Properties

Label 12.21.c.a.5.1
Level $12$
Weight $21$
Character 12.5
Self dual yes
Analytic conductor $30.422$
Analytic rank $0$
Dimension $1$
CM discriminant -3
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] = [12,21,Mod(5,12)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(12, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 21, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("12.5");
 
S:= CuspForms(chi, 21);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 21 \)
Character orbit: \([\chi]\) \(=\) 12.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: \(30.4216518123\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+59049.0 q^{3} -7.73358e7 q^{7} +3.48678e9 q^{9} +O(q^{10})\) \(q+59049.0 q^{3} -7.73358e7 q^{7} +3.48678e9 q^{9} +2.17393e11 q^{13} -3.84300e12 q^{19} -4.56660e12 q^{21} +9.53674e13 q^{25} +2.05891e14 q^{27} +7.93864e14 q^{31} +8.66582e15 q^{37} +1.28368e16 q^{39} +3.67421e16 q^{43} -7.38114e16 q^{49} -2.26925e17 q^{57} -1.38779e18 q^{61} -2.69653e17 q^{63} -1.57943e18 q^{67} +8.57782e18 q^{73} +5.63135e18 q^{75} -3.26283e18 q^{79} +1.21577e19 q^{81} -1.68123e19 q^{91} +4.68769e19 q^{93} -1.46702e20 q^{97} +O(q^{100})\)

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
\(3\) 59049.0 1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −7.73358e7 −0.273779 −0.136889 0.990586i \(-0.543711\pi\)
−0.136889 + 0.990586i \(0.543711\pi\)
\(8\) 0 0
\(9\) 3.48678e9 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 2.17393e11 1.57693 0.788465 0.615080i \(-0.210876\pi\)
0.788465 + 0.615080i \(0.210876\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −3.84300e12 −0.626808 −0.313404 0.949620i \(-0.601469\pi\)
−0.313404 + 0.949620i \(0.601469\pi\)
\(20\) 0 0
\(21\) −4.56660e12 −0.273779
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 9.53674e13 1.00000
\(26\) 0 0
\(27\) 2.05891e14 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 7.93864e14 0.968566 0.484283 0.874911i \(-0.339080\pi\)
0.484283 + 0.874911i \(0.339080\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.66582e15 1.80216 0.901078 0.433658i \(-0.142777\pi\)
0.901078 + 0.433658i \(0.142777\pi\)
\(38\) 0 0
\(39\) 1.28368e16 1.57693
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 3.67421e16 1.70012 0.850060 0.526686i \(-0.176565\pi\)
0.850060 + 0.526686i \(0.176565\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −7.38114e16 −0.925045
\(50\) 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
\(57\) −2.26925e17 −0.626808
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −1.38779e18 −1.94547 −0.972736 0.231916i \(-0.925500\pi\)
−0.972736 + 0.231916i \(0.925500\pi\)
\(62\) 0 0
\(63\) −2.69653e17 −0.273779
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.57943e18 −0.866470 −0.433235 0.901281i \(-0.642628\pi\)
−0.433235 + 0.901281i \(0.642628\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 8.57782e18 1.99594 0.997972 0.0636525i \(-0.0202749\pi\)
0.997972 + 0.0636525i \(0.0202749\pi\)
\(74\) 0 0
\(75\) 5.63135e18 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.26283e18 −0.344606 −0.172303 0.985044i \(-0.555121\pi\)
−0.172303 + 0.985044i \(0.555121\pi\)
\(80\) 0 0
\(81\) 1.21577e19 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
\(91\) −1.68123e19 −0.431730
\(92\) 0 0
\(93\) 4.68769e19 0.968566
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.46702e20 −1.98938 −0.994690 0.102912i \(-0.967184\pi\)
−0.994690 + 0.102912i \(0.967184\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −2.57677e20 −1.91736 −0.958678 0.284494i \(-0.908174\pi\)
−0.958678 + 0.284494i \(0.908174\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −1.61373e20 −0.681656 −0.340828 0.940126i \(-0.610707\pi\)
−0.340828 + 0.940126i \(0.610707\pi\)
\(110\) 0 0
\(111\) 5.11708e20 1.80216
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 7.58003e20 1.57693
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 6.72750e20 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2.15534e21 −1.97460 −0.987299 0.158874i \(-0.949214\pi\)
−0.987299 + 0.158874i \(0.949214\pi\)
