Properties

Label 384.4.f.b.191.1
Level $384$
Weight $4$
Character 384.191
Analytic conductor $22.657$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,4,Mod(191,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.191");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.f (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: no
Analytic conductor: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 191.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 384.191
Dual form 384.4.f.b.191.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.41421 - 5.00000i) q^{3} +(-23.0000 + 14.1421i) q^{9} +O(q^{10})\) \(q+(-1.41421 - 5.00000i) q^{3} +(-23.0000 + 14.1421i) q^{9} +18.0000i q^{11} +107.480i q^{17} +127.279 q^{19} -125.000 q^{25} +(103.238 + 95.0000i) q^{27} +(90.0000 - 25.4558i) q^{33} -56.5685i q^{41} +483.661 q^{43} +343.000 q^{49} +(537.401 - 152.000i) q^{51} +(-180.000 - 636.396i) q^{57} +846.000i q^{59} +1094.60 q^{67} +430.000 q^{73} +(176.777 + 625.000i) q^{75} +(329.000 - 650.538i) q^{81} -1350.00i q^{83} +1329.36i q^{89} -1910.00 q^{97} +(-254.558 - 414.000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 92 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 92 q^{9} - 500 q^{25} + 360 q^{33} + 1372 q^{49} - 720 q^{57} + 1720 q^{73} + 1316 q^{81} - 7640 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) −1.41421 5.00000i −0.272166 0.962250i
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −23.0000 + 14.1421i −0.851852 + 0.523783i
\(10\) 0 0
\(11\) 18.0000i 0.493382i 0.969094 + 0.246691i \(0.0793433\pi\)
−0.969094 + 0.246691i \(0.920657\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 107.480i 1.53340i 0.642006 + 0.766700i \(0.278102\pi\)
−0.642006 + 0.766700i \(0.721898\pi\)
\(18\) 0 0
\(19\) 127.279 1.53683 0.768417 0.639949i \(-0.221045\pi\)
0.768417 + 0.639949i \(0.221045\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −125.000 −1.00000
\(26\) 0 0
\(27\) 103.238 + 95.0000i 0.735855 + 0.677139i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 90.0000 25.4558i 0.474757 0.134282i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 56.5685i 0.215476i −0.994179 0.107738i \(-0.965639\pi\)
0.994179 0.107738i \(-0.0343608\pi\)
\(42\) 0 0
\(43\) 483.661 1.71529 0.857647 0.514239i \(-0.171926\pi\)
0.857647 + 0.514239i \(0.171926\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 343.000 1.00000
\(50\) 0 0
\(51\) 537.401 152.000i 1.47551 0.417338i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −180.000 636.396i −0.418273 1.47882i
\(58\) 0 0
\(59\) 846.000i 1.86678i 0.358868 + 0.933388i \(0.383163\pi\)
−0.358868 + 0.933388i \(0.616837\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1094.60 1.99592 0.997961 0.0638199i \(-0.0203283\pi\)
0.997961 + 0.0638199i \(0.0203283\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 430.000 0.689420 0.344710 0.938709i \(-0.387977\pi\)
0.344710 + 0.938709i \(0.387977\pi\)
\(74\) 0 0
\(75\) 176.777 + 625.000i 0.272166 + 0.962250i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 329.000 650.538i 0.451303 0.892371i
\(82\) 0 0
\(83\) 1350.00i 1.78532i −0.450728 0.892661i \(-0.648836\pi\)
0.450728 0.892661i \(-0.351164\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1329.36i 1.58328i 0.610988 + 0.791640i \(0.290773\pi\)
−0.610988 + 0.791640i \(0.709227\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1910.00 −1.99929 −0.999645 0.0266459i \(-0.991517\pi\)
−0.999645 + 0.0266459i \(0.991517\pi\)
\(98\) 0 0
\(99\) −254.558 414.000i −0.258425 0.420289i
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1710.00i 1.54497i 0.635032 + 0.772486i \(0.280987\pi\)
−0.635032 + 0.772486i \(0.719013\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2387.19i 1.98733i 0.112387 + 0.993665i \(0.464150\pi\)
−0.112387 + 0.993665i \(0.535850\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1007.00 0.756574
\(122\) 0 0
\(123\) −282.843 + 80.0000i −0.207342 + 0.0586452i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) −684.000 2418.31i −0.466844 1.65054i
