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Properties

Label 588.5.c.b.197.1
Level $588$
Weight $5$
Character 588.197
Self dual yes
Analytic conductor $60.782$
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] = [588,5,Mod(197,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.197");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 588.c (of order \(2\), degree \(1\), minimal)

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: \(60.7815382933\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 197.1
Character \(\chi\) \(=\) 588.197

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} +81.0000 q^{9} +O(q^{10})\) \(q-9.00000 q^{3} +81.0000 q^{9} +337.000 q^{13} -647.000 q^{19} +625.000 q^{25} -729.000 q^{27} -1559.00 q^{31} -529.000 q^{37} -3033.00 q^{39} +3191.00 q^{43} +5823.00 q^{57} +1966.00 q^{61} +2903.00 q^{67} -9791.00 q^{73} -5625.00 q^{75} +4679.00 q^{79} +6561.00 q^{81} +14031.0 q^{93} +18814.0 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) −9.00000 −1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 81.0000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 337.000 1.99408 0.997041 − 0.0768662i \(-0.0244914\pi\)
0.997041 + 0.0768662i \(0.0244914\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −647.000 −1.79224 −0.896122 − 0.443808i \(-0.853627\pi\)
−0.896122 + 0.443808i \(0.853627\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) 0 0
\(27\) −729.000 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1559.00 −1.62227 −0.811134 − 0.584860i \(-0.801149\pi\)
−0.811134 + 0.584860i \(0.801149\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −529.000 −0.386413 −0.193207 − 0.981158i \(-0.561889\pi\)
−0.193207 + 0.981158i \(0.561889\pi\)
\(38\) 0 0
\(39\) −3033.00 −1.99408
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 3191.00 1.72580 0.862899 − 0.505377i \(-0.168646\pi\)
0.862899 + 0.505377i \(0.168646\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5823.00 1.79224
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 1966.00 0.528353 0.264176 − 0.964474i \(-0.414900\pi\)
0.264176 + 0.964474i \(0.414900\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2903.00 0.646692 0.323346 − 0.946281i \(-0.395192\pi\)
0.323346 + 0.946281i \(0.395192\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −9791.00 −1.83731 −0.918653 − 0.395066i \(-0.870722\pi\)
−0.918653 + 0.395066i \(0.870722\pi\)
\(74\) 0 0
\(75\) −5625.00 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4679.00 0.749720 0.374860 − 0.927082i \(-0.377691\pi\)
0.374860 + 0.927082i \(0.377691\pi\)
\(80\) 0 0
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 14031.0 1.62227
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18814.0 1.99957 0.999787 − 0.0206175i \(-0.00656321\pi\)
0.999787 + 0.0206175i \(0.00656321\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 19849.0 1.87096 0.935479 − 0.353381i \(-0.114968\pi\)
0.935479 + 0.353381i \(0.114968\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −3313.00 −0.278849 −0.139424 − 0.990233i \(-0.544525\pi\)
−0.139424 + 0.990233i \(0.544525\pi\)
\(110\) 0 0
\(111\) 4761.00 0.386413
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 27297.0 1.99408
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 14641.0 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 31751.0 1.96857 0.984283 − 0.176599i \(-0.0565094\pi\)
0.984283 + 0.176599i \(0.0565094\pi\)
\(128\) 0 0
\(129\) −28719.0 −1.72580
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −24359.0 −1.26075 −0.630376 − 0.776290i \(-0.717099\pi\)
−0.630376 + 0.776290i \(0.717099\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 36194.0 1.58739 0.793693 − 0.608318i \(-0.208156\pi\)
0.793693 + 0.608318i \(0.208156\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 35374.0 1.43511 0.717554 − 0.696502i \(-0.245261\pi\)
0.717554 + 0.696502i \(0.245261\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 15506.0 0.583612 0.291806 − 0.956477i \(-0.405744\pi\)
0.291806 + 0.956477i \(0.405744\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 85008.0 2.97637
\(170\) 0 0
\(171\) −52407.0 −1.79224
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 65521.0 1.99997 0.999985 − 0.00552484i \(-0.00175862\pi\)
0.999985 + 0.00552484i \(0.00175862\pi\)
\(182\) 0 0
\(183\) −17694.0 −0.528353
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −54049.0 −1.45102 −0.725509 − 0.688212i \(-0.758396\pi\)
