↑

Properties

Label 12.47.c.a.5.1
Level $12$
Weight $47$
Character 12.5
Self dual yes
Analytic conductor $160.822$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [12,47,Mod(5,12)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12.5"); S:= CuspForms(chi, 47); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 47, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 47 \)
Character orbit: \([\chi]\) \(=\) 12.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(160.822042359\)
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 5.1
Character \(\chi\) \(=\) 12.5

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.41432e10 q^{3} +8.47209e18 q^{7} +8.86294e21 q^{9} -7.81183e25 q^{13} +5.14956e29 q^{19} -7.97589e29 q^{21} +1.42109e32 q^{25} -8.34385e32 q^{27} +1.47482e34 q^{31} -2.15447e36 q^{37} +7.35431e36 q^{39} -2.42997e37 q^{43} -6.77272e38 q^{49} -4.84796e40 q^{57} -1.52029e41 q^{61} +7.50876e40 q^{63} -1.85380e42 q^{67} +1.27340e43 q^{73} -1.33786e43 q^{75} +4.66093e43 q^{79} +7.85517e43 q^{81} -6.61825e44 q^{91} -1.38844e45 q^{93} -2.33167e45 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) −9.41432e10 −1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 8.47209e18 0.309553 0.154777 − 0.987949i \(-0.450534\pi\)
0.154777 + 0.987949i \(0.450534\pi\)
\(8\) 0 0
\(9\) 8.86294e21 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −7.81183e25 −1.87092 −0.935461 − 0.353430i \(-0.885015\pi\)
−0.935461 + 0.353430i \(0.885015\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 5.14956e29 1.99727 0.998637 − 0.0521909i \(-0.0166204\pi\)
0.998637 + 0.0521909i \(0.0166204\pi\)
\(20\) 0 0
\(21\) −7.97589e29 −0.309553
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 1.42109e32 1.00000
\(26\) 0 0
\(27\) −8.34385e32 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.47482e34 0.736920 0.368460 − 0.929644i \(-0.379885\pi\)
0.368460 + 0.929644i \(0.379885\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.15447e36 −1.83951 −0.919754 − 0.392495i \(-0.871612\pi\)
−0.919754 + 0.392495i \(0.871612\pi\)
\(38\) 0 0
\(39\) 7.35431e36 1.87092
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −2.42997e37 −0.654374 −0.327187 − 0.944960i \(-0.606101\pi\)
−0.327187 + 0.944960i \(0.606101\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −6.77272e38 −0.904177
\(50\) 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
\(57\) −4.84796e40 −1.99727
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −1.52029e41 −1.31625 −0.658127 − 0.752907i \(-0.728651\pi\)
−0.658127 + 0.752907i \(0.728651\pi\)
\(62\) 0 0
\(63\) 7.50876e40 0.309553
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.85380e42 −1.85499 −0.927497 − 0.373830i \(-0.878044\pi\)
−0.927497 + 0.373830i \(0.878044\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.27340e43 1.77231 0.886154 − 0.463392i \(-0.153368\pi\)
0.886154 + 0.463392i \(0.153368\pi\)
\(74\) 0 0
\(75\) −1.33786e43 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.66093e43 1.05451 0.527253 − 0.849708i \(-0.323222\pi\)
0.527253 + 0.849708i \(0.323222\pi\)
\(80\) 0 0
\(81\) 7.85517e43 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
\(91\) −6.61825e44 −0.579150
\(92\) 0 0
\(93\) −1.38844e45 −0.736920
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.33167e45 −0.469804 −0.234902 − 0.972019i \(-0.575477\pi\)
−0.234902 + 0.972019i \(0.575477\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −1.38840e45 −0.0703492 −0.0351746 − 0.999381i \(-0.511199\pi\)
−0.0351746 + 0.999381i \(0.511199\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 4.28447e46 0.590320 0.295160 − 0.955448i \(-0.404627\pi\)
0.295160 + 0.955448i \(0.404627\pi\)
\(110\) 0 0
\(111\) 2.02829e47 1.83951
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −6.92358e47 −1.87092
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 8.01795e47 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −4.87427e48 −1.99721 −0.998606 − 0.0527804i \(-0.983192\pi\)
−0.998606 + 0.0527804i \(0.983192\pi\)
\(128\) 0 0
\(129\) 2.28765e48 0.654374
