Properties

Label 128.3.d.b.63.1
Level $128$
Weight $3$
Character 128.63
Self dual yes
Analytic conductor $3.488$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [128,3,Mod(63,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.63");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 128.d (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: \(3.48774738381\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 63.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 128.63

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.65685 q^{3} +23.0000 q^{9} +O(q^{10})\) \(q-5.65685 q^{3} +23.0000 q^{9} +16.9706 q^{11} -2.00000 q^{17} +16.9706 q^{19} +25.0000 q^{25} -79.1960 q^{27} -96.0000 q^{33} -46.0000 q^{41} +84.8528 q^{43} +49.0000 q^{49} +11.3137 q^{51} -96.0000 q^{57} +84.8528 q^{59} -118.794 q^{67} +142.000 q^{73} -141.421 q^{75} +241.000 q^{81} -50.9117 q^{83} -146.000 q^{89} +94.0000 q^{97} +390.323 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 46 q^{9} - 4 q^{17} + 50 q^{25} - 192 q^{33} - 92 q^{41} + 98 q^{49} - 192 q^{57} + 284 q^{73} + 482 q^{81} - 292 q^{89} + 188 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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
\(3\) −5.65685 −1.88562 −0.942809 0.333333i \(-0.891827\pi\)
−0.942809 + 0.333333i \(0.891827\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 23.0000 2.55556
\(10\) 0 0
\(11\) 16.9706 1.54278 0.771389 0.636364i \(-0.219562\pi\)
0.771389 + 0.636364i \(0.219562\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.117647 −0.0588235 0.998268i \(-0.518735\pi\)
−0.0588235 + 0.998268i \(0.518735\pi\)
\(18\) 0 0
\(19\) 16.9706 0.893188 0.446594 0.894737i \(-0.352637\pi\)
0.446594 + 0.894737i \(0.352637\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) −79.1960 −2.93318
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) −96.0000 −2.90909
\(34\) 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
\(41\) −46.0000 −1.12195 −0.560976 0.827832i \(-0.689574\pi\)
−0.560976 + 0.827832i \(0.689574\pi\)
\(42\) 0 0
\(43\) 84.8528 1.97332 0.986661 0.162791i \(-0.0520495\pi\)
0.986661 + 0.162791i \(0.0520495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 11.3137 0.221837
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −96.0000 −1.68421
\(58\) 0 0
\(59\) 84.8528 1.43818 0.719092 0.694915i \(-0.244558\pi\)
0.719092 + 0.694915i \(0.244558\pi\)
\(60\) 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
\(67\) −118.794 −1.77304 −0.886522 0.462687i \(-0.846886\pi\)
−0.886522 + 0.462687i \(0.846886\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 142.000 1.94521 0.972603 0.232473i \(-0.0746819\pi\)
0.972603 + 0.232473i \(0.0746819\pi\)
\(74\) 0 0
\(75\) −141.421 −1.88562
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 241.000 2.97531
\(82\) 0 0
\(83\) −50.9117 −0.613394 −0.306697 0.951807i \(-0.599224\pi\)
−0.306697 + 0.951807i \(0.599224\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −146.000 −1.64045 −0.820225 0.572041i \(-0.806152\pi\)
−0.820225 + 0.572041i \(0.806152\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 94.0000 0.969072 0.484536 0.874771i \(-0.338988\pi\)
0.484536 + 0.874771i \(0.338988\pi\)
\(98\) 0 0
\(99\) 390.323 3.94266
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −118.794 −1.11022 −0.555112 0.831776i \(-0.687325\pi\)
−0.555112 + 0.831776i \(0.687325\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 98.0000 0.867257 0.433628 0.901092i \(-0.357233\pi\)
0.433628 + 0.901092i \(0.357233\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 167.000 1.38017
\(122\) 0 0
\(123\) 260.215 2.11557
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) −480.000 −3.72093
\(130\) 0 0
\(131\) −254.558 −1.94319 −0.971597 0.236641i \(-0.923953\pi\)
