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Properties

Label 99.5.c.a.10.1
Level $99$
Weight $5$
Character 99.10
Self dual yes
Analytic conductor $10.234$
Analytic rank $0$
Dimension $1$
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] = [99,5,Mod(10,99)] 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("99.10"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(99, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 99.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.2336263453\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 10.1
Character \(\chi\) \(=\) 99.10

$q$-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)
 
\(f(q)\) \(=\) \(q+16.0000 q^{4} +49.0000 q^{5} -121.000 q^{11} +256.000 q^{16} +784.000 q^{20} -167.000 q^{23} +1776.00 q^{25} -553.000 q^{31} -2113.00 q^{37} -1936.00 q^{44} +1918.00 q^{47} +2401.00 q^{49} +718.000 q^{53} -5929.00 q^{55} -4487.00 q^{59} +4096.00 q^{64} -7753.00 q^{67} -7607.00 q^{71} +12544.0 q^{80} +6433.00 q^{89} -2672.00 q^{92} -9793.00 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0
\(4\) 16.0000 1.00000
\(5\) 49.0000 1.96000 0.980000 − 0.198997i \(-0.0637686\pi\)
0.980000 + 0.198997i \(0.0637686\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −121.000 −1.00000
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 256.000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 784.000 1.96000
\(21\) 0 0
\(22\) 0 0
\(23\) −167.000 −0.315690 −0.157845 − 0.987464i \(-0.550455\pi\)
−0.157845 + 0.987464i \(0.550455\pi\)
\(24\) 0 0
\(25\) 1776.00 2.84160
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −553.000 −0.575442 −0.287721 − 0.957714i \(-0.592898\pi\)
−0.287721 + 0.957714i \(0.592898\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2113.00 −1.54346 −0.771731 − 0.635949i \(-0.780609\pi\)
−0.771731 + 0.635949i \(0.780609\pi\)
\(38\) 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 \(\pi\)
\(44\) −1936.00 −1.00000
\(45\) 0 0
\(46\) 0 0
\(47\) 1918.00 0.868266 0.434133 − 0.900849i \(-0.357055\pi\)
0.434133 + 0.900849i \(0.357055\pi\)
\(48\) 0 0
\(49\) 2401.00 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 718.000 0.255607 0.127803 − 0.991800i \(-0.459207\pi\)
0.127803 + 0.991800i \(0.459207\pi\)
\(54\) 0 0
\(55\) −5929.00 −1.96000
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4487.00 −1.28900 −0.644499 − 0.764605i \(-0.722934\pi\)
−0.644499 + 0.764605i \(0.722934\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4096.00 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −7753.00 −1.72711 −0.863555 − 0.504254i \(-0.831768\pi\)
−0.863555 + 0.504254i \(0.831768\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7607.00 −1.50903 −0.754513 − 0.656285i \(-0.772127\pi\)
−0.754513 + 0.656285i \(0.772127\pi\)
\(72\) 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 \(\pi\)
\(80\) 12544.0 1.96000
\(81\) 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
\(89\) 6433.00 0.812145 0.406072 − 0.913841i \(-0.366898\pi\)
0.406072 + 0.913841i \(0.366898\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2672.00 −0.315690
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9793.00 −1.04081 −0.520406 − 0.853919i \(-0.674219\pi\)
−0.520406 + 0.853919i \(0.674219\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 28416.0 2.84160
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 14882.0 1.40277 0.701386 − 0.712782i \(-0.252565\pi\)
0.701386 + 0.712782i \(0.252565\pi\)
\(104\) 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
\(113\) −20687.0 −1.62010 −0.810048 − 0.586364i \(-0.800559\pi\)
−0.810048 + 0.586364i \(0.800559\pi\)
\(114\) 0 0
\(115\) −8183.00 −0.618752
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 14641.0 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −8848.00 −0.575442
\(125\) 56399.0 3.60954
\(126\) 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
\(137\) −32687.0 −1.74154 −0.870771 − 0.491689i \(-0.836380\pi\)
−0.870771 + 0.491689i \(0.836380\pi\)
\(138\) 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
\(148\) −33808.0 −1.54346