\(128\) 0 0
\(129\) 2.16959e21 1.70012
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 2.97201e20 0.171607
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −3.78173e21 −1.40457 −0.702284 0.711897i \(-0.747836\pi\)
−0.702284 + 0.711897i \(0.747836\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −4.35849e21 −0.925045
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −1.22244e22 −1.98361 −0.991806 0.127757i \(-0.959222\pi\)
−0.991806 + 0.127757i \(0.959222\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.37893e22 1.51546 0.757732 0.652565i \(-0.226307\pi\)
0.757732 + 0.652565i \(0.226307\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.63716e22 1.99187 0.995933 0.0901003i \(-0.0287188\pi\)
0.995933 + 0.0901003i \(0.0287188\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 2.82548e22 1.48671
\(170\) 0 0
\(171\) −1.33997e22 −0.626808
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −7.37531e21 −0.273779
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 3.59188e22 0.951778 0.475889 0.879505i \(-0.342126\pi\)
0.475889 + 0.879505i \(0.342126\pi\)
\(182\) 0 0
\(183\) −8.19475e22 −1.94547
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.59228e22 −0.273779
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 1.85803e22 0.259107 0.129554 0.991572i \(-0.458646\pi\)
0.129554 + 0.991572i \(0.458646\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.51493e23 −1.55547 −0.777734 0.628594i \(-0.783631\pi\)
−0.777734 + 0.628594i \(0.783631\pi\)
\(200\) 0 0
\(201\) −9.32640e22 −0.866470
\(202\) 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
\(211\) 2.17102e23 1.24119 0.620595 0.784131i \(-0.286891\pi\)
0.620595 + 0.784131i \(0.286891\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6.13941e22 −0.265173
\(218\) 0 0
\(219\) 5.06512e23 1.99594
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.12862e23 1.35755 0.678776 0.734345i \(-0.262511\pi\)
0.678776 + 0.734345i \(0.262511\pi\)
\(224\) 0 0
\(225\) 3.32526e23 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −6.68658e23 −1.68597 −0.842983 0.537940i \(-0.819203\pi\)
−0.842983 + 0.537940i \(0.819203\pi\)
\(230\) 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
\(237\) −1.92667e23 −0.344606
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.31875e24 −1.99523 −0.997614 0.0690390i \(-0.978007\pi\)
−0.997614 + 0.0690390i \(0.978007\pi\)
\(242\) 0 0
\(243\) 7.17898e23 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8.35442e23 −0.988432
\(248\) 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
\(259\) −6.70178e23 −0.493392
\(260\) 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
\(271\) 3.58295e23 0.167706 0.0838528 0.996478i \(-0.473277\pi\)
0.0838528 + 0.996478i \(0.473277\pi\)
\(272\) 0 0
\(273\) −9.92748e23 −0.431730
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.21714e24 1.20969 0.604843 0.796345i \(-0.293236\pi\)
0.604843 + 0.796345i \(0.293236\pi\)
\(278\) 0 0
\(279\) 2.76803e24 0.968566
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 6.59011e24 1.99999 0.999996 0.00289763i \(-0.000922346\pi\)
0.999996 + 0.00289763i \(0.000922346\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4.06423e24 1.00000
\(290\) 0 0
\(291\) −8.66259e24 −1.98938
\(292\) 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
\(301\) −2.84148e24 −0.465457
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1.48483e25 −1.99661 −0.998303 0.0582280i \(-0.981455\pi\)
−0.998303 + 0.0582280i \(0.981455\pi\)
\(308\) 0 0
\(309\) −1.52155e25 −1.91736
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 1.18677e25 1.31499 0.657497 0.753457i \(-0.271615\pi\)
0.657497 + 0.753457i \(0.271615\pi\)
\(314\) 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
\(325\) 2.07322e25 1.57693
\(326\) 0 0
\(327\) −9.52890e24 −0.681656
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.77456e25 −1.75758 −0.878789 0.477211i \(-0.841648\pi\)
−0.878789 + 0.477211i \(0.841648\pi\)
\(332\) 0 0
\(333\) 3.02158e25 1.80216
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4.30463e24 0.227843 0.113922 0.993490i \(-0.463659\pi\)