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 4.36276e48 0.618263
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 3.77151e49 1.93720 0.968602 − 0.248618i \(-0.0799762\pi\)
0.968602 + 0.248618i \(0.0799762\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.37605e49 0.904177
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −8.56509e49 −0.655031 −0.327516 − 0.944846i \(-0.606211\pi\)
−0.327516 + 0.944846i \(0.606211\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.42496e50 1.06896 0.534482 − 0.845180i \(-0.320507\pi\)
0.534482 + 0.845180i \(0.320507\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 7.56592e50 0.996660 0.498330 − 0.866987i \(-0.333947\pi\)
0.498330 + 0.866987i \(0.333947\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 4.35908e51 2.50035
\(170\) 0 0
\(171\) 4.56403e51 1.99727
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 1.20396e51 0.309553
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 1.40617e52 1.66506 0.832531 − 0.553978i \(-0.186891\pi\)
0.832531 + 0.553978i \(0.186891\pi\)
\(182\) 0 0
\(183\) 1.43125e52 1.31625
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −7.06899e51 −0.309553
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 7.28383e52 1.97034 0.985171 − 0.171573i \(-0.0548849\pi\)
0.985171 + 0.171573i \(0.0548849\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1.21596e53 1.62667 0.813336 − 0.581794i \(-0.197649\pi\)
0.813336 + 0.581794i \(0.197649\pi\)
\(200\) 0 0
\(201\) 1.74523e53 1.85499
\(202\) 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
\(211\) 7.23132e52 0.251606 0.125803 − 0.992055i \(-0.459849\pi\)
0.125803 + 0.992055i \(0.459849\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.24948e53 0.228116
\(218\) 0 0
\(219\) −1.19882e54 −1.77231
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.64517e54 −1.60398 −0.801990 − 0.597337i \(-0.796225\pi\)
−0.801990 + 0.597337i \(0.796225\pi\)
\(224\) 0 0
\(225\) 1.25950e54 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −3.64656e54 −1.93049 −0.965245 − 0.261347i \(-0.915833\pi\)
−0.965245 + 0.261347i \(0.915833\pi\)
\(230\) 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
\(237\) −4.38794e54 −1.05451
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 1.08867e55 1.78034 0.890171 − 0.455626i \(-0.150585\pi\)
0.890171 + 0.455626i \(0.150585\pi\)
\(242\) 0 0
\(243\) −7.39510e54 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.02275e55 −3.73674
\(248\) 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
\(259\) −1.82529e55 −0.569426
\(260\) 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
\(271\) 1.70661e56 1.87863 0.939315 − 0.343055i \(-0.111462\pi\)
0.939315 + 0.343055i \(0.111462\pi\)
\(272\) 0 0
\(273\) 6.23063e55 0.579150
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.77565e56 1.84642 0.923208 − 0.384301i \(-0.125558\pi\)
0.923208 + 0.384301i \(0.125558\pi\)
\(278\) 0 0
\(279\) 1.30712e56 0.736920
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −1.48967e56 −0.605344 −0.302672 − 0.953095i \(-0.597879\pi\)
−0.302672 + 0.953095i \(0.597879\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3.98704e56 1.00000
\(290\) 0 0
\(291\) 2.19510e56 0.469804
\(292\) 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
\(301\) −2.05869e56 −0.202564
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3.14866e57 1.96763 0.983815 − 0.179186i \(-0.0573464\pi\)
0.983815 + 0.179186i \(0.0573464\pi\)
\(308\) 0 0
\(309\) 1.30709e56 0.0703492
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −4.25674e57 −1.70434 −0.852172 − 0.523262i \(-0.824715\pi\)
−0.852172 + 0.523262i \(0.824715\pi\)
\(314\) 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
\(325\) −1.11013e58 −1.87092
\(326\) 0 0
\(327\) −4.03354e57 −0.590320
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.21977e58 −1.34969 −0.674845 − 0.737959i \(-0.735790\pi\)
−0.674845 + 0.737959i \(0.735790\pi\)
\(332\) 0 0
\(333\) −1.90950e58 −1.83951
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.36425e58 0.998630 0.499315 − 0.866420i \(-0.333585\pi\)