\(149\) 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
\(155\) −27097.0 −1.12787
\(156\) 0 0
\(157\) −3073.00 −0.124670 −0.0623352 − 0.998055i \(-0.519855\pi\)
−0.0623352 + 0.998055i \(0.519855\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −48238.0 −1.81557 −0.907787 − 0.419431i \(-0.862230\pi\)
−0.907787 + 0.419431i \(0.862230\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 28561.0 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
\(176\) −30976.0 −1.00000
\(177\) 0 0
\(178\) 0 0
\(179\) 57193.0 1.78499 0.892497 − 0.451053i \(-0.148952\pi\)
0.892497 + 0.451053i \(0.148952\pi\)
\(180\) 0 0
\(181\) 3647.00 0.111321 0.0556607 − 0.998450i \(-0.482273\pi\)
0.0556607 + 0.998450i \(0.482273\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −103537. −3.02519
\(186\) 0 0
\(187\) 0 0
\(188\) 30688.0 0.868266
\(189\) 0 0
\(190\) 0 0
\(191\) 48313.0 1.32433 0.662167 − 0.749357i \(-0.269637\pi\)
0.662167 + 0.749357i \(0.269637\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 38416.0 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −79198.0 −1.99990 −0.999949 − 0.0100501i \(-0.996801\pi\)
−0.999949 + 0.0100501i \(0.996801\pi\)
\(200\) 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 \(\pi\)
\(212\) 11488.0 0.255607
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −94864.0 −1.96000
\(221\) 0 0
\(222\) 0 0
\(223\) 98567.0 1.98208 0.991041 − 0.133555i \(-0.0426392\pi\)
0.991041 + 0.133555i \(0.0426392\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 82607.0 1.57524 0.787618 − 0.616163i \(-0.211314\pi\)
0.787618 + 0.616163i \(0.211314\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 93982.0 1.70180
\(236\) −71792.0 −1.28900
\(237\) 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
\(245\) 117649. 1.96000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 74473.0 1.18209 0.591046 − 0.806638i \(-0.298715\pi\)
0.591046 + 0.806638i \(0.298715\pi\)
\(252\) 0 0
\(253\) 20207.0 0.315690
\(254\) 0 0
\(255\) 0 0
\(256\) 65536.0 1.00000
\(257\) 95998.0 1.45344 0.726718 − 0.686936i \(-0.241045\pi\)
0.726718 + 0.686936i \(0.241045\pi\)
\(258\) 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
\(265\) 35182.0 0.500990
\(266\) 0 0
\(267\) 0 0
\(268\) −124048. −1.72711
\(269\) 13678.0 0.189024 0.0945122 − 0.995524i \(-0.469871\pi\)
0.0945122 + 0.995524i \(0.469871\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −214896. −2.84160
\(276\) 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 \(\pi\)
\(284\) −121712. −1.50903
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 83521.0 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) −219863. −2.52643
\(296\) 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
\(311\) −35042.0 −0.362300 −0.181150 − 0.983455i \(-0.557982\pi\)
−0.181150 + 0.983455i \(0.557982\pi\)
\(312\) 0 0
\(313\) −125713. −1.28319 −0.641596 − 0.767043i \(-0.721727\pi\)
−0.641596 + 0.767043i \(0.721727\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 191953. 1.91019 0.955095 − 0.296301i \(-0.0957532\pi\)
0.955095 + 0.296301i \(0.0957532\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 200704. 1.96000
\(321\) 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
\(331\) 97847.0 0.893082 0.446541 − 0.894763i \(-0.352656\pi\)
0.446541 + 0.894763i \(0.352656\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −379897. −3.38514
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 66913.0 0.575442
\(342\) 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
\(353\) −141407. −1.13481 −0.567403 − 0.823440i \(-0.692052\pi\)
−0.567403 + 0.823440i \(0.692052\pi\)
\(354\) 0 0
\(355\) −372743. −2.95769
\(356\) 102928. 0.812145
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 130321. 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 133847. 0.993749 0.496874 − 0.867823i \(-0.334481\pi\)
0.496874 + 0.867823i \(0.334481\pi\)
\(368\) −42752.0 −0.315690
\(369\) 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
\(379\) −269593. −1.87685 −0.938426 − 0.345479i \(-0.887716\pi\)
−0.938426 + 0.345479i \(0.887716\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −276647. −1.88594 −0.942971 − 0.332874i \(-0.891981\pi\)