\(130\) 0 0
\(131\) 1242.00i 0.828351i −0.910197 0.414176i \(-0.864070\pi\)
0.910197 0.414176i \(-0.135930\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2285.37i 1.42520i 0.701571 + 0.712599i \(0.252482\pi\)
−0.701571 + 0.712599i \(0.747518\pi\)
\(138\) 0 0
\(139\) −2927.42 −1.78634 −0.893168 0.449723i \(-0.851523\pi\)
−0.893168 + 0.449723i \(0.851523\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −485.075 1715.00i −0.272166 0.962250i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −1520.00 2472.05i −0.803168 1.30623i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4047.48 −1.94493 −0.972463 0.233056i \(-0.925127\pi\)
−0.972463 + 0.233056i \(0.925127\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 2197.00 1.00000
\(170\) 0 0
\(171\) −2927.42 + 1800.00i −1.30916 + 0.804967i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4230.00 1196.42i 1.79631 0.508072i
\(178\) 0 0
\(179\) 3834.00i 1.60093i −0.599379 0.800465i \(-0.704586\pi\)
0.599379 0.800465i \(-0.295414\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1934.64 −0.756552
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −2090.00 −0.779490 −0.389745 0.920923i \(-0.627437\pi\)
−0.389745 + 0.920923i \(0.627437\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) −1548.00 5473.01i −0.543221 1.92058i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2291.03i 0.758247i
\(210\) 0 0
\(211\) 381.838 0.124582 0.0622910 0.998058i \(-0.480159\pi\)
0.0622910 + 0.998058i \(0.480159\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −608.112 2150.00i −0.187636 0.663395i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 2875.00 1767.77i 0.851852 0.523783i
\(226\) 0 0
\(227\) 6570.00i 1.92100i 0.278286 + 0.960498i \(0.410234\pi\)
−0.278286 + 0.960498i \(0.589766\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3773.12i 1.06088i 0.847722 + 0.530441i \(0.177974\pi\)
−0.847722 + 0.530441i \(0.822026\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 1222.00 0.326622 0.163311 0.986575i \(-0.447783\pi\)
0.163311 + 0.986575i \(0.447783\pi\)
\(242\) 0 0
\(243\) −3717.97 725.000i −0.981513 0.191394i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −6750.00 + 1909.19i −1.71793 + 0.485903i
\(250\) 0 0
\(251\) 4302.00i 1.08183i −0.841077 0.540916i \(-0.818078\pi\)
0.841077 0.540916i \(-0.181922\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7274.71i 1.76570i 0.469658 + 0.882849i \(0.344377\pi\)
−0.469658 + 0.882849i \(0.655623\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6646.80 1880.00i 1.52351 0.430914i
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2250.00i 0.493382i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1216.22i 0.258199i −0.991632 0.129099i \(-0.958791\pi\)
0.991632 0.129099i \(-0.0412086\pi\)
\(282\) 0 0
\(283\) 5116.62 1.07474 0.537371 0.843346i \(-0.319418\pi\)
0.537371 + 0.843346i \(0.319418\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −6639.00 −1.35131
\(290\) 0 0
\(291\) 2701.15 + 9550.00i 0.544138 + 1.92382i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1710.00 + 1858.28i −0.334088 + 0.363058i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7204.00 1.33926 0.669632 0.742693i \(-0.266452\pi\)
0.669632 + 0.742693i \(0.266452\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 8390.00 1.51511 0.757557 0.652769i \(-0.226393\pi\)
0.757557 + 0.652769i \(0.226393\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 8550.00 2418.31i 1.48665 0.420488i
\(322\) 0 0
\(323\) 13680.0i 2.35658i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8782.27 1.45836 0.729180 0.684322i \(-0.239902\pi\)
0.729180 + 0.684322i \(0.239902\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −11410.0 −1.84434 −0.922170 0.386786i \(-0.873585\pi\)
−0.922170 + 0.386786i \(0.873585\pi\)
\(338\) 0 0
\(339\) 11936.0 3376.00i 1.91231 0.540882i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6030.00i 0.932874i −0.884554 0.466437i \(-0.845537\pi\)
