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Properties

Label 240.9.c.b.209.1
Level $240$
Weight $9$
Character 240.209
Self dual yes
Analytic conductor $97.771$
Analytic rank $0$
Dimension $1$
CM discriminant -15
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] = [240,9,Mod(209,240)] 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("240.209"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 240.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,81,0,-625,0,0,0,6561] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(97.7708664147\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 209.1
Character \(\chi\) \(=\) 240.209

$q$-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)
 
\(f(q)\) \(=\) \(q+81.0000 q^{3} -625.000 q^{5} +6561.00 q^{9} -50625.0 q^{15} +21118.0 q^{17} +203998. q^{19} -550078. q^{23} +390625. q^{25} +531441. q^{27} -1.83168e6 q^{31} -4.10062e6 q^{45} +8.06592e6 q^{47} +5.76480e6 q^{49} +1.71056e6 q^{51} +1.26197e7 q^{53} +1.65238e7 q^{57} +1.43246e7 q^{61} -4.45563e7 q^{69} +3.16406e7 q^{75} +6.96173e7 q^{79} +4.30467e7 q^{81} +3.84720e6 q^{83} -1.31988e7 q^{85} -1.48366e8 q^{93} -1.27499e8 q^{95} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(3\) 81.0000 1.00000
\(4\) 0 0
\(5\) −625.000 −1.00000
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 6561.00 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −50625.0 −1.00000
\(16\) 0 0
\(17\) 21118.0 0.252847 0.126423 − 0.991976i \(-0.459650\pi\)
0.126423 + 0.991976i \(0.459650\pi\)
\(18\) 0 0
\(19\) 203998. 1.56535 0.782675 − 0.622430i \(-0.213855\pi\)
0.782675 + 0.622430i \(0.213855\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −550078. −1.96568 −0.982840 − 0.184459i \(-0.940947\pi\)
−0.982840 + 0.184459i \(0.940947\pi\)
\(24\) 0 0
\(25\) 390625. 1.00000
\(26\) 0 0
\(27\) 531441. 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.83168e6 −1.98337 −0.991684 − 0.128697i \(-0.958921\pi\)
−0.991684 + 0.128697i \(0.958921\pi\)
\(32\) 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
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −4.10062e6 −1.00000
\(46\) 0 0
\(47\) 8.06592e6 1.65296 0.826480 − 0.562965i \(-0.190340\pi\)
0.826480 + 0.562965i \(0.190340\pi\)
\(48\) 0 0
\(49\) 5.76480e6 1.00000
\(50\) 0 0
\(51\) 1.71056e6 0.252847
\(52\) 0 0
\(53\) 1.26197e7 1.59935 0.799677 − 0.600430i \(-0.205004\pi\)
0.799677 + 0.600430i \(0.205004\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.65238e7 1.56535
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 1.43246e7 1.03458 0.517290 − 0.855810i \(-0.326941\pi\)
0.517290 + 0.855810i \(0.326941\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −4.45563e7 −1.96568
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 3.16406e7 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.96173e7 1.78735 0.893673 − 0.448719i \(-0.148119\pi\)
0.893673 + 0.448719i \(0.148119\pi\)
\(80\) 0 0
\(81\) 4.30467e7 1.00000
\(82\) 0 0
\(83\) 3.84720e6 0.0810649 0.0405324 − 0.999178i \(-0.487095\pi\)
0.0405324 + 0.999178i \(0.487095\pi\)
\(84\) 0 0
\(85\) −1.31988e7 −0.252847
\(86\) 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
\(93\) −1.48366e8 −1.98337
\(94\) 0 0
\(95\) −1.27499e8 −1.56535
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 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
\(107\) −1.26157e8 −0.962445 −0.481223 − 0.876598i \(-0.659807\pi\)
−0.481223 + 0.876598i \(0.659807\pi\)
\(108\) 0 0
\(109\) 2.59549e8 1.83871 0.919355 − 0.393429i \(-0.128711\pi\)
0.919355 + 0.393429i \(0.128711\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.05523e7 0.248715 0.124357 − 0.992237i \(-0.460313\pi\)
0.124357 + 0.992237i \(0.460313\pi\)
\(114\) 0 0
\(115\) 3.43799e8 1.96568
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.14359e8 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.44141e8 −1.00000
\(126\) 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
\(135\) −3.32151e8 −1.00000
\(136\) 0 0
\(137\) 5.21273e8 1.47973 0.739866 − 0.672754i \(-0.234889\pi\)
0.739866 + 0.672754i \(0.234889\pi\)
\(138\) 0 0
\(139\) −1.53518e8 −0.411244 −0.205622 − 0.978631i \(-0.565922\pi\)
−0.205622 + 0.978631i \(0.565922\pi\)
\(140\) 0 0
