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Properties

Label 675.4.a.e.1.1
Level $675$
Weight $4$
Character 675.1
Self dual yes
Analytic conductor $39.826$
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] = [675,4,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: \(1\)
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)
 
\(f(q)\) \(=\) \(q-8.00000 q^{4} -17.0000 q^{7} +O(q^{10})\) \(q-8.00000 q^{4} -17.0000 q^{7} +70.0000 q^{13} +64.0000 q^{16} +107.000 q^{19} +136.000 q^{28} -289.000 q^{31} -323.000 q^{37} -71.0000 q^{43} -54.0000 q^{49} -560.000 q^{52} -901.000 q^{61} -512.000 q^{64} +880.000 q^{67} +919.000 q^{73} -856.000 q^{76} -1387.00 q^{79} -1190.00 q^{91} -1853.00 q^{97} +O(q^{100})\)

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
\(7\) −17.0000 −0.917914 −0.458957 − 0.888459i \(-0.651777\pi\)
−0.458957 + 0.888459i \(0.651777\pi\)
\(8\) 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
\(19\) 107.000 1.29197 0.645986 − 0.763349i \(-0.276446\pi\)
0.645986 + 0.763349i \(0.276446\pi\)
\(20\) 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
\(28\) 136.000 0.917914
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −289.000 −1.67438 −0.837192 − 0.546908i \(-0.815805\pi\)
−0.837192 + 0.546908i \(0.815805\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −323.000 −1.43516 −0.717579 − 0.696477i \(-0.754750\pi\)
−0.717579 + 0.696477i \(0.754750\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −71.0000 −0.251800 −0.125900 − 0.992043i \(-0.540182\pi\)
−0.125900 + 0.992043i \(0.540182\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −54.0000 −0.157434
\(50\) 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
\(61\) −901.000 −1.89117 −0.945584 − 0.325379i \(-0.894508\pi\)
−0.945584 + 0.325379i \(0.894508\pi\)
\(62\) 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
\(73\) 919.000 1.47344 0.736718 − 0.676200i \(-0.236375\pi\)
0.736718 + 0.676200i \(0.236375\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −856.000 −1.29197
\(77\) 0 0
\(78\) 0 0
\(79\) −1387.00 −1.97531 −0.987656 − 0.156637i \(-0.949935\pi\)
−0.987656 + 0.156637i \(0.949935\pi\)
\(80\) 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
\(91\) −1190.00 −1.37083
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1853.00 −1.93963 −0.969813 − 0.243851i \(-0.921589\pi\)
−0.969813 + 0.243851i \(0.921589\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 1801.00 1.72289 0.861446 − 0.507850i \(-0.169560\pi\)
0.861446 + 0.507850i \(0.169560\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −1567.00 −1.37699 −0.688493 − 0.725243i \(-0.741727\pi\)
−0.688493 + 0.725243i \(0.741727\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1088.00 −0.917914
\(113\) 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
\(124\) 2312.00 1.67438
\(125\) 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
\(133\) −1819.00 −1.18592
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 467.000 0.284967 0.142484 − 0.989797i \(-0.454491\pi\)
0.142484 + 0.989797i \(0.454491\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 2584.00 1.43516
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −3709.00 −1.99890 −0.999451 − 0.0331378i \(-0.989450\pi\)
−0.999451 + 0.0331378i \(0.989450\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2627.00 −1.33540 −0.667699 − 0.744432i \(-0.732721\pi\)
−0.667699 + 0.744432i \(0.732721\pi\)
\(158\) 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
\(172\) 568.000 0.251800
\(173\) 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
\(193\) −5111.00 −1.90621 −0.953103 − 0.302646i \(-0.902130\pi\)
−0.953103 + 0.302646i \(0.902130\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 432.000 0.157434
\(197\) 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
\(217\) 4913.00 1.53694
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −6659.00 −1.99964 −0.999820 − 0.0189844i \(-0.993957\pi\)
−0.999820 + 0.0189844i \(0.993957\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −6823.00 −1.96889 −0.984447 − 0.175684i \(-0.943786\pi\)
−0.984447 + 0.175684i \(0.943786\pi\)
\(230\) 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
\(244\) 7208.00 1.89117
\(245\) 0 0
\(246\) 0 0
\(247\) 7490.00 1.92946
\(248\) 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
\(259\) 5491.00 1.31735
\(260\) 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
\(271\) 7289.00 1.63386 0.816928 − 0.576739i \(-0.195675\pi\)
0.816928 + 0.576739i \(0.195675\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5167.00 1.12078 0.560388 − 0.828230i \(-0.310652\pi\)
0.560388 + 0.828230i \(0.310652\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −3869.00 −0.812679 −0.406340 − 0.913722i \(-0.633195\pi\)
−0.406340 + 0.913722i \(0.633195\pi\)