0.499315 + 0.866420i \(0.333585\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.20839e58 −0.589444
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −2.31533e58 −0.757928 −0.378964 − 0.925411i \(-0.623720\pi\)
−0.378964 + 0.925411i \(0.623720\pi\)
\(350\) 0 0
\(351\) 6.51808e58 1.87092
\(352\) 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
\(361\) 1.98704e59 2.98910
\(362\) 0 0
\(363\) −7.54836e58 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.91483e58 −0.197157 −0.0985783 − 0.995129i \(-0.531429\pi\)
−0.0985783 + 0.995129i \(0.531429\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.20565e59 −0.854905 −0.427453 − 0.904038i \(-0.640589\pi\)
−0.427453 + 0.904038i \(0.640589\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 4.05314e59 1.99108 0.995542 − 0.0943237i \(-0.0300689\pi\)
0.995542 + 0.0943237i \(0.0300689\pi\)
\(380\) 0 0
\(381\) 4.58880e59 1.99721
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.15366e59 −0.654374
\(388\) 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
\(397\) 1.17682e60 1.98851 0.994253 − 0.107055i \(-0.0341421\pi\)
0.994253 + 0.107055i \(0.0341421\pi\)
\(398\) 0 0
\(399\) −4.10724e59 −0.618263
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −1.15211e60 −1.37872
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −2.18350e60 −1.86002 −0.930010 − 0.367535i \(-0.880202\pi\)
−0.930010 + 0.367535i \(0.880202\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3.55062e60 −1.93720
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −2.85627e60 −1.25116 −0.625578 − 0.780162i \(-0.715137\pi\)
−0.625578 + 0.780162i \(0.715137\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.28800e60 −0.407451
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 5.20908e60 1.19547 0.597736 − 0.801693i \(-0.296067\pi\)
0.597736 + 0.801693i \(0.296067\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 7.46833e60 1.24893 0.624467 − 0.781051i \(-0.285316\pi\)
0.624467 + 0.781051i \(0.285316\pi\)
\(440\) 0 0
\(441\) −6.00262e60 −0.904177
\(442\) 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
\(453\) 8.06344e60 0.655031
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.08716e60 0.271237 0.135618 − 0.990761i \(-0.456698\pi\)
0.135618 + 0.990761i \(0.456698\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 3.94908e61 1.94148 0.970740 − 0.240135i \(-0.0771916\pi\)
0.970740 + 0.240135i \(0.0771916\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −1.57056e61 −0.574220
\(470\) 0 0
\(471\) −3.22436e61 −1.06896
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7.31797e61 1.99727
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 1.68304e62 3.44158
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.15823e62 −1.78085 −0.890427 − 0.455126i \(-0.849594\pi\)
−0.890427 + 0.455126i \(0.849594\pi\)
\(488\) 0 0
\(489\) −7.12280e61 −0.996660
\(490\) 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
\(499\) −1.28425e62 −1.12807 −0.564037 − 0.825750i \(-0.690752\pi\)
−0.564037 + 0.825750i \(0.690752\pi\)
\(500\) 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
\(507\) −4.10378e62 −2.50035
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 1.07884e62 0.548624
\(512\) 0 0
\(513\) −4.29672e62 −1.99727
\(514\) 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
\(523\) 1.55122e62 0.462526 0.231263 − 0.972891i \(-0.425714\pi\)
0.231263 + 0.972891i \(0.425714\pi\)
\(524\) 0 0
\(525\) −1.13344e62 −0.309553
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 4.35994e62 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
\(541\) 1.21106e63 1.65816 0.829082 − 0.559127i \(-0.188864\pi\)
0.829082 + 0.559127i \(0.188864\pi\)
\(542\) 0 0
\(543\) −1.32381e63 −1.66506
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.87870e63 1.99594 0.997971 − 0.0636705i \(-0.0202807\pi\)
0.997971 + 0.0636705i \(0.0202807\pi\)
\(548\) 0 0
\(549\) −1.34742e63 −1.31625
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 3.94878e62 0.326426
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 1.89825e63 1.22428
