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Properties

Label 480.5.i.b.209.1
Level $480$
Weight $5$
Character 480.209
Self dual yes
Analytic conductor $49.618$
Analytic rank $0$
Dimension $1$
CM discriminant -120
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] = [480,5,Mod(209,480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(480, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("480.209");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 480.i (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: \(49.6175822802\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 209.1
Character \(\chi\) \(=\) 480.209

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} +25.0000 q^{5} +81.0000 q^{9} +238.000 q^{11} -142.000 q^{13} -225.000 q^{15} +98.0000 q^{17} +862.000 q^{23} +625.000 q^{25} -729.000 q^{27} -238.000 q^{29} -1442.00 q^{31} -2142.00 q^{33} -1582.00 q^{37} +1278.00 q^{39} -1778.00 q^{43} +2025.00 q^{45} +3262.00 q^{47} +2401.00 q^{49} -882.000 q^{51} +5950.00 q^{55} -2642.00 q^{59} -3550.00 q^{65} +8302.00 q^{67} -7758.00 q^{69} -5625.00 q^{75} +11038.0 q^{79} +6561.00 q^{81} +2450.00 q^{85} +2142.00 q^{87} +12978.0 q^{93} +19278.0 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(421\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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
\(11\) 238.000 1.96694 0.983471 − 0.181065i \(-0.0579545\pi\)
0.983471 + 0.181065i \(0.0579545\pi\)
\(12\) 0 0
\(13\) −142.000 −0.840237 −0.420118 − 0.907469i \(-0.638011\pi\)
−0.420118 + 0.907469i \(0.638011\pi\)
\(14\) 0 0
\(15\) −225.000 −1.00000
\(16\) 0 0
\(17\) 98.0000 0.339100 0.169550 − 0.985522i \(-0.445769\pi\)
0.169550 + 0.985522i \(0.445769\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 862.000 1.62949 0.814745 − 0.579820i \(-0.196877\pi\)
0.814745 + 0.579820i \(0.196877\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) 0 0
\(27\) −729.000 −1.00000
\(28\) 0 0
\(29\) −238.000 −0.282996 −0.141498 − 0.989939i \(-0.545192\pi\)
−0.141498 + 0.989939i \(0.545192\pi\)
\(30\) 0 0
\(31\) −1442.00 −1.50052 −0.750260 − 0.661143i \(-0.770072\pi\)
−0.750260 + 0.661143i \(0.770072\pi\)
\(32\) 0 0
\(33\) −2142.00 −1.96694
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1582.00 −1.15559 −0.577794 − 0.816183i \(-0.696086\pi\)
−0.577794 + 0.816183i \(0.696086\pi\)
\(38\) 0 0
\(39\) 1278.00 0.840237
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −1778.00 −0.961601 −0.480800 − 0.876830i \(-0.659654\pi\)
−0.480800 + 0.876830i \(0.659654\pi\)
\(44\) 0 0
\(45\) 2025.00 1.00000
\(46\) 0 0
\(47\) 3262.00 1.47669 0.738343 − 0.674425i \(-0.235609\pi\)
0.738343 + 0.674425i \(0.235609\pi\)
\(48\) 0 0
\(49\) 2401.00 1.00000
\(50\) 0 0
\(51\) −882.000 −0.339100
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 5950.00 1.96694
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2642.00 −0.758977 −0.379489 − 0.925196i \(-0.623900\pi\)
−0.379489 + 0.925196i \(0.623900\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3550.00 −0.840237
\(66\) 0 0
\(67\) 8302.00 1.84941 0.924705 − 0.380685i \(-0.124312\pi\)
0.924705 + 0.380685i \(0.124312\pi\)
\(68\) 0 0
\(69\) −7758.00 −1.62949
\(70\) 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
\(79\) 11038.0 1.76863 0.884313 − 0.466894i \(-0.154627\pi\)
0.884313 + 0.466894i \(0.154627\pi\)
\(80\) 0 0
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 2450.00 0.339100
\(86\) 0 0
\(87\) 2142.00 0.282996
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 12978.0 1.50052
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 19278.0 1.96694
\(100\) 0 0
\(101\) 12722.0 1.24713 0.623566 − 0.781770i \(-0.285683\pi\)
0.623566 + 0.781770i \(0.285683\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 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
\(111\) 14238.0 1.15559
\(112\) 0 0
\(113\) 2018.00 0.158039 0.0790195 − 0.996873i \(-0.474821\pi\)
0.0790195 + 0.996873i \(0.474821\pi\)
\(114\) 0 0
\(115\) 21550.0 1.62949
\(116\) 0 0
\(117\) −11502.0 −0.840237
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 42003.0 2.86886
\(122\) 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
\(129\) 16002.0 0.961601
\(130\) 0 0
\(131\) −22322.0 −1.30074 −0.650370 − 0.759618i \(-0.725386\pi\)
−0.650370 + 0.759618i \(0.725386\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −18225.0 −1.00000
\(136\) 0 0
\(137\) 33218.0 1.76983 0.884917 − 0.465749i \(-0.154215\pi\)
0.884917 + 0.465749i \(0.154215\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −29358.0 −1.47669
\(142\) 0 0
\(143\) −33796.0 −1.65270
\(144\) 0 0
\(145\) −5950.00 −0.282996
\(146\) 0 0
\(147\) −21609.0 −1.00000
\(148\) 0 0
\(149\) −3598.00 −0.162065 −0.0810324 − 0.996711i \(-0.525822\pi\)
−0.0810324 + 0.996711i \(0.525822\pi\)
\(150\) 0 0
\(151\) 12478.0 0.547257 0.273628 − 0.961835i \(-0.411776\pi\)
0.273628 + 0.961835i \(0.411776\pi\)
\(152\) 0 0