\(284\) 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
\(292\) −7352.00 −1.47344
\(293\) 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
\(301\) 1207.00 0.231131
\(302\) 0 0
\(303\) 0 0
\(304\) 6848.00 1.29197
\(305\) 0 0
\(306\) 0 0
\(307\) 3943.00 0.733026 0.366513 − 0.930413i \(-0.380552\pi\)
0.366513 + 0.930413i \(0.380552\pi\)
\(308\) 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
\(316\) 11096.0 1.97531
\(317\) 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
\(331\) 9899.00 1.64380 0.821901 − 0.569631i \(-0.192914\pi\)
0.821901 + 0.569631i \(0.192914\pi\)
\(332\) 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
\(343\) 6749.00 1.06242
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 10547.0 1.61767 0.808837 − 0.588033i \(-0.200098\pi\)
0.808837 + 0.588033i \(0.200098\pi\)
\(350\) 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
\(361\) 4590.00 0.669194
\(362\) 0 0
\(363\) 0 0
\(364\) 9520.00 1.37083
\(365\) 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
\(373\) 12601.0 1.74921 0.874605 − 0.484837i \(-0.161121\pi\)
0.874605 + 0.484837i \(0.161121\pi\)
\(374\) 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
\(388\) 14824.0 1.93963
\(389\) 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
\(397\) 14257.0 1.80236 0.901182 − 0.433441i \(-0.142701\pi\)
0.901182 + 0.433441i \(0.142701\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −20230.0 −2.50057
\(404\) 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
\(412\) −14408.0 −1.72289
\(413\) 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
\(421\) −10459.0 −1.21078 −0.605392 − 0.795927i \(-0.706984\pi\)
−0.605392 + 0.795927i \(0.706984\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 15317.0 1.73593
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −16739.0 −1.85779 −0.928897 − 0.370338i \(-0.879242\pi\)
−0.928897 + 0.370338i \(0.879242\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 12536.0 1.37699
\(437\) 0 0
\(438\) 0 0
\(439\) −16777.0 −1.82397 −0.911985 − 0.410224i \(-0.865450\pi\)
−0.911985 + 0.410224i \(0.865450\pi\)
\(440\) 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
\(448\) 8704.00 0.917914
\(449\) 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
\(463\) −7811.00 −0.784034 −0.392017 − 0.919958i \(-0.628223\pi\)
−0.392017 + 0.919958i \(0.628223\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −14960.0 −1.47290
\(470\) 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
\(481\) −22610.0 −2.14330
\(482\) 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
\(496\) −18496.0 −1.67438
\(497\) 0 0
\(498\) 0 0
\(499\) −6607.00 −0.592725 −0.296363 − 0.955075i \(-0.595774\pi\)
−0.296363 + 0.955075i \(0.595774\pi\)
\(500\) 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
\(511\) −15623.0 −1.35249
\(512\) 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
\(523\) 11881.0 0.993346 0.496673 − 0.867938i \(-0.334555\pi\)
0.496673 + 0.867938i \(0.334555\pi\)
\(524\) 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
\(532\) 14552.0 1.18592
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1889.00 0.150119 0.0750596 − 0.997179i \(-0.476085\pi\)
0.0750596 + 0.997179i \(0.476085\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 22933.0 1.79259 0.896293 − 0.443463i \(-0.146250\pi\)
0.896293 + 0.443463i \(0.146250\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 23579.0 1.81317
\(554\) 0 0
\(555\) 0 0
\(556\) −3736.00 −0.284967
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −4970.00 −0.376044
\(560\) 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
\(571\) −23941.0 −1.75464 −0.877320 − 0.479905i \(-0.840671\pi\)
−0.877320 + 0.479905i \(0.840671\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −27323.0 −1.97135 −0.985677 − 0.168644i \(-0.946061\pi\)
−0.985677 + 0.168644i \(0.946061\pi\)
\(578\) 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
\(589\) −30923.0 −2.16326
\(590\) 0 0
\(591\) 0 0
\(592\) −20672.0 −1.43516
\(593\) 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
\(601\) 11951.0 0.811134 0.405567 − 0.914065i \(-0.367074\pi\)
0.405567 + 0.914065i \(0.367074\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 29672.0 1.99890
\(605\) 0 0
\(606\) 0 0
\(607\) −22283.0 −1.49001 −0.745007 − 0.667056i \(-0.767554\pi\)
−0.745007 + 0.667056i \(0.767554\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 30241.0 1.99253 0.996266 − 0.0863334i \(-0.0275150\pi\)
0.996266 + 0.0863334i \(0.0275150\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −37.0000 −0.00240251 −0.00120126 − 0.999999i \(-0.500382\pi\)
−0.00120126 + 0.999999i \(0.500382\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 21016.0 1.33540
\(629\) 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
\(637\) −3780.00 −0.235116