−0.971597 + 0.236641i \(0.923953\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −238.000 −1.73723 −0.868613 0.495491i \(-0.834988\pi\)
−0.868613 + 0.495491i \(0.834988\pi\)
\(138\) 0 0
\(139\) −186.676 −1.34299 −0.671497 0.741007i \(-0.734348\pi\)
−0.671497 + 0.741007i \(0.734348\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −277.186 −1.88562
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −46.0000 −0.300654
\(154\) 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
\(163\) −50.9117 −0.312342 −0.156171 0.987730i \(-0.549915\pi\)
−0.156171 + 0.987730i \(0.549915\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 390.323 2.28259
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −480.000 −2.71186
\(178\) 0 0
\(179\) 356.382 1.99096 0.995480 0.0949721i \(-0.0302762\pi\)
0.995480 + 0.0949721i \(0.0302762\pi\)
\(180\) 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
\(187\) −33.9411 −0.181503
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −98.0000 −0.507772 −0.253886 0.967234i \(-0.581709\pi\)
−0.253886 + 0.967234i \(0.581709\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 672.000 3.34328
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 288.000 1.37799
\(210\) 0 0
\(211\) 356.382 1.68901 0.844507 0.535545i \(-0.179894\pi\)
0.844507 + 0.535545i \(0.179894\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −803.273 −3.66791
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 575.000 2.55556
\(226\) 0 0
\(227\) 84.8528 0.373801 0.186900 0.982379i \(-0.440156\pi\)
0.186900 + 0.982379i \(0.440156\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −434.000 −1.86266 −0.931330 0.364175i \(-0.881351\pi\)
−0.931330 + 0.364175i \(0.881351\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −194.000 −0.804979 −0.402490 0.915425i \(-0.631855\pi\)
−0.402490 + 0.915425i \(0.631855\pi\)
\(242\) 0 0
\(243\) −650.538 −2.67711
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 288.000 1.15663
\(250\) 0 0
\(251\) −186.676 −0.743730 −0.371865 0.928287i \(-0.621282\pi\)
−0.371865 + 0.928287i \(0.621282\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 386.000 1.50195 0.750973 0.660333i \(-0.229585\pi\)
0.750973 + 0.660333i \(0.229585\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 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
\(267\) 825.901 3.09326
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 424.264 1.54278
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 238.000 0.846975 0.423488 0.905902i \(-0.360806\pi\)
0.423488 + 0.905902i \(0.360806\pi\)
\(282\) 0 0
\(283\) 560.029 1.97890 0.989450 0.144876i \(-0.0462784\pi\)
0.989450 + 0.144876i \(0.0462784\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −285.000 −0.986159
\(290\) 0 0
\(291\) −531.744 −1.82730
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1344.00 −4.52525
\(298\) 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
\(307\) 288.500 0.939738 0.469869 0.882736i \(-0.344301\pi\)
0.469869 + 0.882736i \(0.344301\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −526.000 −1.68051 −0.840256 0.542191i \(-0.817595\pi\)
−0.840256 + 0.542191i \(0.817595\pi\)
\(314\) 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
\(321\) 672.000 2.09346
\(322\) 0 0
\(323\) −33.9411 −0.105081
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −661.852 −1.99955 −0.999776 0.0211480i \(-0.993268\pi\)
−0.999776 + 0.0211480i \(0.993268\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −478.000 −1.41840 −0.709199 0.705009i \(-0.750943\pi\)
−0.709199 + 0.705009i \(0.750943\pi\)
\(338\) 0 0
\(339\) −554.372 −1.63531
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 220.617 0.635785 0.317892 0.948127i \(-0.397025\pi\)