−0.942971 + 0.332874i \(0.891981\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −156688. −1.04081
\(389\) −3167.00 −0.0209290 −0.0104645 − 0.999945i \(-0.503331\pi\)
−0.0104645 + 0.999945i \(0.503331\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 308882. 1.95980 0.979900 − 0.199491i \(-0.0639289\pi\)
0.979900 + 0.199491i \(0.0639289\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 454656. 2.84160
\(401\) 311998. 1.94027 0.970137 − 0.242558i \(-0.0779863\pi\)
0.970137 + 0.242558i \(0.0779863\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 255673. 1.54346
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 238112. 1.40277
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 282478. 1.60900 0.804501 − 0.593951i \(-0.202433\pi\)
0.804501 + 0.593951i \(0.202433\pi\)
\(420\) 0 0
\(421\) 196082. 1.10630 0.553151 − 0.833081i \(-0.313425\pi\)
0.553151 + 0.833081i \(0.313425\pi\)
\(422\) 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
\(433\) −374353. −1.99667 −0.998333 − 0.0577127i \(-0.981619\pi\)
−0.998333 + 0.0577127i \(0.981619\pi\)
\(434\) 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
\(443\) 194473. 0.990950 0.495475 − 0.868622i \(-0.334994\pi\)
0.495475 + 0.868622i \(0.334994\pi\)
\(444\) 0 0
\(445\) 315217. 1.59180
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15073.0 0.0747665 0.0373832 − 0.999301i \(-0.488098\pi\)
0.0373832 + 0.999301i \(0.488098\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −330992. −1.62010
\(453\) 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
\(460\) −130928. −0.618752
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −427513. −1.99429 −0.997143 − 0.0755399i \(-0.975932\pi\)
−0.997143 + 0.0755399i \(0.975932\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −419447. −1.92328 −0.961642 − 0.274308i \(-0.911551\pi\)
−0.961642 + 0.274308i \(0.911551\pi\)
\(468\) 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
\(484\) 234256. 1.00000
\(485\) −479857. −2.03999
\(486\) 0 0
\(487\) 291287. 1.22818 0.614092 − 0.789235i \(-0.289523\pi\)
0.614092 + 0.789235i \(0.289523\pi\)
\(488\) 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
\(496\) −141568. −0.575442
\(497\) 0 0
\(498\) 0 0
\(499\) −135598. −0.544568 −0.272284 − 0.962217i \(-0.587779\pi\)
−0.272284 + 0.962217i \(0.587779\pi\)
\(500\) 902384. 3.60954
\(501\) 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
\(509\) −495887. −1.91402 −0.957012 − 0.290050i \(-0.906328\pi\)
−0.957012 + 0.290050i \(0.906328\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 729218. 2.74943
\(516\) 0 0
\(517\) −232078. −0.868266
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −124607. −0.459057 −0.229529 − 0.973302i \(-0.573718\pi\)
−0.229529 + 0.973302i \(0.573718\pi\)
\(522\) 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
\(529\) −251952. −0.900340
\(530\) 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
\(539\) −290521. −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 \(\pi\)
\(548\) −522992. −1.74154
\(549\) 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
\(565\) −1.01366e6 −3.17539
\(566\) 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
\(575\) −296592. −0.897065
\(576\) 0 0
\(577\) 447167. 1.34313 0.671565 − 0.740946i \(-0.265622\pi\)
0.671565 + 0.740946i \(0.265622\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −86878.0 −0.255607
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −587762. −1.70579 −0.852894 − 0.522083i \(-0.825155\pi\)
−0.852894 + 0.522083i \(0.825155\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −540928. −1.54346
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 707998. 1.97323 0.986617 − 0.163058i \(-0.0521357\pi\)
0.986617 + 0.163058i \(0.0521357\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 717409. 1.96000
\(606\) 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
\(617\) 480478. 1.26213 0.631064 − 0.775731i \(-0.282619\pi\)
0.631064 + 0.775731i \(0.282619\pi\)
\(618\) 0 0
\(619\) 763847. 1.99354 0.996770 − 0.0803056i \(-0.0255896\pi\)
