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Properties

Label 675.4.a.g.1.1
Level $675$
Weight $4$
Character 675.1
Self dual yes
Analytic conductor $39.826$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(39.8262892539\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \(q-8.00000 q^{4} +37.0000 q^{7} +O(q^{10})\) \(q-8.00000 q^{4} +37.0000 q^{7} +70.0000 q^{13} +64.0000 q^{16} -163.000 q^{19} -296.000 q^{28} -19.0000 q^{31} +433.000 q^{37} -449.000 q^{43} +1026.00 q^{49} -560.000 q^{52} +719.000 q^{61} -512.000 q^{64} +880.000 q^{67} +271.000 q^{73} +1304.00 q^{76} +503.000 q^{79} +2590.00 q^{91} +523.000 q^{97} +O(q^{100})\)

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
\(4\) −8.00000 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 37.0000 1.99781 0.998906 − 0.0467610i \(-0.0148899\pi\)
0.998906 + 0.0467610i \(0.0148899\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 70.0000 1.49342 0.746712 − 0.665148i \(-0.231631\pi\)
0.746712 + 0.665148i \(0.231631\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −163.000 −1.96815 −0.984073 − 0.177766i \(-0.943113\pi\)
−0.984073 + 0.177766i \(0.943113\pi\)
\(20\) 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
\(28\) −296.000 −1.99781
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −19.0000 −0.110081 −0.0550403 − 0.998484i \(-0.517529\pi\)
−0.0550403 + 0.998484i \(0.517529\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 433.000 1.92391 0.961956 − 0.273204i \(-0.0880833\pi\)
0.961956 + 0.273204i \(0.0880833\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −449.000 −1.59237 −0.796184 − 0.605054i \(-0.793151\pi\)
−0.796184 + 0.605054i \(0.793151\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1026.00 2.99125
\(50\) 0 0
\(51\) 0 0
\(52\) −560.000 −1.49342
\(53\) 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
\(61\) 719.000 1.50916 0.754578 − 0.656210i \(-0.227842\pi\)
0.754578 + 0.656210i \(0.227842\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 880.000 1.60461 0.802307 − 0.596912i \(-0.203606\pi\)
0.802307 + 0.596912i \(0.203606\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 271.000 0.434495 0.217248 − 0.976117i \(-0.430292\pi\)
0.217248 + 0.976117i \(0.430292\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 1304.00 1.96815
\(77\) 0 0
\(78\) 0 0
\(79\) 503.000 0.716353 0.358177 − 0.933654i \(-0.383399\pi\)
0.358177 + 0.933654i \(0.383399\pi\)
\(80\) 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
\(91\) 2590.00 2.98358
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 523.000 0.547450 0.273725 − 0.961808i \(-0.411744\pi\)
0.273725 + 0.961808i \(0.411744\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 19.0000 0.0181760 0.00908799 − 0.999959i \(-0.497107\pi\)
0.00908799 + 0.999959i \(0.497107\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 2213.00 1.94465 0.972325 − 0.233630i \(-0.0750606\pi\)
0.972325 + 0.233630i \(0.0750606\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2368.00 1.99781
\(113\) 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
\(121\) −1331.00 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 152.000 0.110081
\(125\) 0 0
\(126\) 0 0
\(127\) −380.000 −0.265508 −0.132754 − 0.991149i \(-0.542382\pi\)
−0.132754 + 0.991149i \(0.542382\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −6031.00 −3.93199
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −3043.00 −1.85686 −0.928431 − 0.371504i \(-0.878842\pi\)
−0.928431 + 0.371504i \(0.878842\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −3464.00 −1.92391
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 1961.00 1.05685 0.528424 − 0.848981i \(-0.322783\pi\)
0.528424 + 0.848981i \(0.322783\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1223.00 −0.621694 −0.310847 − 0.950460i \(-0.600613\pi\)
−0.310847 + 0.950460i \(0.600613\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3400.00 1.63379 0.816897 − 0.576783i \(-0.195692\pi\)
0.816897 + 0.576783i \(0.195692\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 2703.00 1.23031
\(170\) 0 0
\(171\) 0 0
\(172\) 3592.00 1.59237
\(173\) 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
\(181\) 3458.00 1.42006 0.710031 − 0.704171i \(-0.248681\pi\)
0.710031 + 0.704171i \(0.248681\pi\)
\(182\) 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
\(193\) 3961.00 1.47730 0.738650 − 0.674089i \(-0.235464\pi\)
0.738650 + 0.674089i \(0.235464\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −8208.00 −2.99125
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −5236.00 −1.86518 −0.932588 − 0.360942i \(-0.882455\pi\)
