Properties

Label 16.7.c.a.15.1
Level $16$
Weight $7$
Character 16.15
Self dual yes
Analytic conductor $3.681$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [16,7,Mod(15,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.15");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 16.c (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: \(3.68086533792\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 15.1
Character \(\chi\) \(=\) 16.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+234.000 q^{5} +729.000 q^{9} +O(q^{10})\) \(q+234.000 q^{5} +729.000 q^{9} -4070.00 q^{13} -990.000 q^{17} +39131.0 q^{25} -31878.0 q^{29} -55510.0 q^{37} -84942.0 q^{41} +170586. q^{45} +117649. q^{49} -29430.0 q^{53} +234938. q^{61} -952380. q^{65} +427570. q^{73} +531441. q^{81} -231660. q^{85} +1.37896e6 q^{89} -1.47251e6 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 234.000 1.87200 0.936000 0.352000i \(-0.114498\pi\)
0.936000 + 0.352000i \(0.114498\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 729.000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −4070.00 −1.85253 −0.926263 0.376878i \(-0.876998\pi\)
−0.926263 + 0.376878i \(0.876998\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −990.000 −0.201506 −0.100753 0.994911i \(-0.532125\pi\)
−0.100753 + 0.994911i \(0.532125\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 39131.0 2.50438
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −31878.0 −1.30706 −0.653532 0.756899i \(-0.726714\pi\)
−0.653532 + 0.756899i \(0.726714\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −55510.0 −1.09589 −0.547944 0.836515i \(-0.684589\pi\)
−0.547944 + 0.836515i \(0.684589\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −84942.0 −1.23245 −0.616227 0.787568i \(-0.711340\pi\)
−0.616227 + 0.787568i \(0.711340\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 170586. 1.87200
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 117649. 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −29430.0 −0.197680 −0.0988400 0.995103i \(-0.531513\pi\)
−0.0988400 + 0.995103i \(0.531513\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 234938. 1.03506 0.517528 0.855666i \(-0.326852\pi\)
0.517528 + 0.855666i \(0.326852\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −952380. −3.46793
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 427570. 1.09910 0.549552 0.835460i \(-0.314798\pi\)
0.549552 + 0.835460i \(0.314798\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 531441. 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −231660. −0.377220
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.37896e6 1.95606 0.978030 0.208463i \(-0.0668461\pi\)
0.978030 + 0.208463i \(0.0668461\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.47251e6 −1.61340 −0.806702 0.590959i \(-0.798750\pi\)
−0.806702 + 0.590959i \(0.798750\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.70300e6 −1.65291 −0.826457 0.563001i \(-0.809647\pi\)
−0.826457 + 0.563001i \(0.809647\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 458458. 0.354014 0.177007 0.984210i \(-0.443359\pi\)
0.177007 + 0.984210i \(0.443359\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.12221e6 0.777748 0.388874 0.921291i \(-0.372864\pi\)
0.388874 + 0.921291i \(0.372864\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.96703e6 −1.85253
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.77156e6 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.50040e6 2.81621
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.56347e6 −0.996934 −0.498467 0.866909i \(-0.666104\pi\)
−0.498467 + 0.866909i \(0.666104\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −7.45945e6 −2.44683
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.73230e6 1.73289 0.866443 0.499276i \(-0.166401\pi\)
0.866443 + 0.499276i \(0.166401\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −721710. −0.201506
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −7.65799e6 −1.97886 −0.989432 0.144999i \(-0.953682\pi\)
−0.989432 + 0.144999i \(0.953682\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.17381e7 2.43185
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.30729e6 1.21816 0.609080 0.793109i \(-0.291539\pi\)
0.609080 + 0.793109i \(0.291539\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 3.67988e6 0.620581 0.310290 0.950642i \(-0.399574\pi\)
0.310290 + 0.950642i \(0.399574\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.29893e7 −2.05150
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 1.43729e7 1.99928 0.999639 0.0268508i \(-0.00854791\pi\)
0.999639 + 0.0268508i \(0.00854791\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.39125e7 −1.81972 −0.909862 0.414911i \(-0.863813\pi\)