0.113922 + 0.993490i \(0.463659\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.18791e25 0.527037
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 9.20209e24 0.343267 0.171634 0.985161i \(-0.445095\pi\)
0.171634 + 0.985161i \(0.445095\pi\)
\(350\) 0 0
\(351\) 4.47593e25 1.57693
\(352\) 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
\(361\) −2.28213e25 −0.607112
\(362\) 0 0
\(363\) 3.97252e25 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4.41264e25 −0.995492 −0.497746 0.867323i \(-0.665839\pi\)
−0.497746 + 0.867323i \(0.665839\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −9.31800e25 −1.78745 −0.893726 0.448614i \(-0.851918\pi\)
−0.893726 + 0.448614i \(0.851918\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −6.63654e25 −1.08530 −0.542649 0.839959i \(-0.682579\pi\)
−0.542649 + 0.839959i \(0.682579\pi\)
\(380\) 0 0
\(381\) −1.27271e26 −1.97460
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.28112e26 1.70012
\(388\) 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
\(397\) −4.21278e24 −0.0433176 −0.0216588 0.999765i \(-0.506895\pi\)
−0.0216588 + 0.999765i \(0.506895\pi\)
\(398\) 0 0
\(399\) 1.75494e25 0.171607
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 1.72581e26 1.52736
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.98674e26 1.51673 0.758363 0.651832i \(-0.225999\pi\)
0.758363 + 0.651832i \(0.225999\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2.23308e26 −1.40457
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 1.10310e26 0.630655 0.315328 0.948983i \(-0.397886\pi\)
0.315328 + 0.948983i \(0.397886\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.07326e26 0.532629
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 2.65167e26 1.14457 0.572283 0.820056i \(-0.306058\pi\)
0.572283 + 0.820056i \(0.306058\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −2.54040e26 −0.955559 −0.477780 0.878480i \(-0.658558\pi\)
−0.477780 + 0.878480i \(0.658558\pi\)
\(440\) 0 0
\(441\) −2.57365e26 −0.925045
\(442\) 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
\(453\) −7.21836e26 −1.98361
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 7.27017e24 0.0182971 0.00914856 0.999958i \(-0.497088\pi\)
0.00914856 + 0.999958i \(0.497088\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −8.79010e26 −1.94171 −0.970854 0.239672i \(-0.922960\pi\)
−0.970854 + 0.239672i \(0.922960\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 1.22147e26 0.237221
\(470\) 0 0
\(471\) 8.14245e26 1.51546
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −3.66497e26 −0.626808
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 1.88389e27 2.84187
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −8.31548e23 −0.00110815 −0.000554074 1.00000i \(-0.500176\pi\)
−0.000554074 1.00000i \(0.500176\pi\)
\(488\) 0 0
\(489\) 1.55721e27 1.99187
\(490\) 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
\(499\) −1.84980e27 −1.93250 −0.966248 0.257613i \(-0.917064\pi\)
−0.966248 + 0.257613i \(0.917064\pi\)
\(500\) 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
\(507\) 1.66842e27 1.48671
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −6.63372e26 −0.546447
\(512\) 0 0
\(513\) −7.91240e26 −0.626808
\(514\) 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
\(523\) −2.84988e27 −1.86127 −0.930636 0.365946i \(-0.880745\pi\)
−0.930636 + 0.365946i \(0.880745\pi\)
\(524\) 0 0
\(525\) −4.35505e26 −0.273779
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.71616e27 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
\(541\) −4.24599e27 −1.97700 −0.988500 0.151220i \(-0.951680\pi\)
−0.988500 + 0.151220i \(0.951680\pi\)
\(542\) 0 0
\(543\) 2.12097e27 0.951778
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.59721e26 −0.191699 −0.0958497 0.995396i \(-0.530557\pi\)
−0.0958497 + 0.995396i \(0.530557\pi\)
\(548\) 0 0
\(549\) −4.83892e27 −1.94547
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 2.52333e26 0.0943460
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 7.98748e27 2.68097