\(141\) 6.53340e8 1.65296
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.66949e8 1.00000
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 9.17845e8 1.76548 0.882738 − 0.469866i \(-0.155698\pi\)
0.882738 + 0.469866i \(0.155698\pi\)
\(152\) 0 0
\(153\) 1.38555e8 0.252847
\(154\) 0 0
\(155\) 1.14480e9 1.98337
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 1.02219e9 1.59935
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.47924e9 −1.90183 −0.950917 − 0.309445i \(-0.899857\pi\)
−0.950917 + 0.309445i \(0.899857\pi\)
\(168\) 0 0
\(169\) 8.15731e8 1.00000
\(170\) 0 0
\(171\) 1.33843e9 1.56535
\(172\) 0 0
\(173\) 4.85246e8 0.541723 0.270862 − 0.962618i \(-0.412691\pi\)
0.270862 + 0.962618i \(0.412691\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 4.17641e8 0.389125 0.194562 − 0.980890i \(-0.437671\pi\)
0.194562 + 0.980890i \(0.437671\pi\)
\(182\) 0 0
\(183\) 1.16030e9 1.03458
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.63782e8 −0.573507 −0.286754 − 0.958004i \(-0.592576\pi\)
−0.286754 + 0.958004i \(0.592576\pi\)
\(198\) 0 0
\(199\) −3.49007e8 −0.222547 −0.111274 − 0.993790i \(-0.535493\pi\)
−0.111274 + 0.993790i \(0.535493\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.60906e9 −1.96568
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 2.20007e9 1.10996 0.554979 − 0.831864i \(-0.312726\pi\)
0.554979 + 0.831864i \(0.312726\pi\)
\(212\) 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
\(225\) 2.56289e9 1.00000
\(226\) 0 0
\(227\) 1.93187e9 0.727571 0.363786 − 0.931483i \(-0.381484\pi\)
0.363786 + 0.931483i \(0.381484\pi\)
\(228\) 0 0
\(229\) −2.21018e9 −0.803684 −0.401842 − 0.915709i \(-0.631630\pi\)
−0.401842 + 0.915709i \(0.631630\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.64490e9 −1.91528 −0.957640 − 0.287969i \(-0.907020\pi\)
−0.957640 + 0.287969i \(0.907020\pi\)
\(234\) 0 0
\(235\) −5.04120e9 −1.65296
\(236\) 0 0
\(237\) 5.63900e9 1.78735
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 5.86943e9 1.73991 0.869956 − 0.493129i \(-0.164147\pi\)
0.869956 + 0.493129i \(0.164147\pi\)
\(242\) 0 0
\(243\) 3.48678e9 1.00000
\(244\) 0 0
\(245\) −3.60300e9 −1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 3.11623e8 0.0810649
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −1.06910e9 −0.252847
\(256\) 0 0
\(257\) 1.49008e9 0.341567 0.170784 − 0.985309i \(-0.445370\pi\)
0.170784 + 0.985309i \(0.445370\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −5.89901e9 −1.23298 −0.616490 − 0.787363i \(-0.711446\pi\)
−0.616490 + 0.787363i \(0.711446\pi\)
\(264\) 0 0
\(265\) −7.88730e9 −1.59935
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 3.48683e9 0.646477 0.323238 − 0.946318i \(-0.395228\pi\)
0.323238 + 0.946318i \(0.395228\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −1.20177e10 −1.98337
\(280\) 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
\(285\) −1.03274e10 −1.56535
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −6.52979e9 −0.936069
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.44691e10 1.96323 0.981615 − 0.190872i \(-0.0611315\pi\)
0.981615 + 0.190872i \(0.0611315\pi\)
\(294\) 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
\(305\) −8.95290e9 −1.03458
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.33010e10 −1.31718 −0.658592 − 0.752500i \(-0.728848\pi\)
−0.658592 + 0.752500i \(0.728848\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1.02187e10 −0.962445
\(322\) 0 0
\(323\) 4.30803e9 0.395793
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.10235e10 1.83871
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.77059e10 −1.47505 −0.737525 − 0.675320i \(-0.764006\pi\)
−0.737525 + 0.675320i \(0.764006\pi\)
\(332\) 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
\(339\) 3.28474e9 0.248715
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2.78477e10 1.96568
\(346\) 0 0
\(347\) −1.84744e10 −1.27424 −0.637122 − 0.770763i \(-0.719875\pi\)
−0.637122 + 0.770763i \(0.719875\pi\)
\(348\) 0 0
\(349\) −2.85260e10 −1.92282 −0.961410 − 0.275121i \(-0.911282\pi\)
−0.961410 + 0.275121i \(0.911282\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.29370e8 0.0469731 0.0234865 − 0.999724i \(-0.492523\pi\)
