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Properties

Label 3.19.b.a.2.1
Level $3$
Weight $19$
Character 3.2
Self dual yes
Analytic conductor $6.162$
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] = [3,19,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 19 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.16158413129\)
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 2.1
Character \(\chi\) \(=\) 3.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-19683.0 q^{3} +262144. q^{4} +7.75492e7 q^{7} +3.87420e8 q^{9} +O(q^{10})\) \(q-19683.0 q^{3} +262144. q^{4} +7.75492e7 q^{7} +3.87420e8 q^{9} -5.15978e9 q^{12} -7.19754e9 q^{13} +6.87195e10 q^{16} +3.08560e11 q^{19} -1.52640e12 q^{21} +3.81470e12 q^{25} -7.62560e12 q^{27} +2.03291e13 q^{28} -5.00190e13 q^{31} +1.01560e14 q^{36} -2.32409e13 q^{37} +1.41669e14 q^{39} -7.30386e14 q^{43} -1.35261e15 q^{48} +4.38546e15 q^{49} -1.88679e15 q^{52} -6.07338e15 q^{57} -9.48716e15 q^{61} +3.00441e16 q^{63} +1.80144e16 q^{64} -4.17473e16 q^{67} -2.99090e16 q^{73} -7.50847e16 q^{75} +8.08871e16 q^{76} +1.40656e17 q^{79} +1.50095e17 q^{81} -4.00137e17 q^{84} -5.58164e17 q^{91} +9.84524e17 q^{93} +1.40874e17 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −19683.0 −1.00000
\(4\) 262144. 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 7.75492e7 1.92174 0.960871 − 0.276998i \(-0.0893395\pi\)
0.960871 + 0.276998i \(0.0893395\pi\)
\(8\) 0 0
\(9\) 3.87420e8 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −5.15978e9 −1.00000
\(13\) −7.19754e9 −0.678725 −0.339363 − 0.940656i \(-0.610211\pi\)
−0.339363 + 0.940656i \(0.610211\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 6.87195e10 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 3.08560e11 0.956218 0.478109 − 0.878301i \(-0.341322\pi\)
0.478109 + 0.878301i \(0.341322\pi\)
\(20\) 0 0
\(21\) −1.52640e12 −1.92174
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 3.81470e12 1.00000
\(26\) 0 0
\(27\) −7.62560e12 −1.00000
\(28\) 2.03291e13 1.92174
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −5.00190e13 −1.89182 −0.945910 − 0.324430i \(-0.894828\pi\)
−0.945910 + 0.324430i \(0.894828\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.01560e14 1.00000
\(37\) −2.32409e13 −0.178829 −0.0894146 − 0.995994i \(-0.528500\pi\)
−0.0894146 + 0.995994i \(0.528500\pi\)
\(38\) 0 0
\(39\) 1.41669e14 0.678725
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −7.30386e14 −1.45324 −0.726618 − 0.687042i \(-0.758909\pi\)
−0.726618 + 0.687042i \(0.758909\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −1.35261e15 −1.00000
\(49\) 4.38546e15 2.69309
\(50\) 0 0
\(51\) 0 0
\(52\) −1.88679e15 −0.678725
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.07338e15 −0.956218
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −9.48716e15 −0.811274 −0.405637 − 0.914034i \(-0.632950\pi\)
−0.405637 + 0.914034i \(0.632950\pi\)
\(62\) 0 0
\(63\) 3.00441e16 1.92174
\(64\) 1.80144e16 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.17473e16 −1.53446 −0.767229 − 0.641373i \(-0.778365\pi\)
−0.767229 + 0.641373i \(0.778365\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −2.99090e16 −0.508038 −0.254019 − 0.967199i \(-0.581753\pi\)
−0.254019 + 0.967199i \(0.581753\pi\)
\(74\) 0 0
\(75\) −7.50847e16 −1.00000
\(76\) 8.08871e16 0.956218
\(77\) 0 0
\(78\) 0 0
\(79\) 1.40656e17 1.17358 0.586791 − 0.809739i \(-0.300391\pi\)
0.586791 + 0.809739i \(0.300391\pi\)
\(80\) 0 0
\(81\) 1.50095e17 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −4.00137e17 −1.92174
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −5.58164e17 −1.30433
\(92\) 0 0
\(93\) 9.84524e17 1.89182
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.40874e17 0.185304 0.0926521 − 0.995699i \(-0.470466\pi\)
0.0926521 + 0.995699i \(0.470466\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e18 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −2.60567e18 −1.99703 −0.998514 − 0.0545017i \(-0.982643\pi\)
