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Properties

Label 1584.3.j.c.1297.1
Level $1584$
Weight $3$
Character 1584.1297
Self dual yes
Analytic conductor $43.161$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1584,3,Mod(1297,1584)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1584.1297"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1584.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-1,0,0,0,0,0,-22,0,0,0,0,0,0,0,0,0,0,0,-35,0,99] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(43.1608738747\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 44)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1297.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 1584.1297

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.11684 q^{5} -11.0000 q^{11} -43.3505 q^{23} +58.1168 q^{25} -61.5842 q^{31} -47.8179 q^{37} +50.0000 q^{47} +49.0000 q^{49} +70.0000 q^{53} +100.285 q^{55} -10.4158 q^{59} -94.5190 q^{67} +109.584 q^{71} +80.7527 q^{89} +98.9863 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - 22 q^{11} - 35 q^{23} + 99 q^{25} - 37 q^{31} + 25 q^{37} + 100 q^{47} + 98 q^{49} + 140 q^{53} + 11 q^{55} - 107 q^{59} + 35 q^{67} + 133 q^{71} - 97 q^{89} - 95 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(353\) \(991\) \(1189\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) −9.11684 −1.82337 −0.911684 − 0.410891i \(-0.865218\pi\)
−0.911684 + 0.410891i \(0.865218\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −11.0000 −1.00000
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −43.3505 −1.88481 −0.942403 − 0.334480i \(-0.891439\pi\)
−0.942403 + 0.334480i \(0.891439\pi\)
\(24\) 0 0
\(25\) 58.1168 2.32467
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −61.5842 −1.98659 −0.993294 − 0.115617i \(-0.963115\pi\)
−0.993294 + 0.115617i \(0.963115\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −47.8179 −1.29238 −0.646188 − 0.763178i \(-0.723638\pi\)
−0.646188 + 0.763178i \(0.723638\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 50.0000 1.06383 0.531915 − 0.846798i \(-0.321473\pi\)
0.531915 + 0.846798i \(0.321473\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 70.0000 1.32075 0.660377 − 0.750934i \(-0.270396\pi\)
0.660377 + 0.750934i \(0.270396\pi\)
\(54\) 0 0
\(55\) 100.285 1.82337
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.4158 −0.176539 −0.0882693 − 0.996097i \(-0.528134\pi\)
−0.0882693 + 0.996097i \(0.528134\pi\)
\(60\) 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
\(67\) −94.5190 −1.41073 −0.705365 − 0.708844i \(-0.749217\pi\)
−0.705365 + 0.708844i \(0.749217\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 109.584 1.54344 0.771720 − 0.635963i \(-0.219397\pi\)
0.771720 + 0.635963i \(0.219397\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 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
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 80.7527 0.907333 0.453667 − 0.891171i \(-0.350116\pi\)
0.453667 + 0.891171i \(0.350116\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 98.9863 1.02048 0.510239 − 0.860033i \(-0.329557\pi\)
0.510239 + 0.860033i \(0.329557\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 190.000 1.84466 0.922330 − 0.386403i \(-0.126283\pi\)
0.922330 + 0.386403i \(0.126283\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 167.818 1.48511 0.742557 − 0.669783i \(-0.233613\pi\)
0.742557 + 0.669783i \(0.233613\pi\)
\(114\) 0 0
\(115\) 395.220 3.43670
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −301.921 −2.41537
\(126\) 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
\(137\) −72.1821 −0.526877 −0.263438 − 0.964676i \(-0.584857\pi\)
−0.263438 + 0.964676i \(0.584857\pi\)
\(138\) 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
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 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
