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Properties

Label 139.7.b.a.138.1
Level $139$
Weight $7$
Character 139.138
Self dual yes
Analytic conductor $31.978$
Analytic rank $0$
Dimension $1$
CM discriminant -139
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [139,7,Mod(138,139)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(139, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("139.138");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 139 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 139.b (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: \(31.9775176232\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 138.1
Character \(\chi\) \(=\) 139.138

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+64.0000 q^{4} +111.000 q^{5} -565.000 q^{7} +729.000 q^{9} +O(q^{10})\) \(q+64.0000 q^{4} +111.000 q^{5} -565.000 q^{7} +729.000 q^{9} -813.000 q^{11} +4255.00 q^{13} +4096.00 q^{16} +7104.00 q^{20} -3304.00 q^{25} -36160.0 q^{28} -24753.0 q^{29} +59443.0 q^{31} -62715.0 q^{35} +46656.0 q^{36} -7670.00 q^{37} +102258. q^{41} -52032.0 q^{44} +80919.0 q^{45} +207090. q^{47} +201576. q^{49} +272320. q^{52} -90243.0 q^{55} -411885. q^{63} +262144. q^{64} +472305. q^{65} -600685. q^{67} -385197. q^{71} +459345. q^{77} +204203. q^{79} +454656. q^{80} +531441. q^{81} +909915. q^{83} +989463. q^{89} -2.40408e6 q^{91} -592677. q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/139\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 64.0000 1.00000
\(5\) 111.000 0.888000 0.444000 − 0.896027i \(-0.353559\pi\)
0.444000 + 0.896027i \(0.353559\pi\)
\(6\) 0 0
\(7\) −565.000 −1.64723 −0.823615 − 0.567149i \(-0.808046\pi\)
−0.823615 + 0.567149i \(0.808046\pi\)
\(8\) 0 0
\(9\) 729.000 1.00000
\(10\) 0 0
\(11\) −813.000 −0.610819 −0.305409 − 0.952221i \(-0.598793\pi\)
−0.305409 + 0.952221i \(0.598793\pi\)
\(12\) 0 0
\(13\) 4255.00 1.93673 0.968366 − 0.249534i \(-0.0802775\pi\)
0.968366 + 0.249534i \(0.0802775\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4096.00 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 7104.00 0.888000
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −3304.00 −0.211456
\(26\) 0 0
\(27\) 0 0
\(28\) −36160.0 −1.64723
\(29\) −24753.0 −1.01492 −0.507462 − 0.861674i \(-0.669416\pi\)
−0.507462 + 0.861674i \(0.669416\pi\)
\(30\) 0 0
\(31\) 59443.0 1.99533 0.997667 − 0.0682671i \(-0.0217470\pi\)
0.997667 + 0.0682671i \(0.0217470\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −62715.0 −1.46274
\(36\) 46656.0 1.00000
\(37\) −7670.00 −0.151422 −0.0757112 − 0.997130i \(-0.524123\pi\)
−0.0757112 + 0.997130i \(0.524123\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 102258. 1.48370 0.741849 − 0.670567i \(-0.233949\pi\)
0.741849 + 0.670567i \(0.233949\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −52032.0 −0.610819
\(45\) 80919.0 0.888000
\(46\) 0 0
\(47\) 207090. 1.99464 0.997322 − 0.0731307i \(-0.0232990\pi\)
0.997322 + 0.0731307i \(0.0232990\pi\)
\(48\) 0 0
\(49\) 201576. 1.71337
\(50\) 0 0
\(51\) 0 0
\(52\) 272320. 1.93673
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −90243.0 −0.542407
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −411885. −1.64723
\(64\) 262144. 1.00000
\(65\) 472305. 1.71982
\(66\) 0 0
\(67\) −600685. −1.99720 −0.998602 − 0.0528608i \(-0.983166\pi\)
−0.998602 + 0.0528608i \(0.983166\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −385197. −1.07624 −0.538118 − 0.842869i \(-0.680865\pi\)
−0.538118 + 0.842869i \(0.680865\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 459345. 1.00616
\(78\) 0 0
\(79\) 204203. 0.414172 0.207086 − 0.978323i \(-0.433602\pi\)