\(560\) 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
\(567\) −9.40222e26 −0.273779
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 6.30327e27 1.71082 0.855411 0.517950i \(-0.173305\pi\)
0.855411 + 0.517950i \(0.173305\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 7.43062e27 1.81663 0.908315 0.418286i \(-0.137369\pi\)
0.908315 + 0.418286i \(0.137369\pi\)
\(578\) 0 0
\(579\) 1.09715e27 0.259107
\(580\) 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
\(589\) −3.05082e27 −0.607105
\(590\) 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
\(597\) −8.94549e27 −1.55547
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 2.23849e27 0.364091 0.182046 0.983290i \(-0.441728\pi\)
0.182046 + 0.983290i \(0.441728\pi\)
\(602\) 0 0
\(603\) −5.50714e27 −0.866470
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −6.57051e27 −0.967636 −0.483818 0.875169i \(-0.660750\pi\)
−0.483818 + 0.875169i \(0.660750\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −9.01397e27 −1.20313 −0.601564 0.798825i \(-0.705455\pi\)
−0.601564 + 0.798825i \(0.705455\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 2.73783e27 0.331512 0.165756 0.986167i \(-0.446994\pi\)
0.165756 + 0.986167i \(0.446994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.09495e27 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.59436e28 −1.59328 −0.796642 0.604452i \(-0.793392\pi\)
−0.796642 + 0.604452i \(0.793392\pi\)
\(632\) 0 0
\(633\) 1.28196e28 1.24119
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.60461e28 −1.45873
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 1.88641e28 1.56144 0.780721 0.624879i \(-0.214852\pi\)
0.780721 + 0.624879i \(0.214852\pi\)
\(644\) 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
\(651\) −3.62526e27 −0.265173
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.99090e28 1.99594
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.91869e28 −1.20501 −0.602506 0.798114i \(-0.705831\pi\)
−0.602506 + 0.798114i \(0.705831\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 2.43791e28 1.35755
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.43979e28 1.27998 0.639989 0.768384i \(-0.278939\pi\)
0.639989 + 0.768384i \(0.278939\pi\)
\(674\) 0 0
\(675\) 1.96353e28 1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 1.13453e28 0.544651
\(680\) 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
\(687\) −3.94836e28 −1.68597
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −4.94525e28 −1.99254 −0.996272 0.0862731i \(-0.972504\pi\)
−0.996272 + 0.0862731i \(0.972504\pi\)
\(692\) 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
\(703\) −3.33027e28 −1.12961
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.95318e28 −1.23164 −0.615821 0.787886i \(-0.711176\pi\)
−0.615821 + 0.787886i \(0.711176\pi\)
\(710\) 0 0
\(711\) −1.13768e28 −0.344606
\(712\) 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
\(721\) 1.99276e28 0.524932
\(722\) 0 0
\(723\) −7.78710e28 −1.99523
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 8.01727e28 1.94394 0.971969 0.235109i \(-0.0755446\pi\)
0.971969 + 0.235109i \(0.0755446\pi\)
\(728\) 0 0
\(729\) 4.23912e28 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −2.25900e27 −0.0504519 −0.0252259 0.999682i \(-0.508031\pi\)
−0.0252259 + 0.999682i \(0.508031\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 9.25542e28 1.90525 0.952625 0.304148i \(-0.0983719\pi\)
0.952625 + 0.304148i \(0.0983719\pi\)
\(740\) 0 0
\(741\) −4.93320e28 −0.988432
\(742\) 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
\(751\) 1.12900e29 1.97832 0.989158 0.146853i \(-0.0469144\pi\)
0.989158 + 0.146853i \(0.0469144\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.12972e29 −1.82816 −0.914078 0.405540i \(-0.867084\pi\)
−0.914078 + 0.405540i \(0.867084\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.24799e28 0.186623
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.08611e29 1.50180 0.750898 0.660418i \(-0.229621\pi\)
0.750898 + 0.660418i \(0.229621\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 7.57088e28 0.968566