0.317892 + 0.948127i \(0.397025\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 194.000 0.549575 0.274788 0.961505i \(-0.411392\pi\)
0.274788 + 0.961505i \(0.411392\pi\)
\(354\) 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
\(361\) −73.0000 −0.202216
\(362\) 0 0
\(363\) −944.695 −2.60246
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −1058.00 −2.86721
\(370\) 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
\(379\) −322.441 −0.850767 −0.425383 0.905013i \(-0.639861\pi\)
−0.425383 + 0.905013i \(0.639861\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1951.61 5.04293
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 1440.00 3.66412
\(394\) 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
\(401\) 766.000 1.91022 0.955112 0.296244i \(-0.0957342\pi\)
0.955112 + 0.296244i \(0.0957342\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −334.000 −0.816626 −0.408313 0.912842i \(-0.633883\pi\)
−0.408313 + 0.912842i \(0.633883\pi\)
\(410\) 0 0
\(411\) 1346.33 3.27575
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1056.00 2.53237
\(418\) 0 0
\(419\) −661.852 −1.57960 −0.789799 0.613365i \(-0.789815\pi\)
−0.789799 + 0.613365i \(0.789815\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −50.0000 −0.117647
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −578.000 −1.33487 −0.667436 0.744667i \(-0.732608\pi\)
−0.667436 + 0.744667i \(0.732608\pi\)
\(434\) 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
\(441\) 1127.00 2.55556
\(442\) 0 0
\(443\) −118.794 −0.268158 −0.134079 0.990971i \(-0.542808\pi\)
−0.134079 + 0.990971i \(0.542808\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −866.000 −1.92873 −0.964365 0.264574i \(-0.914769\pi\)
−0.964365 + 0.264574i \(0.914769\pi\)
\(450\) 0 0
\(451\) −780.646 −1.73092
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −238.000 −0.520788 −0.260394 0.965502i \(-0.583852\pi\)
−0.260394 + 0.965502i \(0.583852\pi\)
\(458\) 0 0
\(459\) 158.392 0.345080
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −933.381 −1.99867 −0.999337 0.0364026i \(-0.988410\pi\)
−0.999337 + 0.0364026i \(0.988410\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1440.00 3.04440
\(474\) 0 0
\(475\) 424.264 0.893188
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 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
\(489\) 288.000 0.588957
\(490\) 0 0
\(491\) −593.970 −1.20971 −0.604857 0.796334i \(-0.706770\pi\)
−0.604857 + 0.796334i \(0.706770\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −593.970 −1.19032 −0.595160 0.803607i \(-0.702911\pi\)
−0.595160 + 0.803607i \(0.702911\pi\)
\(500\) 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
\(507\) −956.008 −1.88562
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1344.00 −2.61988
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1006.00 −1.93090 −0.965451 0.260584i \(-0.916085\pi\)
−0.965451 + 0.260584i \(0.916085\pi\)
\(522\) 0 0
\(523\) 967.322 1.84956 0.924782 0.380497i \(-0.124247\pi\)
0.924782 + 0.380497i \(0.124247\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 1951.61 3.67536
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −2016.00 −3.75419
\(538\) 0 0
\(539\) 831.558 1.54278
\(540\) 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
\(547\) −390.323 −0.713570 −0.356785 0.934186i \(-0.616127\pi\)
−0.356785 + 0.934186i \(0.616127\pi\)
\(548\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 192.000 0.342246
\(562\) 0 0
\(563\) 1103.09 1.95930 0.979651 0.200710i \(-0.0643251\pi\)
0.979651 + 0.200710i \(0.0643251\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 626.000 1.10018 0.550088 0.835107i \(-0.314594\pi\)
0.550088 + 0.835107i \(0.314594\pi\)
\(570\) 0 0
\(571\) −933.381 −1.63464 −0.817321 0.576182i \(-0.804542\pi\)
