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Properties

Label 32.7.d.a.15.1
Level $32$
Weight $7$
Character 32.15
Self dual yes
Analytic conductor $7.362$
Analytic rank $0$
Dimension $1$
CM discriminant -8
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] = [32,7,Mod(15,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.15");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 32.d (of order \(2\), degree \(1\), not 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: \(7.36173067584\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 15.1
Character \(\chi\) \(=\) 32.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-46.0000 q^{3} +1387.00 q^{9} +O(q^{10})\) \(q-46.0000 q^{3} +1387.00 q^{9} +2338.00 q^{11} -1726.00 q^{17} +2482.00 q^{19} +15625.0 q^{25} -30268.0 q^{27} -107548. q^{33} +134642. q^{41} +74914.0 q^{43} +117649. q^{49} +79396.0 q^{51} -114172. q^{57} -304958. q^{59} +596626. q^{67} -593134. q^{73} -718750. q^{75} +381205. q^{81} -678926. q^{83} -357262. q^{89} +1.82275e6 q^{97} +3.24281e6 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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
\(3\) −46.0000 −1.70370 −0.851852 − 0.523783i \(-0.824520\pi\)
−0.851852 + 0.523783i \(0.824520\pi\)
\(4\) 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
\(9\) 1387.00 1.90261
\(10\) 0 0
\(11\) 2338.00 1.75657 0.878287 − 0.478134i \(-0.158687\pi\)
0.878287 + 0.478134i \(0.158687\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1726.00 −0.351313 −0.175656 − 0.984452i \(-0.556205\pi\)
−0.175656 + 0.984452i \(0.556205\pi\)
\(18\) 0 0
\(19\) 2482.00 0.361860 0.180930 − 0.983496i \(-0.442089\pi\)
0.180930 + 0.983496i \(0.442089\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 15625.0 1.00000
\(26\) 0 0
\(27\) −30268.0 −1.53777
\(28\) 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
\(33\) −107548. −2.99268
\(34\) 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
\(41\) 134642. 1.95357 0.976785 − 0.214222i \(-0.0687216\pi\)
0.976785 + 0.214222i \(0.0687216\pi\)
\(42\) 0 0
\(43\) 74914.0 0.942232 0.471116 − 0.882071i \(-0.343851\pi\)
0.471116 + 0.882071i \(0.343851\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 117649. 1.00000
\(50\) 0 0
\(51\) 79396.0 0.598533
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −114172. −0.616503
\(58\) 0 0
\(59\) −304958. −1.48485 −0.742427 − 0.669927i \(-0.766326\pi\)
−0.742427 + 0.669927i \(0.766326\pi\)
\(60\) 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
\(67\) 596626. 1.98371 0.991854 − 0.127380i \(-0.0406566\pi\)
0.991854 + 0.127380i \(0.0406566\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −593134. −1.52470 −0.762350 − 0.647165i \(-0.775954\pi\)
−0.762350 + 0.647165i \(0.775954\pi\)
\(74\) 0 0
\(75\) −718750. −1.70370
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 381205. 0.717304
\(82\) 0 0
\(83\) −678926. −1.18738 −0.593688 − 0.804695i \(-0.702329\pi\)
−0.593688 + 0.804695i \(0.702329\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −357262. −0.506777 −0.253388 − 0.967365i \(-0.581545\pi\)
−0.253388 + 0.967365i \(0.581545\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.82275e6 1.99716 0.998580 − 0.0532728i \(-0.0169653\pi\)
0.998580 + 0.0532728i \(0.0169653\pi\)
\(98\) 0 0
\(99\) 3.24281e6 3.34207
\(100\) 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
\(107\) −474014. −0.386937 −0.193468 − 0.981107i \(-0.561974\pi\)
−0.193468 + 0.981107i \(0.561974\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.81289e6 −1.94948 −0.974738 − 0.223350i \(-0.928301\pi\)
−0.974738 + 0.223350i \(0.928301\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.69468e6 2.08555
\(122\) 0 0
\(123\) −6.19353e6 −3.32830
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) −3.44604e6 −1.60528
\(130\) 0 0
\(131\) 2.95362e6 1.31383 0.656917 − 0.753963i \(-0.271860\pi\)
