Properties

Label 2736.2.k.p.2431.7
Level $2736$
Weight $2$
Character 2736.2431
Analytic conductor $21.847$
Analytic rank $0$
Dimension $8$
CM discriminant -19
Inner twists $8$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(2431,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.2431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.k (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2702336256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 2431.7
Root \(0.656712 + 2.13746i\) of defining polynomial
Character \(\chi\) \(=\) 2736.2431
Dual form 2736.2.k.p.2431.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.04547 q^{5} -0.418627i q^{7} +O(q^{10})\) \(q+3.04547 q^{5} -0.418627i q^{7} +1.27492i q^{11} +3.88273 q^{17} +4.35890i q^{19} +4.00000i q^{23} +4.27492 q^{25} -1.27492i q^{35} +5.67232i q^{43} +2.72508i q^{47} +6.82475 q^{49} +3.88273i q^{55} +11.2749 q^{61} +5.82475 q^{73} +0.533714 q^{77} -16.0000i q^{83} +11.8248 q^{85} +13.2749i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{25} - 36 q^{49} + 60 q^{61} - 44 q^{73} + 4 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3.04547 1.36198 0.680989 0.732294i \(-0.261550\pi\)
0.680989 + 0.732294i \(0.261550\pi\)
\(6\) 0 0
\(7\) − 0.418627i − 0.158226i −0.996866 0.0791130i \(-0.974791\pi\)
0.996866 0.0791130i \(-0.0252088\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.27492i 0.384402i 0.981356 + 0.192201i \(0.0615626\pi\)
−0.981356 + 0.192201i \(0.938437\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.88273 0.941700 0.470850 0.882213i \(-0.343947\pi\)
0.470850 + 0.882213i \(0.343947\pi\)
\(18\) 0 0
\(19\) 4.35890i 1.00000i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 4.27492 0.854983
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 1.27492i − 0.215500i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 5.67232i 0.865021i 0.901629 + 0.432511i \(0.142372\pi\)
−0.901629 + 0.432511i \(0.857628\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.72508i 0.397494i 0.980051 + 0.198747i \(0.0636872\pi\)
−0.980051 + 0.198747i \(0.936313\pi\)
\(48\) 0 0
\(49\) 6.82475 0.974965
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 3.88273i 0.523547i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 11.2749 1.44361 0.721803 0.692099i \(-0.243314\pi\)
0.721803 + 0.692099i \(0.243314\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 5.82475 0.681736 0.340868 0.940111i \(-0.389279\pi\)
0.340868 + 0.940111i \(0.389279\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.533714 0.0608224
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 16.0000i − 1.75623i −0.478451 0.878114i \(-0.658802\pi\)
0.478451 0.878114i \(-0.341198\pi\)
\(84\) 0 0
\(85\) 11.8248 1.28257
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 13.2749i 1.36198i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −17.4356 −1.73491 −0.867453 0.497519i \(-0.834245\pi\)
−0.867453 + 0.497519i \(0.834245\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 12.1819i 1.13597i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 1.62541i − 0.149001i
\(120\) 0 0
\(121\) 9.37459 0.852235
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.20822 −0.197509
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 22.3746i − 1.95488i −0.211221 0.977438i \(-0.567744\pi\)
0.211221 0.977438i \(-0.432256\pi\)
\(132\) 0 0
\(133\) 1.82475 0.158226
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −17.7391 −1.51556 −0.757778 0.652512i \(-0.773715\pi\)
−0.757778 + 0.652512i \(0.773715\pi\)
\(138\) 0 0
\(139\) 18.6915i 1.58539i 0.609618 + 0.792695i \(0.291323\pi\)
−0.609618 + 0.792695i \(0.708677\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.37097 −0.112314 −0.0561570 0.998422i \(-0.517885\pi\)
−0.0561570 + 0.998422i \(0.517885\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.67451 0.131970
\(162\) 0 0
\(163\) 8.71780i 0.682831i 0.939913 + 0.341415i \(0.110906\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) − 1.78959i − 0.135281i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 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
\(187\) 4.95016i 0.361991i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 10.3746i − 0.750679i −0.926887 0.375339i \(-0.877526\pi\)
0.926887 0.375339i \(-0.122474\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.4356 1.24223 0.621117 0.783718i \(-0.286679\pi\)
0.621117 + 0.783718i \(0.286679\pi\)
\(198\) 0 0
