Properties

Label 115.3.c.a.114.1
Level $115$
Weight $3$
Character 115.114
Self dual yes
Analytic conductor $3.134$
Analytic rank $0$
Dimension $1$
CM discriminant -115
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] = [115,3,Mod(114,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.114");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(3.13352304014\)
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 114.1
Character \(\chi\) \(=\) 115.114

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{4} -5.00000 q^{5} +9.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+4.00000 q^{4} -5.00000 q^{5} +9.00000 q^{7} +9.00000 q^{9} +16.0000 q^{16} -11.0000 q^{17} -20.0000 q^{20} +23.0000 q^{23} +25.0000 q^{25} +36.0000 q^{28} -57.0000 q^{29} -53.0000 q^{31} -45.0000 q^{35} +36.0000 q^{36} -51.0000 q^{37} -33.0000 q^{41} +6.00000 q^{43} -45.0000 q^{45} +32.0000 q^{49} +101.000 q^{53} +3.00000 q^{59} +81.0000 q^{63} +64.0000 q^{64} -111.000 q^{67} -44.0000 q^{68} +27.0000 q^{71} -80.0000 q^{80} +81.0000 q^{81} +41.0000 q^{83} +55.0000 q^{85} +92.0000 q^{92} +174.000 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 4.00000 1.00000
\(5\) −5.00000 −1.00000
\(6\) 0 0
\(7\) 9.00000 1.28571 0.642857 0.765986i \(-0.277749\pi\)
0.642857 + 0.765986i \(0.277749\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) −11.0000 −0.647059 −0.323529 0.946218i \(-0.604869\pi\)
−0.323529 + 0.946218i \(0.604869\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −20.0000 −1.00000
\(21\) 0 0
\(22\) 0 0
\(23\) 23.0000 1.00000
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 36.0000 1.28571
\(29\) −57.0000 −1.96552 −0.982759 0.184893i \(-0.940806\pi\)
−0.982759 + 0.184893i \(0.940806\pi\)
\(30\) 0 0
\(31\) −53.0000 −1.70968 −0.854839 0.518894i \(-0.826344\pi\)
−0.854839 + 0.518894i \(0.826344\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −45.0000 −1.28571
\(36\) 36.0000 1.00000
\(37\) −51.0000 −1.37838 −0.689189 0.724581i \(-0.742033\pi\)
−0.689189 + 0.724581i \(0.742033\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −33.0000 −0.804878 −0.402439 0.915447i \(-0.631837\pi\)
−0.402439 + 0.915447i \(0.631837\pi\)
\(42\) 0 0
\(43\) 6.00000 0.139535 0.0697674 0.997563i \(-0.477774\pi\)
0.0697674 + 0.997563i \(0.477774\pi\)
\(44\) 0 0
\(45\) −45.0000 −1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 32.0000 0.653061
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 101.000 1.90566 0.952830 0.303504i \(-0.0981565\pi\)
0.952830 + 0.303504i \(0.0981565\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.00000 0.0508475 0.0254237 0.999677i \(-0.491907\pi\)
0.0254237 + 0.999677i \(0.491907\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 81.0000 1.28571
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −111.000 −1.65672 −0.828358 0.560199i \(-0.810725\pi\)
−0.828358 + 0.560199i \(0.810725\pi\)
\(68\) −44.0000 −0.647059
\(69\) 0 0
\(70\) 0 0
\(71\) 27.0000 0.380282 0.190141 0.981757i \(-0.439106\pi\)
0.190141 + 0.981757i \(0.439106\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −80.0000 −1.00000
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 41.0000 0.493976 0.246988 0.969019i \(-0.420559\pi\)
0.246988 + 0.969019i \(0.420559\pi\)
\(84\) 0 0
\(85\) 55.0000 0.647059
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 92.0000 1.00000
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 174.000 1.79381 0.896907 0.442219i \(-0.145809\pi\)
0.896907 + 0.442219i \(0.145809\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 100.000 1.00000
\(101\) 87.0000 0.861386 0.430693 0.902498i \(-0.358269\pi\)