\(560\) 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
\(567\) 6.65497e62 0.309553
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 5.05425e63 1.99998 0.999990 − 0.00439516i \(-0.00139903\pi\)
0.999990 + 0.00439516i \(0.00139903\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 4.03397e63 1.25513 0.627566 − 0.778563i \(-0.284051\pi\)
0.627566 + 0.778563i \(0.284051\pi\)
\(578\) 0 0
\(579\) −6.85723e63 −1.97034
\(580\) 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
\(589\) 7.59469e63 1.47183
\(590\) 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
\(597\) −1.14475e64 −1.62667
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.37442e64 −1.67497 −0.837485 − 0.546461i \(-0.815975\pi\)
−0.837485 + 0.546461i \(0.815975\pi\)
\(602\) 0 0
\(603\) −1.64301e64 −1.85499
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −1.91821e64 −1.86019 −0.930093 − 0.367323i \(-0.880274\pi\)
−0.930093 + 0.367323i \(0.880274\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 2.57947e64 1.99498 0.997492 − 0.0707754i \(-0.0225474\pi\)
0.997492 + 0.0707754i \(0.0225474\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −2.43966e64 −1.50815 −0.754075 − 0.656788i \(-0.771915\pi\)
−0.754075 + 0.656788i \(0.771915\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.01948e64 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.92262e64 −0.764218 −0.382109 − 0.924117i \(-0.624802\pi\)
−0.382109 + 0.924117i \(0.624802\pi\)
\(632\) 0 0
\(633\) −6.80779e63 −0.251606
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5.29073e64 1.69164
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 4.44546e64 1.14568 0.572842 − 0.819666i \(-0.305841\pi\)
0.572842 + 0.819666i \(0.305841\pi\)
\(644\) 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
\(651\) −1.17630e64 −0.228116
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.12861e65 1.77231
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.24478e65 −1.70004 −0.850022 − 0.526747i \(-0.823411\pi\)
−0.850022 + 0.526747i \(0.823411\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1.54882e65 1.60398
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.19616e65 1.98298 0.991489 − 0.130193i \(-0.0415596\pi\)
0.991489 + 0.130193i \(0.0415596\pi\)
\(674\) 0 0
\(675\) −1.18573e65 −1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −1.97541e64 −0.145429
\(680\) 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
\(687\) 3.43299e65 1.93049
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −2.99119e65 −1.47180 −0.735899 − 0.677091i \(-0.763240\pi\)
−0.735899 + 0.677091i \(0.763240\pi\)
\(692\) 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
\(703\) −1.10946e66 −3.67400
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −5.42041e65 −1.47628 −0.738140 − 0.674647i \(-0.764296\pi\)
−0.738140 + 0.674647i \(0.764296\pi\)
\(710\) 0 0
\(711\) 4.13095e65 1.05451
\(712\) 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
\(721\) −1.17627e64 −0.0217768
\(722\) 0 0
\(723\) −1.02491e66 −1.78034
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.30270e66 −1.99320 −0.996602 − 0.0823652i \(-0.973753\pi\)
−0.996602 + 0.0823652i \(0.973753\pi\)
\(728\) 0 0
\(729\) 6.96199e65 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.17950e66 1.49385 0.746924 − 0.664909i \(-0.231530\pi\)
0.746924 + 0.664909i \(0.231530\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −7.35874e65 −0.772648 −0.386324 − 0.922363i \(-0.626255\pi\)
−0.386324 + 0.922363i \(0.626255\pi\)
\(740\) 0 0
\(741\) 3.78715e66 3.73674
\(742\) 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
\(751\) −1.67660e66 −1.21538 −0.607689 − 0.794175i \(-0.707903\pi\)
−0.607689 + 0.794175i \(0.707903\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.64501e66 1.59670 0.798348 − 0.602196i \(-0.205708\pi\)
0.798348 + 0.602196i \(0.205708\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 3.62984e65 0.182736
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 4.19812e65 0.176502 0.0882509 − 0.996098i \(-0.471872\pi\)