\(153\) 0 0
\(154\) 0 0
\(155\) 561.454 3.62228
\(156\) 0 0
\(157\) −305.687 −1.94705 −0.973527 − 0.228573i \(-0.926594\pi\)
−0.973527 + 0.228573i \(0.926594\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 70.0000 0.429448 0.214724 − 0.976675i \(-0.431115\pi\)
0.214724 + 0.976675i \(0.431115\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −343.090 −1.91670 −0.958351 − 0.285594i \(-0.907809\pi\)
−0.958351 + 0.285594i \(0.907809\pi\)
\(180\) 0 0
\(181\) 83.9211 0.463652 0.231826 − 0.972757i \(-0.425530\pi\)
0.231826 + 0.972757i \(0.425530\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 435.948 2.35648
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −223.090 −1.16801 −0.584004 − 0.811751i \(-0.698515\pi\)
−0.584004 + 0.811751i \(0.698515\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.0100503 −0.00502513 − 0.999987i \(-0.501600\pi\)
−0.00502513 + 0.999987i \(0.501600\pi\)
\(200\) 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
\(209\) 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
\(221\) 0 0
\(222\) 0 0
\(223\) −196.649 −0.881836 −0.440918 − 0.897547i \(-0.645347\pi\)
−0.440918 + 0.897547i \(0.645347\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 87.2473 0.380993 0.190496 − 0.981688i \(-0.438990\pi\)
0.190496 + 0.981688i \(0.438990\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −455.842 −1.93975
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −446.725 −1.82337
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 274.258 1.09266 0.546331 − 0.837570i \(-0.316024\pi\)
0.546331 + 0.837570i \(0.316024\pi\)
\(252\) 0 0
\(253\) 476.856 1.88481
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 190.000 0.739300 0.369650 − 0.929171i \(-0.379478\pi\)
0.369650 + 0.929171i \(0.379478\pi\)
\(258\) 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
\(265\) −638.179 −2.40822
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −362.000 −1.34572 −0.672862 − 0.739768i \(-0.734935\pi\)
−0.672862 + 0.739768i \(0.734935\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −639.285 −2.32467
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 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
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 94.9590 0.321895
\(296\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −478.000 −1.53698 −0.768489 − 0.639863i \(-0.778991\pi\)
−0.768489 + 0.639863i \(0.778991\pi\)
\(312\) 0 0
\(313\) 623.660 1.99252 0.996262 − 0.0863813i \(-0.0275303\pi\)
0.996262 + 0.0863813i \(0.0275303\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 590.361 1.86234 0.931169 − 0.364588i \(-0.118790\pi\)
0.931169 + 0.364588i \(0.118790\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 583.090 1.76160 0.880800 − 0.473489i \(-0.157006\pi\)
0.880800 + 0.473489i \(0.157006\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 861.715 2.57228
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 677.426 1.98659
\(342\) 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
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −596.856 −1.69081 −0.845405 − 0.534126i \(-0.820641\pi\)
−0.845405 + 0.534126i \(0.820641\pi\)
\(354\) 0 0
\(355\) −999.062 −2.81426
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.32323 −0.00360552 −0.00180276 − 0.999998i \(-0.500574\pi\)
−0.00180276 + 0.999998i \(0.500574\pi\)
\(368\) 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
\(379\) 579.763 1.52972 0.764859 − 0.644197i \(-0.222808\pi\)
0.764859 + 0.644197i \(0.222808\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −265.481 −0.693162 −0.346581 − 0.938020i \(-0.612657\pi\)
−0.346581 + 0.938020i \(0.612657\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 197.426 0.507523 0.253761 − 0.967267i \(-0.418332\pi\)
