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Properties

Label 240.5.c.a.209.1
Level $240$
Weight $5$
Character 240.209
Self dual yes
Analytic conductor $24.809$
Analytic rank $0$
Dimension $1$
CM discriminant -15
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,5,Mod(209,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.209");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 240.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(24.8087911401\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} +25.0000 q^{5} +81.0000 q^{9} +O(q^{10})\) \(q-9.00000 q^{3} +25.0000 q^{5} +81.0000 q^{9} -225.000 q^{15} -382.000 q^{17} +238.000 q^{19} -98.0000 q^{23} +625.000 q^{25} -729.000 q^{27} +1918.00 q^{31} +2025.00 q^{45} +4222.00 q^{47} +2401.00 q^{49} +3438.00 q^{51} +1778.00 q^{53} -2142.00 q^{57} +6482.00 q^{61} +882.000 q^{69} -5625.00 q^{75} +2878.00 q^{79} +6561.00 q^{81} -9938.00 q^{83} -9550.00 q^{85} -17262.0 q^{93} +5950.00 q^{95} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(3\) −9.00000 −1.00000
\(4\) 0 0
\(5\) 25.0000 1.00000
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 81.0000 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
\(15\) −225.000 −1.00000
\(16\) 0 0
\(17\) −382.000 −1.32180 −0.660900 − 0.750474i \(-0.729825\pi\)
−0.660900 + 0.750474i \(0.729825\pi\)
\(18\) 0 0
\(19\) 238.000 0.659280 0.329640 − 0.944107i \(-0.393073\pi\)
0.329640 + 0.944107i \(0.393073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −98.0000 −0.185255 −0.0926276 − 0.995701i \(-0.529527\pi\)
−0.0926276 + 0.995701i \(0.529527\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) 0 0
\(27\) −729.000 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1918.00 1.99584 0.997919 − 0.0644826i \(-0.0205397\pi\)
0.997919 + 0.0644826i \(0.0205397\pi\)
\(32\) 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
\(45\) 2025.00 1.00000
\(46\) 0 0
\(47\) 4222.00 1.91127 0.955636 − 0.294550i \(-0.0951698\pi\)
0.955636 + 0.294550i \(0.0951698\pi\)
\(48\) 0 0
\(49\) 2401.00 1.00000
\(50\) 0 0
\(51\) 3438.00 1.32180
\(52\) 0 0
\(53\) 1778.00 0.632965 0.316483 − 0.948598i \(-0.397498\pi\)
0.316483 + 0.948598i \(0.397498\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2142.00 −0.659280
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 6482.00 1.74200 0.871002 − 0.491279i \(-0.163470\pi\)
0.871002 + 0.491279i \(0.163470\pi\)
\(62\) 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
\(69\) 882.000 0.185255
\(70\) 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
\(75\) −5625.00 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2878.00 0.461144 0.230572 − 0.973055i \(-0.425940\pi\)
0.230572 + 0.973055i \(0.425940\pi\)
\(80\) 0 0
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) −9938.00 −1.44259 −0.721295 − 0.692628i \(-0.756453\pi\)
−0.721295 + 0.692628i \(0.756453\pi\)
\(84\) 0 0
\(85\) −9550.00 −1.32180
\(86\) 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
\(93\) −17262.0 −1.99584
\(94\) 0 0
\(95\) 5950.00 0.659280
\(96\) 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
\(107\) 11662.0 1.01860 0.509302 − 0.860588i \(-0.329904\pi\)
0.509302 + 0.860588i \(0.329904\pi\)
\(108\) 0 0
\(109\) −23278.0 −1.95926 −0.979631 − 0.200804i \(-0.935644\pi\)
−0.979631 + 0.200804i \(0.935644\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16898.0 1.32336 0.661681 − 0.749786i \(-0.269843\pi\)
0.661681 + 0.749786i \(0.269843\pi\)
\(114\) 0 0
\(115\) −2450.00 −0.185255
\(116\) 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
\(124\) 0 0
\(125\) 15625.0 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
\(135\) −18225.0 −1.00000
\(136\) 0 0
\(137\) 13538.0 0.721296 0.360648 − 0.932702i \(-0.382556\pi\)
0.360648 + 0.932702i \(0.382556\pi\)
\(138\) 0 0
\(139\) −30002.0 −1.55282 −0.776409 − 0.630229i \(-0.782961\pi\)
−0.776409 + 0.630229i \(0.782961\pi\)
\(140\) 0 0
\(141\) −37998.0 −1.91127
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −21609.0 −1.00000