0.430693 + 0.902498i \(0.358269\pi\)
\(102\) 0 0
\(103\) −114.000 −1.10680 −0.553398 0.832917i \(-0.686669\pi\)
−0.553398 + 0.832917i \(0.686669\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −191.000 −1.78505 −0.892523 0.451001i \(-0.851067\pi\)
−0.892523 + 0.451001i \(0.851067\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 144.000 1.28571
\(113\) −19.0000 −0.168142 −0.0840708 0.996460i \(-0.526792\pi\)
−0.0840708 + 0.996460i \(0.526792\pi\)
\(114\) 0 0
\(115\) −115.000 −1.00000
\(116\) −228.000 −1.96552
\(117\) 0 0
\(118\) 0 0
\(119\) −99.0000 −0.831933
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −212.000 −1.70968
\(125\) −125.000 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −198.000 −1.51145 −0.755725 0.654889i \(-0.772715\pi\)
−0.755725 + 0.654889i \(0.772715\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 94.0000 0.686131 0.343066 0.939311i \(-0.388535\pi\)
0.343066 + 0.939311i \(0.388535\pi\)
\(138\) 0 0
\(139\) 163.000 1.17266 0.586331 0.810072i \(-0.300572\pi\)
0.586331 + 0.810072i \(0.300572\pi\)
\(140\) −180.000 −1.28571
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 144.000 1.00000
\(145\) 285.000 1.96552
\(146\) 0 0
\(147\) 0 0
\(148\) −204.000 −1.37838
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −158.000 −1.04636 −0.523179 0.852223i \(-0.675254\pi\)
−0.523179 + 0.852223i \(0.675254\pi\)
\(152\) 0 0
\(153\) −99.0000 −0.647059
\(154\) 0 0
\(155\) 265.000 1.70968
\(156\) 0 0
\(157\) −291.000 −1.85350 −0.926752 0.375675i \(-0.877411\pi\)
−0.926752 + 0.375675i \(0.877411\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 207.000 1.28571
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −132.000 −0.804878
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 24.0000 0.139535
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 225.000 1.28571
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −102.000 −0.569832 −0.284916 0.958552i \(-0.591966\pi\)
−0.284916 + 0.958552i \(0.591966\pi\)
\(180\) −180.000 −1.00000
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 255.000 1.37838
\(186\) 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
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 128.000 0.653061
\(197\) 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
\(203\) −513.000 −2.52709
\(204\) 0 0
\(205\) 165.000 0.804878
\(206\) 0 0
\(207\) 207.000 1.00000
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 307.000 1.45498 0.727488 0.686120i \(-0.240688\pi\)
0.727488 + 0.686120i \(0.240688\pi\)
\(212\) 404.000 1.90566
\(213\) 0 0
\(214\) 0 0
\(215\) −30.0000 −0.139535
\(216\) 0 0
\(217\) −477.000 −2.19816
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 225.000 1.00000
\(226\) 0 0
\(227\) 374.000 1.64758 0.823789 0.566897i \(-0.191856\pi\)
0.823789 + 0.566897i \(0.191856\pi\)
\(228\) 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
\(235\) 0 0
\(236\) 12.0000 0.0508475
\(237\) 0 0
\(238\) 0 0
\(239\) 363.000 1.51883 0.759414 0.650607i \(-0.225486\pi\)
0.759414 + 0.650607i \(0.225486\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −160.000 −0.653061
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 324.000 1.28571
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −459.000 −1.77220
\(260\) 0 0
\(261\) −513.000 −1.96552
\(262\) 0 0
\(263\) −319.000 −1.21293 −0.606464 0.795111i \(-0.707413\pi\)
−0.606464 + 0.795111i \(0.707413\pi\)
\(264\) 0 0
\(265\) −505.000 −1.90566
\(266\) 0 0
\(267\) 0 0
\(268\) −444.000 −1.65672
\(269\) 423.000 1.57249 0.786245 0.617914i \(-0.212022\pi\)
0.786245 + 0.617914i \(0.212022\pi\)
\(270\) 0 0
