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Properties

Label 128.5.d.b.63.2
Level $128$
Weight $5$
Character 128.63
Self dual yes
Analytic conductor $13.231$
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] = [128,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.63");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 5 \)
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: \(13.2313552747\)
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: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 63.2
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)
 
\(f(q)\) \(=\) \(q+11.3137 q^{3} +47.0000 q^{9} +O(q^{10})\) \(q+11.3137 q^{3} +47.0000 q^{9} +237.588 q^{11} +574.000 q^{17} -576.999 q^{19} +625.000 q^{25} -384.666 q^{27} +2688.00 q^{33} -1246.00 q^{41} +1187.94 q^{43} +2401.00 q^{49} +6494.07 q^{51} -6528.00 q^{57} -6957.93 q^{59} -7365.22 q^{67} -9506.00 q^{73} +7071.07 q^{75} -8159.00 q^{81} -8044.05 q^{83} -5474.00 q^{89} +9982.00 q^{97} +11166.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 94 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 94 q^{9} + 1148 q^{17} + 1250 q^{25} + 5376 q^{33} - 2492 q^{41} + 4802 q^{49} - 13056 q^{57} - 19012 q^{73} - 16318 q^{81} - 10948 q^{89} + 19964 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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
\(3\) 11.3137 1.25708 0.628539 − 0.777778i \(-0.283653\pi\)
0.628539 + 0.777778i \(0.283653\pi\)
\(4\) 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
\(9\) 47.0000 0.580247
\(10\) 0 0
\(11\) 237.588 1.96354 0.981768 − 0.190083i \(-0.0608756\pi\)
0.981768 + 0.190083i \(0.0608756\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 574.000 1.98616 0.993080 − 0.117443i \(-0.0374699\pi\)
0.993080 + 0.117443i \(0.0374699\pi\)
\(18\) 0 0
\(19\) −576.999 −1.59834 −0.799168 − 0.601108i \(-0.794726\pi\)
−0.799168 + 0.601108i \(0.794726\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) 0 0
\(27\) −384.666 −0.527663
\(28\) 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
\(33\) 2688.00 2.46832
\(34\) 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
\(41\) −1246.00 −0.741225 −0.370613 − 0.928787i \(-0.620852\pi\)
−0.370613 + 0.928787i \(0.620852\pi\)
\(42\) 0 0
\(43\) 1187.94 0.642477 0.321238 − 0.946998i \(-0.395901\pi\)
0.321238 + 0.946998i \(0.395901\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 2401.00 1.00000
\(50\) 0 0
\(51\) 6494.07 2.49676
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6528.00 −2.00923
\(58\) 0 0
\(59\) −6957.93 −1.99883 −0.999416 − 0.0341856i \(-0.989116\pi\)
−0.999416 + 0.0341856i \(0.989116\pi\)
\(60\) 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
\(67\) −7365.22 −1.64073 −0.820364 − 0.571842i \(-0.806229\pi\)
−0.820364 + 0.571842i \(0.806229\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −9506.00 −1.78382 −0.891912 − 0.452209i \(-0.850636\pi\)
−0.891912 + 0.452209i \(0.850636\pi\)
\(74\) 0 0
\(75\) 7071.07 1.25708
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −8159.00 −1.24356
\(82\) 0 0
\(83\) −8044.05 −1.16767 −0.583833 − 0.811874i \(-0.698448\pi\)
−0.583833 + 0.811874i \(0.698448\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5474.00 −0.691074 −0.345537 − 0.938405i \(-0.612303\pi\)
−0.345537 + 0.938405i \(0.612303\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9982.00 1.06090 0.530450 − 0.847716i \(-0.322023\pi\)
0.530450 + 0.847716i \(0.322023\pi\)
\(98\) 0 0
\(99\) 11166.6 1.13934
\(100\) 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
\(107\) 21145.3 1.84691 0.923457 − 0.383702i \(-0.125351\pi\)
0.923457 + 0.383702i \(0.125351\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −15934.0 −1.24787 −0.623933 − 0.781478i \(-0.714466\pi\)
−0.623933 + 0.781478i \(0.714466\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 41807.0 2.85547
\(122\) 0 0