\(199\) − 15.1123i − 1.07128i −0.844446 0.535641i \(-0.820070\pi\)
0.844446 0.535641i \(-0.179930\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −5.55724 −0.384402
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 17.2749i 1.17814i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 8.37459 0.553408 0.276704 0.960955i \(-0.410758\pi\)
0.276704 + 0.960955i \(0.410758\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.3901 0.942728 0.471364 0.881939i \(-0.343762\pi\)
0.471364 + 0.881939i \(0.343762\pi\)
\(234\) 0 0
\(235\) 8.29917i 0.541378i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 28.9244i 1.87097i 0.353373 + 0.935483i \(0.385035\pi\)
−0.353373 + 0.935483i \(0.614965\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 20.7846 1.32788
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 30.3746i 1.91723i 0.284711 + 0.958613i \(0.408102\pi\)
−0.284711 + 0.958613i \(0.591898\pi\)
\(252\) 0 0
\(253\) −5.09967 −0.320613
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 20.9244i − 1.29026i −0.764075 0.645128i \(-0.776804\pi\)
0.764075 0.645128i \(-0.223196\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) − 26.1534i − 1.58871i −0.607457 0.794353i \(-0.707810\pi\)
0.607457 0.794353i \(-0.292190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.45017i 0.328657i
\(276\) 0 0
\(277\) −12.7251 −0.764576 −0.382288 0.924043i \(-0.624864\pi\)
−0.382288 + 0.924043i \(0.624864\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 33.3851i 1.98454i 0.124096 + 0.992270i \(0.460397\pi\)
−0.124096 + 0.992270i \(0.539603\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.92442 −0.113201
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 2.37459 0.136869
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 34.3375 1.96616
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 21.2749i − 1.20639i −0.797594 0.603195i \(-0.793894\pi\)
0.797594 0.603195i \(-0.206106\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 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
\(321\) 0 0
\(322\) 0 0
\(323\) 16.9244i 0.941700i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.14079 0.0628939
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 5.78741i − 0.312491i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 14.7251i 0.790484i 0.918577 + 0.395242i \(0.129339\pi\)
−0.918577 + 0.395242i \(0.870661\pi\)
\(348\) 0 0
\(349\) −6.17525 −0.330553 −0.165277 0.986247i \(-0.552852\pi\)
−0.165277 + 0.986247i \(0.552852\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −34.8712 −1.85601 −0.928003 0.372572i \(-0.878476\pi\)
−0.928003 + 0.372572i \(0.878476\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 34.3746i − 1.81422i −0.420892 0.907111i \(-0.638283\pi\)
0.420892 0.907111i \(-0.361717\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 17.7391 0.928509
\(366\) 0 0
\(367\) − 26.1534i − 1.36520i −0.730794 0.682598i \(-0.760850\pi\)
0.730794 0.682598i \(-0.239150\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 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
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 1.62541 0.0828388
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.39449 −0.324213 −0.162107 0.986773i \(-0.551829\pi\)
−0.162107 + 0.986773i \(0.551829\pi\)
\(390\) 0 0
\(391\) 15.5309i 0.785432i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −37.4743 −1.88078 −0.940389 0.340099i \(-0.889539\pi\)
−0.940389 + 0.340099i \(0.889539\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 48.7276i − 2.39194i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 40.0000i − 1.95413i −0.212946 0.977064i \(-0.568306\pi\)
0.212946 0.977064i \(-0.431694\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 16.5983 0.805138
\(426\) 0 0
\(427\) − 4.71998i − 0.228416i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −17.4356 −0.834058
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 40.9244i 1.94438i 0.234198 + 0.972189i \(0.424754\pi\)
−0.234198 + 0.972189i \(0.575246\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −25.4743 −1.19164 −0.595818 0.803120i \(-0.703172\pi\)
−0.595818 + 0.803120i \(0.703172\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −42.9402 −1.99992 −0.999962 0.00872311i \(-0.997223\pi\)
−0.999962 + 0.00872311i \(0.997223\pi\)
\(462\) 0 0
\(463\) − 28.9687i − 1.34629i −0.739511 0.673145i \(-0.764943\pi\)
