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Properties

Label 2592.3.b.a.1135.2
Level $2592$
Weight $3$
Character 2592.1135
Self dual yes
Analytic conductor $70.627$
Analytic rank $0$
Dimension $2$
CM discriminant -8
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] = [2592,3,Mod(1135,2592)] 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("2592.1135"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2592.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(70.6268845222\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1135.2
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 2592.1135

$q$-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)
 
\(f(q)\) \(=\) \(q+7.69694 q^{11} +30.3939 q^{17} -31.6969 q^{19} +25.0000 q^{25} -81.7878 q^{41} +80.4847 q^{43} +49.0000 q^{49} -32.4847 q^{59} -71.8786 q^{67} +41.6061 q^{73} +158.000 q^{83} -146.000 q^{89} +193.969 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{11} + 2 q^{17} - 34 q^{19} + 50 q^{25} - 46 q^{41} + 14 q^{43} + 98 q^{49} + 82 q^{59} + 62 q^{67} + 142 q^{73} + 316 q^{83} - 292 q^{89} + 94 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 7.69694 0.699722 0.349861 − 0.936802i \(-0.386229\pi\)
0.349861 + 0.936802i \(0.386229\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 30.3939 1.78788 0.893938 − 0.448192i \(-0.147932\pi\)
0.893938 + 0.448192i \(0.147932\pi\)
\(18\) 0 0
\(19\) −31.6969 −1.66826 −0.834130 − 0.551568i \(-0.814030\pi\)
−0.834130 + 0.551568i \(0.814030\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 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
\(41\) −81.7878 −1.99482 −0.997412 − 0.0719030i \(-0.977093\pi\)
−0.997412 + 0.0719030i \(0.977093\pi\)
\(42\) 0 0
\(43\) 80.4847 1.87174 0.935869 − 0.352349i \(-0.114617\pi\)
0.935869 + 0.352349i \(0.114617\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −32.4847 −0.550588 −0.275294 − 0.961360i \(-0.588775\pi\)
−0.275294 + 0.961360i \(0.588775\pi\)
\(60\) 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
\(67\) −71.8786 −1.07281 −0.536407 − 0.843959i \(-0.680219\pi\)
−0.536407 + 0.843959i \(0.680219\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 41.6061 0.569947 0.284973 − 0.958535i \(-0.408015\pi\)
0.284973 + 0.958535i \(0.408015\pi\)
\(74\) 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
\(83\) 158.000 1.90361 0.951807 − 0.306697i \(-0.0992238\pi\)
0.951807 + 0.306697i \(0.0992238\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −146.000 −1.64045 −0.820225 − 0.572041i \(-0.806152\pi\)
−0.820225 + 0.572041i \(0.806152\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 193.969 1.99968 0.999842 − 0.0177651i \(-0.00565510\pi\)
0.999842 + 0.0177651i \(0.00565510\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 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
\(107\) 191.879 1.79326 0.896629 − 0.442783i \(-0.146009\pi\)
0.896629 + 0.442783i \(0.146009\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −98.0000 −0.867257 −0.433628 − 0.901092i \(-0.642767\pi\)
−0.433628 + 0.901092i \(0.642767\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −61.7571 −0.510390
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 62.0000 0.473282 0.236641 − 0.971597i \(-0.423953\pi\)
0.236641 + 0.971597i \(0.423953\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.42449 −0.0103978 −0.00519888 − 0.999986i \(-0.501655\pi\)
−0.00519888 + 0.999986i \(0.501655\pi\)
\(138\) 0 0
\(139\) 264.666 1.90407 0.952037 − 0.305983i \(-0.0989849\pi\)
0.952037 + 0.305983i \(0.0989849\pi\)
\(140\) 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
\(155\) 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
\(163\) 322.000 1.97546 0.987730 − 0.156171i \(-0.0499150\pi\)
0.987730 + 0.156171i \(0.0499150\pi\)
\(164\) 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
\(179\) −34.0000 −0.189944 −0.0949721 − 0.995480i \(-0.530276\pi\)
−0.0949721 + 0.995480i \(0.530276\pi\)
\(180\) 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
\(187\) 233.940 1.25101
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 274.333 1.42141 0.710706 − 0.703489i \(-0.248375\pi\)
0.710706 + 0.703489i \(0.248375\pi\)
\(194\) 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
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −243.969 −1.16732
\(210\) 0 0
\(211\) 226.000 1.07109 0.535545 − 0.844507i \(-0.320106\pi\)
0.535545 + 0.844507i \(0.320106\pi\)