−0.817321 + 0.576182i \(0.804542\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000 0.00346620 0.00173310 0.999998i \(-0.499448\pi\)
0.00173310 + 0.999998i \(0.499448\pi\)
\(578\) 0 0
\(579\) 554.372 0.957464
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 288.500 0.491481 0.245741 0.969336i \(-0.420969\pi\)
0.245741 + 0.969336i \(0.420969\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −862.000 −1.45363 −0.726813 0.686836i \(-0.758999\pi\)
−0.726813 + 0.686836i \(0.758999\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −914.000 −1.52080 −0.760399 0.649456i \(-0.774997\pi\)
−0.760399 + 0.649456i \(0.774997\pi\)
\(602\) 0 0
\(603\) −2732.26 −4.53111
\(604\) 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
\(617\) 334.000 0.541329 0.270665 0.962674i \(-0.412757\pi\)
0.270665 + 0.962674i \(0.412757\pi\)
\(618\) 0 0
\(619\) 1103.09 1.78205 0.891023 0.453958i \(-0.149988\pi\)
0.891023 + 0.453958i \(0.149988\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) −1629.17 −2.59836
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −2016.00 −3.18483
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −482.000 −0.751950 −0.375975 0.926630i \(-0.622692\pi\)
−0.375975 + 0.926630i \(0.622692\pi\)
\(642\) 0 0
\(643\) 424.264 0.659820 0.329910 0.944012i \(-0.392982\pi\)
0.329910 + 0.944012i \(0.392982\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 1440.00 2.21880
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3266.00 4.97108
\(658\) 0 0
\(659\) −865.499 −1.31335 −0.656676 0.754173i \(-0.728038\pi\)
−0.656676 + 0.754173i \(0.728038\pi\)
\(660\) 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
\(673\) 1246.00 1.85141 0.925706 0.378244i \(-0.123472\pi\)
0.925706 + 0.378244i \(0.123472\pi\)
\(674\) 0 0
\(675\) −1979.90 −2.93318
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −480.000 −0.704846
\(682\) 0 0
\(683\) 1306.73 1.91323 0.956613 0.291362i \(-0.0941083\pi\)
0.956613 + 0.291362i \(0.0941083\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1170.97 1.69460 0.847300 0.531114i \(-0.178227\pi\)
0.847300 + 0.531114i \(0.178227\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 92.0000 0.131994
\(698\) 0 0
\(699\) 2455.07 3.51227
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1097.43 1.51788
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1511.00 2.07270
\(730\) 0 0
\(731\) −169.706 −0.232155
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2016.00 −2.73541
\(738\) 0 0
\(739\) 1442.50 1.95196 0.975980 0.217862i \(-0.0699083\pi\)
0.975980 + 0.217862i \(0.0699083\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −1170.97 −1.56756
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 1056.00 1.40239
\(754\) 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
\(761\) 1394.00 1.83180 0.915900 0.401406i \(-0.131478\pi\)
0.915900 + 0.401406i \(0.131478\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1054.00 1.37061 0.685306 0.728256i \(-0.259669\pi\)
0.685306 + 0.728256i \(0.259669\pi\)
\(770\) 0 0
\(771\) −2183.55 −2.83210
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −780.646 −1.00211
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1272.79 −1.61727 −0.808635 0.588310i \(-0.799793\pi\)
−0.808635 + 0.588310i \(0.799793\pi\)
\(788\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −3358.00 −4.19226
\(802\) 0 0
\(803\) 2409.82 3.00102
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1582.00 −1.95550 −0.977750 0.209772i \(-0.932728\pi\)
−0.977750 + 0.209772i \(0.932728\pi\)
\(810\) 0 0
\(811\) −1612.20 −1.98792 −0.993960 0.109741i \(-0.964998\pi\)
−0.993960 + 0.109741i \(0.964998\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1440.00 1.76255
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) −2400.00 −2.90909
\(826\) 0 0