\(123\) −14096.9 −0.931779
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 13440.0 0.807644
\(130\) 0 0
\(131\) −15782.6 −0.919680 −0.459840 − 0.888002i \(-0.652093\pi\)
−0.459840 + 0.888002i \(0.652093\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19106.0 1.01796 0.508978 − 0.860780i \(-0.330024\pi\)
0.508978 + 0.860780i \(0.330024\pi\)
\(138\) 0 0
\(139\) −38455.3 −1.99034 −0.995168 − 0.0981833i \(-0.968697\pi\)
−0.995168 + 0.0981833i \(0.968697\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 27164.2 1.25708
\(148\) 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
\(153\) 26978.0 1.15246
\(154\) 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
\(163\) 16393.6 0.617018 0.308509 − 0.951221i \(-0.400170\pi\)
0.308509 + 0.951221i \(0.400170\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 28561.0 1.00000
\(170\) 0 0
\(171\) −27119.0 −0.927429
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −78720.0 −2.51269
\(178\) 0 0
\(179\) −12117.0 −0.378171 −0.189086 − 0.981961i \(-0.560552\pi\)
−0.189086 + 0.981961i \(0.560552\pi\)
\(180\) 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
\(187\) 136375. 3.89990
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 64894.0 1.74217 0.871084 − 0.491134i \(-0.163418\pi\)
0.871084 + 0.491134i \(0.163418\pi\)
\(194\) 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
\(201\) −83328.0 −2.06252
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −137088. −3.13839
\(210\) 0 0
\(211\) −80542.3 −1.80909 −0.904543 − 0.426383i \(-0.859788\pi\)
−0.904543 + 0.426383i \(0.859788\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −107548. −2.24241
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 29375.0 0.580247
\(226\) 0 0
\(227\) 37844.4 0.734428 0.367214 − 0.930136i \(-0.380312\pi\)
0.367214 + 0.930136i \(0.380312\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −79778.0 −1.46951 −0.734753 − 0.678335i \(-0.762702\pi\)
−0.734753 + 0.678335i \(0.762702\pi\)
\(234\) 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
\(241\) 78526.0 1.35201 0.676004 − 0.736898i \(-0.263710\pi\)
0.676004 + 0.736898i \(0.263710\pi\)
\(242\) 0 0
\(243\) −61150.6 −1.03559
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −91008.0 −1.46785
\(250\) 0 0
\(251\) 86991.1 1.38079 0.690395 − 0.723433i \(-0.257437\pi\)
0.690395 + 0.723433i \(0.257437\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16898.0 0.255840 0.127920 − 0.991784i \(-0.459170\pi\)
0.127920 + 0.991784i \(0.459170\pi\)
\(258\) 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
\(267\) −61931.2 −0.868735
\(268\) 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
\(275\) 148492. 1.96354
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 101278. 1.28263 0.641317 − 0.767276i \(-0.278389\pi\)
0.641317 + 0.767276i \(0.278389\pi\)
\(282\) 0 0
\(283\) −45922.3 −0.573391 −0.286696 − 0.958022i \(-0.592557\pi\)
−0.286696 + 0.958022i \(0.592557\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 245955. 2.94483
\(290\) 0 0
\(291\) 112933. 1.33363
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −91392.0 −1.03608
\(298\) 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
\(307\) 156367. 1.65908 0.829541 − 0.558446i \(-0.188602\pi\)
0.829541 + 0.558446i \(0.188602\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 80738.0 0.824118 0.412059 − 0.911157i \(-0.364810\pi\)
0.412059 + 0.911157i \(0.364810\pi\)
\(314\) 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
\(321\) 239232. 2.32172
\(322\) 0 0
\(323\) −331198. −3.17455
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −9265.93 −0.0845732 −0.0422866 − 0.999106i \(-0.513464\pi\)
−0.0422866 + 0.999106i \(0.513464\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1346.00 0.0118518 0.00592591 − 0.999982i \(-0.498114\pi\)
0.00592591 + 0.999982i \(0.498114\pi\)
\(338\) 0 0
\(339\) −180273. −1.56867