0.656917 + 0.753963i \(0.271860\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −80206.0 −0.0311921 −0.0155961 − 0.999878i \(-0.504965\pi\)
−0.0155961 + 0.999878i \(0.504965\pi\)
\(138\) 0 0
\(139\) 3.19856e6 1.19100 0.595498 − 0.803357i \(-0.296955\pi\)
0.595498 + 0.803357i \(0.296955\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −5.41185e6 −1.70370
\(148\) 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
\(153\) −2.39396e6 −0.668410
\(154\) 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
\(163\) 7.72059e6 1.78274 0.891370 − 0.453277i \(-0.149745\pi\)
0.891370 + 0.453277i \(0.149745\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 4.82681e6 1.00000
\(170\) 0 0
\(171\) 3.44253e6 0.688478
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.40281e7 2.52975
\(178\) 0 0
\(179\) −3.22888e6 −0.562979 −0.281490 − 0.959564i \(-0.590829\pi\)
−0.281490 + 0.959564i \(0.590829\pi\)
\(180\) 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
\(187\) −4.03539e6 −0.617107
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −1.00100e7 −1.39240 −0.696198 − 0.717850i \(-0.745126\pi\)
−0.696198 + 0.717850i \(0.745126\pi\)
\(194\) 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
\(201\) −2.74448e7 −3.37965
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.80292e6 0.635634
\(210\) 0 0
\(211\) −1.86421e7 −1.98448 −0.992240 − 0.124340i \(-0.960319\pi\)
−0.992240 + 0.124340i \(0.960319\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2.72842e7 2.59764
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 2.16719e7 1.90261
\(226\) 0 0
\(227\) −1.97707e7 −1.69023 −0.845114 − 0.534586i \(-0.820467\pi\)
−0.845114 + 0.534586i \(0.820467\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.10622e7 0.874530 0.437265 − 0.899333i \(-0.355947\pi\)
0.437265 + 0.899333i \(0.355947\pi\)
\(234\) 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
\(241\) −2.65018e7 −1.89332 −0.946659 − 0.322237i \(-0.895565\pi\)
−0.946659 + 0.322237i \(0.895565\pi\)
\(242\) 0 0
\(243\) 4.52994e6 0.315699
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 3.12306e7 2.02294
\(250\) 0 0
\(251\) 1.31193e7 0.829640 0.414820 − 0.909904i \(-0.363845\pi\)
0.414820 + 0.909904i \(0.363845\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.89723e7 −1.11769 −0.558844 − 0.829273i \(-0.688755\pi\)
−0.558844 + 0.829273i \(0.688755\pi\)
\(258\) 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
\(267\) 1.64341e7 0.863398
\(268\) 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
\(275\) 3.65312e7 1.75657
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.28969e7 1.93333 0.966667 − 0.256038i \(-0.0824172\pi\)
0.966667 + 0.256038i \(0.0824172\pi\)
\(282\) 0 0
\(283\) −1.91505e7 −0.844931 −0.422466 − 0.906379i \(-0.638835\pi\)
−0.422466 + 0.906379i \(0.638835\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.11585e7 −0.876579
\(290\) 0 0
\(291\) −8.38467e7 −3.40257
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −7.07666e7 −2.70121
\(298\) 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
\(307\) −5.97121e6 −0.206370 −0.103185 − 0.994662i \(-0.532903\pi\)
−0.103185 + 0.994662i \(0.532903\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 9.06351e6 0.295572 0.147786 − 0.989019i \(-0.452785\pi\)
0.147786 + 0.989019i \(0.452785\pi\)
\(314\) 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
\(321\) 2.18046e7 0.659225
\(322\) 0 0
\(323\) −4.28393e6 −0.127126
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.59882e6 0.126813 0.0634063 − 0.997988i \(-0.479804\pi\)
0.0634063 + 0.997988i \(0.479804\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 5.36426e7 1.40159 0.700794 − 0.713364i \(-0.252829\pi\)
0.700794 + 0.713364i \(0.252829\pi\)
\(338\) 0 0
\(339\) 1.29393e8 3.32133
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.72029e7 1.12975 0.564873 − 0.825178i \(-0.308925\pi\)