0.0882509 + 0.996098i \(0.471872\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 2.09585e66 0.736920
\(776\) 0 0
\(777\) 1.71839e66 0.569426
\(778\) 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
\(787\) −7.73829e66 −1.91085 −0.955425 − 0.295235i \(-0.904602\pi\)
−0.955425 + 0.295235i \(0.904602\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.18762e67 2.46261
\(794\) 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
\(811\) −8.94562e66 −1.10696 −0.553478 − 0.832864i \(-0.686700\pi\)
−0.553478 + 0.832864i \(0.686700\pi\)
\(812\) 0 0
\(813\) −1.60666e67 −1.87863
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.25133e67 −1.30696
\(818\) 0 0
\(819\) −5.86572e66 −0.579150
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 7.73854e66 0.683066 0.341533 − 0.939870i \(-0.389054\pi\)
0.341533 + 0.939870i \(0.389054\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −4.10657e66 −0.306708 −0.153354 − 0.988171i \(-0.549007\pi\)
−0.153354 + 0.988171i \(0.549007\pi\)
\(830\) 0 0
\(831\) −2.61308e67 −1.84642
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.23057e67 −0.736920
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.86341e67 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 6.79288e66 0.309553
\(848\) 0 0
\(849\) 1.40242e67 0.605344
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 3.54334e67 1.37273 0.686366 − 0.727257i \(-0.259205\pi\)
0.686366 + 0.727257i \(0.259205\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 2.01655e67 0.664914 0.332457 − 0.943118i \(-0.392122\pi\)
0.332457 + 0.943118i \(0.392122\pi\)
\(860\) 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
\(867\) −3.75352e67 −1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 1.44816e68 3.47055
\(872\) 0 0
\(873\) −2.06654e67 −0.469804
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 9.66603e67 1.97814 0.989071 − 0.147439i \(-0.0471030\pi\)
0.989071 + 0.147439i \(0.0471030\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1.14023e68 −1.99478 −0.997392 − 0.0721752i \(-0.977006\pi\)
−0.997392 + 0.0721752i \(0.977006\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −4.12953e67 −0.618244
\(890\) 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
\(903\) 1.93812e67 0.202564
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 8.67488e67 0.819023 0.409511 − 0.912305i \(-0.365699\pi\)
0.409511 + 0.912305i \(0.365699\pi\)
\(908\) 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
\(919\) 2.74765e68 1.91736 0.958682 − 0.284479i \(-0.0918207\pi\)
0.958682 + 0.284479i \(0.0918207\pi\)
\(920\) 0 0
\(921\) −2.96425e68 −1.96763
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −3.06169e68 −1.83951
\(926\) 0 0
\(927\) −1.23053e67 −0.0703492
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −3.48766e68 −1.80589
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.08077e68 −0.482749 −0.241375 − 0.970432i \(-0.577598\pi\)
−0.241375 + 0.970432i \(0.577598\pi\)
\(938\) 0 0
\(939\) 4.00743e68 1.70434
\(940\) 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
\(949\) −9.94758e68 −3.31585
\(950\) 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
\(961\) −1.83023e68 −0.456949
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 7.37119e68 1.59488 0.797442 − 0.603396i \(-0.206186\pi\)
0.797442 + 0.603396i \(0.206186\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 3.19526e68 0.599668
\(974\) 0 0
\(975\) 1.04511e69 1.87092
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 3.79730e68 0.590320
\(982\) 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
\(991\) 8.26640e68 1.01771 0.508853 − 0.860853i \(-0.330070\pi\)
0.508853 + 0.860853i \(0.330070\pi\)
\(992\) 0 0
\(993\) 1.14833e69 1.34969
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −9.58041e68 −1.02659 −0.513293 − 0.858213i \(-0.671575\pi\)
−0.513293 + 0.858213i \(0.671575\pi\)
\(998\) 0 0
\(999\) 1.79766e69 1.83951
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 12.47.c.a.5.1 ✓ 1
3.2 odd 2 CM 12.47.c.a.5.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.47.c.a.5.1 ✓ 1 1.1 even 1 trivial
12.47.c.a.5.1 ✓ 1 3.2 odd 2 CM