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −145166. −1.20561 −0.602805 − 0.797889i \(-0.705950\pi\)
−0.602805 + 0.797889i \(0.705950\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −211582. −1.69797 −0.848984 − 0.528419i \(-0.822785\pi\)
−0.848984 + 0.528419i \(0.822785\pi\)
\(354\) 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
\(361\) 202607. 1.55468
\(362\) 0 0
\(363\) 472992. 3.58956
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −58562.0 −0.430094
\(370\) 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
\(379\) −221194. −1.53991 −0.769955 − 0.638098i \(-0.779722\pi\)
−0.769955 + 0.638098i \(0.779722\pi\)
\(380\) 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
\(387\) 55833.2 0.372795
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −178560. −1.15611
\(394\) 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
\(401\) −265154. −1.64896 −0.824479 − 0.565893i \(-0.808532\pi\)
−0.824479 + 0.565893i \(0.808532\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −223006. −1.33312 −0.666561 − 0.745450i \(-0.732234\pi\)
−0.666561 + 0.745450i \(0.732234\pi\)
\(410\) 0 0
\(411\) 216160. 1.27965
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −435072. −2.50201
\(418\) 0 0
\(419\) 340192. 1.93774 0.968871 − 0.247566i \(-0.0796308\pi\)
0.968871 + 0.247566i \(0.0796308\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 358750. 1.98616
\(426\) 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
\(433\) 40894.0 0.218114 0.109057 − 0.994035i \(-0.465217\pi\)
0.109057 + 0.994035i \(0.465217\pi\)
\(434\) 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
\(441\) 112847. 0.580247
\(442\) 0 0
\(443\) −104301. −0.531473 −0.265737 − 0.964046i \(-0.585615\pi\)
−0.265737 + 0.964046i \(0.585615\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −346754. −1.72000 −0.860001 − 0.510293i \(-0.829537\pi\)
−0.860001 + 0.510293i \(0.829537\pi\)
\(450\) 0 0
\(451\) −296034. −1.45542
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −361054. −1.72878 −0.864390 − 0.502822i \(-0.832295\pi\)
−0.864390 + 0.502822i \(0.832295\pi\)
\(458\) 0 0
\(459\) −220798. −1.04802
\(460\) 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
\(467\) 31735.0 0.145514 0.0727569 − 0.997350i \(-0.476820\pi\)
0.0727569 + 0.997350i \(0.476820\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 282240. 1.26153
\(474\) 0 0
\(475\) −360624. −1.59834
\(476\) 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
\(489\) 185472. 0.775641
\(490\) 0 0
\(491\) −464484. −1.92667 −0.963337 − 0.268296i \(-0.913540\pi\)
−0.963337 + 0.268296i \(0.913540\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 476364. 1.91310 0.956550 − 0.291569i \(-0.0941773\pi\)
0.956550 + 0.291569i \(0.0941773\pi\)
\(500\) 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
\(507\) 323131. 1.25708
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 221952. 0.843382
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 469154. 1.72838 0.864191 − 0.503163i \(-0.167830\pi\)
0.864191 + 0.503163i \(0.167830\pi\)
\(522\) 0 0
\(523\) 384994. 1.40751 0.703754 − 0.710444i \(-0.251506\pi\)
0.703754 + 0.710444i \(0.251506\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 279841. 1.00000
\(530\) 0 0
\(531\) −327023. −1.15982
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −137088. −0.475391
\(538\) 0 0
\(539\) 570448. 1.96354
\(540\) 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
\(547\) −398910. −1.33322 −0.666608 − 0.745409i \(-0.732254\pi\)
−0.666608 + 0.745409i \(0.732254\pi\)
\(548\) 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
\(561\) 1.54291e6 4.90248
\(562\) 0 0
\(563\) −249298. −0.786505 −0.393252 − 0.919431i \(-0.628650\pi\)
−0.393252 + 0.919431i \(0.628650\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −255646. −0.789613 −0.394807 − 0.918764i \(-0.629188\pi\)