−0.725509 + 0.688212i \(0.758396\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −69794.0 −1.76243 −0.881215 − 0.472715i \(-0.843274\pi\)
−0.881215 + 0.472715i \(0.843274\pi\)
\(200\) 0 0
\(201\) −26127.0 −0.646692
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −61486.0 −1.38106 −0.690528 − 0.723306i \(-0.742622\pi\)
−0.690528 + 0.723306i \(0.742622\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 88119.0 1.83731
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −14786.0 −0.297332 −0.148666 − 0.988888i \(-0.547498\pi\)
−0.148666 + 0.988888i \(0.547498\pi\)
\(224\) 0 0
\(225\) 50625.0 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −41807.0 −0.797220 −0.398610 − 0.917121i \(-0.630507\pi\)
−0.398610 + 0.917121i \(0.630507\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −42111.0 −0.749720
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 34366.0 0.591691 0.295845 − 0.955236i \(-0.404399\pi\)
0.295845 + 0.955236i \(0.404399\pi\)
\(242\) 0 0
\(243\) −59049.0 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −218039. −3.57388
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 88318.0 1.20257 0.601285 − 0.799034i \(-0.294655\pi\)
0.601285 + 0.799034i \(0.294655\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 126383. 1.64713 0.823567 − 0.567218i \(-0.191980\pi\)
0.823567 + 0.567218i \(0.191980\pi\)
\(278\) 0 0
\(279\) −126279. −1.62227
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 156697. 1.95654 0.978268 − 0.207345i \(-0.0664822\pi\)
0.978268 + 0.207345i \(0.0664822\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 83521.0 1.00000
\(290\) 0 0
\(291\) −169326. −1.99957
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 124489. 1.32085 0.660426 − 0.750891i \(-0.270376\pi\)
0.660426 + 0.750891i \(0.270376\pi\)
\(308\) 0 0
\(309\) −178641. −1.87096
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −12911.0 −0.131787 −0.0658933 − 0.997827i \(-0.520990\pi\)
−0.0658933 + 0.997827i \(0.520990\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 210625. 1.99408
\(326\) 0 0
\(327\) 29817.0 0.278849
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −129721. −1.18401 −0.592004 − 0.805935i \(-0.701663\pi\)
−0.592004 + 0.805935i \(0.701663\pi\)
\(332\) 0 0
\(333\) −42849.0 −0.386413
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −199249. −1.75443 −0.877216 − 0.480097i \(-0.840602\pi\)
−0.877216 + 0.480097i \(0.840602\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −8402.00 −0.0689814 −0.0344907 − 0.999405i \(-0.510981\pi\)
−0.0344907 + 0.999405i \(0.510981\pi\)
\(350\) 0 0
\(351\) −245673. −1.99408
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 288288. 2.21214
\(362\) 0 0
\(363\) −131769. −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 217849. 1.61742 0.808711 − 0.588206i \(-0.200166\pi\)
0.808711 + 0.588206i \(0.200166\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −263617. −1.89477 −0.947383 − 0.320101i \(-0.896283\pi\)
−0.947383 + 0.320101i \(0.896283\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −280393. −1.95204 −0.976020 − 0.217681i \(-0.930151\pi\)
−0.976020 + 0.217681i \(0.930151\pi\)
\(380\) 0 0
\(381\) −285759. −1.96857
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 258471. 1.72580
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −313631. −1.98993 −0.994965 − 0.100219i \(-0.968046\pi\)
−0.994965 + 0.100219i \(0.968046\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −525383. −3.23494
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 314113. 1.87776 0.938878 − 0.344249i \(-0.111867\pi\)
0.938878 + 0.344249i \(0.111867\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 219231. 1.26075
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 349439. 1.97155 0.985774 − 0.168079i \(-0.0537562\pi\)
0.985774 + 0.168079i \(0.0537562\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 121969. 0.650539 0.325270 − 0.945621i \(-0.394545\pi\)
0.325270 + 0.945621i \(0.394545\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 376606. 1.95415 0.977076 − 0.212892i \(-0.0682884\pi\)
0.977076 + 0.212892i \(0.0682884\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −325746. −1.58739
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −415489. −1.98942 −0.994711 − 0.102709i \(-0.967249\pi\)