0.564873 + 0.825178i \(0.308925\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.52211e7 −1.48274 −0.741368 − 0.671099i \(-0.765823\pi\)
−0.741368 + 0.671099i \(0.765823\pi\)
\(354\) 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
\(361\) −4.08856e7 −0.869057
\(362\) 0 0
\(363\) −1.69955e8 −3.55316
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 1.86748e8 3.71687
\(370\) 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
\(379\) −2.72157e7 −0.499921 −0.249961 − 0.968256i \(-0.580418\pi\)
−0.249961 + 0.968256i \(0.580418\pi\)
\(380\) 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
\(387\) 1.03906e8 1.79270
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −1.35866e8 −2.23838
\(394\) 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
\(401\) −7.99344e7 −1.23965 −0.619827 − 0.784738i \(-0.712797\pi\)
−0.619827 + 0.784738i \(0.712797\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.30356e8 1.90529 0.952644 − 0.304088i \(-0.0983516\pi\)
0.952644 + 0.304088i \(0.0983516\pi\)
\(410\) 0 0
\(411\) 3.68948e6 0.0531422
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.47134e8 −2.02910
\(418\) 0 0
\(419\) −1.34918e8 −1.83412 −0.917062 − 0.398744i \(-0.869446\pi\)
−0.917062 + 0.398744i \(0.869446\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.69688e7 −0.351313
\(426\) 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
\(433\) −1.32005e8 −1.62603 −0.813014 − 0.582244i \(-0.802175\pi\)
−0.813014 + 0.582244i \(0.802175\pi\)
\(434\) 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
\(441\) 1.63179e8 1.90261
\(442\) 0 0
\(443\) −1.59916e8 −1.83942 −0.919711 − 0.392595i \(-0.871577\pi\)
−0.919711 + 0.392595i \(0.871577\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.25702e8 1.38869 0.694344 − 0.719643i \(-0.255694\pi\)
0.694344 + 0.719643i \(0.255694\pi\)
\(450\) 0 0
\(451\) 3.14793e8 3.43159
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.35637e8 1.42112 0.710558 − 0.703639i \(-0.248443\pi\)
0.710558 + 0.703639i \(0.248443\pi\)
\(458\) 0 0
\(459\) 5.22426e7 0.540240
\(460\) 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
\(467\) −2.22058e7 −0.218030 −0.109015 − 0.994040i \(-0.534770\pi\)
−0.109015 + 0.994040i \(0.534770\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.75149e8 1.65510
\(474\) 0 0
\(475\) 3.87812e7 0.361860
\(476\) 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
\(489\) −3.55147e8 −3.03726
\(490\) 0 0
\(491\) 8.73643e7 0.738056 0.369028 − 0.929418i \(-0.379691\pi\)
0.369028 + 0.929418i \(0.379691\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −8.32468e7 −0.669986 −0.334993 − 0.942221i \(-0.608734\pi\)
−0.334993 + 0.942221i \(0.608734\pi\)
\(500\) 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
\(507\) −2.22033e8 −1.70370
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −7.51252e7 −0.556459
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.98899e8 −1.40644 −0.703218 − 0.710974i \(-0.748254\pi\)
−0.703218 + 0.710974i \(0.748254\pi\)
\(522\) 0 0
\(523\) 2.63549e8 1.84228 0.921141 − 0.389229i \(-0.127259\pi\)
0.921141 + 0.389229i \(0.127259\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.48036e8 1.00000
\(530\) 0 0
\(531\) −4.22977e8 −2.82509
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.48528e8 0.959150
\(538\) 0 0
\(539\) 2.75063e8 1.75657
\(540\) 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
\(547\) −1.50088e8 −0.917030 −0.458515 − 0.888687i \(-0.651618\pi\)
−0.458515 + 0.888687i \(0.651618\pi\)
\(548\) 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
\(561\) 1.85628e8 1.05137
\(562\) 0 0
\(563\) −2.03362e8 −1.13958 −0.569789 − 0.821791i \(-0.692975\pi\)
−0.569789 + 0.821791i \(0.692975\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.62709e8 −1.96889 −0.984445 − 0.175695i \(-0.943783\pi\)
−0.984445 + 0.175695i \(0.943783\pi\)