−0.394807 + 0.918764i \(0.629188\pi\)
\(570\) 0 0
\(571\) 614165. 1.88370 0.941852 − 0.336028i \(-0.109084\pi\)
0.941852 + 0.336028i \(0.109084\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −665854. −1.99999 −0.999994 − 0.00346620i \(-0.998897\pi\)
−0.999994 + 0.00346620i \(0.998897\pi\)
\(578\) 0 0
\(579\) 734192. 2.19004
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −328313. −0.952821 −0.476410 − 0.879223i \(-0.658062\pi\)
−0.476410 + 0.879223i \(0.658062\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 39746.0 0.113027 0.0565137 − 0.998402i \(-0.482002\pi\)
0.0565137 + 0.998402i \(0.482002\pi\)
\(594\) 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
\(601\) −112994. −0.312829 −0.156414 − 0.987692i \(-0.549993\pi\)
−0.156414 + 0.987692i \(0.549993\pi\)
\(602\) 0 0
\(603\) −346166. −0.952027
\(604\) 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
\(617\) 649822. 1.70696 0.853481 − 0.521123i \(-0.174487\pi\)
0.853481 + 0.521123i \(0.174487\pi\)
\(618\) 0 0
\(619\) −619935. −1.61795 −0.808974 − 0.587844i \(-0.799977\pi\)
−0.808974 + 0.587844i \(0.799977\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 390625. 1.00000
\(626\) 0 0
\(627\) −1.55097e6 −3.94520
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −911232. −2.27416
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 589438. 1.43457 0.717286 − 0.696779i \(-0.245384\pi\)
0.717286 + 0.696779i \(0.245384\pi\)
\(642\) 0 0
\(643\) 515057. 1.24576 0.622878 − 0.782319i \(-0.285963\pi\)
0.622878 + 0.782319i \(0.285963\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −1.65312e6 −3.92478
\(650\) 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
\(657\) −446782. −1.03506
\(658\) 0 0
\(659\) 860306. 1.98099 0.990494 − 0.137554i \(-0.0439240\pi\)
0.990494 + 0.137554i \(0.0439240\pi\)
\(660\) 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
\(673\) −646658. −1.42772 −0.713862 − 0.700286i \(-0.753056\pi\)
−0.713862 + 0.700286i \(0.753056\pi\)
\(674\) 0 0
\(675\) −240416. −0.527663
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 428160. 0.923234
\(682\) 0 0
\(683\) 520080. 1.11488 0.557441 − 0.830217i \(-0.311783\pi\)
0.557441 + 0.830217i \(0.311783\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 859491. 1.80005 0.900027 − 0.435835i \(-0.143547\pi\)
0.900027 + 0.435835i \(0.143547\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −715204. −1.47219
\(698\) 0 0
\(699\) −902585. −1.84728
\(700\) 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
\(723\) 888420. 1.69958
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −30961.0 −0.0582586
\(730\) 0 0
\(731\) 681877. 1.27606
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.74989e6 −3.22163
\(738\) 0 0
\(739\) −464484. −0.850515 −0.425258 − 0.905072i \(-0.639817\pi\)
−0.425258 + 0.905072i \(0.639817\pi\)
\(740\) 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
\(747\) −378070. −0.677534
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 984192. 1.73576
\(754\) 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
\(761\) 784994. 1.35549 0.677746 − 0.735296i \(-0.262957\pi\)
0.677746 + 0.735296i \(0.262957\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 71806.0 0.121425 0.0607125 − 0.998155i \(-0.480663\pi\)
0.0607125 + 0.998155i \(0.480663\pi\)
\(770\) 0 0
\(771\) 191179. 0.321611
\(772\) 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
\(779\) 718941. 1.18473
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1.17861e6 −1.90291 −0.951457 − 0.307783i \(-0.900413\pi\)
−0.951457 + 0.307783i \(0.900413\pi\)
\(788\) 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
\(801\) −257278. −0.400994
\(802\) 0 0
\(803\) −2.25851e6 −3.50260
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.19376e6 1.82398 0.911991 − 0.410209i \(-0.134544\pi\)
0.911991 + 0.410209i \(0.134544\pi\)
\(810\) 0 0