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −11042.0 −0.484277 −0.242139 − 0.970242i \(-0.577849\pi\)
−0.242139 + 0.970242i \(0.577849\pi\)
\(152\) 0 0
\(153\) −30942.0 −1.32180
\(154\) 0 0
\(155\) 47950.0 1.99584
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −16002.0 −0.632965
\(160\) 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
\(167\) −8738.00 −0.313313 −0.156657 − 0.987653i \(-0.550072\pi\)
−0.156657 + 0.987653i \(0.550072\pi\)
\(168\) 0 0
\(169\) 28561.0 1.00000
\(170\) 0 0
\(171\) 19278.0 0.659280
\(172\) 0 0
\(173\) −36142.0 −1.20759 −0.603796 − 0.797139i \(-0.706346\pi\)
−0.603796 + 0.797139i \(0.706346\pi\)
\(174\) 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
\(181\) −50638.0 −1.54568 −0.772840 − 0.634601i \(-0.781164\pi\)
−0.772840 + 0.634601i \(0.781164\pi\)
\(182\) 0 0
\(183\) −58338.0 −1.74200
\(184\) 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
\(197\) 62258.0 1.60422 0.802108 − 0.597179i \(-0.203712\pi\)
0.802108 + 0.597179i \(0.203712\pi\)
\(198\) 0 0
\(199\) 59038.0 1.49082 0.745410 − 0.666606i \(-0.232254\pi\)
0.745410 + 0.666606i \(0.232254\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −7938.00 −0.185255
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −42002.0 −0.943420 −0.471710 − 0.881754i \(-0.656363\pi\)
−0.471710 + 0.881754i \(0.656363\pi\)
\(212\) 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
\(225\) 50625.0 1.00000
\(226\) 0 0
\(227\) 85102.0 1.65154 0.825768 − 0.564010i \(-0.190742\pi\)
0.825768 + 0.564010i \(0.190742\pi\)
\(228\) 0 0
\(229\) −57358.0 −1.09376 −0.546881 − 0.837210i \(-0.684185\pi\)
−0.546881 + 0.837210i \(0.684185\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −107422. −1.97871 −0.989353 − 0.145534i \(-0.953510\pi\)
−0.989353 + 0.145534i \(0.953510\pi\)
\(234\) 0 0
\(235\) 105550. 1.91127
\(236\) 0 0
\(237\) −25902.0 −0.461144
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 112322. 1.93389 0.966943 − 0.254994i \(-0.0820734\pi\)
0.966943 + 0.254994i \(0.0820734\pi\)
\(242\) 0 0
\(243\) −59049.0 −1.00000
\(244\) 0 0
\(245\) 60025.0 1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 89442.0 1.44259
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 85950.0 1.32180
\(256\) 0 0
\(257\) 85058.0 1.28780 0.643901 − 0.765109i \(-0.277315\pi\)
0.643901 + 0.765109i \(0.277315\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −60578.0 −0.875797 −0.437898 − 0.899024i \(-0.644277\pi\)
−0.437898 + 0.899024i \(0.644277\pi\)
\(264\) 0 0
\(265\) 44450.0 0.632965
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −85442.0 −1.16341 −0.581705 − 0.813400i \(-0.697614\pi\)
−0.581705 + 0.813400i \(0.697614\pi\)
\(272\) 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
\(279\) 155358. 1.99584
\(280\) 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
\(285\) −53550.0 −0.659280
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 62403.0 0.747153
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −16462.0 −0.191755 −0.0958776 − 0.995393i \(-0.530566\pi\)
−0.0958776 + 0.995393i \(0.530566\pi\)
\(294\) 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
\(305\) 162050. 1.74200
\(306\) 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
\(317\) −183022. −1.82131 −0.910657 − 0.413163i \(-0.864424\pi\)
−0.910657 + 0.413163i \(0.864424\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −104958. −1.01860
\(322\) 0 0
\(323\) −90916.0 −0.871436
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 209502. 1.95926
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 204238. 1.86415 0.932074 − 0.362267i \(-0.117997\pi\)
0.932074 + 0.362267i \(0.117997\pi\)
\(332\) 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
\(339\) −152082. −1.32336
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 22050.0 0.185255
\(346\) 0 0
\(347\) −102578. −0.851913 −0.425957 − 0.904744i \(-0.640062\pi\)
−0.425957 + 0.904744i \(0.640062\pi\)
\(348\) 0 0
\(349\) −33838.0 −0.277814 −0.138907 − 0.990305i \(-0.544359\pi\)