\(570\) 0 0
\(571\) −3.58715e8 −1.92682 −0.963409 − 0.268035i \(-0.913626\pi\)
−0.963409 + 0.268035i \(0.913626\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.99757e6 −0.0103986 −0.00519929 − 0.999986i \(-0.501655\pi\)
−0.00519929 + 0.999986i \(0.501655\pi\)
\(578\) 0 0
\(579\) 4.60461e8 2.37223
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.97402e8 1.47038 0.735189 − 0.677862i \(-0.237094\pi\)
0.735189 + 0.677862i \(0.237094\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.68860e8 1.28933 0.644663 − 0.764467i \(-0.276998\pi\)
0.644663 + 0.764467i \(0.276998\pi\)
\(594\) 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
\(601\) −2.26861e8 −1.04505 −0.522525 − 0.852624i \(-0.675010\pi\)
−0.522525 + 0.852624i \(0.675010\pi\)
\(602\) 0 0
\(603\) 8.27520e8 3.77422
\(604\) 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
\(617\) 3.44191e8 1.46536 0.732679 − 0.680575i \(-0.238270\pi\)
0.732679 + 0.680575i \(0.238270\pi\)
\(618\) 0 0
\(619\) −4.68505e8 −1.97534 −0.987671 − 0.156543i \(-0.949965\pi\)
−0.987671 + 0.156543i \(0.949965\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.44141e8 1.00000
\(626\) 0 0
\(627\) −2.66934e8 −1.08293
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 8.57535e8 3.38096
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4.82154e8 −1.83068 −0.915338 − 0.402687i \(-0.868076\pi\)
−0.915338 + 0.402687i \(0.868076\pi\)
\(642\) 0 0
\(643\) −2.83407e8 −1.06605 −0.533025 − 0.846099i \(-0.678945\pi\)
−0.533025 + 0.846099i \(0.678945\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −7.12992e8 −2.60826
\(650\) 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
\(657\) −8.22677e8 −2.90090
\(658\) 0 0
\(659\) −3.12918e8 −1.09339 −0.546694 − 0.837332i \(-0.684114\pi\)
−0.546694 + 0.837332i \(0.684114\pi\)
\(660\) 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
\(673\) −2.41386e8 −0.791895 −0.395947 − 0.918273i \(-0.629584\pi\)
−0.395947 + 0.918273i \(0.629584\pi\)
\(674\) 0 0
\(675\) −4.72938e8 −1.53777
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 9.09454e8 2.87965
\(682\) 0 0
\(683\) 4.93943e8 1.55030 0.775148 − 0.631779i \(-0.217675\pi\)
0.775148 + 0.631779i \(0.217675\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 6.55966e8 1.98814 0.994071 − 0.108734i \(-0.0346796\pi\)
0.994071 + 0.108734i \(0.0346796\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −2.32392e8 −0.686314
\(698\) 0 0
\(699\) −5.08862e8 −1.48994
\(700\) 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
\(723\) 1.21908e9 3.22565
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −4.86276e8 −1.25516
\(730\) 0 0
\(731\) −1.29302e8 −0.331018
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.39491e9 3.48453
\(738\) 0 0
\(739\) −4.94167e8 −1.22445 −0.612224 − 0.790685i \(-0.709725\pi\)
−0.612224 + 0.790685i \(0.709725\pi\)
\(740\) 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
\(747\) −9.41670e8 −2.25911
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −6.03488e8 −1.41346
\(754\) 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
\(761\) 2.86987e8 0.651191 0.325595 − 0.945509i \(-0.394435\pi\)
0.325595 + 0.945509i \(0.394435\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.98978e8 1.53704 0.768519 − 0.639827i \(-0.220994\pi\)
0.768519 + 0.639827i \(0.220994\pi\)
\(770\) 0 0
\(771\) 8.72725e8 1.90421
\(772\) 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
\(779\) 3.34181e8 0.706919
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 9.26584e8 1.90091 0.950453 − 0.310868i \(-0.100620\pi\)
0.950453 + 0.310868i \(0.100620\pi\)
\(788\) 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
\(801\) −4.95522e8 −0.964197
\(802\) 0 0
\(803\) −1.38675e9 −2.67825
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −8.53143e8 −1.61130 −0.805649 − 0.592393i \(-0.798183\pi\)
−0.805649 + 0.592393i \(0.798183\pi\)