\(271\) −493.000 −1.81919 −0.909594 0.415498i \(-0.863607\pi\)
−0.909594 + 0.415498i \(0.863607\pi\)
\(272\) −176.000 −0.647059
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −477.000 −1.70968
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 561.000 1.98233 0.991166 0.132627i \(-0.0423411\pi\)
0.991166 + 0.132627i \(0.0423411\pi\)
\(284\) 108.000 0.380282
\(285\) 0 0
\(286\) 0 0
\(287\) −297.000 −1.03484
\(288\) 0 0
\(289\) −168.000 −0.581315
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 541.000 1.84642 0.923208 0.384300i \(-0.125557\pi\)
0.923208 + 0.384300i \(0.125557\pi\)
\(294\) 0 0
\(295\) −15.0000 −0.0508475
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 54.0000 0.179402
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 162.000 0.520900 0.260450 0.965487i \(-0.416129\pi\)
0.260450 + 0.965487i \(0.416129\pi\)
\(312\) 0 0
\(313\) 501.000 1.60064 0.800319 0.599574i \(-0.204663\pi\)
0.800319 + 0.599574i \(0.204663\pi\)
\(314\) 0 0
\(315\) −405.000 −1.28571
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −320.000 −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 324.000 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −373.000 −1.12689 −0.563444 0.826154i \(-0.690524\pi\)
−0.563444 + 0.826154i \(0.690524\pi\)
\(332\) 164.000 0.493976
\(333\) −459.000 −1.37838
\(334\) 0 0
\(335\) 555.000 1.65672
\(336\) 0 0
\(337\) −306.000 −0.908012 −0.454006 0.890999i \(-0.650006\pi\)
−0.454006 + 0.890999i \(0.650006\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 220.000 0.647059
\(341\) 0 0
\(342\) 0 0
\(343\) −153.000 −0.446064
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −337.000 −0.965616 −0.482808 0.875726i \(-0.660383\pi\)
−0.482808 + 0.875726i \(0.660383\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −135.000 −0.380282
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −711.000 −1.93733 −0.968665 0.248372i \(-0.920105\pi\)
−0.968665 + 0.248372i \(0.920105\pi\)
\(368\) 368.000 1.00000
\(369\) −297.000 −0.804878
\(370\) 0 0
\(371\) 909.000 2.45013
\(372\) 0 0
\(373\) 726.000 1.94638 0.973190 0.230001i \(-0.0738730\pi\)
0.973190 + 0.230001i \(0.0738730\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 361.000 0.942559 0.471279 0.881984i \(-0.343792\pi\)
0.471279 + 0.881984i \(0.343792\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 54.0000 0.139535
\(388\) 696.000 1.79381
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) −253.000 −0.647059
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 400.000 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 348.000 0.861386
\(405\) −405.000 −1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 703.000 1.71883 0.859413 0.511282i \(-0.170829\pi\)
0.859413 + 0.511282i \(0.170829\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −456.000 −1.10680
\(413\) 27.0000 0.0653753
\(414\) 0 0
\(415\) −205.000 −0.493976
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −275.000 −0.647059
\(426\) 0 0
\(427\) 0 0
\(428\) −764.000 −1.78505
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 261.000 0.602771 0.301386 0.953502i \(-0.402551\pi\)
0.301386 + 0.953502i \(0.402551\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 418.000 0.952164 0.476082 0.879401i \(-0.342057\pi\)
0.476082 + 0.879401i \(0.342057\pi\)
\(440\) 0 0
\(441\) 288.000 0.653061
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 576.000 1.28571
\(449\) 783.000 1.74388 0.871938 0.489617i \(-0.162863\pi\)
0.871938 + 0.489617i \(0.162863\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −76.0000 −0.168142
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −891.000 −1.94967 −0.974836 0.222924i \(-0.928440\pi\)