0.739511 0.673145i \(-0.235057\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.2749i 1.16958i 0.811183 + 0.584792i \(0.198824\pi\)
−0.811183 + 0.584792i \(0.801176\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −7.23174 −0.332516
\(474\) 0 0
\(475\) 18.6339i 0.854983i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 4.00000i − 0.182765i −0.995816 0.0913823i \(-0.970871\pi\)
0.995816 0.0913823i \(-0.0291285\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 8.00000i − 0.361035i −0.983572 0.180517i \(-0.942223\pi\)
0.983572 0.180517i \(-0.0577772\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 22.8777i − 1.02415i −0.858941 0.512074i \(-0.828877\pi\)
0.858941 0.512074i \(-0.171123\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 44.0000i − 1.96186i −0.194354 0.980932i \(-0.562261\pi\)
0.194354 0.980932i \(-0.437739\pi\)
\(504\) 0 0
\(505\) −53.0997 −2.36290
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) − 2.43840i − 0.107868i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −3.47425 −0.152798
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 8.70099i 0.374778i
\(540\) 0 0
\(541\) −21.4743 −0.923250 −0.461625 0.887075i \(-0.652733\pi\)
−0.461625 + 0.887075i \(0.652733\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 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
\(557\) −38.5237 −1.63230 −0.816152 0.577838i \(-0.803897\pi\)
−0.816152 + 0.577838i \(0.803897\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 26.1534i 1.09449i 0.836974 + 0.547243i \(0.184323\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 17.0997i 0.713105i
\(576\) 0 0
\(577\) −40.0241 −1.66622 −0.833112 0.553104i \(-0.813443\pi\)
−0.833112 + 0.553104i \(0.813443\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.69803 −0.277881
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 38.0241i − 1.56942i −0.619862 0.784711i \(-0.712811\pi\)
0.619862 0.784711i \(-0.287189\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −34.8712 −1.43199 −0.715994 0.698106i \(-0.754026\pi\)
−0.715994 + 0.698106i \(0.754026\pi\)
\(594\) 0 0
\(595\) − 4.95016i − 0.202937i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 28.5501 1.16073
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −6.92442 −0.279675 −0.139837 0.990174i \(-0.544658\pi\)
−0.139837 + 0.990174i \(0.544658\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.7156 −0.511911 −0.255956 0.966689i \(-0.582390\pi\)
−0.255956 + 0.966689i \(0.582390\pi\)
\(618\) 0 0
\(619\) 43.5890i 1.75199i 0.482321 + 0.875995i \(0.339794\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −28.0997 −1.12399
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) − 36.9643i − 1.47153i −0.677239 0.735763i \(-0.736824\pi\)
0.677239 0.735763i \(-0.263176\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) − 35.8969i − 1.41564i −0.706395 0.707818i \(-0.749680\pi\)
0.706395 0.707818i \(-0.250320\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 50.0241i − 1.96665i −0.181857 0.983325i \(-0.558211\pi\)
0.181857 0.983325i \(-0.441789\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20.2509 0.792479 0.396239 0.918147i \(-0.370315\pi\)
0.396239 + 0.918147i \(0.370315\pi\)
\(654\) 0 0
\(655\) − 68.1412i − 2.66250i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.55724 0.215500
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 14.3746i 0.554925i
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −54.0241 −2.06415
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 10.6958i 0.406889i 0.979086 + 0.203445i \(0.0652137\pi\)
−0.979086 + 0.203445i \(0.934786\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 56.9244i 2.15927i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 17.4356 0.658533 0.329267 0.944237i \(-0.393198\pi\)
0.329267 + 0.944237i \(0.393198\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.29901i 0.274507i
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 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
\(719\) − 5.62541i − 0.209793i −0.994483 0.104896i \(-0.966549\pi\)
0.994483 0.104896i \(-0.0334511\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 51.6580i − 1.91589i −0.286954 0.957944i \(-0.592643\pi\)
0.286954 0.957944i \(-0.407357\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 22.0241i 0.814590i
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 23.7150i 0.872370i 0.899857 + 0.436185i \(0.143671\pi\)
−0.899857 + 0.436185i \(0.856329\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −4.17525 −0.152969