−0.998514 + 0.0545017i \(0.982643\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −1.99900e18 −1.00000
\(109\) 6.74620e17 0.310614 0.155307 − 0.987866i \(-0.450363\pi\)
0.155307 + 0.987866i \(0.450363\pi\)
\(110\) 0 0
\(111\) 4.57452e17 0.178829
\(112\) 5.32914e18 1.92174
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.78848e18 −0.678725
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.55992e18 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −1.31122e19 −1.89182
\(125\) 0 0
\(126\) 0 0
\(127\) −1.19898e19 −1.39501 −0.697507 − 0.716577i \(-0.745708\pi\)
−0.697507 + 0.716577i \(0.745708\pi\)
\(128\) 0 0
\(129\) 1.43762e19 1.45324
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 2.39286e19 1.83760
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −2.53396e19 −1.30818 −0.654088 − 0.756419i \(-0.726947\pi\)
−0.654088 + 0.756419i \(0.726947\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 2.66233e19 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) −8.63191e19 −2.69309
\(148\) −6.09247e18 −0.178829
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 8.00162e19 1.96058 0.980292 − 0.197555i \(-0.0633001\pi\)
0.980292 + 0.197555i \(0.0633001\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 3.71377e19 0.678725
\(157\) 3.70895e19 0.639962 0.319981 − 0.947424i \(-0.396323\pi\)
0.319981 + 0.947424i \(0.396323\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.38736e20 −1.70805 −0.854027 − 0.520228i \(-0.825847\pi\)
−0.854027 + 0.520228i \(0.825847\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −6.06508e19 −0.539332
\(170\) 0 0
\(171\) 1.19542e20 0.956218
\(172\) −1.91466e20 −1.45324
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 2.95827e20 1.92174
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.03664e19 −0.0497187 −0.0248593 − 0.999691i \(-0.507914\pi\)
−0.0248593 + 0.999691i \(0.507914\pi\)
\(182\) 0 0
\(183\) 1.86736e20 0.811274
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −5.91359e20 −1.92174
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −3.54577e20 −1.00000
\(193\) −2.01075e20 −0.541181 −0.270591 − 0.962695i \(-0.587219\pi\)
−0.270591 + 0.962695i \(0.587219\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.14962e21 2.69309
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −5.88368e20 −1.20219 −0.601093 − 0.799179i \(-0.705268\pi\)
−0.601093 + 0.799179i \(0.705268\pi\)
\(200\) 0 0
\(201\) 8.21712e20 1.53446
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −4.94611e20 −0.678725
\(209\) 0 0
\(210\) 0 0
\(211\) 7.90505e20 0.953592 0.476796 − 0.879014i \(-0.341798\pi\)
0.476796 + 0.879014i \(0.341798\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3.87893e21 −3.63559
\(218\) 0 0
\(219\) 5.88699e20 0.508038
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2.70988e21 1.98704 0.993519 − 0.113667i \(-0.0362597\pi\)
0.993519 + 0.113667i \(0.0362597\pi\)
\(224\) 0 0
\(225\) 1.47789e21 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) −1.59210e21 −0.956218
\(229\) 1.69454e21 0.978434 0.489217 − 0.872162i \(-0.337283\pi\)
0.489217 + 0.872162i \(0.337283\pi\)
\(230\) 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
\(237\) −2.76852e21 −1.17358
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 2.94205e21 1.07275 0.536373 − 0.843981i \(-0.319794\pi\)
0.536373 + 0.843981i \(0.319794\pi\)
\(242\) 0 0
\(243\) −2.95431e21 −1.00000
\(244\) −2.48700e21 −0.811274
\(245\) 0 0
\(246\) 0 0
\(247\) −2.22087e21 −0.649009
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 7.87589e21 1.92174
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4.72237e21 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −1.80232e21 −0.343663
\(260\) 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
\(268\) −1.09438e22 −1.53446
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −1.34682e22 −1.70839 −0.854195 − 0.519952i \(-0.825950\pi\)