0.0234865 + 0.999724i \(0.492523\pi\)
\(354\) 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
\(361\) 2.46316e10 1.45032
\(362\) 0 0
\(363\) 1.73631e10 1.00000
\(364\) 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
\(375\) −1.97754e10 −1.00000
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −2.80519e10 −1.35958 −0.679792 − 0.733405i \(-0.737930\pi\)
−0.679792 + 0.733405i \(0.737930\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.16291e10 −0.540445 −0.270222 − 0.962798i \(-0.587097\pi\)
−0.270222 + 0.962798i \(0.587097\pi\)
\(384\) 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
\(391\) −1.16165e10 −0.497016
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.35108e10 −1.78735
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 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
\(405\) −2.69042e10 −1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.28802e10 1.53237 0.766186 − 0.642619i \(-0.222152\pi\)
0.766186 + 0.642619i \(0.222152\pi\)
\(410\) 0 0
\(411\) 4.22231e10 1.47973
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −2.40450e9 −0.0810649
\(416\) 0 0
\(417\) −1.24350e10 −0.411244
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −6.27660e10 −1.99800 −0.999001 − 0.0446850i \(-0.985772\pi\)
−0.999001 + 0.0446850i \(0.985772\pi\)
\(422\) 0 0
\(423\) 5.29205e10 1.65296
\(424\) 0 0
\(425\) 8.24922e9 0.252847
\(426\) 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
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.12215e11 −3.07698
\(438\) 0 0
\(439\) 7.42807e10 1.99994 0.999972 − 0.00748227i \(-0.00238170\pi\)
0.999972 + 0.00748227i \(0.00238170\pi\)
\(440\) 0 0
\(441\) 3.78229e10 1.00000
\(442\) 0 0
\(443\) −7.18229e10 −1.86487 −0.932433 − 0.361342i \(-0.882319\pi\)
−0.932433 + 0.361342i \(0.882319\pi\)
\(444\) 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
\(453\) 7.43455e10 1.76548
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 1.12230e10 0.252847
\(460\) 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
\(465\) 9.27289e10 1.98337
\(466\) 0 0
\(467\) 5.02257e9 0.105599 0.0527994 − 0.998605i \(-0.483186\pi\)
0.0527994 + 0.998605i \(0.483186\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7.96867e10 1.56535
\(476\) 0 0
\(477\) 8.27977e10 1.59935
\(478\) 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
\(491\) 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
\(499\) −2.18003e10 −0.351610 −0.175805 − 0.984425i \(-0.556253\pi\)
−0.175805 + 0.984425i \(0.556253\pi\)
\(500\) 0 0
\(501\) −1.19818e11 −1.90183
\(502\) 0 0
\(503\) 1.04314e11 1.62956 0.814782 − 0.579767i \(-0.196856\pi\)
0.814782 + 0.579767i \(0.196856\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.60742e10 1.00000
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1.08413e11 1.56535
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 3.93049e10 0.541723
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.86815e10 −0.501488
\(528\) 0 0
\(529\) 2.24275e11 2.86390
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 7.88481e10 0.962445
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.61284e11 1.88279 0.941395 − 0.337305i \(-0.109515\pi\)
0.941395 + 0.337305i \(0.109515\pi\)
\(542\) 0 0
\(543\) 3.38289e10 0.389125
\(544\) 0 0
\(545\) −1.62218e11 −1.83871
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 9.39840e10 1.03458
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.33216e11 1.38399 0.691997 − 0.721900i \(-0.256731\pi\)
0.691997 + 0.721900i \(0.256731\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.71432e11 1.70631 0.853157 − 0.521655i \(-0.174685\pi\)
0.853157 + 0.521655i \(0.174685\pi\)
\(564\) 0 0
\(565\) −2.53452e10 −0.248715
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −6.18846e10 −0.582154 −0.291077 − 0.956700i \(-0.594014\pi\)
−0.291077 + 0.956700i \(0.594014\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.14874e11 −1.96568
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 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
\(587\) −2.35841e11 −1.98640 −0.993202 − 0.116405i \(-0.962863\pi\)
−0.993202 + 0.116405i \(0.962863\pi\)
\(588\) 0 0
\(589\) −3.73659e11 −3.10467
\(590\) 0 0
\(591\) −6.99663e10 −0.573507
