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Properties

Label 12.15.c.a.5.1
Level $12$
Weight $15$
Character 12.5
Self dual yes
Analytic conductor $14.919$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [12,15,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, 15); 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, 15, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 15 \)
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: \(14.9194761782\)
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)
 
\(f(q)\) \(=\) \(q-2187.00 q^{3} -1.38902e6 q^{7} +4.78297e6 q^{9} +8.80720e7 q^{13} +1.51253e9 q^{19} +3.03779e9 q^{21} +6.10352e9 q^{25} -1.04604e10 q^{27} -2.30143e10 q^{31} -1.11626e11 q^{37} -1.92613e11 q^{39} +5.28052e11 q^{43} +1.25116e12 q^{49} -3.30790e12 q^{57} +6.21460e12 q^{61} -6.64365e12 q^{63} -1.19731e13 q^{67} +1.72784e13 q^{73} -1.33484e13 q^{75} +3.83831e13 q^{79} +2.28768e13 q^{81} -1.22334e14 q^{91} +5.03323e13 q^{93} -1.16517e13 q^{97} +O(q^{100})\)

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
\(3\) −2187.00 −1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −1.38902e6 −1.68664 −0.843321 − 0.537410i \(-0.819403\pi\)
−0.843321 + 0.537410i \(0.819403\pi\)
\(8\) 0 0
\(9\) 4.78297e6 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 8.80720e7 1.40357 0.701785 − 0.712389i \(-0.252387\pi\)
0.701785 + 0.712389i \(0.252387\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 1.51253e9 1.69211 0.846055 − 0.533096i \(-0.178971\pi\)
0.846055 + 0.533096i \(0.178971\pi\)
\(20\) 0 0
\(21\) 3.03779e9 1.68664
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 6.10352e9 1.00000
\(26\) 0 0
\(27\) −1.04604e10 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −2.30143e10 −0.836500 −0.418250 − 0.908332i \(-0.637356\pi\)
−0.418250 + 0.908332i \(0.637356\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.11626e11 −1.17585 −0.587927 − 0.808914i \(-0.700056\pi\)
−0.587927 + 0.808914i \(0.700056\pi\)
\(38\) 0 0
\(39\) −1.92613e11 −1.40357
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 5.28052e11 1.94266 0.971331 − 0.237732i \(-0.0764042\pi\)
0.971331 + 0.237732i \(0.0764042\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.25116e12 1.84476
\(50\) 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
\(57\) −3.30790e12 −1.69211
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 6.21460e12 1.97744 0.988722 − 0.149761i \(-0.0478503\pi\)
0.988722 + 0.149761i \(0.0478503\pi\)
\(62\) 0 0
\(63\) −6.64365e12 −1.68664
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.19731e13 −1.97553 −0.987767 − 0.155937i \(-0.950160\pi\)
−0.987767 + 0.155937i \(0.950160\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.72784e13 1.56403 0.782014 − 0.623261i \(-0.214192\pi\)
0.782014 + 0.623261i \(0.214192\pi\)
\(74\) 0 0
\(75\) −1.33484e13 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 3.83831e13 1.99871 0.999357 − 0.0358548i \(-0.0114154\pi\)
0.999357 + 0.0358548i \(0.0114154\pi\)
\(80\) 0 0
\(81\) 2.28768e13 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
\(91\) −1.22334e14 −2.36732
\(92\) 0 0
\(93\) 5.03323e13 0.836500
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.16517e13 −0.144207 −0.0721036 − 0.997397i \(-0.522971\pi\)
−0.0721036 + 0.997397i \(0.522971\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −1.81555e14 −1.47620 −0.738102 − 0.674689i \(-0.764278\pi\)
−0.738102 + 0.674689i \(0.764278\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −8.25601e13 −0.451632 −0.225816 − 0.974170i \(-0.572505\pi\)
−0.225816 + 0.974170i \(0.572505\pi\)
\(110\) 0 0
\(111\) 2.44126e14 1.17585
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.21245e14 1.40357
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.79750e14 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 9.70570e14 1.82138 0.910690 − 0.413090i \(-0.135550\pi\)
0.910690 + 0.413090i \(0.135550\pi\)
\(128\) 0 0
\(129\) −1.15485e15 −1.94266
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −2.10094e15 −2.85398