−0.854195 + 0.519952i \(0.825950\pi\)
\(272\) 0 0
\(273\) 1.09863e22 1.30433
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.74737e22 1.81998 0.909992 − 0.414627i \(-0.136088\pi\)
0.909992 + 0.414627i \(0.136088\pi\)
\(278\) 0 0
\(279\) −1.93784e22 −1.89182
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 1.88036e22 1.61496 0.807481 − 0.589893i \(-0.200830\pi\)
0.807481 + 0.589893i \(0.200830\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.40631e22 1.00000
\(290\) 0 0
\(291\) −2.77282e21 −0.185304
\(292\) −7.84046e21 −0.508038
\(293\) 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
\(300\) −1.96830e22 −1.00000
\(301\) −5.66408e22 −2.79274
\(302\) 0 0
\(303\) 0 0
\(304\) 2.12041e22 0.956218
\(305\) 0 0
\(306\) 0 0
\(307\) 3.04921e22 1.25876 0.629379 − 0.777099i \(-0.283309\pi\)
0.629379 + 0.777099i \(0.283309\pi\)
\(308\) 0 0
\(309\) 5.12874e22 1.99703
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −5.09179e22 −1.76592 −0.882961 − 0.469446i \(-0.844454\pi\)
−0.882961 + 0.469446i \(0.844454\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 3.68720e22 1.17358
\(317\) 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
\(324\) 3.93464e22 1.00000
\(325\) −2.74564e22 −0.678725
\(326\) 0 0
\(327\) −1.32785e22 −0.310614
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −8.39476e22 −1.76018 −0.880089 − 0.474809i \(-0.842517\pi\)
−0.880089 + 0.474809i \(0.842517\pi\)
\(332\) 0 0
\(333\) −9.00402e21 −0.178829
\(334\) 0 0
\(335\) 0 0
\(336\) −1.04893e23 −1.92174
\(337\) 8.69978e22 1.55181 0.775907 − 0.630848i \(-0.217293\pi\)
0.775907 + 0.630848i \(0.217293\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 2.13807e23 3.25368
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −1.24118e23 −1.61587 −0.807934 − 0.589273i \(-0.799414\pi\)
−0.807934 + 0.589273i \(0.799414\pi\)
\(350\) 0 0
\(351\) 5.48856e22 0.678725
\(352\) 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
\(361\) −8.91827e21 −0.0856478
\(362\) 0 0
\(363\) −1.09436e23 −1.00000
\(364\) −1.46319e23 −1.30433
\(365\) 0 0
\(366\) 0 0
\(367\) 7.41082e22 0.613581 0.306790 − 0.951777i \(-0.400745\pi\)
0.306790 + 0.951777i \(0.400745\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 2.58087e23 1.89182
\(373\) −2.77992e23 −1.98908 −0.994541 − 0.104343i \(-0.966726\pi\)
−0.994541 + 0.104343i \(0.966726\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.69326e23 1.66927 0.834634 − 0.550805i \(-0.185679\pi\)
0.834634 + 0.550805i \(0.185679\pi\)
\(380\) 0 0
\(381\) 2.35995e23 1.39501
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.82966e23 −1.45324
\(388\) 3.69293e22 0.185304
\(389\) 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
\(397\) −4.40794e23 −1.79937 −0.899687 − 0.436536i \(-0.856205\pi\)
−0.899687 + 0.436536i \(0.856205\pi\)
\(398\) 0 0
\(399\) −4.70986e23 −1.83760
\(400\) 2.62144e23 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 3.60014e23 1.28403
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.40494e23 1.99989 0.999943 − 0.0106792i \(-0.00339935\pi\)
0.999943 + 0.0106792i \(0.00339935\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −6.83060e23 −1.99703
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.98759e23 1.30818
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 6.02317e23 1.44973 0.724863 − 0.688893i \(-0.241903\pi\)
0.724863 + 0.688893i \(0.241903\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −7.35722e23 −1.55906
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −5.24027e23 −1.00000
\(433\) −6.93813e23 −1.29674 −0.648368 − 0.761327i \(-0.724548\pi\)
−0.648368 + 0.761327i \(0.724548\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.76848e23 0.310614
\(437\) 0 0
\(438\) 0 0
\(439\) −9.95932e23 −1.64456 −0.822279 − 0.569085i \(-0.807298\pi\)
−0.822279 + 0.569085i \(0.807298\pi\)
\(440\) 0 0
\(441\) 1.69902e24 2.69309
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 1.19918e23 0.178829