−0.909862 + 0.414911i \(0.863813\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1.98764e7 −2.30715
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.02930e6 0.373296
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 2.85265e7 2.50438
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 1.68141e7 1.40013 0.700064 0.714080i \(-0.253155\pi\)
0.700064 + 0.714080i \(0.253155\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.49411e7 −1.97173 −0.985865 0.167543i \(-0.946417\pi\)
−0.985865 + 0.167543i \(0.946417\pi\)
\(234\) 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
\(241\) 201058. 0.0143638 0.00718191 0.999974i \(-0.497714\pi\)
0.00718191 + 0.999974i \(0.497714\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.75299e7 1.87200
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.15960e7 −1.86137 −0.930686 0.365819i \(-0.880789\pi\)
−0.930686 + 0.365819i \(0.880789\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.32391e7 −1.30706
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −6.88662e6 −0.370057
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.73294e7 −1.40402 −0.702009 0.712168i \(-0.747714\pi\)
−0.702009 + 0.712168i \(0.747714\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4.07760e7 1.91852 0.959258 0.282532i \(-0.0911742\pi\)
0.959258 + 0.282532i \(0.0911742\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.08288e7 0.488048 0.244024 0.969769i \(-0.421533\pi\)
0.244024 + 0.969769i \(0.421533\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.31575e7 −0.959395
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.83912e7 1.52626 0.763131 0.646244i \(-0.223661\pi\)
0.763131 + 0.646244i \(0.223661\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.49755e7 1.93762
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −1.45704e7 −0.475157 −0.237578 0.971368i \(-0.576354\pi\)
−0.237578 + 0.971368i \(0.576354\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.18450e7 1.31361 0.656805 0.754061i \(-0.271908\pi\)
0.656805 + 0.754061i \(0.271908\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.59263e8 −4.63944
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) −4.04668e7 −1.09589
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7.63724e7 1.99548 0.997739 0.0672052i \(-0.0214082\pi\)
0.997739 + 0.0672052i \(0.0214082\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 4.66380e6 0.109714 0.0548572 0.998494i \(-0.482530\pi\)
0.0548572 + 0.998494i \(0.482530\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7.70972e7 −1.75273 −0.876363 0.481651i \(-0.840037\pi\)
−0.876363 + 0.481651i \(0.840037\pi\)
\(354\) 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
\(361\) 4.70459e7 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.00051e8 2.05752
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −6.19227e7 −1.23245
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6.31878e7 1.21761 0.608803 0.793321i \(-0.291650\pi\)
0.608803 + 0.793321i \(0.291650\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.29743e8 2.42137
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 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
\(387\) 0 0
\(388\) 0 0
\(389\) −1.17588e8 −1.99762 −0.998812 0.0487324i \(-0.984482\pi\)
−0.998812 + 0.0487324i \(0.984482\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −3.27126e7 −0.522809 −0.261404 0.965229i \(-0.584186\pi\)
−0.261404 + 0.965229i \(0.584186\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.23212e8 −1.91082 −0.955410 0.295282i \(-0.904586\pi\)
−0.955410 + 0.295282i \(0.904586\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.24357e8 1.87200
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −8.57705e7 −1.25363 −0.626813 0.779169i \(-0.715641\pi\)
−0.626813 + 0.779169i \(0.715641\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −3.06448e7 −0.410687 −0.205343 0.978690i \(-0.565831\pi\)
−0.205343 + 0.978690i \(0.565831\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.87397e7 −0.504649
\(426\) 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
\(433\) −1.38726e8 −1.70882 −0.854408 0.519602i \(-0.826080\pi\)
−0.854408 + 0.519602i \(0.826080\pi\)
\(434\) 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
\(441\) 8.57661e7 1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 3.22677e8 3.66175
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7.86233e7 0.868585 0.434292 0.900772i \(-0.356998\pi\)
0.434292 + 0.900772i \(0.356998\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.15600e7 0.854533 0.427267 0.904126i \(-0.359477\pi\)
0.427267 + 0.904126i \(0.359477\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.90571e8 −1.94516 −0.972578 0.232576i \(-0.925285\pi\)
−0.972578 + 0.232576i \(0.925285\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2.14545e7 −0.197680