\(827\) −1069.15 −1.29280 −0.646400 0.762999i \(-0.723726\pi\)
−0.646400 + 0.762999i \(0.723726\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −98.0000 −0.117647
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) −1346.33 −1.59707
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −3168.00 −3.73145
\(850\) 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
\(857\) 1202.00 1.40257 0.701284 0.712882i \(-0.252611\pi\)
0.701284 + 0.712882i \(0.252611\pi\)
\(858\) 0 0
\(859\) 492.146 0.572929 0.286465 0.958091i \(-0.407520\pi\)
0.286465 + 0.958091i \(0.407520\pi\)
\(860\) 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
\(867\) 1612.20 1.85952
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2162.00 2.47652
\(874\) 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
\(881\) −1438.00 −1.63224 −0.816118 0.577885i \(-0.803878\pi\)
−0.816118 + 0.577885i \(0.803878\pi\)
\(882\) 0 0
\(883\) −118.794 −0.134534 −0.0672672 0.997735i \(-0.521428\pi\)
−0.0672672 + 0.997735i \(0.521428\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4089.91 4.59024
\(892\) 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
\(907\) −593.970 −0.654873 −0.327436 0.944873i \(-0.606185\pi\)
−0.327436 + 0.944873i \(0.606185\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −864.000 −0.946331
\(914\) 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
\(921\) −1632.00 −1.77199
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1058.00 −1.13886 −0.569429 0.822040i \(-0.692836\pi\)
−0.569429 + 0.822040i \(0.692836\pi\)
\(930\) 0 0
\(931\) 831.558 0.893188
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 718.000 0.766275 0.383138 0.923691i \(-0.374843\pi\)
0.383138 + 0.923691i \(0.374843\pi\)
\(938\) 0 0
\(939\) 2975.51 3.16880
\(940\) 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
\(947\) −1612.20 −1.70243 −0.851216 0.524815i \(-0.824134\pi\)
−0.851216 + 0.524815i \(0.824134\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −142.000 −0.149003 −0.0745016 0.997221i \(-0.523737\pi\)
−0.0745016 + 0.997221i \(0.523737\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 0 0
\(963\) −2732.26 −2.83724
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 192.000 0.198142
\(970\) 0 0
\(971\) −1680.09 −1.73026 −0.865132 0.501545i \(-0.832765\pi\)
−0.865132 + 0.501545i \(0.832765\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1918.00 1.96315 0.981576 0.191071i \(-0.0611960\pi\)
0.981576 + 0.191071i \(0.0611960\pi\)
\(978\) 0 0
\(979\) −2477.70 −2.53085
\(980\) 0 0
\(981\) 0 0
\(982\) 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
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 3744.00 3.77039
\(994\) 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 128.3.d.b.63.1 2
3.2 odd 2 1152.3.b.c.703.1 2
4.3 odd 2 inner 128.3.d.b.63.2 yes 2
8.3 odd 2 CM 128.3.d.b.63.1 2
8.5 even 2 inner 128.3.d.b.63.2 yes 2
12.11 even 2 1152.3.b.c.703.2 2
16.3 odd 4 256.3.c.d.255.2 2
16.5 even 4 256.3.c.d.255.2 2
16.11 odd 4 256.3.c.d.255.1 2
16.13 even 4 256.3.c.d.255.1 2
24.5 odd 2 1152.3.b.c.703.2 2
24.11 even 2 1152.3.b.c.703.1 2
48.5 odd 4 2304.3.g.k.1279.1 2
48.11 even 4 2304.3.g.k.1279.2 2
48.29 odd 4 2304.3.g.k.1279.2 2
48.35 even 4 2304.3.g.k.1279.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.3.d.b.63.1 2 1.1 even 1 trivial
128.3.d.b.63.1 2 8.3 odd 2 CM
128.3.d.b.63.2 yes 2 4.3 odd 2 inner
128.3.d.b.63.2 yes 2 8.5 even 2 inner
256.3.c.d.255.1 2 16.11 odd 4
256.3.c.d.255.1 2 16.13 even 4
256.3.c.d.255.2 2 16.3 odd 4
256.3.c.d.255.2 2 16.5 even 4
1152.3.b.c.703.1 2 3.2 odd 2
1152.3.b.c.703.1 2 24.11 even 2
1152.3.b.c.703.2 2 12.11 even 2
1152.3.b.c.703.2 2 24.5 odd 2
2304.3.g.k.1279.1 2 48.5 odd 4
2304.3.g.k.1279.1 2 48.35 even 4
2304.3.g.k.1279.2 2 48.11 even 4
2304.3.g.k.1279.2 2 48.29 odd 4