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 1.39700e24 1.92174
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −1.57496e24 −1.96058
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 7.76662e23 0.893278 0.446639 − 0.894714i \(-0.352621\pi\)
0.446639 + 0.894714i \(0.352621\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 1.46414e24 1.49746 0.748730 − 0.662876i \(-0.230664\pi\)
0.748730 + 0.662876i \(0.230664\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −7.30982e23 −0.678725
\(469\) −3.23747e24 −2.94883
\(470\) 0 0
\(471\) −7.30033e23 −0.639962
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.17706e24 0.956218
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 1.67278e23 0.121376
\(482\) 0 0
\(483\) 0 0
\(484\) 1.45750e24 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 4.80567e23 0.311884 0.155942 − 0.987766i \(-0.450159\pi\)
0.155942 + 0.987766i \(0.450159\pi\)
\(488\) 0 0
\(489\) 2.73075e24 1.70805
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −3.43728e24 −1.89182
\(497\) 0 0
\(498\) 0 0
\(499\) 8.91419e23 0.464705 0.232352 − 0.972632i \(-0.425358\pi\)
0.232352 + 0.972632i \(0.425358\pi\)
\(500\) 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
\(507\) 1.19379e24 0.539332
\(508\) −3.14306e24 −1.39501
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −2.31942e24 −0.976317
\(512\) 0 0
\(513\) −2.35295e24 −0.956218
\(514\) 0 0
\(515\) 0 0
\(516\) 3.76863e24 1.45324
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 5.85369e24 1.99947 0.999735 − 0.0230233i \(-0.00732920\pi\)
0.999735 + 0.0230233i \(0.00732920\pi\)
\(524\) 0 0
\(525\) −5.82276e24 −1.92174
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.24415e24 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 6.27273e24 1.83760
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.14660e24 −1.80021 −0.900107 − 0.435669i \(-0.856512\pi\)
−0.900107 + 0.435669i \(0.856512\pi\)
\(542\) 0 0
\(543\) 2.04041e23 0.0497187
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −8.12698e24 −1.85372 −0.926858 − 0.375412i \(-0.877501\pi\)
−0.926858 + 0.375412i \(0.877501\pi\)
\(548\) 0 0
\(549\) −3.67552e24 −0.811274
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.09077e25 2.25532
\(554\) 0 0
\(555\) 0 0
\(556\) −6.64261e24 −1.30818
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 5.25698e24 0.986348
\(560\) 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
\(567\) 1.16397e25 1.92174
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −1.27817e25 −1.98090 −0.990450 − 0.137870i \(-0.955974\pi\)
−0.990450 + 0.137870i \(0.955974\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 6.97915e24 1.00000
\(577\) 7.45982e24 1.05232 0.526158 − 0.850387i \(-0.323632\pi\)
0.526158 + 0.850387i \(0.323632\pi\)
\(578\) 0 0
\(579\) 3.95776e24 0.541181
\(580\) 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 \(\pi\)
\(588\) −2.26280e25 −2.69309
\(589\) −1.54338e25 −1.80899
\(590\) 0 0
\(591\) 0 0
\(592\) −1.59711e24 −0.178829
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.15808e25 1.20219
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1.64594e25 1.60896 0.804479 − 0.593981i \(-0.202445\pi\)
0.804479 + 0.593981i \(0.202445\pi\)
\(602\) 0 0
\(603\) −1.61738e25 −1.53446
\(604\) 2.09758e25 1.96058
\(605\) 0 0
\(606\) 0 0
\(607\) −7.26866e24 −0.649765 −0.324882 − 0.945754i \(-0.605325\pi\)
−0.324882 + 0.945754i \(0.605325\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 2.12295e25 1.73698 0.868491 − 0.495705i \(-0.165090\pi\)
0.868491 + 0.495705i \(0.165090\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −2.66830e25 −1.99995 −0.999974 − 0.00720748i \(-0.997706\pi\)
−0.999974 + 0.00720748i \(0.997706\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 9.73543e24 0.678725
\(625\) 1.45519e25 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 9.72279e24 0.639962
\(629\) 0 0
\(630\) 0 0