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 1.05578e15 1.05310 0.526549 − 0.850144i \(-0.323485\pi\)
0.526549 + 0.850144i \(0.323485\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2.73628e15 −1.84476
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 2.35511e15 1.31575 0.657875 − 0.753127i \(-0.271456\pi\)
0.657875 + 0.753127i \(0.271456\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.02338e15 0.860559 0.430279 − 0.902696i \(-0.358415\pi\)
0.430279 + 0.902696i \(0.358415\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.51863e15 0.496752 0.248376 − 0.968664i \(-0.420103\pi\)
0.248376 + 0.968664i \(0.420103\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 3.81929e15 0.970010
\(170\) 0 0
\(171\) 7.23438e15 1.69211
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −8.47792e15 −1.68664
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.11443e16 −1.75106 −0.875531 − 0.483161i \(-0.839489\pi\)
−0.875531 + 0.483161i \(0.839489\pi\)
\(182\) 0 0
\(183\) −1.35913e16 −1.97744
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.45297e16 1.68664
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −1.06337e16 −1.06607 −0.533034 − 0.846094i \(-0.678948\pi\)
−0.533034 + 0.846094i \(0.678948\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −3.75474e15 −0.303815 −0.151907 − 0.988395i \(-0.548542\pi\)
−0.151907 + 0.988395i \(0.548542\pi\)
\(200\) 0 0
\(201\) 2.61853e16 1.97553
\(202\) 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
\(211\) 1.14160e16 0.613110 0.306555 − 0.951853i \(-0.400824\pi\)
0.306555 + 0.951853i \(0.400824\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.19674e16 1.41088
\(218\) 0 0
\(219\) −3.77880e16 −1.56403
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.42664e16 0.520212 0.260106 − 0.965580i \(-0.416242\pi\)
0.260106 + 0.965580i \(0.416242\pi\)
\(224\) 0 0
\(225\) 2.91929e16 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −2.55749e16 −0.774402 −0.387201 − 0.921995i \(-0.626558\pi\)
−0.387201 + 0.921995i \(0.626558\pi\)
\(230\) 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
\(237\) −8.39439e16 −1.99871
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −9.11223e16 −1.92977 −0.964884 − 0.262675i \(-0.915395\pi\)
−0.964884 + 0.262675i \(0.915395\pi\)
\(242\) 0 0
\(243\) −5.00315e16 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.33211e17 2.37500
\(248\) 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
\(259\) 1.55051e17 1.98325
\(260\) 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
\(271\) 2.10731e16 0.196311 0.0981554 − 0.995171i \(-0.468706\pi\)
0.0981554 + 0.995171i \(0.468706\pi\)
\(272\) 0 0
\(273\) 2.67544e17 2.36732
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.50224e17 −1.99973 −0.999863 − 0.0165399i \(-0.994735\pi\)
−0.999863 + 0.0165399i \(0.994735\pi\)
\(278\) 0 0
\(279\) −1.10077e17 −0.836500
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −9.04106e16 −0.621891 −0.310945 − 0.950428i \(-0.600646\pi\)
−0.310945 + 0.950428i \(0.600646\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.68378e17 1.00000
\(290\) 0 0
\(291\) 2.54823e16 0.144207
\(292\) 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
\(301\) −7.33475e17 −3.27657
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4.82000e17 −1.87533 −0.937667 − 0.347536i \(-0.887018\pi\)
−0.937667 + 0.347536i \(0.887018\pi\)
\(308\) 0 0
\(309\) 3.97060e17 1.47620
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 5.88344e17 1.99904 0.999519 − 0.0310007i \(-0.00986940\pi\)
0.999519 + 0.0310007i \(0.00986940\pi\)
\(314\) 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
\(325\) 5.37549e17 1.40357
\(326\) 0 0
\(327\) 1.80559e17 0.451632
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6.86499e17 1.57704 0.788522 − 0.615007i \(-0.210847\pi\)
0.788522 + 0.615007i \(0.210847\pi\)
\(332\) 0 0
\(333\) −5.33904e17 −1.17585
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 6.40150e17 1.29680 0.648399 − 0.761301i \(-0.275439\pi\)
0.648399 + 0.761301i \(0.275439\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −7.95821e17 −1.42481