\(811\) 286972. 0.436313 0.218156 − 0.975914i \(-0.429996\pi\)
0.218156 + 0.975914i \(0.429996\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −685440. −1.02689
\(818\) 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
\(825\) 1.68000e6 2.46832
\(826\) 0 0
\(827\) −1.34926e6 −1.97281 −0.986405 − 0.164334i \(-0.947452\pi\)
−0.986405 + 0.164334i \(0.947452\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.37817e6 1.98616
\(834\) 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
\(841\) 707281. 1.00000
\(842\) 0 0
\(843\) 1.14583e6 1.61237
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −519552. −0.720798
\(850\) 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
\(857\) −24094.0 −0.0328055 −0.0164028 − 0.999865i \(-0.505221\pi\)
−0.0164028 + 0.999865i \(0.505221\pi\)
\(858\) 0 0
\(859\) 810073. 1.09784 0.548918 − 0.835876i \(-0.315040\pi\)
0.548918 + 0.835876i \(0.315040\pi\)
\(860\) 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
\(867\) 2.78266e6 3.70188
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 469154. 0.615583
\(874\) 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
\(881\) 515522. 0.664195 0.332097 − 0.943245i \(-0.392244\pi\)
0.332097 + 0.943245i \(0.392244\pi\)
\(882\) 0 0
\(883\) 209315. 0.268460 0.134230 − 0.990950i \(-0.457144\pi\)
0.134230 + 0.990950i \(0.457144\pi\)
\(884\) 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
\(891\) −1.93848e6 −2.44178
\(892\) 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
\(907\) 1.01806e6 1.23754 0.618772 − 0.785571i \(-0.287631\pi\)
0.618772 + 0.785571i \(0.287631\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −1.91117e6 −2.29275
\(914\) 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
\(921\) 1.76909e6 2.08560
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 606718. 0.703000 0.351500 − 0.936188i \(-0.385672\pi\)
0.351500 + 0.936188i \(0.385672\pi\)
\(930\) 0 0
\(931\) −1.38537e6 −1.59834
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.24041e6 1.41282 0.706411 − 0.707802i \(-0.250313\pi\)
0.706411 + 0.707802i \(0.250313\pi\)
\(938\) 0 0
\(939\) 913446. 1.03598
\(940\) 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
\(947\) 1.60253e6 1.78692 0.893462 − 0.449138i \(-0.148269\pi\)
0.893462 + 0.449138i \(0.148269\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.79625e6 −1.97780 −0.988899 − 0.148589i \(-0.952527\pi\)
−0.988899 + 0.148589i \(0.952527\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 923521. 1.00000
\(962\) 0 0
\(963\) 993830. 1.07167
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) −3.74707e6 −3.99066
\(970\) 0 0
\(971\) −1.63640e6 −1.73561 −0.867805 − 0.496906i \(-0.834470\pi\)
−0.867805 + 0.496906i \(0.834470\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.76967e6 −1.85397 −0.926984 − 0.375101i \(-0.877608\pi\)
−0.926984 + 0.375101i \(0.877608\pi\)
\(978\) 0 0
\(979\) −1.30056e6 −1.35695
\(980\) 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
\(993\) −104832. −0.106315
\(994\) 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 128.5.d.b.63.2 yes 2
3.2 odd 2 1152.5.b.a.703.1 2
4.3 odd 2 inner 128.5.d.b.63.1 ✓ 2
8.3 odd 2 CM 128.5.d.b.63.2 yes 2
8.5 even 2 inner 128.5.d.b.63.1 ✓ 2
12.11 even 2 1152.5.b.a.703.2 2
16.3 odd 4 256.5.c.e.255.1 2
16.5 even 4 256.5.c.e.255.1 2
16.11 odd 4 256.5.c.e.255.2 2
16.13 even 4 256.5.c.e.255.2 2
24.5 odd 2 1152.5.b.a.703.2 2
24.11 even 2 1152.5.b.a.703.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.5.d.b.63.1 ✓ 2 4.3 odd 2 inner
128.5.d.b.63.1 ✓ 2 8.5 even 2 inner
128.5.d.b.63.2 yes 2 1.1 even 1 trivial
128.5.d.b.63.2 yes 2 8.3 odd 2 CM
256.5.c.e.255.1 2 16.3 odd 4
256.5.c.e.255.1 2 16.5 even 4
256.5.c.e.255.2 2 16.11 odd 4
256.5.c.e.255.2 2 16.13 even 4
1152.5.b.a.703.1 2 3.2 odd 2
1152.5.b.a.703.1 2 24.11 even 2
1152.5.b.a.703.2 2 12.11 even 2
1152.5.b.a.703.2 2 24.5 odd 2