\(212\) 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
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −296.485 −1.30610 −0.653050 − 0.757315i \(-0.726511\pi\)
−0.653050 + 0.757315i \(0.726511\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 70.0306 0.300561 0.150280 − 0.988643i \(-0.451982\pi\)
0.150280 + 0.988643i \(0.451982\pi\)
\(234\) 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
\(241\) −479.120 −1.98805 −0.994026 − 0.109146i \(-0.965188\pi\)
−0.994026 + 0.109146i \(0.965188\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 71.3337 0.284198 0.142099 − 0.989852i \(-0.454615\pi\)
0.142099 + 0.989852i \(0.454615\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 486.939 1.89470 0.947352 − 0.320195i \(-0.103748\pi\)
0.947352 + 0.320195i \(0.103748\pi\)
\(258\) 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
\(267\) 0 0
\(268\) 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
\(275\) 192.423 0.699722
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 238.000 0.846975 0.423488 − 0.905902i \(-0.360806\pi\)
0.423488 + 0.905902i \(0.360806\pi\)
\(282\) 0 0
\(283\) 82.0000 0.289753 0.144876 − 0.989450i \(-0.453722\pi\)
0.144876 + 0.989450i \(0.453722\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 634.788 2.19650
\(290\) 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
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 520.848 1.69657 0.848287 − 0.529537i \(-0.177634\pi\)
0.848287 + 0.529537i \(0.177634\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −30.9388 −0.0988459 −0.0494230 − 0.998778i \(-0.515738\pi\)
−0.0494230 + 0.998778i \(0.515738\pi\)
\(314\) 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
\(323\) −963.393 −2.98264
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −14.0000 −0.0422961 −0.0211480 − 0.999776i \(-0.506732\pi\)
−0.0211480 + 0.999776i \(0.506732\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 650.514 1.93031 0.965155 − 0.261680i \(-0.0842765\pi\)
0.965155 + 0.261680i \(0.0842765\pi\)
\(338\) 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
\(347\) 520.060 1.49873 0.749366 − 0.662156i \(-0.230358\pi\)
0.749366 + 0.662156i \(0.230358\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −490.878 −1.39059 −0.695294 − 0.718725i \(-0.744726\pi\)
−0.695294 + 0.718725i \(0.744726\pi\)
\(354\) 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
\(361\) 643.696 1.78309
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 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
\(379\) 63.7582 0.168227 0.0841137 − 0.996456i \(-0.473194\pi\)
0.0841137 + 0.996456i \(0.473194\pi\)
\(380\) 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
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 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
\(401\) −177.243 −0.442002 −0.221001 − 0.975274i \(-0.570932\pi\)
−0.221001 + 0.975274i \(0.570932\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −479.665 −1.17278 −0.586388 − 0.810030i \(-0.699451\pi\)
−0.586388 + 0.810030i \(0.699451\pi\)
\(410\) 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
\(419\) −514.000 −1.22673 −0.613365 − 0.789799i \(-0.710185\pi\)
−0.613365 + 0.789799i \(0.710185\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 759.847 1.78788
\(426\) 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
\(433\) −847.484 −1.95724 −0.978619 − 0.205684i \(-0.934058\pi\)
−0.978619 + 0.205684i \(0.934058\pi\)
\(434\) 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
\(443\) −336.121 −0.758739 −0.379370 − 0.925245i \(-0.623859\pi\)
−0.379370 + 0.925245i \(0.623859\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 638.757 1.42262 0.711311 − 0.702878i \(-0.248102\pi\)
0.711311 + 0.702878i \(0.248102\pi\)
\(450\) 0 0
\(451\) −629.515 −1.39582
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 883.241 1.93269 0.966347 − 0.257244i \(-0.0828143\pi\)
0.966347 + 0.257244i \(0.0828143\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 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
\(467\) 825.332 1.76731 0.883653 − 0.468143i \(-0.155077\pi\)
0.883653 + 0.468143i \(0.155077\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 619.486 1.30969
\(474\) 0 0
\(475\) −792.423 −1.66826
\(476\) 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
\(489\) 0 0
\(490\) 0 0
\(491\) −905.393 −1.84398 −0.921989 − 0.387217i \(-0.873436\pi\)