\(638\) 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
\(661\) −13249.0 −0.779616 −0.389808 − 0.920896i \(-0.627459\pi\)
−0.389808 + 0.920896i \(0.627459\pi\)
\(662\) 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
\(673\) 33949.0 1.94448 0.972242 − 0.233977i \(-0.0751742\pi\)
0.972242 + 0.233977i \(0.0751742\pi\)
\(674\) 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
\(679\) 31501.0 1.78041
\(680\) 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
\(688\) −4544.00 −0.251800
\(689\) 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
\(703\) −34561.0 −1.85419
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −8623.00 −0.456761 −0.228381 − 0.973572i \(-0.573343\pi\)
−0.228381 + 0.973572i \(0.573343\pi\)
\(710\) 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
\(721\) −30617.0 −1.58147
\(722\) 0 0
\(723\) 0 0
\(724\) −27664.0 −1.42006
\(725\) 0 0
\(726\) 0 0
\(727\) 27253.0 1.39031 0.695157 − 0.718858i \(-0.255335\pi\)
0.695157 + 0.718858i \(0.255335\pi\)
\(728\) 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
\(751\) 41021.0 1.99318 0.996590 − 0.0825179i \(-0.0262962\pi\)
0.996590 + 0.0825179i \(0.0262962\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −24137.0 −1.15888 −0.579441 − 0.815014i \(-0.696729\pi\)
−0.579441 + 0.815014i \(0.696729\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 26639.0 1.26395
\(764\) 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
\(772\) 40888.0 1.90621
\(773\) 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
\(784\) −3456.00 −0.157434
\(785\) 0 0
\(786\) 0 0
\(787\) 14653.0 0.663689 0.331844 − 0.943334i \(-0.392329\pi\)
0.331844 + 0.943334i \(0.392329\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −63070.0 −2.82431
\(794\) 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
\(811\) 1241.00 0.0537329 0.0268665 − 0.999639i \(-0.491447\pi\)
0.0268665 + 0.999639i \(0.491447\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −7597.00 −0.325319
\(818\) 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
\(829\) −47143.0 −1.97508 −0.987542 − 0.157358i \(-0.949702\pi\)
−0.987542 + 0.157358i \(0.949702\pi\)
\(830\) 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
\(847\) 22627.0 0.917914
\(848\) 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
\(859\) 18503.0 0.734941 0.367470 − 0.930035i \(-0.380224\pi\)
0.367470 + 0.930035i \(0.380224\pi\)
\(860\) 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
\(868\) −39304.0 −1.53694
\(869\) 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
\(877\) 36793.0 1.41666 0.708330 − 0.705881i \(-0.249449\pi\)
0.708330 + 0.705881i \(0.249449\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −52109.0 −1.98597 −0.992983 − 0.118260i \(-0.962269\pi\)
−0.992983 + 0.118260i \(0.962269\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 6460.00 0.243714
\(890\) 0 0
\(891\) 0 0
\(892\) 53272.0 1.99964
\(893\) 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
\(907\) 49447.0 1.81021 0.905105 − 0.425188i \(-0.139792\pi\)
0.905105 + 0.425188i \(0.139792\pi\)
\(908\) 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
\(916\) 54584.0 1.96889
\(917\) 0 0
\(918\) 0 0
\(919\) 46817.0 1.68047 0.840234 − 0.542224i \(-0.182417\pi\)
0.840234 + 0.542224i \(0.182417\pi\)
\(920\) 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
\(931\) −5778.00 −0.203401
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −15227.0 −0.530891 −0.265445 − 0.964126i \(-0.585519\pi\)
−0.265445 + 0.964126i \(0.585519\pi\)
\(938\) 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
\(949\) 64330.0 2.20046
\(950\) 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
\(961\) 53730.0 1.80356
\(962\) 0 0
\(963\) 0 0
\(964\) 59024.0 1.97203
\(965\) 0 0
\(966\) 0 0
\(967\) 3907.00 0.129928 0.0649641 − 0.997888i \(-0.479307\pi\)
0.0649641 + 0.997888i \(0.479307\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −7939.00 −0.261575
\(974\) 0 0
\(975\) 0 0
\(976\) −57664.0 −1.89117
\(977\) 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
\(988\) −59920.0 −1.92946
\(989\) 0 0
\(990\) 0 0
\(991\) 59669.0 1.91266 0.956331 − 0.292286i \(-0.0944158\pi\)
0.956331 + 0.292286i \(0.0944158\pi\)
\(992\) 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.e.1.1 ✓ 1
3.2 odd 2 CM 675.4.a.e.1.1 ✓ 1
5.2 odd 4 675.4.b.g.649.1 2
5.3 odd 4 675.4.b.g.649.2 2
5.4 even 2 675.4.a.f.1.1 yes 1
15.2 even 4 675.4.b.g.649.1 2
15.8 even 4 675.4.b.g.649.2 2
15.14 odd 2 675.4.a.f.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
675.4.a.e.1.1 ✓ 1 1.1 even 1 trivial
675.4.a.e.1.1 ✓ 1 3.2 odd 2 CM
675.4.a.f.1.1 yes 1 5.4 even 2
675.4.a.f.1.1 yes 1 15.14 odd 2
675.4.b.g.649.1 2 5.2 odd 4
675.4.b.g.649.1 2 15.2 even 4
675.4.b.g.649.2 2 5.3 odd 4
675.4.b.g.649.2 2 15.8 even 4