0.253761 + 0.967267i \(0.418332\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −790.000 −1.98992 −0.994962 − 0.100251i \(-0.968036\pi\)
−0.994962 + 0.100251i \(0.968036\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −98.0000 −0.244389 −0.122195 − 0.992506i \(-0.538993\pi\)
−0.122195 + 0.992506i \(0.538993\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 525.997 1.29238
\(408\) 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
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −262.000 −0.625298 −0.312649 − 0.949869i \(-0.601216\pi\)
−0.312649 + 0.949869i \(0.601216\pi\)
\(420\) 0 0
\(421\) −742.000 −1.76247 −0.881235 − 0.472678i \(-0.843287\pi\)
−0.881235 + 0.472678i \(0.843287\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 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
\(433\) −737.165 −1.70246 −0.851230 − 0.524792i \(-0.824143\pi\)
−0.851230 + 0.524792i \(0.824143\pi\)
\(434\) 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
\(441\) 0 0
\(442\) 0 0
\(443\) 885.997 1.99999 0.999997 − 0.00260851i \(-0.000830316\pi\)
0.999997 + 0.00260851i \(0.000830316\pi\)
\(444\) 0 0
\(445\) −736.209 −1.65440
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −248.595 −0.553663 −0.276832 − 0.960918i \(-0.589284\pi\)
−0.276832 + 0.960918i \(0.589284\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −783.866 −1.69302 −0.846508 − 0.532376i \(-0.821299\pi\)
−0.846508 + 0.532376i \(0.821299\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 574.519 1.23023 0.615117 − 0.788436i \(-0.289109\pi\)
0.615117 + 0.788436i \(0.289109\pi\)
\(468\) 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
\(477\) 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
\(485\) −902.443 −1.86071
\(486\) 0 0
\(487\) 808.024 1.65919 0.829594 − 0.558367i \(-0.188572\pi\)
0.829594 + 0.558367i \(0.188572\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −602.000 −1.20641 −0.603206 − 0.797585i \(-0.706110\pi\)
−0.603206 + 0.797585i \(0.706110\pi\)
\(500\) 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
\(507\) 0 0
\(508\) 0 0
\(509\) 632.753 1.24313 0.621564 − 0.783363i \(-0.286497\pi\)
0.621564 + 0.783363i \(0.286497\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1732.20 −3.36350
\(516\) 0 0
\(517\) −550.000 −1.06383
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −968.595 −1.85911 −0.929554 − 0.368687i \(-0.879807\pi\)
−0.929554 + 0.368687i \(0.879807\pi\)
\(522\) 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
\(529\) 1350.27 2.55249
\(530\) 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
\(539\) −539.000 −1.00000
\(540\) 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
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 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
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) −1529.97 −2.70791
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2519.40 −4.38156
\(576\) 0 0
\(577\) −932.492 −1.61610 −0.808052 − 0.589112i \(-0.799478\pi\)
−0.808052 + 0.589112i \(0.799478\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −770.000 −1.32075
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1130.00 1.92504 0.962521 − 0.271206i \(-0.0874225\pi\)
0.962521 + 0.271206i \(0.0874225\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 98.0000 0.163606 0.0818030 − 0.996649i \(-0.473932\pi\)
0.0818030 + 0.996649i \(0.473932\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1103.14 −1.82337
\(606\) 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
\(617\) −530.000 −0.858995 −0.429498 − 0.903068i \(-0.641309\pi\)
−0.429498 + 0.903068i \(0.641309\pi\)
\(618\) 0 0
\(619\) −661.584 −1.06880 −0.534398 − 0.845233i \(-0.679461\pi\)
−0.534398 + 0.845233i \(0.679461\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1299.65 2.07943