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −28.0241 −1.01855 −0.509276 0.860603i \(-0.670087\pi\)
−0.509276 + 0.860603i \(0.670087\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 45.4519 1.64763 0.823816 0.566857i \(-0.191841\pi\)
0.823816 + 0.566857i \(0.191841\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 44.3746 1.60019 0.800094 0.599874i \(-0.204783\pi\)
0.800094 + 0.599874i \(0.204783\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 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
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 54.8185 1.95656
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 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
\(799\) 10.5808i 0.374320i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7.42608i 0.262061i
\(804\) 0 0
\(805\) 5.09967 0.179740
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −32.6630 −1.14837 −0.574184 0.818726i \(-0.694681\pi\)
−0.574184 + 0.818726i \(0.694681\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 26.5498i 0.930000i
\(816\) 0 0
\(817\) −24.7251 −0.865021
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 56.7966 1.98221 0.991107 0.133064i \(-0.0424817\pi\)
0.991107 + 0.133064i \(0.0424817\pi\)
\(822\) 0 0
\(823\) 36.1271i 1.25931i 0.776875 + 0.629655i \(0.216804\pi\)
−0.776875 + 0.629655i \(0.783196\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 26.4987 0.918124
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 39.5912 1.36198
\(846\) 0 0
\(847\) − 3.92445i − 0.134846i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 15.3425i 0.523478i 0.965139 + 0.261739i \(0.0842960\pi\)
−0.965139 + 0.261739i \(0.915704\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0.924421i 0.0312511i
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −54.2848 −1.82890 −0.914451 0.404696i \(-0.867377\pi\)
−0.914451 + 0.404696i \(0.867377\pi\)
\(882\) 0 0
\(883\) − 59.4234i − 1.99976i −0.0155546 0.999879i \(-0.504951\pi\)
0.0155546 0.999879i \(-0.495049\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −11.8784 −0.397494
\(894\) 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
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 20.3987 0.675098
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.36660 −0.309312
\(918\) 0 0
\(919\) − 8.71780i − 0.287574i −0.989609 0.143787i \(-0.954072\pi\)
0.989609 0.143787i \(-0.0459280\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 34.8712 1.14409 0.572043 0.820223i \(-0.306151\pi\)
0.572043 + 0.820223i \(0.306151\pi\)
\(930\) 0 0
\(931\) 29.7484i 0.974965i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 15.0756i 0.493024i
\(936\) 0 0
\(937\) −57.4743 −1.87760 −0.938801 0.344460i \(-0.888062\pi\)
−0.938801 + 0.344460i \(0.888062\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 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
\(947\) − 8.00000i − 0.259965i −0.991516 0.129983i \(-0.958508\pi\)
0.991516 0.129983i \(-0.0414921\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) − 31.5955i − 1.02241i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 7.42608i 0.239801i
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 61.0246i 1.96242i 0.192947 + 0.981209i \(0.438195\pi\)
−0.192947 + 0.981209i \(0.561805\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 7.82475 0.250850
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 53.0997 1.69190
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −22.6893 −0.721478
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 46.0241i − 1.45906i
\(996\) 0 0
\(997\) 35.2749 1.11717 0.558584 0.829448i \(-0.311345\pi\)
0.558584 + 0.829448i \(0.311345\pi\)
\(998\) 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 2736.2.k.p.2431.7 yes 8
3.2 odd 2 inner 2736.2.k.p.2431.1 8
4.3 odd 2 inner 2736.2.k.p.2431.8 yes 8
12.11 even 2 inner 2736.2.k.p.2431.2 yes 8
19.18 odd 2 CM 2736.2.k.p.2431.7 yes 8
57.56 even 2 inner 2736.2.k.p.2431.1 8
76.75 even 2 inner 2736.2.k.p.2431.8 yes 8
228.227 odd 2 inner 2736.2.k.p.2431.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2736.2.k.p.2431.1 8 3.2 odd 2 inner
2736.2.k.p.2431.1 8 57.56 even 2 inner
2736.2.k.p.2431.2 yes 8 12.11 even 2 inner
2736.2.k.p.2431.2 yes 8 228.227 odd 2 inner
2736.2.k.p.2431.7 yes 8 1.1 even 1 trivial
2736.2.k.p.2431.7 yes 8 19.18 odd 2 CM
2736.2.k.p.2431.8 yes 8 4.3 odd 2 inner
2736.2.k.p.2431.8 yes 8 76.75 even 2 inner