−0.974836 + 0.222924i \(0.928440\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −460.000 −1.00000
\(461\) −918.000 −1.99132 −0.995662 0.0930482i \(-0.970339\pi\)
−0.995662 + 0.0930482i \(0.970339\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −912.000 −1.96552
\(465\) 0 0
\(466\) 0 0
\(467\) 929.000 1.98929 0.994647 0.103334i \(-0.0329512\pi\)
0.994647 + 0.103334i \(0.0329512\pi\)
\(468\) 0 0
\(469\) −999.000 −2.13006
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −396.000 −0.831933
\(477\) 909.000 1.90566
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 484.000 1.00000
\(485\) −870.000 −1.79381
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 867.000 1.76578 0.882892 0.469576i \(-0.155593\pi\)
0.882892 + 0.469576i \(0.155593\pi\)
\(492\) 0 0
\(493\) 627.000 1.27181
\(494\) 0 0
\(495\) 0 0
\(496\) −848.000 −1.70968
\(497\) 243.000 0.488934
\(498\) 0 0
\(499\) −37.0000 −0.0741483 −0.0370741 0.999313i \(-0.511804\pi\)
−0.0370741 + 0.999313i \(0.511804\pi\)
\(500\) −500.000 −1.00000
\(501\) 0 0
\(502\) 0 0
\(503\) −799.000 −1.58847 −0.794235 0.607611i \(-0.792128\pi\)
−0.794235 + 0.607611i \(0.792128\pi\)
\(504\) 0 0
\(505\) −435.000 −0.861386
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −822.000 −1.61493 −0.807466 0.589915i \(-0.799161\pi\)
−0.807466 + 0.589915i \(0.799161\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 570.000 1.10680
\(516\) 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
\(523\) −954.000 −1.82409 −0.912046 0.410088i \(-0.865498\pi\)
−0.912046 + 0.410088i \(0.865498\pi\)
\(524\) −792.000 −1.51145
\(525\) 0 0
\(526\) 0 0
\(527\) 583.000 1.10626
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 27.0000 0.0508475
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 955.000 1.78505
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −758.000 −1.40111 −0.700555 0.713599i \(-0.747064\pi\)
−0.700555 + 0.713599i \(0.747064\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 376.000 0.686131
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 652.000 1.17266
\(557\) −1091.00 −1.95871 −0.979354 0.202154i \(-0.935206\pi\)
−0.979354 + 0.202154i \(0.935206\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −720.000 −1.28571
\(561\) 0 0
\(562\) 0 0
\(563\) 1.00000 0.00177620 0.000888099 1.00000i \(-0.499717\pi\)
0.000888099 1.00000i \(0.499717\pi\)
\(564\) 0 0
\(565\) 95.0000 0.168142
\(566\) 0 0
\(567\) 729.000 1.28571
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 575.000 1.00000
\(576\) 576.000 1.00000
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 1140.00 1.96552
\(581\) 369.000 0.635112
\(582\) 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
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −816.000 −1.37838
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 495.000 0.831933
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 738.000 1.23205 0.616027 0.787725i \(-0.288741\pi\)
0.616027 + 0.787725i \(0.288741\pi\)
\(600\) 0 0
\(601\) 167.000 0.277870 0.138935 0.990301i \(-0.455632\pi\)
0.138935 + 0.990301i \(0.455632\pi\)
\(602\) 0 0
\(603\) −999.000 −1.65672
\(604\) −632.000 −1.04636
\(605\) −605.000 −1.00000
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −396.000 −0.647059
\(613\) 246.000 0.401305 0.200653 0.979662i \(-0.435694\pi\)
0.200653 + 0.979662i \(0.435694\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 629.000 1.01945 0.509724 0.860338i \(-0.329747\pi\)
0.509724 + 0.860338i \(0.329747\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 1060.00 1.70968