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −304.910 −0.483218 −0.241609 − 0.970374i \(-0.577675\pi\)
−0.241609 + 0.970374i \(0.577675\pi\)
\(632\) 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
\(641\) −533.269 −0.831932 −0.415966 − 0.909380i \(-0.636557\pi\)
−0.415966 + 0.909380i \(0.636557\pi\)
\(642\) 0 0
\(643\) 1257.37 1.95548 0.977739 − 0.209827i \(-0.0672900\pi\)
0.977739 + 0.209827i \(0.0672900\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1290.67 1.99485 0.997427 − 0.0716872i \(-0.0228383\pi\)
0.997427 + 0.0716872i \(0.0228383\pi\)
\(648\) 0 0
\(649\) 114.574 0.176539
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 501.014 0.767249 0.383625 − 0.923489i \(-0.374676\pi\)
0.383625 + 0.923489i \(0.374676\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1136.59 1.71951 0.859754 − 0.510708i \(-0.170617\pi\)
0.859754 + 0.510708i \(0.170617\pi\)
\(662\) 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
\(673\) 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
\(683\) −790.000 −1.15666 −0.578331 − 0.815802i \(-0.696296\pi\)
−0.578331 + 0.815802i \(0.696296\pi\)
\(684\) 0 0
\(685\) 658.073 0.960690
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1278.93 −1.85084 −0.925421 − 0.378941i \(-0.876288\pi\)
−0.925421 + 0.378941i \(0.876288\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 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
\(709\) −290.100 −0.409168 −0.204584 − 0.978849i \(-0.565584\pi\)
−0.204584 + 0.978849i \(0.565584\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2669.71 3.74433
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 150.932 0.209919 0.104959 − 0.994476i \(-0.466529\pi\)
0.104959 + 0.994476i \(0.466529\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 220.807 0.303724 0.151862 − 0.988402i \(-0.451473\pi\)
0.151862 + 0.988402i \(0.451473\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1039.71 1.41073
\(738\) 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
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 504.437 0.671687 0.335844 − 0.941918i \(-0.390979\pi\)
0.335844 + 0.941918i \(0.390979\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −70.0000 −0.0924703 −0.0462351 − 0.998931i \(-0.514722\pi\)
−0.0462351 + 0.998931i \(0.514722\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1370.00 −1.77232 −0.886158 − 0.463384i \(-0.846635\pi\)
−0.886158 + 0.463384i \(0.846635\pi\)
\(774\) 0 0
\(775\) −3579.08 −4.61817
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −1205.43 −1.54344
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2786.90 3.55020
\(786\) 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
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −938.986 −1.17815 −0.589076 − 0.808078i \(-0.700508\pi\)
−0.589076 + 0.808078i \(0.700508\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −638.179 −0.783042
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1644.18 1.99778 0.998892 − 0.0470635i \(-0.0149863\pi\)
0.998892 + 0.0470635i \(0.0149863\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 1657.94 1.99993 0.999965 − 0.00833724i \(-0.00265386\pi\)
0.999965 + 0.00833724i \(0.00265386\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1248.28 1.48782 0.743909 − 0.668281i \(-0.232970\pi\)
0.743909 + 0.668281i \(0.232970\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1540.75 −1.82337
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2072.93 2.43588
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 957.111 1.11422 0.557108 − 0.830440i \(-0.311911\pi\)
0.557108 + 0.830440i \(0.311911\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −430.000 −0.498262 −0.249131 − 0.968470i \(-0.580145\pi\)
−0.249131 + 0.968470i \(0.580145\pi\)
\(864\) 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 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1562.10 1.77310 0.886549 − 0.462634i \(-0.153096\pi\)