−0.932588 + 0.360942i \(0.882455\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 4480.00 1.49342
\(209\) 0 0
\(210\) 0 0
\(211\) 6032.00 1.96806 0.984028 − 0.178011i \(-0.0569664\pi\)
0.984028 + 0.178011i \(0.0569664\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −703.000 −0.219921
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 3439.00 1.03270 0.516351 − 0.856377i \(-0.327290\pi\)
0.516351 + 0.856377i \(0.327290\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 2357.00 0.680153 0.340076 − 0.940398i \(-0.389547\pi\)
0.340076 + 0.940398i \(0.389547\pi\)
\(230\) 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
\(241\) −7378.00 −1.97203 −0.986014 − 0.166662i \(-0.946701\pi\)
−0.986014 + 0.166662i \(0.946701\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −5752.00 −1.50916
\(245\) 0 0
\(246\) 0 0
\(247\) −11410.0 −2.93927
\(248\) 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
\(256\) 4096.00 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 16021.0 3.84362
\(260\) 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
\(268\) −7040.00 −1.60461
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −8101.00 −1.81587 −0.907935 − 0.419111i \(-0.862342\pi\)
−0.907935 + 0.419111i \(0.862342\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −9197.00 −1.99492 −0.997462 − 0.0711951i \(-0.977319\pi\)
−0.997462 + 0.0711951i \(0.977319\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 9469.00 1.98895 0.994476 − 0.104961i \(-0.0334717\pi\)
0.994476 + 0.104961i \(0.0334717\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4913.00 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −2168.00 −0.434495
\(293\) 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
\(301\) −16613.0 −3.18125
\(302\) 0 0
\(303\) 0 0
\(304\) −10432.0 −1.96815
\(305\) 0 0
\(306\) 0 0
\(307\) 6697.00 1.24501 0.622505 − 0.782616i \(-0.286115\pi\)
0.622505 + 0.782616i \(0.286115\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −10010.0 −1.80766 −0.903832 − 0.427888i \(-0.859258\pi\)
−0.903832 + 0.427888i \(0.859258\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −4024.00 −0.716353
\(317\) 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
\(331\) −10891.0 −1.80853 −0.904265 − 0.426971i \(-0.859580\pi\)
−0.904265 + 0.426971i \(0.859580\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4930.00 0.796897 0.398448 − 0.917191i \(-0.369549\pi\)
0.398448 + 0.917191i \(0.369549\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 25271.0 3.97815
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 1367.00 0.209667 0.104834 − 0.994490i \(-0.466569\pi\)
0.104834 + 0.994490i \(0.466569\pi\)
\(350\) 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
\(361\) 19710.0 2.87360
\(362\) 0 0
\(363\) 0 0
\(364\) −20720.0 −2.98358
\(365\) 0 0
\(366\) 0 0
\(367\) −4340.00 −0.617292 −0.308646 − 0.951177i \(-0.599876\pi\)
−0.308646 + 0.951177i \(0.599876\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −251.000 −0.0348426 −0.0174213 − 0.999848i \(-0.505546\pi\)
−0.0174213 + 0.999848i \(0.505546\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −8584.00 −1.16340 −0.581702 − 0.813402i \(-0.697613\pi\)
−0.581702 + 0.813402i \(0.697613\pi\)
\(380\) 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
\(388\) −4184.00 −0.547450
\(389\) 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
\(397\) −13067.0 −1.65192 −0.825962 − 0.563726i \(-0.809368\pi\)
−0.825962 + 0.563726i \(0.809368\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −1330.00 −0.164397
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 8246.00 0.996916 0.498458 − 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −152.000 −0.0181760
\(413\) 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
\(421\) −6679.00 −0.773194 −0.386597 − 0.922249i \(-0.626350\pi\)
−0.386597 + 0.922249i \(0.626350\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 26603.0 3.01501
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 14149.0 1.57034 0.785170 − 0.619280i \(-0.212575\pi\)
0.785170 + 0.619280i \(0.212575\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −17704.0 −1.94465
\(437\) 0 0
\(438\) 0 0
\(439\) 1853.00 0.201455 0.100728 − 0.994914i \(-0.467883\pi\)
0.100728 + 0.994914i \(0.467883\pi\)
\(440\) 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
\(448\) −18944.0 −1.99781
\(449\) 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
\(457\) −12710.0 −1.30098 −0.650491 − 0.759514i \(-0.725437\pi\)