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −1164.00 −1.85350
\(629\) 561.000 0.891892
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 243.000 0.380282
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −159.000 −0.247278 −0.123639 0.992327i \(-0.539457\pi\)
−0.123639 + 0.992327i \(0.539457\pi\)
\(644\) 828.000 1.28571
\(645\) 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
\(655\) 990.000 1.51145
\(656\) −528.000 −0.804878
\(657\) 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
\(667\) −1311.00 −1.96552
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 676.000 1.00000
\(677\) 509.000 0.751846 0.375923 0.926651i \(-0.377326\pi\)
0.375923 + 0.926651i \(0.377326\pi\)
\(678\) 0 0
\(679\) 1566.00 2.30633
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −470.000 −0.686131
\(686\) 0 0
\(687\) 0 0
\(688\) 96.0000 0.139535
\(689\) 0 0
\(690\) 0 0
\(691\) 922.000 1.33430 0.667149 0.744924i \(-0.267514\pi\)
0.667149 + 0.744924i \(0.267514\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −815.000 −1.17266
\(696\) 0 0
\(697\) 363.000 0.520803
\(698\) 0 0
\(699\) 0 0
\(700\) 900.000 1.28571
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 783.000 1.10750
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1219.00 −1.70968
\(714\) 0 0
\(715\) 0 0
\(716\) −408.000 −0.569832
\(717\) 0 0
\(718\) 0 0
\(719\) −1437.00 −1.99861 −0.999305 0.0372872i \(-0.988128\pi\)
−0.999305 + 0.0372872i \(0.988128\pi\)
\(720\) −720.000 −1.00000
\(721\) −1026.00 −1.42302
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1425.00 −1.96552
\(726\) 0 0
\(727\) 1329.00 1.82806 0.914030 0.405646i \(-0.132953\pi\)
0.914030 + 0.405646i \(0.132953\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) −66.0000 −0.0902873
\(732\) 0 0
\(733\) −339.000 −0.462483 −0.231241 0.972896i \(-0.574279\pi\)
−0.231241 + 0.972896i \(0.574279\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1397.00 −1.89039 −0.945196 0.326503i \(-0.894130\pi\)
−0.945196 + 0.326503i \(0.894130\pi\)
\(740\) 1020.00 1.37838
\(741\) 0 0
\(742\) 0 0
\(743\) −1394.00 −1.87618 −0.938089 0.346395i \(-0.887406\pi\)
−0.938089 + 0.346395i \(0.887406\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 369.000 0.493976
\(748\) 0 0
\(749\) −1719.00 −2.29506
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 790.000 1.04636
\(756\) 0 0
\(757\) 1269.00 1.67635 0.838177 0.545398i \(-0.183622\pi\)
0.838177 + 0.545398i \(0.183622\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1353.00 −1.77792 −0.888962 0.457981i \(-0.848573\pi\)
−0.888962 + 0.457981i \(0.848573\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 495.000 0.647059
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −74.0000 −0.0957309 −0.0478655 0.998854i \(-0.515242\pi\)
−0.0478655 + 0.998854i \(0.515242\pi\)
\(774\) 0 0
\(775\) −1325.00 −1.70968
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 512.000 0.653061
\(785\) 1455.00 1.85350
\(786\) 0 0
\(787\) −1551.00 −1.97078 −0.985388 0.170327i \(-0.945518\pi\)
−0.985388 + 0.170327i \(0.945518\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −171.000 −0.216182
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1189.00 1.49184 0.745922 0.666033i \(-0.232009\pi\)
0.745922 + 0.666033i \(0.232009\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −1035.00 −1.28571
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1257.00 −1.55377 −0.776885 0.629642i \(-0.783201\pi\)
−0.776885 + 0.629642i \(0.783201\pi\)
\(810\) 0 0
\(811\) 587.000 0.723798 0.361899 0.932217i \(-0.382129\pi\)
0.361899 + 0.932217i \(0.382129\pi\)
\(812\) −2052.00 −2.52709