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −7.77937e17 −1.23358 −0.616789 − 0.787129i \(-0.711567\pi\)
−0.616789 + 0.787129i \(0.711567\pi\)
\(350\) 0 0
\(351\) −9.21264e17 −1.40357
\(352\) 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
\(361\) 1.48874e18 1.86324
\(362\) 0 0
\(363\) −8.30513e17 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.90714e17 −0.212677 −0.106339 − 0.994330i \(-0.533913\pi\)
−0.106339 + 0.994330i \(0.533913\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.63882e18 −1.63143 −0.815717 − 0.578451i \(-0.803657\pi\)
−0.815717 + 0.578451i \(0.803657\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.17971e18 1.94055 0.970275 − 0.242005i \(-0.0778051\pi\)
0.970275 + 0.242005i \(0.0778051\pi\)
\(380\) 0 0
\(381\) −2.12264e18 −1.82138
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.52565e18 1.94266
\(388\) 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
\(397\) 5.32786e17 0.342783 0.171392 − 0.985203i \(-0.445174\pi\)
0.171392 + 0.985203i \(0.445174\pi\)
\(398\) 0 0
\(399\) 4.59475e18 2.85398
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −2.02691e18 −1.17409
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.96209e17 0.363644 0.181822 − 0.983331i \(-0.441801\pi\)
0.181822 + 0.983331i \(0.441801\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2.30899e18 −1.05310
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 3.16127e17 0.134861 0.0674305 − 0.997724i \(-0.478520\pi\)
0.0674305 + 0.997724i \(0.478520\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −8.63222e18 −3.33524
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −4.46280e18 −1.56384 −0.781920 − 0.623379i \(-0.785760\pi\)
−0.781920 + 0.623379i \(0.785760\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 4.28786e18 1.36455 0.682275 − 0.731096i \(-0.260991\pi\)
0.682275 + 0.731096i \(0.260991\pi\)
\(440\) 0 0
\(441\) 5.98425e18 1.84476
\(442\) 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
\(453\) −5.15063e18 −1.31575
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.21512e18 1.97333 0.986666 − 0.162756i \(-0.0520385\pi\)
0.986666 + 0.162756i \(0.0520385\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −8.07617e18 −1.77067 −0.885337 − 0.464951i \(-0.846072\pi\)
−0.885337 + 0.464951i \(0.846072\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 1.66310e19 3.33202
\(470\) 0 0
\(471\) −4.42514e18 −0.860559
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 9.23175e18 1.69211
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −9.83113e18 −1.65039
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 5.89452e18 0.907290 0.453645 − 0.891183i \(-0.350123\pi\)
0.453645 + 0.891183i \(0.350123\pi\)
\(488\) 0 0
\(489\) −3.32125e18 −0.496752
\(490\) 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
\(499\) −2.79456e18 −0.362752 −0.181376 − 0.983414i \(-0.558055\pi\)
−0.181376 + 0.983414i \(0.558055\pi\)
\(500\) 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
\(507\) −8.35280e18 −0.970010
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −2.40001e19 −2.63796
\(512\) 0 0
\(513\) −1.58216e19 −1.69211
\(514\) 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
\(523\) 1.66420e19 1.55486 0.777432 − 0.628966i \(-0.216522\pi\)
0.777432 + 0.628966i \(0.216522\pi\)
\(524\) 0 0
\(525\) 1.85412e19 1.68664
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.15928e19 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
\(541\) −1.35275e19 −0.997328 −0.498664 − 0.866795i \(-0.666176\pi\)
−0.498664 + 0.866795i \(0.666176\pi\)
\(542\) 0 0
\(543\) 2.43725e19 1.75106
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 6.51749e18 0.444804 0.222402 − 0.974955i \(-0.428610\pi\)
0.222402 + 0.974955i \(0.428610\pi\)
\(548\) 0 0
\(549\) 2.97242e19 1.97744
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −5.33150e19 −3.37111
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 4.65065e19 2.72666
\(560\) 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
\(567\) −3.17764e19 −1.68664