−0.994711 + 0.102709i \(0.967249\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 423191. 1.97412 0.987062 − 0.160339i \(-0.0512587\pi\)
0.987062 + 0.160339i \(0.0512587\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −318366. −1.43511
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −404375. −1.79224
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −178273. −0.770540
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −98569.0 −0.415607 −0.207803 − 0.978171i \(-0.566631\pi\)
−0.207803 + 0.978171i \(0.566631\pi\)
\(488\) 0 0
\(489\) −139554. −0.583612
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 271127. 1.08886 0.544430 − 0.838807i \(-0.316746\pi\)
0.544430 + 0.838807i \(0.316746\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −765072. −2.97637
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 471663. 1.79224
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −97751.0 −0.357370 −0.178685 − 0.983906i \(-0.557184\pi\)
−0.178685 + 0.983906i \(0.557184\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 279841. 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 43487.0 0.148582 0.0742908 − 0.997237i \(-0.476331\pi\)
0.0742908 + 0.997237i \(0.476331\pi\)
\(542\) 0 0
\(543\) −589689. −1.99997
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −342382. −1.14429 −0.572145 − 0.820152i \(-0.693889\pi\)
−0.572145 + 0.820152i \(0.693889\pi\)
\(548\) 0 0
\(549\) 159246. 0.528353
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 1.07537e6 3.44138
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 486407. 1.49186 0.745929 − 0.666025i \(-0.232006\pi\)
0.745929 + 0.666025i \(0.232006\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −401231. −1.20515 −0.602577 − 0.798060i \(-0.705860\pi\)
−0.602577 + 0.798060i \(0.705860\pi\)
\(578\) 0 0
\(579\) 486441. 1.45102
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 1.00867e6 2.90750
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 628146. 1.76243
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 269473. 0.746047 0.373024 − 0.927822i \(-0.378321\pi\)
0.373024 + 0.927822i \(0.378321\pi\)
\(602\) 0 0
\(603\) 235143. 0.646692
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 597769. 1.62239 0.811196 − 0.584774i \(-0.198817\pi\)
0.811196 + 0.584774i \(0.198817\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 516338. 1.37408 0.687042 − 0.726618i \(-0.258909\pi\)
0.687042 + 0.726618i \(0.258909\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −134279. −0.350451 −0.175225 − 0.984528i \(-0.556065\pi\)
−0.175225 + 0.984528i \(0.556065\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 390625. 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −342046. −0.859065 −0.429532 − 0.903052i \(-0.641322\pi\)
−0.429532 + 0.903052i \(0.641322\pi\)
\(632\) 0 0
\(633\) 553374. 1.38106
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −25031.0 −0.0605419 −0.0302710 − 0.999542i \(-0.509637\pi\)
−0.0302710 + 0.999542i \(0.509637\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −793071. −1.83731
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −568559. −1.30129 −0.650643 − 0.759384i \(-0.725500\pi\)
−0.650643 + 0.759384i \(0.725500\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 133074. 0.297332
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 479471. 1.05860 0.529300 − 0.848434i \(-0.322454\pi\)
0.529300 + 0.848434i \(0.322454\pi\)
\(674\) 0 0
\(675\) −455625. −1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 376263. 0.797220
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 865561. 1.81277 0.906383 − 0.422458i \(-0.138832\pi\)
0.906383 + 0.422458i \(0.138832\pi\)
\(692\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 342263. 0.692547
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −133006. −0.264593 −0.132297 − 0.991210i \(-0.542235\pi\)
−0.132297 + 0.991210i \(0.542235\pi\)
\(710\) 0 0
\(711\) 378999. 0.749720
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −309294. −0.591691
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −984983. −1.86363 −0.931815 − 0.362932i \(-0.881776\pi\)
−0.931815 + 0.362932i \(0.881776\pi\)
\(728\) 0 0
\(729\) 531441. 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 927889. 1.72698 0.863492 − 0.504363i \(-0.168273\pi\)
0.863492 + 0.504363i \(0.168273\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.08036e6 −1.97824 −0.989122 − 0.147095i \(-0.953008\pi\)