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 2.25926e8 2.03016
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.44567e8 −3.02029
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 3.15592e7 0.263382
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −3.98502e8 −3.09425
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.01116e7 −0.455832 −0.227916 0.973681i \(-0.573191\pi\)
−0.227916 + 0.973681i \(0.573191\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.80651e8 1.98451 0.992256 0.124212i \(-0.0396402\pi\)
0.992256 + 0.124212i \(0.0396402\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.48036e8 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.45714e8 2.28315
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.81611e8 −1.77851 −0.889257 0.457407i \(-0.848778\pi\)
−0.889257 + 0.457407i \(0.848778\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.07279e8 0.662714
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 1.71270e8 1.03506
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.71210e8 −1.56942 −0.784710 0.619863i \(-0.787188\pi\)
−0.784710 + 0.619863i \(0.787188\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 2.62597e8 1.45594
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.50122e8 1.90056 0.950281 0.311393i \(-0.100796\pi\)
0.950281 + 0.311393i \(0.100796\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −3.72270e8 −1.93790 −0.968948 0.247264i \(-0.920469\pi\)
−0.968948 + 0.247264i \(0.920469\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −6.94285e8 −3.46793
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.76744e8 −0.847578 −0.423789 0.905761i \(-0.639300\pi\)
−0.423789 + 0.905761i \(0.639300\pi\)
\(594\) 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
\(601\) −1.44143e8 −0.664002 −0.332001 0.943279i \(-0.607724\pi\)
−0.332001 + 0.943279i \(0.607724\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.14545e8 1.87200
\(606\) 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
\(613\) 7.85685e7 0.341088 0.170544 0.985350i \(-0.445447\pi\)
0.170544 + 0.985350i \(0.445447\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.30573e8 −0.981642 −0.490821 0.871260i \(-0.663303\pi\)
−0.490821 + 0.871260i \(0.663303\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 6.75673e8 2.76756
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.49549e7 0.220828
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.78831e8 −1.85253
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.05573e8 1.16022 0.580111 0.814538i \(-0.303009\pi\)
0.580111 + 0.814538i \(0.303009\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.55866e8 1.99632 0.998161 0.0606193i \(-0.0193076\pi\)
0.998161 + 0.0606193i \(0.0193076\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.11699e8 1.09910
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 9.06129e7 0.313752 0.156876 0.987618i \(-0.449858\pi\)
0.156876 + 0.987618i \(0.449858\pi\)
\(662\) 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
\(673\) −5.89733e8 −1.93468 −0.967342 0.253473i \(-0.918427\pi\)
−0.967342 + 0.253473i \(0.918427\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.04143e8 −1.94703 −0.973517 0.228616i \(-0.926580\pi\)
−0.973517 + 0.228616i \(0.926580\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −5.99852e8 −1.86626
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.19780e8 0.366207
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 8.40926e7 0.248347
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.87743e8 −0.835317 −0.417658 0.908604i \(-0.637149\pi\)
−0.417658 + 0.908604i \(0.637149\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.42174e8 1.80183 0.900916 0.433993i \(-0.142896\pi\)
0.900916 + 0.433993i \(0.142896\pi\)
\(710\) 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
\(723\) 0 0
\(724\) 0 0
\(725\) −1.24742e9 −3.27339
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 3.87420e8 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 7.11418e8 1.80640 0.903198 0.429225i \(-0.141213\pi\)
0.903198 + 0.429225i \(0.141213\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 1.34136e9 3.24396
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3.60626e8 0.831322 0.415661 0.909520i \(-0.363550\pi\)
0.415661 + 0.909520i \(0.363550\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.35040e8 0.306413 0.153207 0.988194i \(-0.451040\pi\)
0.153207 + 0.988194i \(0.451040\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.68880e8 −0.377220
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −8.16391e8 −1.79523 −0.897613 0.440785i \(-0.854700\pi\)
−0.897613 + 0.440785i \(0.854700\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.39790e8 1.38516 0.692578 0.721343i \(-0.256475\pi\)