−0.138907 + 0.990305i \(0.544359\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −174142. −1.39751 −0.698754 − 0.715362i \(-0.746262\pi\)
−0.698754 + 0.715362i \(0.746262\pi\)
\(354\) 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
\(361\) −73677.0 −0.565350
\(362\) 0 0
\(363\) −131769. −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
\(375\) −140625. −1.00000
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −263282. −1.83292 −0.916458 − 0.400130i \(-0.868965\pi\)
−0.916458 + 0.400130i \(0.868965\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −177218. −1.20812 −0.604060 − 0.796939i \(-0.706451\pi\)
−0.604060 + 0.796939i \(0.706451\pi\)
\(384\) 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
\(391\) 37436.0 0.244870
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 71950.0 0.461144
\(396\) 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
\(405\) 164025. 1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −314398. −1.87946 −0.939730 − 0.341917i \(-0.888924\pi\)
−0.939730 + 0.341917i \(0.888924\pi\)
\(410\) 0 0
\(411\) −121842. −0.721296
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −248450. −1.44259
\(416\) 0 0
\(417\) 270018. 1.55282
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 7922.00 0.0446962 0.0223481 − 0.999750i \(-0.492886\pi\)
0.0223481 + 0.999750i \(0.492886\pi\)
\(422\) 0 0
\(423\) 341982. 1.91127
\(424\) 0 0
\(425\) −238750. −1.32180
\(426\) 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
\(437\) −23324.0 −0.122135
\(438\) 0 0
\(439\) −1442.00 −0.00748232 −0.00374116 − 0.999993i \(-0.501191\pi\)
−0.00374116 + 0.999993i \(0.501191\pi\)
\(440\) 0 0
\(441\) 194481. 1.00000
\(442\) 0 0
\(443\) 72142.0 0.367604 0.183802 − 0.982963i \(-0.441159\pi\)
0.183802 + 0.982963i \(0.441159\pi\)
\(444\) 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
\(453\) 99378.0 0.484277
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 278478. 1.32180
\(460\) 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
\(465\) −431550. −1.99584
\(466\) 0 0
\(467\) 316462. 1.45107 0.725534 − 0.688186i \(-0.241593\pi\)
0.725534 + 0.688186i \(0.241593\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 148750. 0.659280
\(476\) 0 0
\(477\) 144018. 0.632965
\(478\) 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
\(499\) −381842. −1.53350 −0.766748 − 0.641948i \(-0.778126\pi\)
−0.766748 + 0.641948i \(0.778126\pi\)
\(500\) 0 0
\(501\) 78642.0 0.313313
\(502\) 0 0
\(503\) −482018. −1.90514 −0.952571 − 0.304317i \(-0.901572\pi\)
−0.952571 + 0.304317i \(0.901572\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −257049. −1.00000
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −173502. −0.659280
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 325278. 1.20759
\(520\) 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
\(527\) −732676. −2.63810
\(528\) 0 0
\(529\) −270237. −0.965681
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 291550. 1.01860
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 576722. 1.97048 0.985240 − 0.171179i \(-0.0547577\pi\)
0.985240 + 0.171179i \(0.0547577\pi\)
\(542\) 0 0
\(543\) 455742. 1.54568
\(544\) 0 0
\(545\) −581950. −1.95926
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 525042. 1.74200
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −243502. −0.784860 −0.392430 − 0.919782i \(-0.628365\pi\)
−0.392430 + 0.919782i \(0.628365\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 610222. 1.92518 0.962589 − 0.270964i \(-0.0873425\pi\)
0.962589 + 0.270964i \(0.0873425\pi\)
\(564\) 0 0
\(565\) 422450. 1.32336
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 523918. 1.60691 0.803454 − 0.595367i \(-0.202993\pi\)
0.803454 + 0.595367i \(0.202993\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −61250.0 −0.185255
\(576\) 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
\(587\) −40178.0 −0.116604 −0.0583018 − 0.998299i \(-0.518569\pi\)
−0.0583018 + 0.998299i \(0.518569\pi\)
\(588\) 0 0
\(589\) 456484. 1.31582
\(590\) 0 0