−0.921989 + 0.387217i \(0.873436\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 113.393 0.227240 0.113620 − 0.993524i \(-0.463755\pi\)
0.113620 + 0.993524i \(0.463755\pi\)
\(500\) 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
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 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
\(521\) −738.151 −1.41680 −0.708398 − 0.705813i \(-0.750582\pi\)
−0.708398 + 0.705813i \(0.750582\pi\)
\(522\) 0 0
\(523\) −398.000 −0.760994 −0.380497 − 0.924782i \(-0.624247\pi\)
−0.380497 + 0.924782i \(0.624247\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 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
\(539\) 377.150 0.699722
\(540\) 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
\(547\) 849.030 1.55216 0.776078 − 0.630637i \(-0.217206\pi\)
0.776078 + 0.630637i \(0.217206\pi\)
\(548\) 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
\(563\) −842.301 −1.49609 −0.748047 − 0.663646i \(-0.769008\pi\)
−0.748047 + 0.663646i \(0.769008\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1136.03 1.99654 0.998268 − 0.0588367i \(-0.0187391\pi\)
0.998268 + 0.0588367i \(0.0187391\pi\)
\(570\) 0 0
\(571\) −1137.33 −1.99182 −0.995912 − 0.0903277i \(-0.971209\pi\)
−0.995912 + 0.0903277i \(0.971209\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1000.39 −1.73378 −0.866891 − 0.498498i \(-0.833885\pi\)
−0.866891 + 0.498498i \(0.833885\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 319.152 0.543700 0.271850 − 0.962340i \(-0.412365\pi\)
0.271850 + 0.962340i \(0.412365\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 862.000 1.45363 0.726813 − 0.686836i \(-0.241001\pi\)
0.726813 + 0.686836i \(0.241001\pi\)
\(594\) 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
\(601\) 219.059 0.364491 0.182246 − 0.983253i \(-0.441663\pi\)
0.182246 + 0.983253i \(0.441663\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 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
\(617\) −1195.79 −1.93806 −0.969032 − 0.246935i \(-0.920577\pi\)
−0.969032 + 0.246935i \(0.920577\pi\)
\(618\) 0 0
\(619\) 674.301 1.08934 0.544670 − 0.838651i \(-0.316655\pi\)
0.544670 + 0.838651i \(0.316655\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 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
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −787.786 −1.22899 −0.614497 − 0.788919i \(-0.710641\pi\)
−0.614497 + 0.788919i \(0.710641\pi\)
\(642\) 0 0
\(643\) 239.577 0.372592 0.186296 − 0.982494i \(-0.440352\pi\)
0.186296 + 0.982494i \(0.440352\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −250.033 −0.385258
\(650\) 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
\(657\) 0 0
\(658\) 0 0
\(659\) −994.000 −1.50835 −0.754173 − 0.656676i \(-0.771962\pi\)
−0.754173 + 0.656676i \(0.771962\pi\)
\(660\) 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
\(673\) −1246.00 −1.85141 −0.925706 − 0.378244i \(-0.876528\pi\)
−0.925706 + 0.378244i \(0.876528\pi\)
\(674\) 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
\(683\) −1330.66 −1.94826 −0.974132 − 0.225980i \(-0.927442\pi\)
−0.974132 + 0.225980i \(0.927442\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −734.000 −1.06223 −0.531114 − 0.847300i \(-0.678227\pi\)
−0.531114 + 0.847300i \(0.678227\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −2485.85 −3.56649
\(698\) 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
\(709\) 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
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2446.24 3.34643
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −553.245 −0.750672
\(738\) 0 0
\(739\) −1410.24 −1.90831 −0.954154 − 0.299316i \(-0.903242\pi\)
−0.954154 + 0.299316i \(0.903242\pi\)
\(740\) 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
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 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
\(761\) −1394.00 −1.83180 −0.915900 − 0.401406i \(-0.868522\pi\)
−0.915900 + 0.401406i \(0.868522\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1054.00 −1.37061 −0.685306 − 0.728256i \(-0.740331\pi\)
−0.685306 + 0.728256i \(0.740331\pi\)
\(770\) 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
\(779\) 2592.42 3.32788
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −926.000 −1.17662 −0.588310 − 0.808635i \(-0.700207\pi\)
−0.588310 + 0.808635i \(0.700207\pi\)