−0.650491 + 0.759514i \(0.725437\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −11969.0 −1.20140 −0.600698 − 0.799476i \(-0.705111\pi\)
−0.600698 + 0.799476i \(0.705111\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 32560.0 3.20572
\(470\) 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
\(481\) 30310.0 2.87322
\(482\) 0 0
\(483\) 0 0
\(484\) 10648.0 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −20900.0 −1.94470 −0.972351 − 0.233526i \(-0.924974\pi\)
−0.972351 + 0.233526i \(0.924974\pi\)
\(488\) 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
\(496\) −1216.00 −0.110081
\(497\) 0 0
\(498\) 0 0
\(499\) 21743.0 1.95060 0.975301 − 0.220880i \(-0.0708930\pi\)
0.975301 + 0.220880i \(0.0708930\pi\)
\(500\) 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
\(508\) 3040.00 0.265508
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 10027.0 0.868040
\(512\) 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
\(523\) −23921.0 −1.99999 −0.999993 − 0.00383755i \(-0.998778\pi\)
−0.999993 + 0.00383755i \(0.998778\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 48248.0 3.93199
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 20789.0 1.65211 0.826053 − 0.563593i \(-0.190581\pi\)
0.826053 + 0.563593i \(0.190581\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −21293.0 −1.66439 −0.832196 − 0.554481i \(-0.812917\pi\)
−0.832196 + 0.554481i \(0.812917\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 18611.0 1.43114
\(554\) 0 0
\(555\) 0 0
\(556\) 24344.0 1.85686
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −31430.0 −2.37808
\(560\) 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
\(571\) 629.000 0.0460995 0.0230498 − 0.999734i \(-0.492662\pi\)
0.0230498 + 0.999734i \(0.492662\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 9613.00 0.693578 0.346789 − 0.937943i \(-0.387272\pi\)
0.346789 + 0.937943i \(0.387272\pi\)
\(578\) 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
\(589\) 3097.00 0.216655
\(590\) 0 0
\(591\) 0 0
\(592\) 27712.0 1.92391
\(593\) 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
\(601\) 17351.0 1.17764 0.588820 − 0.808264i \(-0.299593\pi\)
0.588820 + 0.808264i \(0.299593\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −15688.0 −1.05685
\(605\) 0 0
\(606\) 0 0
\(607\) −6137.00 −0.410368 −0.205184 − 0.978723i \(-0.565779\pi\)
−0.205184 + 0.978723i \(0.565779\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −12851.0 −0.846732 −0.423366 − 0.905959i \(-0.639152\pi\)
−0.423366 + 0.905959i \(0.639152\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 26693.0 1.73325 0.866625 − 0.498959i \(-0.166284\pi\)
0.866625 + 0.498959i \(0.166284\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 9784.00 0.621694
\(629\) 0 0
\(630\) 0 0
\(631\) 1892.00 0.119365 0.0596825 − 0.998217i \(-0.480991\pi\)
0.0596825 + 0.998217i \(0.480991\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 71820.0 4.46721
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −13160.0 −0.807122 −0.403561 − 0.914953i \(-0.632228\pi\)
−0.403561 + 0.914953i \(0.632228\pi\)
\(644\) 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
\(652\) −27200.0 −1.63379
\(653\) 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
\(661\) 33731.0 1.98485 0.992423 − 0.122864i \(-0.0392080\pi\)
0.992423 + 0.122864i \(0.0392080\pi\)
\(662\) 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
\(673\) −9899.00 −0.566981 −0.283491 − 0.958975i \(-0.591492\pi\)
−0.283491 + 0.958975i \(0.591492\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −21624.0 −1.23031
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 19351.0 1.09370
\(680\) 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
\(688\) −28736.0 −1.59237
\(689\) 0 0
\(690\) 0 0
\(691\) −16072.0 −0.884816 −0.442408 − 0.896814i \(-0.645876\pi\)
−0.442408 + 0.896814i \(0.645876\pi\)
\(692\) 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
\(703\) −70579.0 −3.78654
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −27523.0 −1.45790 −0.728948 − 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(710\) 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
\(721\) 703.000 0.0363122
\(722\) 0 0
\(723\) 0 0
\(724\) −27664.0 −1.42006
\(725\) 0 0
\(726\) 0 0
\(727\) −38033.0 −1.94026 −0.970128 − 0.242594i \(-0.922002\pi\)
−0.970128 + 0.242594i \(0.922002\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −15050.0 −0.758369 −0.379184 − 0.925321i \(-0.623795\pi\)
−0.379184 + 0.925321i \(0.623795\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 31376.0 1.56182 0.780910 − 0.624644i \(-0.214756\pi\)
0.780910 + 0.624644i \(0.214756\pi\)