−0.989122 + 0.147095i \(0.953008\pi\)
\(740\) 0 0
\(741\) 1.96235e6 3.57388
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 779159. 1.38149 0.690743 − 0.723101i \(-0.257284\pi\)
0.690743 + 0.723101i \(0.257284\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −443854. −0.774548 −0.387274 − 0.921965i \(-0.626583\pi\)
−0.387274 + 0.921965i \(0.626583\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 732481. 1.23864 0.619318 − 0.785140i \(-0.287409\pi\)
0.619318 + 0.785140i \(0.287409\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −974375. −1.62227
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1.20111e6 −1.93924 −0.969621 − 0.244613i \(-0.921339\pi\)
−0.969621 + 0.244613i \(0.921339\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 662542. 1.05358
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −976754. −1.48506 −0.742529 − 0.669814i \(-0.766374\pi\)
−0.742529 + 0.669814i \(0.766374\pi\)
\(812\) 0 0
\(813\) −794862. −1.20257
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2.06458e6 −3.09305
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −235294. −0.347385 −0.173693 − 0.984800i \(-0.555570\pi\)
−0.173693 + 0.984800i \(0.555570\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 49681.0 0.0722905 0.0361453 − 0.999347i \(-0.488492\pi\)
0.0361453 + 0.999347i \(0.488492\pi\)
\(830\) 0 0
\(831\) −1.13745e6 −1.64713
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.13651e6 1.62227
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 707281. 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −1.41027e6 −1.95654
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 69889.0 0.0960530 0.0480265 − 0.998846i \(-0.484707\pi\)
0.0480265 + 0.998846i \(0.484707\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −534962. −0.724998 −0.362499 − 0.931984i \(-0.618076\pi\)
−0.362499 + 0.931984i \(0.618076\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −751689. −1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 978311. 1.28956
\(872\) 0 0
\(873\) 1.52393e6 1.99957
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.18065e6 −1.53505 −0.767527 − 0.641017i \(-0.778513\pi\)
−0.767527 + 0.641017i \(0.778513\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 25703.0 0.0329657 0.0164829 − 0.999864i \(-0.494753\pi\)
0.0164829 + 0.999864i \(0.494753\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 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
\(907\) 465911. 0.566355 0.283177 − 0.959068i \(-0.408612\pi\)
0.283177 + 0.959068i \(0.408612\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −746281. −0.883632 −0.441816 − 0.897106i \(-0.645666\pi\)
−0.441816 + 0.897106i \(0.645666\pi\)
\(920\) 0 0
\(921\) −1.12040e6 −1.32085
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −330625. −0.386413
\(926\) 0 0
\(927\) 1.60777e6 1.87096
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.65547e6 −1.88557 −0.942784 − 0.333403i \(-0.891803\pi\)
−0.942784 + 0.333403i \(0.891803\pi\)
\(938\) 0 0
\(939\) 116199. 0.131787
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −3.29957e6 −3.66374
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.50696e6 1.63175
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.80617e6 −1.93155 −0.965774 − 0.259386i \(-0.916480\pi\)
−0.965774 + 0.259386i \(0.916480\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −1.89562e6 −1.99408
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −268353. −0.278849
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.05996e6 1.07930 0.539649 − 0.841890i \(-0.318557\pi\)
0.539649 + 0.841890i \(0.318557\pi\)
\(992\) 0 0
\(993\) 1.16749e6 1.18401
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −223151. −0.224496 −0.112248 − 0.993680i \(-0.535805\pi\)
−0.112248 + 0.993680i \(0.535805\pi\)
\(998\) 0 0
\(999\) 385641. 0.386413
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 588.5.c.b.197.1 1
3.2 odd 2 CM 588.5.c.b.197.1 1
7.3 odd 6 84.5.p.a.65.1 yes 2
7.5 odd 6 84.5.p.a.53.1 ✓ 2
7.6 odd 2 588.5.c.c.197.1 1
21.5 even 6 84.5.p.a.53.1 ✓ 2
21.17 even 6 84.5.p.a.65.1 yes 2
21.20 even 2 588.5.c.c.197.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.5.p.a.53.1 ✓ 2 7.5 odd 6
84.5.p.a.53.1 ✓ 2 21.5 even 6
84.5.p.a.65.1 yes 2 7.3 odd 6
84.5.p.a.65.1 yes 2 21.17 even 6
588.5.c.b.197.1 1 1.1 even 1 trivial
588.5.c.b.197.1 1 3.2 odd 2 CM
588.5.c.c.197.1 1 7.6 odd 2
588.5.c.c.197.1 1 21.20 even 2