0.884554 0.466437i \(-0.154463\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12377.2i 1.86621i −0.359605 0.933104i \(-0.617089\pi\)
0.359605 0.933104i \(-0.382911\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 9341.00 1.36186
\(362\) 0 0
\(363\) −1424.11 5035.00i −0.205913 0.728014i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 800.000 + 1301.08i 0.112863 + 0.183554i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 9036.82 1.22478 0.612389 0.790557i \(-0.290209\pi\)
0.612389 + 0.790557i \(0.290209\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −11124.2 + 6840.00i −1.46118 + 0.898441i
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −6210.00 + 1756.45i −0.797082 + 0.225449i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14453.3i 1.79990i −0.435989 0.899952i \(-0.643601\pi\)
0.435989 0.899952i \(-0.356399\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 16346.0 1.97618 0.988090 0.153877i \(-0.0491758\pi\)
0.988090 + 0.153877i \(0.0491758\pi\)
\(410\) 0 0
\(411\) 11426.8 3232.00i 1.37140 0.387890i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4140.00 + 14637.1i 0.486179 + 1.71890i
\(418\) 0 0
\(419\) 16794.0i 1.95809i 0.203639 + 0.979046i \(0.434723\pi\)
−0.203639 + 0.979046i \(0.565277\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 13435.0i 1.53340i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −5510.00 −0.611533 −0.305766 0.952107i \(-0.598913\pi\)
−0.305766 + 0.952107i \(0.598913\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −7889.00 + 4850.75i −0.851852 + 0.523783i
\(442\) 0 0
\(443\) 18270.0i 1.95944i −0.200361 0.979722i \(-0.564211\pi\)
0.200361 0.979722i \(-0.435789\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7438.76i 0.781864i 0.920420 + 0.390932i \(0.127847\pi\)
−0.920420 + 0.390932i \(0.872153\pi\)
\(450\) 0 0
\(451\) 1018.23 0.106312
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −18070.0 −1.84963 −0.924813 0.380422i \(-0.875779\pi\)
−0.924813 + 0.380422i \(0.875779\pi\)
\(458\) 0 0
\(459\) −10210.6 + 11096.0i −1.03832 + 1.12836i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15030.0i 1.48931i −0.667452 0.744653i \(-0.732615\pi\)
0.667452 0.744653i \(-0.267385\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8705.90i 0.846295i
\(474\) 0 0
\(475\) −15909.9 −1.53683
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 5724.00 + 20237.4i 0.529342 + 1.87151i
\(490\) 0 0
\(491\) 12222.0i 1.12336i 0.827354 + 0.561681i \(0.189845\pi\)
−0.827354 + 0.561681i \(0.810155\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −12855.2 −1.15326 −0.576631 0.817005i \(-0.695633\pi\)
−0.576631 + 0.817005i \(0.695633\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3107.03 10985.0i −0.272166 0.962250i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 13140.0 + 12091.5i 1.13089 + 1.04065i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21948.6i 1.84565i −0.385215 0.922827i \(-0.625873\pi\)
0.385215 0.922827i \(-0.374127\pi\)
\(522\) 0 0
\(523\) −23444.8 −1.96017 −0.980087 0.198569i \(-0.936371\pi\)
−0.980087 + 0.198569i \(0.936371\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) −11964.2 19458.0i −0.977785 1.59022i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −19170.0 + 5422.09i −1.54050 + 0.435718i
\(538\) 0 0
\(539\) 6174.00i 0.493382i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −13313.4 −1.04066 −0.520329 0.853966i \(-0.674191\pi\)
−0.520329 + 0.853966i \(0.674191\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 2736.00 + 9673.22i 0.205907 + 0.727992i
\(562\) 0 0
\(563\) 23670.0i 1.77189i −0.463795 0.885943i \(-0.653512\pi\)
0.463795 0.885943i \(-0.346488\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27039.8i 1.99221i −0.0881913 0.996104i \(-0.528109\pi\)
0.0881913 0.996104i \(-0.471891\pi\)
\(570\) 0 0
\(571\) 3691.10 0.270521 0.135261 0.990810i \(-0.456813\pi\)
0.135261 + 0.990810i \(0.456813\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 19550.0 1.41053 0.705266 0.708943i \(-0.250827\pi\)