\(591\) −560322. −1.60422
\(592\) 0 0
\(593\) 656258. 1.86623 0.933115 − 0.359578i \(-0.117079\pi\)
0.933115 + 0.359578i \(0.117079\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −531342. −1.49082
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −663838. −1.83786 −0.918932 − 0.394417i \(-0.870947\pi\)
−0.918932 + 0.394417i \(0.870947\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 366025. 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
\(617\) 645218. 1.69487 0.847435 − 0.530900i \(-0.178146\pi\)
0.847435 + 0.530900i \(0.178146\pi\)
\(618\) 0 0
\(619\) 279118. 0.728461 0.364231 − 0.931309i \(-0.381332\pi\)
0.364231 + 0.931309i \(0.381332\pi\)
\(620\) 0 0
\(621\) 71442.0 0.185255
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 390625. 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 739678. 1.85774 0.928868 − 0.370411i \(-0.120783\pi\)
0.928868 + 0.370411i \(0.120783\pi\)
\(632\) 0 0
\(633\) 378018. 0.943420
\(634\) 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
\(647\) 338782. 0.809304 0.404652 − 0.914471i \(-0.367393\pi\)
0.404652 + 0.914471i \(0.367393\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 388178. 0.910342 0.455171 − 0.890404i \(-0.349578\pi\)
0.455171 + 0.890404i \(0.349578\pi\)
\(654\) 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
\(661\) −171598. −0.392744 −0.196372 − 0.980529i \(-0.562916\pi\)
−0.196372 + 0.980529i \(0.562916\pi\)
\(662\) 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
\(675\) −455625. −1.00000
\(676\) 0 0
\(677\) −776782. −1.69481 −0.847407 − 0.530945i \(-0.821837\pi\)
−0.847407 + 0.530945i \(0.821837\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −765918. −1.65154
\(682\) 0 0
\(683\) 925582. 1.98415 0.992073 − 0.125665i \(-0.0401065\pi\)
0.992073 + 0.125665i \(0.0401065\pi\)
\(684\) 0 0
\(685\) 338450. 0.721296
\(686\) 0 0
\(687\) 516222. 1.09376
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −792722. −1.66022 −0.830108 − 0.557602i \(-0.811722\pi\)
−0.830108 + 0.557602i \(0.811722\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −750050. −1.55282
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 966798. 1.97871
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −949950. −1.91127
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −454798. −0.904745 −0.452372 − 0.891829i \(-0.649422\pi\)
−0.452372 + 0.891829i \(0.649422\pi\)
\(710\) 0 0
\(711\) 233118. 0.461144
\(712\) 0 0
\(713\) −187964. −0.369739
\(714\) 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
\(723\) −1.01090e6 −1.93389
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 531441. 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −540225. −1.00000
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.04520e6 −1.91387 −0.956933 − 0.290310i \(-0.906242\pi\)
−0.956933 + 0.290310i \(0.906242\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 839902. 1.52143 0.760713 − 0.649088i \(-0.224849\pi\)
0.760713 + 0.649088i \(0.224849\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −804978. −1.44259
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −939842. −1.66638 −0.833192 − 0.552984i \(-0.813489\pi\)
−0.833192 + 0.552984i \(0.813489\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −276050. −0.484277
\(756\) 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
\(765\) −773550. −1.32180
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −848638. −1.43506 −0.717530 − 0.696528i \(-0.754727\pi\)
−0.717530 + 0.696528i \(0.754727\pi\)
\(770\) 0 0
\(771\) −765522. −1.28780
\(772\) 0 0
\(773\) 1.13362e6 1.89718 0.948588 − 0.316513i \(-0.102512\pi\)
0.948588 + 0.316513i \(0.102512\pi\)
\(774\) 0 0
\(775\) 1.19875e6 1.99584
\(776\) 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
\(789\) 545202. 0.875797
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −400050. −0.632965
\(796\) 0 0
\(797\) −588142. −0.925903 −0.462952 − 0.886384i \(-0.653210\pi\)
−0.462952 + 0.886384i \(0.653210\pi\)
\(798\) 0 0
\(799\) −1.61280e6 −2.52632