\(788\) 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
\(801\) 0 0
\(802\) 0 0
\(803\) 320.240 0.398804
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −497.061 −0.614414 −0.307207 − 0.951643i \(-0.599394\pi\)
−0.307207 + 0.951643i \(0.599394\pi\)
\(810\) 0 0
\(811\) 1307.21 1.61185 0.805924 − 0.592019i \(-0.201669\pi\)
0.805924 + 0.592019i \(0.201669\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2551.12 −3.12254
\(818\) 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
\(825\) 0 0
\(826\) 0 0
\(827\) 1262.00 1.52600 0.762999 − 0.646400i \(-0.223726\pi\)
0.762999 + 0.646400i \(0.223726\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1489.30 1.78788
\(834\) 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
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 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
\(857\) −1202.00 −1.40257 −0.701284 − 0.712882i \(-0.747389\pi\)
−0.701284 + 0.712882i \(0.747389\pi\)
\(858\) 0 0
\(859\) 1249.21 1.45426 0.727131 − 0.686499i \(-0.240853\pi\)
0.727131 + 0.686499i \(0.240853\pi\)
\(860\) 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
\(867\) 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
\(881\) 1438.00 1.63224 0.816118 − 0.577885i \(-0.196122\pi\)
0.816118 + 0.577885i \(0.196122\pi\)
\(882\) 0 0
\(883\) −983.879 −1.11425 −0.557123 − 0.830430i \(-0.688095\pi\)
−0.557123 + 0.830430i \(0.688095\pi\)
\(884\) 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
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −342.607 −0.377737 −0.188868 − 0.982002i \(-0.560482\pi\)
−0.188868 + 0.982002i \(0.560482\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 1216.12 1.33200
\(914\) 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
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1058.00 −1.13886 −0.569429 − 0.822040i \(-0.692836\pi\)
−0.569429 + 0.822040i \(0.692836\pi\)
\(930\) 0 0
\(931\) −1553.15 −1.66826
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −718.000 −0.766275 −0.383138 − 0.923691i \(-0.625157\pi\)
−0.383138 + 0.923691i \(0.625157\pi\)
\(938\) 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
\(947\) −899.209 −0.949534 −0.474767 − 0.880111i \(-0.657468\pi\)
−0.474767 + 0.880111i \(0.657468\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1717.06 −1.80174 −0.900869 − 0.434090i \(-0.857070\pi\)
−0.900869 + 0.434090i \(0.857070\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 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
\(971\) 974.000 1.00309 0.501545 − 0.865132i \(-0.332765\pi\)
0.501545 + 0.865132i \(0.332765\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1282.33 −1.31252 −0.656260 − 0.754535i \(-0.727863\pi\)
−0.656260 + 0.754535i \(0.727863\pi\)
\(978\) 0 0
\(979\) −1123.75 −1.14786
\(980\) 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
\(993\) 0 0
\(994\) 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 2592.3.b.a.1135.2 2
3.2 odd 2 2592.3.b.b.1135.1 2
4.3 odd 2 648.3.b.b.163.1 2
8.3 odd 2 CM 2592.3.b.a.1135.2 2
8.5 even 2 648.3.b.b.163.1 2
9.2 odd 6 288.3.t.a.175.1 4
9.4 even 3 864.3.t.a.559.1 4
9.5 odd 6 288.3.t.a.79.1 4
9.7 even 3 864.3.t.a.847.1 4
12.11 even 2 648.3.b.a.163.2 2
24.5 odd 2 648.3.b.a.163.2 2
24.11 even 2 2592.3.b.b.1135.1 2
36.7 odd 6 216.3.p.a.91.2 4
36.11 even 6 72.3.p.a.67.2 yes 4
36.23 even 6 72.3.p.a.43.2 ✓ 4
36.31 odd 6 216.3.p.a.19.2 4
72.5 odd 6 72.3.p.a.43.2 ✓ 4
72.11 even 6 288.3.t.a.175.1 4
72.13 even 6 216.3.p.a.19.2 4
72.29 odd 6 72.3.p.a.67.2 yes 4
72.43 odd 6 864.3.t.a.847.1 4
72.59 even 6 288.3.t.a.79.1 4
72.61 even 6 216.3.p.a.91.2 4
72.67 odd 6 864.3.t.a.559.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.3.p.a.43.2 ✓ 4 36.23 even 6
72.3.p.a.43.2 ✓ 4 72.5 odd 6
72.3.p.a.67.2 yes 4 36.11 even 6
72.3.p.a.67.2 yes 4 72.29 odd 6
216.3.p.a.19.2 4 36.31 odd 6
216.3.p.a.19.2 4 72.13 even 6
216.3.p.a.91.2 4 36.7 odd 6
216.3.p.a.91.2 4 72.61 even 6
288.3.t.a.79.1 4 9.5 odd 6
288.3.t.a.79.1 4 72.59 even 6
288.3.t.a.175.1 4 9.2 odd 6
288.3.t.a.175.1 4 72.11 even 6
648.3.b.a.163.2 2 12.11 even 2
648.3.b.a.163.2 2 24.5 odd 2
648.3.b.b.163.1 2 4.3 odd 2
648.3.b.b.163.1 2 8.5 even 2
864.3.t.a.559.1 4 9.4 even 3
864.3.t.a.559.1 4 72.67 odd 6
864.3.t.a.847.1 4 9.7 even 3
864.3.t.a.847.1 4 72.43 odd 6
2592.3.b.a.1135.2 2 1.1 even 1 trivial
2592.3.b.a.1135.2 2 8.3 odd 2 CM
2592.3.b.b.1135.1 2 3.2 odd 2
2592.3.b.b.1135.1 2 24.11 even 2