0.705266 + 0.708943i \(0.250827\pi\)
\(578\) 0 0
\(579\) 2955.71 + 10450.0i 0.212150 + 0.750064i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10350.0i 0.727752i 0.931447 + 0.363876i \(0.118547\pi\)
−0.931447 + 0.363876i \(0.881453\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12173.6i 0.843015i −0.906825 0.421507i \(-0.861501\pi\)
0.906825 0.421507i \(-0.138499\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 14398.0 0.977216 0.488608 0.872503i \(-0.337505\pi\)
0.488608 + 0.872503i \(0.337505\pi\)
\(602\) 0 0
\(603\) −25175.8 + 15480.0i −1.70023 + 1.04543i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11206.2i 0.731192i 0.930774 + 0.365596i \(0.119135\pi\)
−0.930774 + 0.365596i \(0.880865\pi\)
\(618\) 0 0
\(619\) −2418.31 −0.157027 −0.0785136 0.996913i \(-0.525017\pi\)
−0.0785136 + 0.996913i \(0.525017\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15625.0 1.00000
\(626\) 0 0
\(627\) 11455.1 3240.00i 0.729623 0.206369i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −540.000 1909.19i −0.0339069 0.119879i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 31763.2i 1.95721i −0.205745 0.978606i \(-0.565962\pi\)
0.205745 0.978606i \(-0.434038\pi\)
\(642\) 0 0
\(643\) −15757.2 −0.966411 −0.483205 0.875507i \(-0.660528\pi\)
−0.483205 + 0.875507i \(0.660528\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −15228.0 −0.921034
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −9890.00 + 6081.12i −0.587284 + 0.361107i
\(658\) 0 0
\(659\) 29754.0i 1.75880i −0.476081 0.879402i \(-0.657943\pi\)
0.476081 0.879402i \(-0.342057\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −19190.0 −1.09914 −0.549569 0.835448i \(-0.685208\pi\)
−0.549569 + 0.835448i \(0.685208\pi\)
\(674\) 0 0
\(675\) −12904.7 11875.0i −0.735855 0.677139i
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 32850.0 9291.38i 1.84848 0.522829i
\(682\) 0 0
\(683\) 11970.0i 0.670599i 0.942112 + 0.335300i \(0.108838\pi\)
−0.942112 + 0.335300i \(0.891162\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 36274.6 1.99703 0.998517 0.0544477i \(-0.0173398\pi\)
0.998517 + 0.0544477i \(0.0173398\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 6080.00 0.330411
\(698\) 0 0
\(699\) 18865.6 5336.00i 1.02083 0.288735i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −1728.17 6110.00i −0.0888953 0.314292i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1633.00 + 19615.1i 0.0829650 + 0.996552i
\(730\) 0 0
\(731\) 51984.0i 2.63023i
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 19702.8i 0.984753i
\(738\) 0 0
\(739\) 17691.8 0.880655 0.440327 0.897837i \(-0.354862\pi\)
0.440327 + 0.897837i \(0.354862\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 19091.9 + 31050.0i 0.935121 + 1.52083i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −21510.0 + 6083.95i −1.04099 + 0.294437i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24381.0i 1.16138i 0.814124 + 0.580691i \(0.197218\pi\)
−0.814124 + 0.580691i \(0.802782\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −40106.0 −1.88070 −0.940351 0.340207i \(-0.889503\pi\)
−0.940351 + 0.340207i \(0.889503\pi\)
\(770\) 0 0
\(771\) 36373.6 10288.0i 1.69904 0.480562i
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7200.00i 0.331151i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 43605.9 1.97507 0.987536 0.157396i \(-0.0503098\pi\)
0.987536 + 0.157396i \(0.0503098\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −18800.0 30575.3i −0.829295 1.34872i
\(802\) 0 0
\(803\) 7740.00i 0.340148i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 43727.5i 1.90034i 0.311730 + 0.950171i \(0.399092\pi\)
−0.311730 + 0.950171i \(0.600908\pi\)
\(810\) 0 0
\(811\) 26855.9 1.16281 0.581405 0.813614i \(-0.302503\pi\)
0.581405 + 0.813614i \(0.302503\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 61560.0 2.63612
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) −11250.0 + 3181.98i −0.474757 + 0.134282i
\(826\) 0 0