\(810\) 0 0
\(811\) −3.45583e8 −0.647873 −0.323937 − 0.946079i \(-0.605006\pi\)
−0.323937 + 0.946079i \(0.605006\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.85937e8 0.340956
\(818\) 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
\(825\) −1.68044e9 −2.99268
\(826\) 0 0
\(827\) 5.79438e8 1.02445 0.512225 − 0.858851i \(-0.328821\pi\)
0.512225 + 0.858851i \(0.328821\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.03062e8 −0.351313
\(834\) 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
\(841\) 5.94823e8 1.00000
\(842\) 0 0
\(843\) −1.97326e9 −3.29383
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 8.80924e8 1.43951
\(850\) 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
\(857\) −9.11769e8 −1.44858 −0.724290 − 0.689496i \(-0.757832\pi\)
−0.724290 + 0.689496i \(0.757832\pi\)
\(858\) 0 0
\(859\) −8.15878e8 −1.28720 −0.643599 − 0.765363i \(-0.722560\pi\)
−0.643599 + 0.765363i \(0.722560\pi\)
\(860\) 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
\(867\) 9.73291e8 1.49343
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.52816e9 3.79981
\(874\) 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
\(881\) 3.74799e8 0.548114 0.274057 − 0.961714i \(-0.411634\pi\)
0.274057 + 0.961714i \(0.411634\pi\)
\(882\) 0 0
\(883\) 1.34895e9 1.95935 0.979676 − 0.200584i \(-0.0642840\pi\)
0.979676 + 0.200584i \(0.0642840\pi\)
\(884\) 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
\(891\) 8.91257e8 1.26000
\(892\) 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
\(907\) 8.05321e8 1.07931 0.539656 − 0.841885i \(-0.318554\pi\)
0.539656 + 0.841885i \(0.318554\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −1.58733e9 −2.08571
\(914\) 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
\(921\) 2.74676e8 0.351594
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.55501e9 −1.93948 −0.969739 − 0.244146i \(-0.921492\pi\)
−0.969739 + 0.244146i \(0.921492\pi\)
\(930\) 0 0
\(931\) 2.92005e8 0.361860
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.52100e9 1.84889 0.924443 − 0.381320i \(-0.124530\pi\)
0.924443 + 0.381320i \(0.124530\pi\)
\(938\) 0 0
\(939\) −4.16921e8 −0.503567
\(940\) 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
\(947\) −1.69218e9 −1.99249 −0.996244 − 0.0865891i \(-0.972403\pi\)
−0.996244 + 0.0865891i \(0.972403\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.84034e8 0.443701 0.221851 − 0.975081i \(-0.428790\pi\)
0.221851 + 0.975081i \(0.428790\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 8.87504e8 1.00000
\(962\) 0 0
\(963\) −6.57457e8 −0.736188
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 1.97061e8 0.216585
\(970\) 0 0
\(971\) 1.83097e9 1.99997 0.999986 − 0.00535408i \(-0.00170426\pi\)
0.999986 + 0.00535408i \(0.00170426\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.56343e9 −1.67647 −0.838234 − 0.545310i \(-0.816412\pi\)
−0.838234 + 0.545310i \(0.816412\pi\)
\(978\) 0 0
\(979\) −8.35279e8 −0.890191
\(980\) 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
\(993\) −2.11546e8 −0.216051
\(994\) 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 32.7.d.a.15.1 1
3.2 odd 2 288.7.b.a.271.1 1
4.3 odd 2 8.7.d.a.3.1 ✓ 1
8.3 odd 2 CM 32.7.d.a.15.1 1
8.5 even 2 8.7.d.a.3.1 ✓ 1
12.11 even 2 72.7.b.a.19.1 1
16.3 odd 4 256.7.c.d.255.2 2
16.5 even 4 256.7.c.d.255.2 2
16.11 odd 4 256.7.c.d.255.1 2
16.13 even 4 256.7.c.d.255.1 2
24.5 odd 2 72.7.b.a.19.1 1
24.11 even 2 288.7.b.a.271.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.7.d.a.3.1 ✓ 1 4.3 odd 2
8.7.d.a.3.1 ✓ 1 8.5 even 2
32.7.d.a.15.1 1 1.1 even 1 trivial
32.7.d.a.15.1 1 8.3 odd 2 CM
72.7.b.a.19.1 1 12.11 even 2
72.7.b.a.19.1 1 24.5 odd 2
256.7.c.d.255.1 2 16.11 odd 4
256.7.c.d.255.1 2 16.13 even 4
256.7.c.d.255.2 2 16.3 odd 4
256.7.c.d.255.2 2 16.5 even 4
288.7.b.a.271.1 1 3.2 odd 2
288.7.b.a.271.1 1 24.11 even 2