\(631\) −2.97028e25 −1.87298 −0.936492 − 0.350690i \(-0.885947\pi\)
−0.936492 + 0.350690i \(0.885947\pi\)
\(632\) 0 0
\(633\) −1.55595e25 −0.953592
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.15646e25 −1.82787
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 2.99340e25 1.59319 0.796593 − 0.604516i \(-0.206633\pi\)
0.796593 + 0.604516i \(0.206633\pi\)
\(644\) 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
\(651\) 7.63490e25 3.63559
\(652\) −3.63689e25 −1.70805
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.15874e25 −0.508038
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 3.56082e25 1.47821 0.739107 − 0.673588i \(-0.235248\pi\)
0.739107 + 0.673588i \(0.235248\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −5.33385e25 −1.98704
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 8.67758e24 0.306382 0.153191 − 0.988197i \(-0.451045\pi\)
0.153191 + 0.988197i \(0.451045\pi\)
\(674\) 0 0
\(675\) −2.90893e25 −1.00000
\(676\) −1.58992e25 −0.539332
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 1.09247e25 0.356107
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 3.13373e25 0.956218
\(685\) 0 0
\(686\) 0 0
\(687\) −3.33536e25 −0.978434
\(688\) −5.01917e25 −1.45324
\(689\) 0 0
\(690\) 0 0
\(691\) 6.63882e25 1.84837 0.924184 − 0.381947i \(-0.124746\pi\)
0.924184 + 0.381947i \(0.124746\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 7.75492e25 1.92174
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −7.17122e24 −0.171000
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.69495e25 −0.374404 −0.187202 − 0.982321i \(-0.559942\pi\)
−0.187202 + 0.982321i \(0.559942\pi\)
\(710\) 0 0
\(711\) 5.44928e25 1.17358
\(712\) 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
\(721\) −2.02067e26 −3.83777
\(722\) 0 0
\(723\) −5.79084e25 −1.07275
\(724\) −2.71748e24 −0.0497187
\(725\) 0 0
\(726\) 0 0
\(727\) 1.13905e25 0.200785 0.100393 − 0.994948i \(-0.467990\pi\)
0.100393 + 0.994948i \(0.467990\pi\)
\(728\) 0 0
\(729\) 5.81497e25 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 4.89517e25 0.811274
\(733\) 8.44039e25 1.38174 0.690871 − 0.722979i \(-0.257227\pi\)
0.690871 + 0.722979i \(0.257227\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −8.10569e25 −1.23308 −0.616538 − 0.787325i \(-0.711465\pi\)
−0.616538 + 0.787325i \(0.711465\pi\)
\(740\) 0 0
\(741\) 4.37134e25 0.649009
\(742\) 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
\(751\) 1.33690e26 1.75930 0.879650 − 0.475622i \(-0.157777\pi\)
0.879650 + 0.475622i \(0.157777\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −1.55021e26 −1.92174
\(757\) 1.37746e26 1.68739 0.843696 − 0.536822i \(-0.180375\pi\)
0.843696 + 0.536822i \(0.180375\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 5.23162e25 0.596919
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −9.29503e25 −1.00000
\(769\) −1.49823e26 −1.59309 −0.796545 − 0.604579i \(-0.793341\pi\)
−0.796545 + 0.604579i \(0.793341\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −5.27107e25 −0.541181
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −1.90807e26 −1.89182
\(776\) 0 0
\(777\) 3.54750e25 0.343663
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.01367e26 2.69309
\(785\) 0 0
\(786\) 0 0
\(787\) 1.02574e26 0.885659 0.442830 − 0.896606i \(-0.353975\pi\)
0.442830 + 0.896606i \(0.353975\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6.82842e25 0.550632
\(794\) 0 0
\(795\) 0 0
\(796\) −1.54237e26 −1.20219
\(797\) 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
\(804\) 2.15407e26 1.53446
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −2.99605e26 −1.97406 −0.987032 − 0.160521i \(-0.948683\pi\)
−0.987032 + 0.160521i \(0.948683\pi\)
\(812\) 0 0
\(813\) 2.65095e26 1.70839
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2.25368e26 −1.38961
\(818\) 0 0
\(819\) −2.16244e26 −1.30433
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −7.02403e22 −0.000405497 0 −0.000202749 − 1.00000i \(-0.500065\pi\)