\(827\) 23490.0i 0.987699i −0.869547 0.493850i \(-0.835589\pi\)
0.869547 0.493850i \(-0.164411\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 36865.7i 1.53340i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −24389.0 −1.00000
\(842\) 0 0
\(843\) −6081.12 + 1720.00i −0.248452 + 0.0702728i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −7236.00 25583.1i −0.292508 1.03417i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 46589.9i 1.85704i −0.371289 0.928518i \(-0.621084\pi\)
0.371289 0.928518i \(-0.378916\pi\)
\(858\) 0 0
\(859\) 21255.6 0.844276 0.422138 0.906532i \(-0.361280\pi\)
0.422138 + 0.906532i \(0.361280\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 9388.96 + 33195.0i 0.367781 + 1.30030i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 43930.0 27011.5i 1.70310 1.04719i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 31508.7i 1.20494i 0.798141 + 0.602471i \(0.205817\pi\)
−0.798141 + 0.602471i \(0.794183\pi\)
\(882\) 0 0
\(883\) −52209.9 −1.98981 −0.994906 0.100806i \(-0.967858\pi\)
−0.994906 + 0.100806i \(0.967858\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 11709.7 + 5922.00i 0.440280 + 0.222665i
\(892\) 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
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 47933.4 1.75480 0.877399 0.479762i \(-0.159277\pi\)
0.877399 + 0.479762i \(0.159277\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 24300.0 0.880846
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −10188.0 36020.0i −0.364502 1.28871i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 56200.8i 1.98481i −0.123007 0.992406i \(-0.539254\pi\)
0.123007 0.992406i \(-0.460746\pi\)
\(930\) 0 0
\(931\) 43656.8 1.53683
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −56270.0 −1.96186 −0.980929 0.194367i \(-0.937735\pi\)
−0.980929 + 0.194367i \(0.937735\pi\)
\(938\) 0 0
\(939\) −11865.3 41950.0i −0.412362 1.45792i
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 58230.0i 1.99812i 0.0433353 + 0.999061i \(0.486202\pi\)
−0.0433353 + 0.999061i \(0.513798\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 36701.7i 1.24752i 0.781617 + 0.623759i \(0.214395\pi\)
−0.781617 + 0.623759i \(0.785605\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 29791.0 1.00000
\(962\) 0 0
\(963\) −24183.1 39330.0i −0.809229 1.31609i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 68400.0 19346.4i 2.26762 0.641380i
\(970\) 0 0
\(971\) 162.000i 0.00535410i 0.999996 + 0.00267705i \(0.000852132\pi\)
−0.999996 + 0.00267705i \(0.999148\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 58554.1i 1.91741i 0.284399 + 0.958706i \(0.408206\pi\)
−0.284399 + 0.958706i \(0.591794\pi\)
\(978\) 0 0
\(979\) −23928.5 −0.781162
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −12420.0 43911.3i −0.396915 1.40331i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 384.4.f.b.191.1 4
3.2 odd 2 inner 384.4.f.b.191.2 yes 4
4.3 odd 2 inner 384.4.f.b.191.4 yes 4
8.3 odd 2 CM 384.4.f.b.191.1 4
8.5 even 2 inner 384.4.f.b.191.4 yes 4
12.11 even 2 inner 384.4.f.b.191.3 yes 4
16.3 odd 4 768.4.c.a.767.2 2
16.5 even 4 768.4.c.a.767.2 2
16.11 odd 4 768.4.c.j.767.1 2
16.13 even 4 768.4.c.j.767.1 2
24.5 odd 2 inner 384.4.f.b.191.3 yes 4
24.11 even 2 inner 384.4.f.b.191.2 yes 4
48.5 odd 4 768.4.c.j.767.2 2
48.11 even 4 768.4.c.a.767.1 2
48.29 odd 4 768.4.c.a.767.1 2
48.35 even 4 768.4.c.j.767.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.f.b.191.1 4 1.1 even 1 trivial
384.4.f.b.191.1 4 8.3 odd 2 CM
384.4.f.b.191.2 yes 4 3.2 odd 2 inner
384.4.f.b.191.2 yes 4 24.11 even 2 inner
384.4.f.b.191.3 yes 4 12.11 even 2 inner
384.4.f.b.191.3 yes 4 24.5 odd 2 inner
384.4.f.b.191.4 yes 4 4.3 odd 2 inner
384.4.f.b.191.4 yes 4 8.5 even 2 inner
768.4.c.a.767.1 2 48.11 even 4
768.4.c.a.767.1 2 48.29 odd 4
768.4.c.a.767.2 2 16.3 odd 4
768.4.c.a.767.2 2 16.5 even 4
768.4.c.j.767.1 2 16.11 odd 4
768.4.c.j.767.1 2 16.13 even 4
768.4.c.j.767.2 2 48.5 odd 4
768.4.c.j.767.2 2 48.35 even 4