0.996770 + 0.0803056i \(0.0255896\pi\)
\(620\) −433552. −1.12787
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.65355e6 4.23309
\(626\) 0 0
\(627\) 0 0
\(628\) −49168.0 −0.124670
\(629\) 0 0
\(630\) 0 0
\(631\) 675047. 1.69541 0.847706 − 0.530466i \(-0.177983\pi\)
0.847706 + 0.530466i \(0.177983\pi\)
\(632\) 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
\(641\) 269713. 0.656426 0.328213 − 0.944604i \(-0.393554\pi\)
0.328213 + 0.944604i \(0.393554\pi\)
\(642\) 0 0
\(643\) −670873. −1.62263 −0.811313 − 0.584612i \(-0.801247\pi\)
−0.811313 + 0.584612i \(0.801247\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 517993. 1.23741 0.618707 − 0.785621i \(-0.287657\pi\)
0.618707 + 0.785621i \(0.287657\pi\)
\(648\) 0 0
\(649\) 542927. 1.28900
\(650\) 0 0
\(651\) 0 0
\(652\) −771808. −1.81557
\(653\) −824207. −1.93290 −0.966451 − 0.256850i \(-0.917315\pi\)
−0.966451 + 0.256850i \(0.917315\pi\)
\(654\) 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
\(661\) 455567. 1.04268 0.521338 − 0.853350i \(-0.325433\pi\)
0.521338 + 0.853350i \(0.325433\pi\)
\(662\) 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
\(676\) 456976. 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 308878. 0.662134 0.331067 − 0.943607i \(-0.392591\pi\)
0.331067 + 0.943607i \(0.392591\pi\)
\(684\) 0 0
\(685\) −1.60166e6 −3.41342
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 239687. 0.501982 0.250991 − 0.967989i \(-0.419243\pi\)
0.250991 + 0.967989i \(0.419243\pi\)
\(692\) 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
\(704\) −495616. −1.00000
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 111887. 0.222581 0.111290 − 0.993788i \(-0.464502\pi\)
0.111290 + 0.993788i \(0.464502\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 92351.0 0.181661
\(714\) 0 0
\(715\) 0 0
\(716\) 915088. 1.78499
\(717\) 0 0
\(718\) 0 0
\(719\) −318647. −0.616385 −0.308192 − 0.951324i \(-0.599724\pi\)
−0.308192 + 0.951324i \(0.599724\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 58352.0 0.111321
\(725\) 0 0
\(726\) 0 0
\(727\) 778967. 1.47384 0.736920 − 0.675980i \(-0.236280\pi\)
0.736920 + 0.675980i \(0.236280\pi\)
\(728\) 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
\(737\) 938113. 1.72711
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) −1.65659e6 −3.02519
\(741\) 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
\(751\) −181273. −0.321405 −0.160703 − 0.987003i \(-0.551376\pi\)
−0.160703 + 0.987003i \(0.551376\pi\)
\(752\) 491008. 0.868266
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.14120e6 −1.99145 −0.995725 − 0.0923714i \(-0.970555\pi\)
−0.995725 + 0.0923714i \(0.970555\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 773008. 1.32433
\(765\) 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
\(773\) −681842. −1.14110 −0.570551 − 0.821262i \(-0.693270\pi\)
−0.570551 + 0.821262i \(0.693270\pi\)
\(774\) 0 0
\(775\) −982128. −1.63518
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 920447. 1.50903
\(782\) 0 0
\(783\) 0 0
\(784\) 614656. 1.00000
\(785\) −150577. −0.244354
\(786\) 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
\(796\) −1.26717e6 −1.99990
\(797\) −1.24181e6 −1.95496 −0.977479 − 0.211032i \(-0.932317\pi\)
−0.977479 + 0.211032i \(0.932317\pi\)
\(798\) 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
\(815\) −2.36366e6 −3.55853
\(816\) 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
\(823\) −784633. −1.15842 −0.579211 − 0.815178i \(-0.696639\pi\)
−0.579211 + 0.815178i \(0.696639\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −706993. −1.02874 −0.514371 − 0.857568i \(-0.671974\pi\)
−0.514371 + 0.857568i \(0.671974\pi\)
\(830\) 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
\(839\) 1.28743e6 1.82895 0.914473 − 0.404648i \(-0.132606\pi\)
0.914473 + 0.404648i \(0.132606\pi\)
\(840\) 0 0
\(841\) 707281. 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.39949e6 1.96000
\(846\) 0 0
\(847\) 0 0
\(848\) 183808. 0.255607
\(849\) 0 0
\(850\) 0 0
\(851\) 352871. 0.487256
\(852\) 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