\(740\) 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
\(751\) −17569.0 −0.853664 −0.426832 − 0.904331i \(-0.640370\pi\)
−0.426832 + 0.904331i \(0.640370\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −17333.0 −0.832204 −0.416102 − 0.909318i \(-0.636604\pi\)
−0.416102 + 0.909318i \(0.636604\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 81881.0 3.88505
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −4606.00 −0.215990 −0.107995 − 0.994151i \(-0.534443\pi\)
−0.107995 + 0.994151i \(0.534443\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −31688.0 −1.47730
\(773\) 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
\(784\) 65664.0 2.99125
\(785\) 0 0
\(786\) 0 0
\(787\) 28747.0 1.30206 0.651029 − 0.759053i \(-0.274338\pi\)
0.651029 + 0.759053i \(0.274338\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 50330.0 2.25381
\(794\) 0 0
\(795\) 0 0
\(796\) 41888.0 1.86518
\(797\) 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
\(811\) −40609.0 −1.75829 −0.879146 − 0.476552i \(-0.841886\pi\)
−0.879146 + 0.476552i \(0.841886\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 73187.0 3.13401
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 12220.0 0.517573 0.258786 − 0.965935i \(-0.416677\pi\)
0.258786 + 0.965935i \(0.416677\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 30077.0 1.26009 0.630047 − 0.776557i \(-0.283036\pi\)
0.630047 + 0.776557i \(0.283036\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −35840.0 −1.49342
\(833\) 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
\(841\) −24389.0 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −48256.0 −1.96806
\(845\) 0 0
\(846\) 0 0
\(847\) −49247.0 −1.99781
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 46690.0 1.87413 0.937066 − 0.349151i \(-0.113530\pi\)
0.937066 + 0.349151i \(0.113530\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −49807.0 −1.97834 −0.989169 − 0.146779i \(-0.953109\pi\)
−0.989169 + 0.146779i \(0.953109\pi\)
\(860\) 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
\(868\) 5624.00 0.219921
\(869\) 0 0
\(870\) 0 0
\(871\) 61600.0 2.39637
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13357.0 0.514292 0.257146 − 0.966373i \(-0.417218\pi\)
0.257146 + 0.966373i \(0.417218\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 31429.0 1.19781 0.598907 − 0.800818i \(-0.295602\pi\)
0.598907 + 0.800818i \(0.295602\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −14060.0 −0.530436
\(890\) 0 0
\(891\) 0 0
\(892\) −27512.0 −1.03270
\(893\) 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
\(907\) −4607.00 −0.168658 −0.0843291 − 0.996438i \(-0.526875\pi\)
−0.0843291 + 0.996438i \(0.526875\pi\)
\(908\) 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
\(916\) −18856.0 −0.680153
\(917\) 0 0
\(918\) 0 0
\(919\) −49573.0 −1.77939 −0.889697 − 0.456552i \(-0.849084\pi\)
−0.889697 + 0.456552i \(0.849084\pi\)
\(920\) 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
\(931\) −167238. −5.88722
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −40283.0 −1.40447 −0.702235 − 0.711945i \(-0.747814\pi\)
−0.702235 + 0.711945i \(0.747814\pi\)
\(938\) 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
\(949\) 18970.0 0.648885
\(950\) 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
\(961\) −29430.0 −0.987882
\(962\) 0 0
\(963\) 0 0
\(964\) 59024.0 1.97203
\(965\) 0 0
\(966\) 0 0
\(967\) −53927.0 −1.79336 −0.896678 − 0.442683i \(-0.854027\pi\)
−0.896678 + 0.442683i \(0.854027\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −112591. −3.70966
\(974\) 0 0
\(975\) 0 0
\(976\) 46016.0 1.50916
\(977\) 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
\(988\) 91280.0 2.93927
\(989\) 0 0
\(990\) 0 0
\(991\) −14041.0 −0.450078 −0.225039 − 0.974350i \(-0.572251\pi\)
−0.225039 + 0.974350i \(0.572251\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −28910.0 −0.918344 −0.459172 − 0.888347i \(-0.651854\pi\)
−0.459172 + 0.888347i \(0.651854\pi\)
\(998\) 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 675.4.a.g.1.1 yes 1
3.2 odd 2 CM 675.4.a.g.1.1 yes 1
5.2 odd 4 675.4.b.h.649.2 2
5.3 odd 4 675.4.b.h.649.1 2
5.4 even 2 675.4.a.d.1.1 ✓ 1
15.2 even 4 675.4.b.h.649.2 2
15.8 even 4 675.4.b.h.649.1 2
15.14 odd 2 675.4.a.d.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
675.4.a.d.1.1 ✓ 1 5.4 even 2
675.4.a.d.1.1 ✓ 1 15.14 odd 2
675.4.a.g.1.1 yes 1 1.1 even 1 trivial
675.4.a.g.1.1 yes 1 3.2 odd 2 CM
675.4.b.h.649.1 2 5.3 odd 4
675.4.b.h.649.1 2 15.8 even 4
675.4.b.h.649.2 2 5.2 odd 4
675.4.b.h.649.2 2 15.2 even 4