0.207086 + 0.978323i \(0.433602\pi\)
\(80\) 454656. 0.888000
\(81\) 531441. 1.00000
\(82\) 0 0
\(83\) 909915. 1.59135 0.795677 − 0.605722i \(-0.207116\pi\)
0.795677 + 0.605722i \(0.207116\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 989463. 1.40356 0.701778 − 0.712396i \(-0.252390\pi\)
0.701778 + 0.712396i \(0.252390\pi\)
\(90\) 0 0
\(91\) −2.40408e6 −3.19024
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) −592677. −0.610819
\(100\) −211456. −0.211456
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.15415e6 −1.75843 −0.879214 − 0.476427i \(-0.841932\pi\)
−0.879214 + 0.476427i \(0.841932\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.31424e6 −1.64723
\(113\) 850695. 0.589574 0.294787 − 0.955563i \(-0.404751\pi\)
0.294787 + 0.955563i \(0.404751\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.58419e6 −1.01492
\(117\) 3.10190e6 1.93673
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.11059e6 −0.626900
\(122\) 0 0
\(123\) 0 0
\(124\) 3.80435e6 1.99533
\(125\) −2.10112e6 −1.07577
\(126\) 0 0
\(127\) 3.90648e6 1.90710 0.953551 − 0.301232i \(-0.0973978\pi\)
0.953551 + 0.301232i \(0.0973978\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.26096e6 −1.89537 −0.947683 − 0.319212i \(-0.896582\pi\)
−0.947683 + 0.319212i \(0.896582\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.03810e6 −1.57042 −0.785210 − 0.619229i \(-0.787445\pi\)
−0.785210 + 0.619229i \(0.787445\pi\)
\(138\) 0 0
\(139\) −2.68562e6 −1.00000
\(140\) −4.01376e6 −1.46274
\(141\) 0 0
\(142\) 0 0
\(143\) −3.45932e6 −1.18299
\(144\) 2.98598e6 1.00000
\(145\) −2.74758e6 −0.901253
\(146\) 0 0
\(147\) 0 0
\(148\) −490880. −0.151422
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.59817e6 1.77186
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −7.50156e6 −1.73216 −0.866082 − 0.499902i \(-0.833369\pi\)
−0.866082 + 0.499902i \(0.833369\pi\)
\(164\) 6.54451e6 1.48370
\(165\) 0 0
\(166\) 0 0
\(167\) −8.90019e6 −1.91095 −0.955476 − 0.295068i \(-0.904658\pi\)
−0.955476 + 0.295068i \(0.904658\pi\)
\(168\) 0 0
\(169\) 1.32782e7 2.75093
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.61066e6 −1.85616 −0.928079 − 0.372382i \(-0.878541\pi\)
−0.928079 + 0.372382i \(0.878541\pi\)
\(174\) 0 0
\(175\) 1.86676e6 0.348317
\(176\) −3.33005e6 −0.610819
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 5.17882e6 0.888000
\(181\) 1.18282e7 1.99473 0.997363 − 0.0725762i \(-0.0231221\pi\)
0.997363 + 0.0725762i \(0.0231221\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −851370. −0.134463
\(186\) 0 0
\(187\) 0 0
\(188\) 1.32538e7 1.99464
\(189\) 0 0
\(190\) 0 0
\(191\) −1.22457e7 −1.75746 −0.878729 − 0.477321i \(-0.841608\pi\)
−0.878729 + 0.477321i \(0.841608\pi\)
\(192\) 0 0
\(193\) −9.10190e6 −1.26608 −0.633039 − 0.774120i \(-0.718193\pi\)
−0.633039 + 0.774120i \(0.718193\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.29009e7 1.71337
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.39854e7 1.67181
\(204\) 0 0
\(205\) 1.13506e7 1.31752
\(206\) 0 0
\(207\) 0 0
\(208\) 1.74285e7 1.93673
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3.35853e7 −3.28677
\(218\) 0 0
\(219\) 0 0
\(220\) −5.77555e6 −0.542407
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −2.40862e6 −0.211456
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 2.29870e7 1.77124
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.28918e7 0.944319 0.472159 − 0.881513i \(-0.343475\pi\)
0.472159 + 0.881513i \(0.343475\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.23749e7 1.52147
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.82197e6 −0.241694 −0.120847 − 0.992671i \(-0.538561\pi\)
−0.120847 + 0.992671i \(0.538561\pi\)