0.692578 + 0.721343i \(0.256475\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.79197e9 −3.70443
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −9.56198e8 −1.91747
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.47615e8 1.47674 0.738368 0.674398i \(-0.235597\pi\)
0.738368 + 0.674398i \(0.235597\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 1.00526e9 1.95606
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.17457e8 −0.977302 −0.488651 0.872479i \(-0.662511\pi\)
−0.488651 + 0.872479i \(0.662511\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.10335e9 −1.99382 −0.996908 0.0785816i \(-0.974961\pi\)
−0.996908 + 0.0785816i \(0.974961\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −6.02782e8 −1.05803 −0.529013 0.848613i \(-0.677438\pi\)
−0.529013 + 0.848613i \(0.677438\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.16473e8 −0.201506
\(834\) 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
\(841\) 4.21384e8 0.708418
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.74671e9 4.55243
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −8.26038e8 −1.33092 −0.665462 0.746432i \(-0.731765\pi\)
−0.665462 + 0.746432i \(0.731765\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 8.56602e8 1.36093 0.680467 0.732779i \(-0.261777\pi\)
0.680467 + 0.732779i \(0.261777\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 1.47591e9 2.28040
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −1.07346e9 −1.61340
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 4.58388e8 0.679570 0.339785 0.940503i \(-0.389646\pi\)
0.339785 + 0.940503i \(0.389646\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.31647e9 −1.92524 −0.962619 0.270860i \(-0.912692\pi\)
−0.962619 + 0.270860i \(0.912692\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 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
\(891\) 0 0
\(892\) 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
\(901\) 2.91357e7 0.0398337
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8.61092e8 1.16173
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −1.24149e9 −1.65291
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 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
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −2.17216e9 −2.74452
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −6.50820e8 −0.811734 −0.405867 0.913932i \(-0.633030\pi\)
−0.405867 + 0.913932i \(0.633030\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.05307e9 1.28009 0.640044 0.768338i \(-0.278916\pi\)
0.640044 + 0.768338i \(0.278916\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −6.81896e8 −0.818370 −0.409185 0.912452i \(-0.634187\pi\)
−0.409185 + 0.912452i \(0.634187\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −1.74021e9 −2.03612
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.49042e9 1.72199 0.860996 0.508612i \(-0.169841\pi\)
0.860996 + 0.508612i \(0.169841\pi\)
\(954\) 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
\(963\) 0 0
\(964\) 0 0
\(965\) 3.36327e9 3.74265
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.33909e9 1.43591 0.717953 0.696092i \(-0.245079\pi\)
0.717953 + 0.696092i \(0.245079\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 3.34216e8 0.354014
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −3.25552e9 −3.40652
\(986\) 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
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.14875e8 0.822253 0.411127 0.911578i \(-0.365135\pi\)
0.411127 + 0.911578i \(0.365135\pi\)
\(998\) 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 16.7.c.a.15.1 1
3.2 odd 2 144.7.g.a.127.1 1
4.3 odd 2 CM 16.7.c.a.15.1 1
5.2 odd 4 400.7.h.a.399.2 2
5.3 odd 4 400.7.h.a.399.1 2
5.4 even 2 400.7.b.a.351.1 1
8.3 odd 2 64.7.c.a.63.1 1
8.5 even 2 64.7.c.a.63.1 1
12.11 even 2 144.7.g.a.127.1 1
16.3 odd 4 256.7.d.b.127.1 2
16.5 even 4 256.7.d.b.127.2 2
16.11 odd 4 256.7.d.b.127.2 2
16.13 even 4 256.7.d.b.127.1 2
20.3 even 4 400.7.h.a.399.1 2
20.7 even 4 400.7.h.a.399.2 2
20.19 odd 2 400.7.b.a.351.1 1
24.5 odd 2 576.7.g.c.127.1 1
24.11 even 2 576.7.g.c.127.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.7.c.a.15.1 1 1.1 even 1 trivial
16.7.c.a.15.1 1 4.3 odd 2 CM
64.7.c.a.63.1 1 8.3 odd 2
64.7.c.a.63.1 1 8.5 even 2
144.7.g.a.127.1 1 3.2 odd 2
144.7.g.a.127.1 1 12.11 even 2
256.7.d.b.127.1 2 16.3 odd 4
256.7.d.b.127.1 2 16.13 even 4
256.7.d.b.127.2 2 16.5 even 4
256.7.d.b.127.2 2 16.11 odd 4
400.7.b.a.351.1 1 5.4 even 2
400.7.b.a.351.1 1 20.19 odd 2
400.7.h.a.399.1 2 5.3 odd 4
400.7.h.a.399.1 2 20.3 even 4
400.7.h.a.399.2 2 5.2 odd 4
400.7.h.a.399.2 2 20.7 even 4
576.7.g.c.127.1 1 24.5 odd 2
576.7.g.c.127.1 1 24.11 even 2