\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) −1.51012e7 −1.52796
\(629\) 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
\(637\) −3.01390e7 −2.94293
\(638\) 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 \(-0.0537129\pi\)
0.985796 + 0.167944i \(0.0537129\pi\)
\(644\) 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
\(661\) 1.97928e7 1.76199 0.880993 − 0.473129i \(-0.156876\pi\)
0.880993 + 0.473129i \(0.156876\pi\)
\(662\) 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
\(673\) −2.24188e7 −1.90798 −0.953991 − 0.299836i \(-0.903068\pi\)
−0.953991 + 0.299836i \(0.903068\pi\)
\(674\) 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
\(679\) −3.17151e7 −2.63993
\(680\) 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
\(688\) −1.95820e7 −1.57719
\(689\) 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
\(703\) 9.53855e6 0.727938
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.27014e7 −1.69604 −0.848021 − 0.529962i \(-0.822206\pi\)
−0.848021 + 0.529962i \(0.822206\pi\)
\(710\) 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
\(721\) −4.99086e7 −3.57550
\(722\) 0 0
\(723\) 0 0
\(724\) 2.72969e7 1.93538
\(725\) 0 0
\(726\) 0 0
\(727\) 1.56893e7 1.10095 0.550474 − 0.834853i \(-0.314447\pi\)
0.550474 + 0.834853i \(0.314447\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1.13028e7 −0.777006 −0.388503 − 0.921448i \(-0.627008\pi\)
−0.388503 + 0.921448i \(0.627008\pi\)
\(734\) 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
\(751\) 2.61832e7 1.69404 0.847019 − 0.531563i \(-0.178395\pi\)
0.847019 + 0.531563i \(0.178395\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.98526e7 1.25915 0.629575 − 0.776940i \(-0.283229\pi\)
0.629575 + 0.776940i \(0.283229\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −5.84909e7 −3.63728
\(764\) 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
\(772\) −1.16737e7 −0.704959
\(773\) 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
\(784\) 3.98223e7 2.31386
\(785\) 0 0
\(786\) 0 0
\(787\) 3.31061e7 1.90534 0.952668 − 0.304012i \(-0.0983263\pi\)
0.952668 + 0.304012i \(0.0983263\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.99352e7 1.69043
\(794\) 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
\(811\) −3.71463e7 −1.98319 −0.991594 − 0.129386i \(-0.958699\pi\)
−0.991594 + 0.129386i \(0.958699\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.73841e7 1.43530
\(818\) 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 \(-0.361129\pi\)
0.422567 + 0.906332i \(0.361129\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −1.67808e7 −0.848060 −0.424030 − 0.905648i \(-0.639385\pi\)
−0.424030 + 0.905648i \(0.639385\pi\)
\(830\) 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
\(847\) −3.80080e7 −1.82040
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −3.00842e6 −0.141568 −0.0707842 − 0.997492i \(-0.522550\pi\)
−0.0707842 + 0.997492i \(0.522550\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 3.58297e7 1.65676 0.828381 − 0.560164i \(-0.189262\pi\)
0.828381 + 0.560164i \(0.189262\pi\)
\(860\) 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
\(868\) −5.74028e7 −2.58603
\(869\) 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
\(877\) −3.83931e6 −0.168560 −0.0842800 − 0.996442i \(-0.526859\pi\)
−0.0842800 + 0.996442i \(0.526859\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 2.89598e7 1.24996 0.624978 − 0.780643i \(-0.285108\pi\)
0.624978 + 0.780643i \(0.285108\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −6.30356e7 −2.67505
\(890\) 0 0
\(891\) 0 0
\(892\) 4.51529e7 1.90009
\(893\) 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
\(907\) 2.62120e7 1.05799 0.528995 − 0.848625i \(-0.322569\pi\)
0.528995 + 0.848625i \(0.322569\pi\)
\(908\) 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
\(916\) 4.67042e7 1.83915
\(917\) 0 0
\(918\) 0 0
\(919\) −4.20857e7 −1.64379 −0.821893 − 0.569642i \(-0.807082\pi\)
−0.821893 + 0.569642i \(0.807082\pi\)
\(920\) 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
\(931\) −5.56890e7 −2.10570
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.36738e7 1.62507 0.812535 − 0.582913i \(-0.198087\pi\)
0.812535 + 0.582913i \(0.198087\pi\)
\(938\) 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
\(949\) −6.16722e7 −2.22292
\(950\) 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
\(961\) 2.91461e7 1.01805
\(962\) 0 0
\(963\) 0 0
\(964\) −1.39705e7 −0.484193
\(965\) 0 0
\(966\) 0 0
\(967\) −3.23453e7 −1.11236 −0.556180 − 0.831062i \(-0.687733\pi\)
−0.556180 + 0.831062i \(0.687733\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −1.07299e8 −3.63340
\(974\) 0 0
\(975\) 0 0
\(976\) −3.95530e7 −1.32909
\(977\) 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
\(988\) −3.55136e7 −1.15745
\(989\) 0 0
\(990\) 0 0
\(991\) −4.11127e7 −1.32982 −0.664908 − 0.746925i \(-0.731529\pi\)
−0.664908 + 0.746925i \(0.731529\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −4.48154e7 −1.42787 −0.713937 − 0.700210i \(-0.753090\pi\)
−0.713937 + 0.700210i \(0.753090\pi\)
\(998\) 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.d.1.1 yes 1
3.2 odd 2 CM 675.6.a.d.1.1 yes 1
5.4 even 2 675.6.a.b.1.1 ✓ 1
15.14 odd 2 675.6.a.b.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
675.6.a.b.1.1 ✓ 1 5.4 even 2
675.6.a.b.1.1 ✓ 1 15.14 odd 2
675.6.a.d.1.1 yes 1 1.1 even 1 trivial
675.6.a.d.1.1 yes 1 3.2 odd 2 CM