\(252\) −2.63606e7 −1.64723
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.67772e7 1.00000
\(257\) 2.20162e7 1.29701 0.648504 − 0.761211i \(-0.275395\pi\)
0.648504 + 0.761211i \(0.275395\pi\)
\(258\) 0 0
\(259\) 4.33355e6 0.249428
\(260\) 3.02275e7 1.71982
\(261\) −1.80449e7 −1.01492
\(262\) 0 0
\(263\) −3.54747e7 −1.95007 −0.975037 − 0.222040i \(-0.928728\pi\)
−0.975037 + 0.222040i \(0.928728\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −3.84438e7 −1.99720
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.68615e6 0.129161
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 4.33339e7 1.99533
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −2.40167e6 −0.105963 −0.0529815 − 0.998595i \(-0.516872\pi\)
−0.0529815 + 0.998595i \(0.516872\pi\)
\(284\) −2.46526e7 −1.07624
\(285\) 0 0
\(286\) 0 0
\(287\) −5.77758e7 −2.44399
\(288\) 0 0
\(289\) 2.41376e7 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 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
\(307\) −2.71330e7 −0.937739 −0.468869 − 0.883268i \(-0.655339\pi\)
−0.468869 + 0.883268i \(0.655339\pi\)
\(308\) 2.93981e7 1.00616
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −6.04488e7 −1.97131 −0.985654 − 0.168781i \(-0.946017\pi\)
−0.985654 + 0.168781i \(0.946017\pi\)
\(314\) 0 0
\(315\) −4.57192e7 −1.46274
\(316\) 1.30690e7 0.414172
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 2.01242e7 0.619935
\(320\) 2.90980e7 0.888000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 3.40122e7 1.00000
\(325\) −1.40585e7 −0.409534
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.17006e8 −3.28564
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 5.82346e7 1.59135
\(333\) −5.59143e6 −0.151422
\(334\) 0 0
\(335\) −6.66760e7 −1.77352
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.83272e7 −1.21879
\(342\) 0 0
\(343\) −4.74188e7 −1.17508
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.17243e7 0.519944 0.259972 − 0.965616i \(-0.416287\pi\)
0.259972 + 0.965616i \(0.416287\pi\)
\(348\) 0 0
\(349\) 3.90602e7 0.918879 0.459440 − 0.888209i \(-0.348050\pi\)
0.459440 + 0.888209i \(0.348050\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −4.27569e7 −0.955698
\(356\) 6.33256e7 1.40356
\(357\) 0 0
\(358\) 0 0
\(359\) 9.04968e6 0.195592 0.0977958 − 0.995207i \(-0.468821\pi\)
0.0977958 + 0.995207i \(0.468821\pi\)
\(360\) 0 0
\(361\) 4.70459e7 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) −1.53861e8 −3.19024
\(365\) 0 0
\(366\) 0 0
\(367\) 9.10542e7 1.84205 0.921026 − 0.389501i \(-0.127352\pi\)
0.921026 + 0.389501i \(0.127352\pi\)
\(368\) 0 0
\(369\) 7.45461e7 1.48370
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.05324e8 −1.96564
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 5.09873e7 0.893470
\(386\) 0 0
\(387\) 0 0
\(388\) 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
\(395\) 2.26665e7 0.367785
\(396\) −3.79313e7 −0.610819
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.35332e7 −0.211456
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 2.52930e8 3.86443
\(404\) 0 0
\(405\) 5.89900e7 0.888000
\(406\) 0 0
\(407\) 6.23571e6 0.0924917
\(408\) 0 0
\(409\) −2.08214e7 −0.304327 −0.152163 − 0.988355i \(-0.548624\pi\)
−0.152163 + 0.988355i \(0.548624\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1.01001e8 1.41312
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −4.59504e7 −0.615804 −0.307902 − 0.951418i \(-0.599627\pi\)
−0.307902 + 0.951418i \(0.599627\pi\)
\(422\) 0 0
\(423\) 1.50969e8 1.99464
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.37866e8 −1.75843
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 1.07021e8 1.31827 0.659137 − 0.752023i \(-0.270922\pi\)