\(592\) 0 0
\(593\) −1.83361e11 −1.48282 −0.741408 − 0.671055i \(-0.765841\pi\)
−0.741408 + 0.671055i \(0.765841\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.82696e10 −0.222547
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1.79749e11 1.37774 0.688870 − 0.724884i \(-0.258107\pi\)
0.688870 + 0.724884i \(0.258107\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.33974e11 −1.00000
\(606\) 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
\(617\) −1.26458e11 −0.872581 −0.436291 − 0.899806i \(-0.643708\pi\)
−0.436291 + 0.899806i \(0.643708\pi\)
\(618\) 0 0
\(619\) 2.15718e11 1.46934 0.734672 − 0.678422i \(-0.237336\pi\)
0.734672 + 0.678422i \(0.237336\pi\)
\(620\) 0 0
\(621\) −2.92334e11 −1.96568
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.52588e11 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −2.30059e11 −1.45118 −0.725591 − 0.688126i \(-0.758434\pi\)
−0.725591 + 0.688126i \(0.758434\pi\)
\(632\) 0 0
\(633\) 1.78206e11 1.10996
\(634\) 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
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.35694e11 −1.34503 −0.672513 − 0.740085i \(-0.734785\pi\)
−0.672513 + 0.740085i \(0.734785\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.12967e11 1.17128 0.585639 − 0.810572i \(-0.300844\pi\)
0.585639 + 0.810572i \(0.300844\pi\)
\(654\) 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
\(661\) −3.52354e11 −1.84575 −0.922876 − 0.385097i \(-0.874168\pi\)
−0.922876 + 0.385097i \(0.874168\pi\)
\(662\) 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
\(675\) 2.07594e11 1.00000
\(676\) 0 0
\(677\) −1.83259e11 −0.872391 −0.436196 − 0.899852i \(-0.643674\pi\)
−0.436196 + 0.899852i \(0.643674\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 1.56482e11 0.727571
\(682\) 0 0
\(683\) 4.21478e11 1.93683 0.968416 − 0.249338i \(-0.0802131\pi\)
0.968416 + 0.249338i \(0.0802131\pi\)
\(684\) 0 0
\(685\) −3.25796e11 −1.47973
\(686\) 0 0
\(687\) −1.79024e11 −0.803684
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.72432e11 −0.756320 −0.378160 − 0.925740i \(-0.623443\pi\)
−0.378160 + 0.925740i \(0.623443\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.59487e10 0.411244
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −4.57236e11 −1.91528
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −4.08337e11 −1.65296
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.98535e11 −1.18144 −0.590718 − 0.806878i \(-0.701156\pi\)
−0.590718 + 0.806878i \(0.701156\pi\)
\(710\) 0 0
\(711\) 4.56759e11 1.78735
\(712\) 0 0
\(713\) 1.00757e12 3.89867
\(714\) 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
\(723\) 4.75424e11 1.73991
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2.82430e11 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −2.91843e11 −1.00000
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −4.95951e11 −1.66288 −0.831440 − 0.555614i \(-0.812483\pi\)
−0.831440 + 0.555614i \(0.812483\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.59192e10 0.314739 0.157369 − 0.987540i \(-0.449699\pi\)
0.157369 + 0.987540i \(0.449699\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2.52415e10 0.0810649
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.47109e11 −0.776834 −0.388417 − 0.921484i \(-0.626978\pi\)
−0.388417 + 0.921484i \(0.626978\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5.73653e11 −1.76548
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 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
\(765\) −8.65970e10 −0.252847
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.07708e10 0.0593947 0.0296973 − 0.999559i \(-0.490546\pi\)
0.0296973 + 0.999559i \(0.490546\pi\)
\(770\) 0 0
\(771\) 1.20696e11 0.341567
\(772\) 0 0
\(773\) −5.71008e11 −1.59928 −0.799639 − 0.600480i \(-0.794976\pi\)
−0.799639 + 0.600480i \(0.794976\pi\)
\(774\) 0 0
\(775\) −7.15501e11 −1.98337
\(776\) 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
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) −4.77820e11 −1.23298
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −6.38871e11 −1.59935
\(796\) 0 0
\(797\) 4.61070e11 1.14270 0.571352 − 0.820705i \(-0.306419\pi\)
0.571352 + 0.820705i \(0.306419\pi\)