−0.000202749 1.00000i \(0.500065\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 2.15708e26 1.16648 0.583238 − 0.812301i \(-0.301786\pi\)
0.583238 + 0.812301i \(0.301786\pi\)
\(830\) 0 0
\(831\) −3.43935e26 −1.81998
\(832\) −1.29659e26 −0.678725
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.81425e26 1.89182
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 2.10457e26 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 2.07226e26 0.953592
\(845\) 0 0
\(846\) 0 0
\(847\) 4.31167e26 1.92174
\(848\) 0 0
\(849\) −3.70111e26 −1.61496
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −2.57705e26 −1.07791 −0.538955 − 0.842335i \(-0.681181\pi\)
−0.538955 + 0.842335i \(0.681181\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 3.22977e26 1.26833 0.634166 − 0.773197i \(-0.281343\pi\)
0.634166 + 0.773197i \(0.281343\pi\)
\(860\) 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
\(867\) −2.76804e26 −1.00000
\(868\) −1.01684e27 −3.63559
\(869\) 0 0
\(870\) 0 0
\(871\) 3.00478e26 1.04148
\(872\) 0 0
\(873\) 5.45775e25 0.185304
\(874\) 0 0
\(875\) 0 0
\(876\) 1.54324e26 0.508038
\(877\) −6.37777e24 −0.0207813 −0.0103906 − 0.999946i \(-0.503308\pi\)
−0.0103906 + 0.999946i \(0.503308\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 4.94420e26 1.51513 0.757565 − 0.652760i \(-0.226389\pi\)
0.757565 + 0.652760i \(0.226389\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −9.29800e26 −2.68086
\(890\) 0 0
\(891\) 0 0
\(892\) 7.10379e26 1.98704
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 3.87420e26 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) 1.11486e27 2.79274
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −7.26459e26 −1.74882 −0.874411 − 0.485186i \(-0.838752\pi\)
−0.874411 + 0.485186i \(0.838752\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −4.17360e26 −0.956218
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 4.44213e26 0.978434
\(917\) 0 0
\(918\) 0 0
\(919\) −8.94212e26 −1.91250 −0.956249 − 0.292553i \(-0.905495\pi\)
−0.956249 + 0.292553i \(0.905495\pi\)
\(920\) 0 0
\(921\) −6.00176e26 −1.25876
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −8.86572e25 −0.178829
\(926\) 0 0
\(927\) −1.00949e27 −1.99703
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1.35318e27 2.57518
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.58605e25 0.0823726 0.0411863 − 0.999151i \(-0.486886\pi\)
0.0411863 + 0.999151i \(0.486886\pi\)
\(938\) 0 0
\(939\) 1.00222e27 1.76592
\(940\) 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 \(\pi\)
\(948\) −7.25752e26 −1.17358
\(949\) 2.15271e26 0.344818
\(950\) 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
\(961\) 1.80285e27 2.57898
\(962\) 0 0
\(963\) 0 0
\(964\) 7.71241e26 1.07275
\(965\) 0 0
\(966\) 0 0
\(967\) −1.36759e27 −1.84977 −0.924886 − 0.380243i \(-0.875840\pi\)
−0.924886 + 0.380243i \(0.875840\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −7.74455e26 −1.00000
\(973\) −1.96506e27 −2.51397
\(974\) 0 0
\(975\) 5.40425e26 0.678725
\(976\) −6.51953e26 −0.811274
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 2.61362e26 0.310614
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −5.82188e26 −0.649009
\(989\) 0 0
\(990\) 0 0
\(991\) −3.82354e26 −0.414765 −0.207383 − 0.978260i \(-0.566495\pi\)
−0.207383 + 0.978260i \(0.566495\pi\)
\(992\) 0 0
\(993\) 1.65234e27 1.76018
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.87125e27 1.92254 0.961272 − 0.275602i \(-0.0888771\pi\)
0.961272 + 0.275602i \(0.0888771\pi\)
\(998\) 0 0
\(999\) 1.77226e26 0.178829
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 3.19.b.a.2.1 ✓ 1
3.2 odd 2 CM 3.19.b.a.2.1 ✓ 1
4.3 odd 2 48.19.e.a.17.1 1
12.11 even 2 48.19.e.a.17.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.19.b.a.2.1 ✓ 1 1.1 even 1 trivial
3.19.b.a.2.1 ✓ 1 3.2 odd 2 CM
48.19.e.a.17.1 1 4.3 odd 2
48.19.e.a.17.1 1 12.11 even 2