0.659137 + 0.752023i \(0.270922\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.46949e8 1.71337
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 1.09830e8 1.24636
\(446\) 0 0
\(447\) 0 0
\(448\) −1.48111e8 −1.64723
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) −8.31358e7 −0.906271
\(452\) 5.44445e7 0.589574
\(453\) 0 0
\(454\) 0 0
\(455\) −2.66852e8 −2.83294
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.95902e8 −1.99957 −0.999786 − 0.0207100i \(-0.993407\pi\)
−0.999786 + 0.0207100i \(0.993407\pi\)
\(462\) 0 0
\(463\) −1.87297e8 −1.88707 −0.943533 − 0.331280i \(-0.892520\pi\)
−0.943533 + 0.331280i \(0.892520\pi\)
\(464\) −1.01388e8 −1.01492
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 1.98521e8 1.93673
\(469\) 3.39387e8 3.28985
\(470\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −3.26358e7 −0.293265
\(482\) 0 0
\(483\) 0 0
\(484\) −7.10779e7 −0.626900
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −6.57871e7 −0.542407
\(496\) 2.43479e8 1.99533
\(497\) 2.17636e8 1.77281
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −1.34472e8 −1.07577
\(501\) 0 0
\(502\) 0 0
\(503\) 2.06305e8 1.62109 0.810543 − 0.585680i \(-0.199172\pi\)
0.810543 + 0.585680i \(0.199172\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 2.50014e8 1.90710
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1.68364e8 −1.21837
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 1.59605e8 1.11569 0.557843 − 0.829946i \(-0.311629\pi\)
0.557843 + 0.829946i \(0.311629\pi\)
\(524\) −2.72701e8 −1.89537
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.48036e8 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.35108e8 2.87353
\(534\) 0 0
\(535\) −2.39111e8 −1.56148
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.63881e8 −1.04656
\(540\) 0 0
\(541\) 1.05673e7 0.0667376 0.0333688 − 0.999443i \(-0.489376\pi\)
0.0333688 + 0.999443i \(0.489376\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −2.58439e8 −1.57042
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.15375e8 −0.682237
\(554\) 0 0
\(555\) 0 0
\(556\) −1.71880e8 −1.00000
\(557\) −3.34502e8 −1.93568 −0.967839 − 0.251572i \(-0.919052\pi\)
−0.967839 + 0.251572i \(0.919052\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2.56881e8 −1.46274
\(561\) 0 0
\(562\) 0 0
\(563\) −1.40140e8 −0.785304 −0.392652 − 0.919687i \(-0.628442\pi\)
−0.392652 + 0.919687i \(0.628442\pi\)
\(564\) 0 0
\(565\) 9.44271e7 0.523542
\(566\) 0 0
\(567\) −3.00264e8 −1.64723
\(568\) 0 0
\(569\) 5.86022e7 0.318110 0.159055 − 0.987270i \(-0.449155\pi\)
0.159055 + 0.987270i \(0.449155\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) −2.21396e8 −1.18299
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.91103e8 1.00000
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −1.75845e8 −0.901253
\(581\) −5.14102e8 −2.62133
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 3.44310e8 1.71982
\(586\) 0 0
\(587\) 3.88311e7 0.191984 0.0959922 − 0.995382i \(-0.469398\pi\)
0.0959922 + 0.995382i \(0.469398\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −3.14163e7 −0.151422
\(593\) −4.16611e8 −1.99787 −0.998933 − 0.0461772i \(-0.985296\pi\)
−0.998933 + 0.0461772i \(0.985296\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 3.88207e8 1.78830 0.894148 − 0.447771i \(-0.147782\pi\)
0.894148 + 0.447771i \(0.147782\pi\)
\(602\) 0 0
\(603\) −4.37899e8 −1.99720
\(604\) 0 0
\(605\) −1.23276e8 −0.556687
\(606\) 0 0
\(607\) −3.02793e8 −1.35388 −0.676939 − 0.736040i \(-0.736694\pi\)
−0.676939 + 0.736040i \(0.736694\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.81168e8 3.86309
\(612\) 0 0