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 660.000 0.804878
\(821\) −198.000 −0.241169 −0.120585 0.992703i \(-0.538477\pi\)
−0.120585 + 0.992703i \(0.538477\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 209.000 0.252721 0.126360 0.991984i \(-0.459670\pi\)
0.126360 + 0.991984i \(0.459670\pi\)
\(828\) 828.000 1.00000
\(829\) −1217.00 −1.46803 −0.734017 0.679131i \(-0.762357\pi\)
−0.734017 + 0.679131i \(0.762357\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −352.000 −0.422569
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 2408.00 2.86326
\(842\) 0 0
\(843\) 0 0
\(844\) 1228.00 1.45498
\(845\) −845.000 −1.00000
\(846\) 0 0
\(847\) 1089.00 1.28571
\(848\) 1616.00 1.90566
\(849\) 0 0
\(850\) 0 0
\(851\) −1173.00 −1.37838
\(852\) 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
\(859\) 683.000 0.795111 0.397555 0.917578i \(-0.369859\pi\)
0.397555 + 0.917578i \(0.369859\pi\)
\(860\) −120.000 −0.139535
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) −1908.00 −2.19816
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1566.00 1.79381
\(874\) 0 0
\(875\) −1125.00 −1.28571
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 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
\(895\) 510.000 0.569832
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3021.00 3.36040
\(900\) 900.000 1.00000
\(901\) −1111.00 −1.23307
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 969.000 1.06836 0.534179 0.845372i \(-0.320621\pi\)
0.534179 + 0.845372i \(0.320621\pi\)
\(908\) 1496.00 1.64758
\(909\) 783.000 0.861386
\(910\) 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
\(917\) −1782.00 −1.94329
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1275.00 −1.37838
\(926\) 0 0
\(927\) −1026.00 −1.10680
\(928\) 0 0
\(929\) −1017.00 −1.09473 −0.547363 0.836895i \(-0.684368\pi\)
−0.547363 + 0.836895i \(0.684368\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1506.00 −1.60726 −0.803629 0.595131i \(-0.797100\pi\)
−0.803629 + 0.595131i \(0.797100\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) −759.000 −0.804878
\(944\) 48.0000 0.0508475
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1406.00 1.47534 0.737671 0.675161i \(-0.235926\pi\)
0.737671 + 0.675161i \(0.235926\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1452.00 1.51883
\(957\) 0 0
\(958\) 0 0
\(959\) 846.000 0.882169
\(960\) 0 0
\(961\) 1848.00 1.92300
\(962\) 0 0
\(963\) −1719.00 −1.78505
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 1467.00 1.50771
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 829.000 0.848516 0.424258 0.905541i \(-0.360535\pi\)
0.424258 + 0.905541i \(0.360535\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −640.000 −0.653061
\(981\) 0 0
\(982\) 0 0
\(983\) 1921.00 1.95422 0.977111 0.212731i \(-0.0682357\pi\)
0.977111 + 0.212731i \(0.0682357\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 138.000 0.139535
\(990\) 0 0
\(991\) −893.000 −0.901110 −0.450555 0.892749i \(-0.648774\pi\)
−0.450555 + 0.892749i \(0.648774\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 115.3.c.a.114.1 1
5.2 odd 4 575.3.d.a.551.2 2
5.3 odd 4 575.3.d.a.551.1 2
5.4 even 2 115.3.c.b.114.1 yes 1
23.22 odd 2 115.3.c.b.114.1 yes 1
115.22 even 4 575.3.d.a.551.1 2
115.68 even 4 575.3.d.a.551.2 2
115.114 odd 2 CM 115.3.c.a.114.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.3.c.a.114.1 1 1.1 even 1 trivial
115.3.c.a.114.1 1 115.114 odd 2 CM
115.3.c.b.114.1 yes 1 5.4 even 2
115.3.c.b.114.1 yes 1 23.22 odd 2
575.3.d.a.551.1 2 5.3 odd 4
575.3.d.a.551.1 2 115.22 even 4
575.3.d.a.551.2 2 5.2 odd 4
575.3.d.a.551.2 2 115.68 even 4