0.886549 + 0.462634i \(0.153096\pi\)
\(882\) 0 0
\(883\) −1370.00 −1.55153 −0.775764 − 0.631023i \(-0.782636\pi\)
−0.775764 + 0.631023i \(0.782636\pi\)
\(884\) 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
\(895\) 3127.89 3.49485
\(896\) 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
\(905\) −765.096 −0.845409
\(906\) 0 0
\(907\) 1750.00 1.92944 0.964719 − 0.263282i \(-0.0848050\pi\)
0.964719 + 0.263282i \(0.0848050\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1778.00 1.95170 0.975851 − 0.218439i \(-0.0700963\pi\)
0.975851 + 0.218439i \(0.0700963\pi\)
\(912\) 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
\(925\) −2779.03 −3.00435
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 958.000 1.03122 0.515608 − 0.856824i \(-0.327566\pi\)
0.515608 + 0.856824i \(0.327566\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 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
\(947\) 468.540 0.494763 0.247381 − 0.968918i \(-0.420430\pi\)
0.247381 + 0.968918i \(0.420430\pi\)
\(948\) 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
\(955\) 2033.87 2.12971
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2831.62 2.94653
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1721.78 −1.77321 −0.886604 − 0.462530i \(-0.846942\pi\)
−0.886604 + 0.462530i \(0.846942\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1930.88 −1.97633 −0.988166 − 0.153386i \(-0.950982\pi\)
−0.988166 + 0.153386i \(0.950982\pi\)
\(978\) 0 0
\(979\) −888.279 −0.907333
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1928.85 −1.96221 −0.981104 − 0.193483i \(-0.938022\pi\)
−0.981104 + 0.193483i \(0.938022\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1582.00 1.59637 0.798184 − 0.602414i \(-0.205794\pi\)
0.798184 + 0.602414i \(0.205794\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 18.2337 0.0183253
\(996\) 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 1584.3.j.c.1297.1 2
3.2 odd 2 176.3.h.b.65.2 2
4.3 odd 2 396.3.f.a.109.1 2
11.10 odd 2 CM 1584.3.j.c.1297.1 2
12.11 even 2 44.3.d.a.21.1 ✓ 2
24.5 odd 2 704.3.h.f.65.1 2
24.11 even 2 704.3.h.c.65.2 2
33.32 even 2 176.3.h.b.65.2 2
44.43 even 2 396.3.f.a.109.1 2
60.23 odd 4 1100.3.e.a.549.2 4
60.47 odd 4 1100.3.e.a.549.3 4
60.59 even 2 1100.3.f.a.901.2 2
84.83 odd 2 2156.3.h.a.197.2 2
132.35 odd 10 484.3.f.b.161.1 8
132.47 even 10 484.3.f.b.233.2 8
132.59 even 10 484.3.f.b.457.2 8
132.71 even 10 484.3.f.b.481.1 8
132.83 odd 10 484.3.f.b.481.1 8
132.95 odd 10 484.3.f.b.457.2 8
132.107 odd 10 484.3.f.b.233.2 8
132.119 even 10 484.3.f.b.161.1 8
132.131 odd 2 44.3.d.a.21.1 ✓ 2
264.131 odd 2 704.3.h.c.65.2 2
264.197 even 2 704.3.h.f.65.1 2
660.263 even 4 1100.3.e.a.549.2 4
660.527 even 4 1100.3.e.a.549.3 4
660.659 odd 2 1100.3.f.a.901.2 2
924.923 even 2 2156.3.h.a.197.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
44.3.d.a.21.1 ✓ 2 12.11 even 2
44.3.d.a.21.1 ✓ 2 132.131 odd 2
176.3.h.b.65.2 2 3.2 odd 2
176.3.h.b.65.2 2 33.32 even 2
396.3.f.a.109.1 2 4.3 odd 2
396.3.f.a.109.1 2 44.43 even 2
484.3.f.b.161.1 8 132.35 odd 10
484.3.f.b.161.1 8 132.119 even 10
484.3.f.b.233.2 8 132.47 even 10
484.3.f.b.233.2 8 132.107 odd 10
484.3.f.b.457.2 8 132.59 even 10
484.3.f.b.457.2 8 132.95 odd 10
484.3.f.b.481.1 8 132.71 even 10
484.3.f.b.481.1 8 132.83 odd 10
704.3.h.c.65.2 2 24.11 even 2
704.3.h.c.65.2 2 264.131 odd 2
704.3.h.f.65.1 2 24.5 odd 2
704.3.h.f.65.1 2 264.197 even 2
1100.3.e.a.549.2 4 60.23 odd 4
1100.3.e.a.549.2 4 660.263 even 4
1100.3.e.a.549.3 4 60.47 odd 4
1100.3.e.a.549.3 4 660.527 even 4
1100.3.f.a.901.2 2 60.59 even 2
1100.3.f.a.901.2 2 660.659 odd 2
1584.3.j.c.1297.1 2 1.1 even 1 trivial
1584.3.j.c.1297.1 2 11.10 odd 2 CM
2156.3.h.a.197.2 2 84.83 odd 2
2156.3.h.a.197.2 2 924.923 even 2