\(613\) 2.70090e8 1.17254 0.586270 − 0.810116i \(-0.300596\pi\)
0.586270 + 0.810116i \(0.300596\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 1.49411e7 0.0629956 0.0314978 − 0.999504i \(-0.489972\pi\)
0.0314978 + 0.999504i \(0.489972\pi\)
\(620\) 4.22283e8 1.77186
\(621\) 0 0
\(622\) 0 0
\(623\) −5.59047e8 −2.31198
\(624\) 0 0
\(625\) −1.81599e8 −0.743830
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.33619e8 1.69351
\(636\) 0 0
\(637\) 8.57706e8 3.31833
\(638\) 0 0
\(639\) −2.80809e8 −1.07624
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5.32783e8 1.96715 0.983575 − 0.180501i \(-0.0577721\pi\)
0.983575 + 0.180501i \(0.0577721\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −4.80100e8 −1.73216
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −4.72966e8 −1.68309
\(656\) 4.18849e8 1.48370
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −5.69612e8 −1.91095
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.03087e8 0.338190 0.169095 − 0.985600i \(-0.445916\pi\)
0.169095 + 0.985600i \(0.445916\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 8.49806e8 2.75093
\(677\) 5.37736e8 1.73302 0.866509 − 0.499162i \(-0.166359\pi\)
0.866509 + 0.499162i \(0.166359\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −5.46292e7 −0.171460 −0.0857300 − 0.996318i \(-0.527322\pi\)
−0.0857300 + 0.996318i \(0.527322\pi\)
\(684\) 0 0
\(685\) −4.48230e8 −1.39453
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) −6.15083e8 −1.85616
\(693\) 3.34863e8 1.00616
\(694\) 0 0
\(695\) −2.98104e8 −0.888000
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.19473e8 0.348317
\(701\) −6.73695e8 −1.95573 −0.977866 − 0.209232i \(-0.932904\pi\)
−0.977866 + 0.209232i \(0.932904\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −2.13123e8 −0.610819
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 1.48864e8 0.414172
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −3.83984e8 −1.05050
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.05776e8 1.62977 0.814884 − 0.579624i \(-0.196801\pi\)
0.814884 + 0.579624i \(0.196801\pi\)
\(720\) 3.31444e8 0.888000
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 7.57005e8 1.99473
\(725\) 8.17839e7 0.214612
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 3.87420e8 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −3.36613e8 −0.854710 −0.427355 − 0.904084i \(-0.640555\pi\)
−0.427355 + 0.904084i \(0.640555\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.88357e8 1.21993
\(738\) 0 0
\(739\) −9.88086e7 −0.244828 −0.122414 − 0.992479i \(-0.539064\pi\)
−0.122414 + 0.992479i \(0.539064\pi\)
\(740\) −5.44877e7 −0.134463
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.63328e8 1.59135
\(748\) 0 0
\(749\) 1.21709e9 2.89654
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 8.48241e8 1.99464
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −8.35954e8 −1.89683 −0.948415 − 0.317031i \(-0.897314\pi\)
−0.948415 + 0.317031i \(0.897314\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −7.83727e8 −1.75746
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −5.82522e8 −1.26608
\(773\) −9.18927e8 −1.98949 −0.994746 − 0.102370i \(-0.967357\pi\)
−0.994746 + 0.102370i \(0.967357\pi\)
\(774\) 0 0
\(775\) −1.96400e8 −0.421925
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 3.13165e8 0.657386
\(782\) 0 0
\(783\) 0 0
\(784\) 8.25655e8 1.71337
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.80643e8 −0.971165
\(792\) 0 0
\(793\) 0 0
\(794\) 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
\(801\) 7.21319e8 1.40356
\(802\) 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
\(811\) 6.26911e8 1.17528 0.587642 − 0.809121i \(-0.300056\pi\)