\(776\) 0 0
\(777\) −3.95733e28 −0.493392
\(778\) 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
\(787\) 5.99663e28 0.657900 0.328950 0.944347i \(-0.393305\pi\)
0.328950 + 0.944347i \(0.393305\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −3.01696e29 −3.06787
\(794\) 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
\(811\) −2.12623e29 −1.72743 −0.863717 0.503977i \(-0.831870\pi\)
−0.863717 + 0.503977i \(0.831870\pi\)
\(812\) 0 0
\(813\) 2.11570e28 0.167706
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.41200e29 −1.06565
\(818\) 0 0
\(819\) −5.86207e28 −0.431730
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −2.18456e29 −1.53238 −0.766189 0.642615i \(-0.777849\pi\)
−0.766189 + 0.642615i \(0.777849\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 2.88760e29 1.88361 0.941806 0.336157i \(-0.109127\pi\)
0.941806 + 0.336157i \(0.109127\pi\)
\(830\) 0 0
\(831\) 1.89969e29 1.20969
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.63450e29 0.968566
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.76995e29 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −5.20276e28 −0.273779
\(848\) 0 0
\(849\) 3.89139e29 1.99999
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −2.94561e29 −1.44439 −0.722197 0.691688i \(-0.756867\pi\)
−0.722197 + 0.691688i \(0.756867\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 4.19966e29 1.91992 0.959959 0.280142i \(-0.0903816\pi\)
0.959959 + 0.280142i \(0.0903816\pi\)
\(860\) 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
\(867\) 2.39989e29 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −3.43358e29 −1.36636
\(872\) 0 0
\(873\) −5.11517e29 −1.98938
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 5.07930e29 1.88716 0.943579 0.331148i \(-0.107436\pi\)
0.943579 + 0.331148i \(0.107436\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −5.21104e29 −1.80850 −0.904248 0.427007i \(-0.859568\pi\)
−0.904248 + 0.427007i \(0.859568\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.66685e29 0.540603
\(890\) 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
\(903\) −1.67787e29 −0.465457
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −2.85320e29 −0.757285 −0.378643 0.925543i \(-0.623609\pi\)
−0.378643 + 0.925543i \(0.623609\pi\)
\(908\) 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
\(919\) −1.65435e29 −0.385010 −0.192505 0.981296i \(-0.561661\pi\)
−0.192505 + 0.981296i \(0.561661\pi\)
\(920\) 0 0
\(921\) −8.76776e29 −1.99661
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 8.26437e29 1.80216
\(926\) 0 0
\(927\) −8.98463e29 −1.91736
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 2.83657e29 0.579826
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −8.29096e29 −1.58931 −0.794656 0.607060i \(-0.792349\pi\)
−0.794656 + 0.607060i \(0.792349\pi\)
\(938\) 0 0
\(939\) 7.00777e29 1.31499
\(940\) 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
\(949\) 1.86476e30 3.14746
\(950\) 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
\(961\) −4.15702e28 −0.0618797
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.38005e30 1.93033 0.965163 0.261651i \(-0.0842669\pi\)
0.965163 + 0.261651i \(0.0842669\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 2.92463e29 0.384541
\(974\) 0 0
\(975\) 1.22422e30 1.57693
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −5.62672e29 −0.681656
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.77812e30 −1.94637 −0.973183 0.230031i \(-0.926117\pi\)
−0.973183 + 0.230031i \(0.926117\pi\)
\(992\) 0 0
\(993\) −1.63835e30 −1.75758
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.93934e30 −1.99849 −0.999246 0.0388167i \(-0.987641\pi\)
−0.999246 + 0.0388167i \(0.987641\pi\)
\(998\) 0 0
\(999\) 1.78421e30 1.80216
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 12.21.c.a.5.1 1
3.2 odd 2 CM 12.21.c.a.5.1 1
4.3 odd 2 48.21.e.a.17.1 1
12.11 even 2 48.21.e.a.17.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.21.c.a.5.1 1 1.1 even 1 trivial
12.21.c.a.5.1 1 3.2 odd 2 CM
48.21.e.a.17.1 1 4.3 odd 2
48.21.e.a.17.1 1 12.11 even 2