\(859\) −902713. −1.22339 −0.611693 − 0.791095i \(-0.709511\pi\)
−0.611693 + 0.791095i \(0.709511\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.30464e6 1.75174 0.875868 − 0.482552i \(-0.160290\pi\)
0.875868 + 0.482552i \(0.160290\pi\)
\(864\) 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
\(880\) −1.51782e6 −1.96000
\(881\) −658847. −0.848854 −0.424427 − 0.905462i \(-0.639524\pi\)
−0.424427 + 0.905462i \(0.639524\pi\)
\(882\) 0 0
\(883\) 317522. 0.407242 0.203621 − 0.979050i \(-0.434729\pi\)
0.203621 + 0.979050i \(0.434729\pi\)
\(884\) 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
\(892\) 1.57707e6 1.98208
\(893\) 0 0
\(894\) 0 0
\(895\) 2.80246e6 3.49859
\(896\) 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
\(905\) 178703. 0.218190
\(906\) 0 0
\(907\) 1.41720e6 1.72273 0.861365 − 0.507987i \(-0.169610\pi\)
0.861365 + 0.507987i \(0.169610\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.50144e6 −1.80914 −0.904569 − 0.426327i \(-0.859807\pi\)
−0.904569 + 0.426327i \(0.859807\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.32171e6 1.57524
\(917\) 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
\(925\) −3.75269e6 −4.38590
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 808318. 0.936593 0.468296 − 0.883571i \(-0.344868\pi\)
0.468296 + 0.883571i \(0.344868\pi\)
\(930\) 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
\(940\) 1.50371e6 1.70180
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.14867e6 −1.28900
\(945\) 0 0
\(946\) 0 0
\(947\) −42407.0 −0.0472865 −0.0236433 − 0.999720i \(-0.507527\pi\)
−0.0236433 + 0.999720i \(0.507527\pi\)
\(948\) 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
\(955\) 2.36734e6 2.59569
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −617712. −0.668866
\(962\) 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
\(971\) 1.87879e6 1.99269 0.996347 − 0.0854008i \(-0.0272171\pi\)
0.996347 + 0.0854008i \(0.0272171\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 408433. 0.427890 0.213945 − 0.976846i \(-0.431369\pi\)
0.213945 + 0.976846i \(0.431369\pi\)
\(978\) 0 0
\(979\) −778393. −0.812145
\(980\) 1.88238e6 1.96000
\(981\) 0 0
\(982\) 0 0
\(983\) 1.52935e6 1.58271 0.791354 − 0.611358i \(-0.209377\pi\)
0.791354 + 0.611358i \(0.209377\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 538562. 0.548389 0.274194 − 0.961674i \(-0.411589\pi\)
0.274194 + 0.961674i \(0.411589\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3.88070e6 −3.91980
\(996\) 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 99.5.c.a.10.1 1
3.2 odd 2 11.5.b.a.10.1 ✓ 1
11.10 odd 2 CM 99.5.c.a.10.1 1
12.11 even 2 176.5.h.a.65.1 1
15.2 even 4 275.5.d.a.274.1 2
15.8 even 4 275.5.d.a.274.2 2
15.14 odd 2 275.5.c.a.76.1 1
24.5 odd 2 704.5.h.a.65.1 1
24.11 even 2 704.5.h.b.65.1 1
33.2 even 10 121.5.d.a.40.1 4
33.5 odd 10 121.5.d.a.118.1 4
33.8 even 10 121.5.d.a.112.1 4
33.14 odd 10 121.5.d.a.112.1 4
33.17 even 10 121.5.d.a.118.1 4
33.20 odd 10 121.5.d.a.40.1 4
33.26 odd 10 121.5.d.a.94.1 4
33.29 even 10 121.5.d.a.94.1 4
33.32 even 2 11.5.b.a.10.1 ✓ 1
132.131 odd 2 176.5.h.a.65.1 1
165.32 odd 4 275.5.d.a.274.1 2
165.98 odd 4 275.5.d.a.274.2 2
165.164 even 2 275.5.c.a.76.1 1
264.131 odd 2 704.5.h.b.65.1 1
264.197 even 2 704.5.h.a.65.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.5.b.a.10.1 ✓ 1 3.2 odd 2
11.5.b.a.10.1 ✓ 1 33.32 even 2
99.5.c.a.10.1 1 1.1 even 1 trivial
99.5.c.a.10.1 1 11.10 odd 2 CM
121.5.d.a.40.1 4 33.2 even 10
121.5.d.a.40.1 4 33.20 odd 10
121.5.d.a.94.1 4 33.26 odd 10
121.5.d.a.94.1 4 33.29 even 10
121.5.d.a.112.1 4 33.8 even 10
121.5.d.a.112.1 4 33.14 odd 10
121.5.d.a.118.1 4 33.5 odd 10
121.5.d.a.118.1 4 33.17 even 10
176.5.h.a.65.1 1 12.11 even 2
176.5.h.a.65.1 1 132.131 odd 2
275.5.c.a.76.1 1 15.14 odd 2
275.5.c.a.76.1 1 165.164 even 2
275.5.d.a.274.1 2 15.2 even 4
275.5.d.a.274.1 2 165.32 odd 4
275.5.d.a.274.2 2 15.8 even 4
275.5.d.a.274.2 2 165.98 odd 4
704.5.h.a.65.1 1 24.5 odd 2
704.5.h.a.65.1 1 264.197 even 2
704.5.h.b.65.1 1 24.11 even 2
704.5.h.b.65.1 1 264.131 odd 2