0.587642 + 0.809121i \(0.300056\pi\)
\(812\) 8.95068e8 1.67181
\(813\) 0 0
\(814\) 0 0
\(815\) −8.32674e8 −1.53816
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −1.75257e9 −3.19024
\(820\) 7.26441e8 1.31752
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.11542e9 1.93673
\(833\) 0 0
\(834\) 0 0
\(835\) −9.87921e8 −1.69693
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.98310e8 1.69036 0.845180 − 0.534481i \(-0.179493\pi\)
0.845180 + 0.534481i \(0.179493\pi\)
\(840\) 0 0
\(841\) 1.78877e7 0.0300723
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.47388e9 2.44283
\(846\) 0 0
\(847\) 6.27484e8 1.03265
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 5.08521e8 0.802287 0.401144 − 0.916015i \(-0.368613\pi\)
0.401144 + 0.916015i \(0.368613\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.27151e9 −1.97827 −0.989136 − 0.147002i \(-0.953038\pi\)
−0.989136 + 0.147002i \(0.953038\pi\)
\(864\) 0 0
\(865\) −1.06678e9 −1.64827
\(866\) 0 0
\(867\) 0 0
\(868\) −2.14946e9 −3.28677
\(869\) −1.66017e8 −0.252984
\(870\) 0 0
\(871\) −2.55591e9 −3.86805
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.18713e9 1.77205
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −3.69635e8 −0.542407
\(881\) −1.03788e9 −1.51782 −0.758910 − 0.651195i \(-0.774268\pi\)
−0.758910 + 0.651195i \(0.774268\pi\)
\(882\) 0 0
\(883\) 1.04341e9 1.51556 0.757782 − 0.652507i \(-0.226283\pi\)
0.757782 + 0.652507i \(0.226283\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −2.20716e9 −3.14144
\(890\) 0 0
\(891\) −4.32062e8 −0.610819
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.47139e9 −2.02511
\(900\) −1.54151e8 −0.211456
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.31293e9 1.77132
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −4.01094e8 −0.530507 −0.265253 − 0.964179i \(-0.585456\pi\)
−0.265253 + 0.964179i \(0.585456\pi\)
\(912\) 0 0
\(913\) −7.39761e8 −0.972029
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.40744e9 3.12211
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.63901e9 −2.08438
\(924\) 0 0
\(925\) 2.53417e7 0.0320192
\(926\) 0 0
\(927\) 0 0
\(928\) 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
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 1.47117e9 1.77124
\(941\) 1.89239e8 0.227113 0.113556 − 0.993532i \(-0.463776\pi\)
0.113556 + 0.993532i \(0.463776\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.51335e9 −1.78193 −0.890963 − 0.454075i \(-0.849970\pi\)
−0.890963 + 0.454075i \(0.849970\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −1.35928e9 −1.56062
\(956\) 8.25073e8 0.944319
\(957\) 0 0
\(958\) 0 0
\(959\) 2.28153e9 2.58684
\(960\) 0 0
\(961\) 2.64597e9 2.98136
\(962\) 0 0
\(963\) −1.57038e9 −1.75843
\(964\) 0 0
\(965\) −1.01031e9 −1.12428
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −7.69085e8 −0.840072 −0.420036 − 0.907507i \(-0.637983\pi\)
−0.420036 + 0.907507i \(0.637983\pi\)
\(972\) 0 0
\(973\) 1.51737e9 1.64723
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.13134e9 −1.21313 −0.606566 − 0.795033i \(-0.707453\pi\)
−0.606566 + 0.795033i \(0.707453\pi\)
\(978\) 0 0
\(979\) −8.04433e8 −0.857318
\(980\) 1.43200e9 1.52147
\(981\) 0 0
\(982\) 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
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 6.73699e8 0.679799 0.339900 − 0.940462i \(-0.389607\pi\)
0.339900 + 0.940462i \(0.389607\pi\)
\(998\) 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 139.7.b.a.138.1 ✓ 1
139.138 odd 2 CM 139.7.b.a.138.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
139.7.b.a.138.1 ✓ 1 1.1 even 1 trivial
139.7.b.a.138.1 ✓ 1 139.138 odd 2 CM