\(798\) 0 0
\(799\) 1.70336e11 0.417946
\(800\) 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
\(811\) 8.44251e11 1.95159 0.975794 − 0.218694i \(-0.0701796\pi\)
0.975794 + 0.218694i \(0.0701796\pi\)
\(812\) 0 0
\(813\) 2.82433e11 0.646477
\(814\) 0 0
\(815\) 0 0
\(816\) 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
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5.72422e11 1.22375 0.611877 − 0.790953i \(-0.290415\pi\)
0.611877 + 0.790953i \(0.290415\pi\)
\(828\) 0 0
\(829\) 3.15356e11 0.667703 0.333852 − 0.942626i \(-0.391652\pi\)
0.333852 + 0.942626i \(0.391652\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.21741e11 0.252847
\(834\) 0 0
\(835\) 9.24525e11 1.90183
\(836\) 0 0
\(837\) −9.73431e11 −1.98337
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.00246e11 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5.09832e11 −1.00000
\(846\) 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
\(855\) −8.36519e11 −1.56535
\(856\) 0 0
\(857\) −6.28524e11 −1.16520 −0.582598 − 0.812761i \(-0.697964\pi\)
−0.582598 + 0.812761i \(0.697964\pi\)
\(858\) 0 0
\(859\) −9.50927e11 −1.74652 −0.873262 − 0.487251i \(-0.838000\pi\)
−0.873262 + 0.487251i \(0.838000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 8.91341e11 1.60694 0.803472 − 0.595343i \(-0.202984\pi\)
0.803472 + 0.595343i \(0.202984\pi\)
\(864\) 0 0
\(865\) −3.03279e11 −0.541723
\(866\) 0 0
\(867\) −5.28913e11 −0.936069
\(868\) 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
\(879\) 1.17200e12 1.96323
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 4.41859e11 0.713821 0.356911 − 0.934139i \(-0.383830\pi\)
0.356911 + 0.934139i \(0.383830\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.64543e12 2.58746
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 2.66502e11 0.404391
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.61026e11 −0.389125
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −7.25185e11 −1.03458
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.42655e12 1.99997 0.999987 − 0.00511270i \(-0.00162743\pi\)
0.999987 + 0.00511270i \(0.00162743\pi\)
\(920\) 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
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1.17601e12 1.56535
\(932\) 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
\(947\) −1.14087e12 −1.41852 −0.709262 − 0.704945i \(-0.750972\pi\)
−0.709262 + 0.704945i \(0.750972\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −1.07738e12 −1.31718
\(952\) 0 0
\(953\) 1.52277e12 1.84613 0.923064 − 0.384646i \(-0.125676\pi\)
0.923064 + 0.384646i \(0.125676\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.50217e12 2.93375
\(962\) 0 0
\(963\) −8.27716e11 −0.962445
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 3.48950e11 0.395793
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.53140e12 −1.68078 −0.840390 − 0.541983i \(-0.817674\pi\)
−0.840390 + 0.541983i \(0.817674\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.70290e12 1.83871
\(982\) 0 0
\(983\) 1.68782e12 1.80765 0.903823 − 0.427907i \(-0.140749\pi\)
0.903823 + 0.427907i \(0.140749\pi\)
\(984\) 0 0
\(985\) 5.39864e11 0.573507
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 8.34017e11 0.864730 0.432365 − 0.901699i \(-0.357679\pi\)
0.432365 + 0.901699i \(0.357679\pi\)
\(992\) 0 0
\(993\) −1.43418e12 −1.47505
\(994\) 0 0
\(995\) 2.18129e11 0.222547
\(996\) 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 240.9.c.b.209.1 1
3.2 odd 2 240.9.c.a.209.1 1
4.3 odd 2 15.9.d.a.14.1 ✓ 1
5.4 even 2 240.9.c.a.209.1 1
12.11 even 2 15.9.d.b.14.1 yes 1
15.14 odd 2 CM 240.9.c.b.209.1 1
20.3 even 4 75.9.c.d.26.2 2
20.7 even 4 75.9.c.d.26.1 2
20.19 odd 2 15.9.d.b.14.1 yes 1
60.23 odd 4 75.9.c.d.26.1 2
60.47 odd 4 75.9.c.d.26.2 2
60.59 even 2 15.9.d.a.14.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.9.d.a.14.1 ✓ 1 4.3 odd 2
15.9.d.a.14.1 ✓ 1 60.59 even 2
15.9.d.b.14.1 yes 1 12.11 even 2
15.9.d.b.14.1 yes 1 20.19 odd 2
75.9.c.d.26.1 2 20.7 even 4
75.9.c.d.26.1 2 60.23 odd 4
75.9.c.d.26.2 2 20.3 even 4
75.9.c.d.26.2 2 60.47 odd 4
240.9.c.a.209.1 1 3.2 odd 2
240.9.c.a.209.1 1 5.4 even 2
240.9.c.b.209.1 1 1.1 even 1 trivial
240.9.c.b.209.1 1 15.14 odd 2 CM