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −2.64825e18 −0.133816 −0.0669078 − 0.997759i \(-0.521313\pi\)
−0.0669078 + 0.997759i \(0.521313\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −2.46134e19 −1.15596 −0.577979 − 0.816052i \(-0.696158\pi\)
−0.577979 + 0.816052i \(0.696158\pi\)
\(578\) 0 0
\(579\) 2.32560e19 1.06607
\(580\) 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
\(589\) −3.48098e19 −1.41545
\(590\) 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
\(597\) 8.21163e18 0.303815
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.63865e18 −0.0578581 −0.0289290 − 0.999581i \(-0.509210\pi\)
−0.0289290 + 0.999581i \(0.509210\pi\)
\(602\) 0 0
\(603\) −5.72672e19 −1.97553
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −5.51469e19 −1.81635 −0.908176 − 0.418588i \(-0.862525\pi\)
−0.908176 + 0.418588i \(0.862525\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 6.22458e19 1.91376 0.956878 − 0.290491i \(-0.0938188\pi\)
0.956878 + 0.290491i \(0.0938188\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 6.53065e19 1.87552 0.937761 − 0.347283i \(-0.112896\pi\)
0.937761 + 0.347283i \(0.112896\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 3.72529e19 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 5.72705e19 1.43789 0.718945 − 0.695067i \(-0.244625\pi\)
0.718945 + 0.695067i \(0.244625\pi\)
\(632\) 0 0
\(633\) −2.49669e19 −0.613110
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.10192e20 2.58925
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −8.97867e19 −1.97576 −0.987882 − 0.155205i \(-0.950396\pi\)
−0.987882 + 0.155205i \(0.950396\pi\)
\(644\) 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
\(651\) −6.99126e19 −1.41088
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.26423e19 1.56403
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −6.64598e19 −1.20545 −0.602725 − 0.797949i \(-0.705918\pi\)
−0.602725 + 0.797949i \(0.705918\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −3.12006e19 −0.520212
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.00073e20 −1.60034 −0.800170 − 0.599774i \(-0.795257\pi\)
−0.800170 + 0.599774i \(0.795257\pi\)
\(674\) 0 0
\(675\) −6.38449e19 −1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 1.61845e19 0.243226
\(680\) 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
\(687\) 5.59324e19 0.774402
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 8.09125e19 1.07565 0.537823 − 0.843058i \(-0.319247\pi\)
0.537823 + 0.843058i \(0.319247\pi\)
\(692\) 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
\(703\) −1.68838e20 −1.98967
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.70916e20 −1.89784 −0.948921 − 0.315514i \(-0.897823\pi\)
−0.948921 + 0.315514i \(0.897823\pi\)
\(710\) 0 0
\(711\) 1.83585e20 1.99871
\(712\) 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
\(721\) 2.52183e20 2.48983
\(722\) 0 0
\(723\) 1.99284e20 1.92977
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.50414e20 −1.40135 −0.700677 − 0.713478i \(-0.747119\pi\)
−0.700677 + 0.713478i \(0.747119\pi\)
\(728\) 0 0
\(729\) 1.09419e20 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.63598e20 1.43897 0.719483 − 0.694510i \(-0.244379\pi\)
0.719483 + 0.694510i \(0.244379\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.21802e20 1.01191 0.505956 − 0.862559i \(-0.331140\pi\)
0.505956 + 0.862559i \(0.331140\pi\)
\(740\) 0 0
\(741\) −2.91333e20 −2.37500
\(742\) 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
\(751\) −5.64342e19 −0.418854 −0.209427 − 0.977824i \(-0.567160\pi\)
−0.209427 + 0.977824i \(0.567160\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.34099e20 −1.64334 −0.821672 − 0.569960i \(-0.806959\pi\)
−0.821672 + 0.569960i \(0.806959\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.14678e20 0.761741
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −2.78367e20 −1.75039 −0.875193 − 0.483775i \(-0.839265\pi\)
−0.875193 + 0.483775i \(0.839265\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −1.40468e20 −0.836500
\(776\) 0 0
\(777\) −3.39097e20 −1.98325