\(800\) 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
\(811\) 144718. 0.220029 0.110015 − 0.993930i \(-0.464910\pi\)
0.110015 + 0.993930i \(0.464910\pi\)
\(812\) 0 0
\(813\) 768978. 1.16341
\(814\) 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
\(827\) 1.22798e6 1.79548 0.897741 − 0.440524i \(-0.145207\pi\)
0.897741 + 0.440524i \(0.145207\pi\)
\(828\) 0 0
\(829\) −1.12248e6 −1.63331 −0.816655 − 0.577126i \(-0.804174\pi\)
−0.816655 + 0.577126i \(0.804174\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −917182. −1.32180
\(834\) 0 0
\(835\) −218450. −0.313313
\(836\) 0 0
\(837\) −1.39822e6 −1.99584
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 707281. 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 714025. 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
\(855\) 481950. 0.659280
\(856\) 0 0
\(857\) 1.30666e6 1.77910 0.889550 − 0.456838i \(-0.151018\pi\)
0.889550 + 0.456838i \(0.151018\pi\)
\(858\) 0 0
\(859\) 1.42824e6 1.93559 0.967797 − 0.251732i \(-0.0810002\pi\)
0.967797 + 0.251732i \(0.0810002\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.41446e6 1.89920 0.949598 − 0.313471i \(-0.101492\pi\)
0.949598 + 0.313471i \(0.101492\pi\)
\(864\) 0 0
\(865\) −903550. −1.20759
\(866\) 0 0
\(867\) −561627. −0.747153
\(868\) 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
\(879\) 148158. 0.191755
\(880\) 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
\(887\) −1.29610e6 −1.64737 −0.823684 − 0.567049i \(-0.808085\pi\)
−0.823684 + 0.567049i \(0.808085\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.00484e6 1.26006
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −679196. −0.836653
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.26595e6 −1.54568
\(906\) 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
\(915\) −1.45845e6 −1.74200
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 4318.00 0.00511272 0.00255636 − 0.999997i \(-0.499186\pi\)
0.00255636 + 0.999997i \(0.499186\pi\)
\(920\) 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
\(931\) 571438. 0.659280
\(932\) 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
\(947\) −683858. −0.762546 −0.381273 − 0.924463i \(-0.624514\pi\)
−0.381273 + 0.924463i \(0.624514\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 1.64720e6 1.82131
\(952\) 0 0
\(953\) 356258. 0.392264 0.196132 − 0.980577i \(-0.437162\pi\)
0.196132 + 0.980577i \(0.437162\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.75520e6 2.98337
\(962\) 0 0
\(963\) 944622. 1.01860
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 818244. 0.871436
\(970\) 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
\(977\) 1.83130e6 1.91854 0.959268 − 0.282498i \(-0.0911631\pi\)
0.959268 + 0.282498i \(0.0911631\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.88552e6 −1.95926
\(982\) 0 0
\(983\) −1.88554e6 −1.95132 −0.975659 − 0.219291i \(-0.929626\pi\)
−0.975659 + 0.219291i \(0.929626\pi\)
\(984\) 0 0
\(985\) 1.55645e6 1.60422
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.04640e6 1.06549 0.532745 − 0.846276i \(-0.321160\pi\)
0.532745 + 0.846276i \(0.321160\pi\)
\(992\) 0 0
\(993\) −1.83814e6 −1.86415
\(994\) 0 0
\(995\) 1.47595e6 1.49082
\(996\) 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 240.5.c.a.209.1 1
3.2 odd 2 240.5.c.b.209.1 1
4.3 odd 2 15.5.d.a.14.1 ✓ 1
5.4 even 2 240.5.c.b.209.1 1
12.11 even 2 15.5.d.b.14.1 yes 1
15.14 odd 2 CM 240.5.c.a.209.1 1
20.3 even 4 75.5.c.e.26.2 2
20.7 even 4 75.5.c.e.26.1 2
20.19 odd 2 15.5.d.b.14.1 yes 1
60.23 odd 4 75.5.c.e.26.1 2
60.47 odd 4 75.5.c.e.26.2 2
60.59 even 2 15.5.d.a.14.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.5.d.a.14.1 ✓ 1 4.3 odd 2
15.5.d.a.14.1 ✓ 1 60.59 even 2
15.5.d.b.14.1 yes 1 12.11 even 2
15.5.d.b.14.1 yes 1 20.19 odd 2
75.5.c.e.26.1 2 20.7 even 4
75.5.c.e.26.1 2 60.23 odd 4
75.5.c.e.26.2 2 20.3 even 4
75.5.c.e.26.2 2 60.47 odd 4
240.5.c.a.209.1 1 1.1 even 1 trivial
240.5.c.a.209.1 1 15.14 odd 2 CM
240.5.c.b.209.1 1 3.2 odd 2
240.5.c.b.209.1 1 5.4 even 2