\(778\) 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
\(787\) 2.42603e20 1.29740 0.648698 − 0.761046i \(-0.275314\pi\)
0.648698 + 0.761046i \(0.275314\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 5.47332e20 2.77548
\(794\) 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
\(811\) −3.87857e20 −1.68084 −0.840419 − 0.541937i \(-0.817691\pi\)
−0.840419 + 0.541937i \(0.817691\pi\)
\(812\) 0 0
\(813\) −4.60869e19 −0.196311
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7.98693e20 3.28720
\(818\) 0 0
\(819\) −5.85119e20 −2.36732
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 5.03724e20 1.96967 0.984835 − 0.173493i \(-0.0555054\pi\)
0.984835 + 0.173493i \(0.0555054\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −4.98089e20 −1.85108 −0.925541 − 0.378648i \(-0.876389\pi\)
−0.925541 + 0.378648i \(0.876389\pi\)
\(830\) 0 0
\(831\) 5.47240e20 1.99973
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2.40738e20 0.836500
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 2.97558e20 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −5.27481e20 −1.68664
\(848\) 0 0
\(849\) 1.97728e20 0.621891
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −6.14482e19 −0.187010 −0.0935052 − 0.995619i \(-0.529807\pi\)
−0.0935052 + 0.995619i \(0.529807\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −3.38702e20 −0.981445 −0.490723 − 0.871316i \(-0.663267\pi\)
−0.490723 + 0.871316i \(0.663267\pi\)
\(860\) 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
\(867\) −3.68242e20 −1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −1.05450e21 −2.77280
\(872\) 0 0
\(873\) −5.57297e19 −0.144207
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 5.17897e20 1.29792 0.648958 − 0.760825i \(-0.275205\pi\)
0.648958 + 0.760825i \(0.275205\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 2.62906e20 0.628169 0.314084 − 0.949395i \(-0.398303\pi\)
0.314084 + 0.949395i \(0.398303\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −1.34814e21 −3.07202
\(890\) 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
\(903\) 1.60411e21 3.27657
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 7.43690e20 1.47279 0.736395 − 0.676551i \(-0.236526\pi\)
0.736395 + 0.676551i \(0.236526\pi\)
\(908\) 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
\(919\) 3.19457e20 0.577038 0.288519 − 0.957474i \(-0.406837\pi\)
0.288519 + 0.957474i \(0.406837\pi\)
\(920\) 0 0
\(921\) 1.05413e21 1.87533
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −6.81311e20 −1.17585
\(926\) 0 0
\(927\) −8.68370e20 −1.47620
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1.89241e21 3.12154
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.25540e21 −1.97973 −0.989865 − 0.142008i \(-0.954644\pi\)
−0.989865 + 0.142008i \(0.954644\pi\)
\(938\) 0 0
\(939\) −1.28671e21 −1.99904
\(940\) 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
\(949\) 1.52175e21 2.19522
\(950\) 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
\(961\) −2.27286e20 −0.300268
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 2.03183e20 0.256981 0.128491 − 0.991711i \(-0.458987\pi\)
0.128491 + 0.991711i \(0.458987\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) −1.46650e21 −1.77620
\(974\) 0 0
\(975\) −1.17562e21 −1.40357
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −3.94882e20 −0.451632
\(982\) 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
\(991\) 3.03686e20 0.323526 0.161763 − 0.986830i \(-0.448282\pi\)
0.161763 + 0.986830i \(0.448282\pi\)
\(992\) 0 0
\(993\) −1.50137e21 −1.57704
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.19373e21 1.21911 0.609553 − 0.792745i \(-0.291349\pi\)
0.609553 + 0.792745i \(0.291349\pi\)
\(998\) 0 0
\(999\) 1.16765e21 1.17585
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 12.15.c.a.5.1 ✓ 1
3.2 odd 2 CM 12.15.c.a.5.1 ✓ 1
4.3 odd 2 48.15.e.a.17.1 1
12.11 even 2 48.15.e.a.17.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.15.c.a.5.1 ✓ 1 1.1 even 1 trivial
12.15.c.a.5.1 ✓ 1 3.2 odd 2 CM
48.15.e.a.17.1 1 4.3 odd 2
48.15.e.a.17.1 1 12.11 even 2