\(153\) 7938.00 0.339100
\(154\) 0 0
\(155\) −36050.0 −1.50052
\(156\) 0 0
\(157\) 25778.0 1.04580 0.522902 − 0.852393i \(-0.324850\pi\)
0.522902 + 0.852393i \(0.324850\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −45458.0 −1.71094 −0.855471 − 0.517851i \(-0.826732\pi\)
−0.855471 + 0.517851i \(0.826732\pi\)
\(164\) 0 0
\(165\) −53550.0 −1.96694
\(166\) 0 0
\(167\) 13342.0 0.478397 0.239198 − 0.970971i \(-0.423115\pi\)
0.239198 + 0.970971i \(0.423115\pi\)
\(168\) 0 0
\(169\) −8397.00 −0.294002
\(170\) 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
\(177\) 23778.0 0.758977
\(178\) 0 0
\(179\) 43918.0 1.37068 0.685341 − 0.728223i \(-0.259653\pi\)
0.685341 + 0.728223i \(0.259653\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −39550.0 −1.15559
\(186\) 0 0
\(187\) 23324.0 0.666991
\(188\) 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
\(195\) 31950.0 0.840237
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1918.00 0.0484331 0.0242166 − 0.999707i \(-0.492291\pi\)
0.0242166 + 0.999707i \(0.492291\pi\)
\(200\) 0 0
\(201\) −74718.0 −1.84941
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 69822.0 1.62949
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −44450.0 −0.961601
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −13916.0 −0.284925
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 50625.0 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −103102. −1.89913 −0.949566 − 0.313567i \(-0.898476\pi\)
−0.949566 + 0.313567i \(0.898476\pi\)
\(234\) 0 0
\(235\) 81550.0 1.47669
\(236\) 0 0
\(237\) −99342.0 −1.76863
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −116158. −1.99993 −0.999966 − 0.00829868i \(-0.997358\pi\)
−0.999966 + 0.00829868i \(0.997358\pi\)
\(242\) 0 0
\(243\) −59049.0 −1.00000
\(244\) 0 0
\(245\) 60025.0 1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −102482. −1.62667 −0.813336 − 0.581794i \(-0.802351\pi\)
−0.813336 + 0.581794i \(0.802351\pi\)
\(252\) 0 0
\(253\) 205156. 3.20511
\(254\) 0 0
\(255\) −22050.0 −0.339100
\(256\) 0 0
\(257\) 74018.0 1.12065 0.560326 − 0.828272i \(-0.310676\pi\)
0.560326 + 0.828272i \(0.310676\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −19278.0 −0.282996
\(262\) 0 0
\(263\) 138142. 1.99717 0.998583 − 0.0532131i \(-0.0169463\pi\)
0.998583 + 0.0532131i \(0.0169463\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 50642.0 0.699852 0.349926 − 0.936777i \(-0.386207\pi\)
0.349926 + 0.936777i \(0.386207\pi\)
\(270\) 0 0
\(271\) −142562. −1.94118 −0.970589 − 0.240744i \(-0.922609\pi\)
−0.970589 + 0.240744i \(0.922609\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 148750. 1.96694
\(276\) 0 0
\(277\) 45458.0 0.592449 0.296224 − 0.955118i \(-0.404272\pi\)
0.296224 + 0.955118i \(0.404272\pi\)
\(278\) 0 0
\(279\) −116802. −1.50052
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −4658.00 −0.0581603 −0.0290801 − 0.999577i \(-0.509258\pi\)
−0.0290801 + 0.999577i \(0.509258\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −73917.0 −0.885011
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) −66050.0 −0.758977
\(296\) 0 0
\(297\) −173502. −1.96694
\(298\) 0 0
\(299\) −122404. −1.36916
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −114498. −1.24713
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 187822. 1.99283 0.996414 − 0.0846146i \(-0.0269659\pi\)
0.996414 + 0.0846146i \(0.0269659\pi\)
\(308\) 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
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) −56644.0 −0.556638
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −88750.0 −0.840237
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) −128142. −1.15559
\(334\) 0 0
\(335\) 207550. 1.84941
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) −18162.0 −0.158039
\(340\) 0 0
\(341\) −343196. −2.95144
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −193950. −1.62949
\(346\) 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
\(351\) 103518. 0.840237
\(352\) 0 0
\(353\) 168098. 1.34900 0.674502 − 0.738273i \(-0.264358\pi\)
0.674502 + 0.738273i \(0.264358\pi\)
\(354\) 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
\(361\) 130321. 1.00000
\(362\) 0 0
\(363\) −378027. −2.86886
\(364\) 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
\(373\) 220178. 1.58255 0.791273 − 0.611463i \(-0.209419\pi\)
0.791273 + 0.611463i \(0.209419\pi\)
\(374\) 0 0
\(375\) −140625. −1.00000
\(376\) 0 0
\(377\) 33796.0 0.237784
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −245378. −1.67278 −0.836389 − 0.548137i \(-0.815337\pi\)
−0.836389 + 0.548137i \(0.815337\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −144018. −0.961601
\(388\) 0 0
\(389\) 271922. 1.79699 0.898494 − 0.438986i \(-0.144662\pi\)
0.898494 + 0.438986i \(0.144662\pi\)
\(390\) 0 0
\(391\) 84476.0 0.552560
\(392\) 0 0
\(393\) 200898. 1.30074
\(394\) 0 0
\(395\) 275950. 1.76863
\(396\) 0 0
\(397\) −272782. −1.73075 −0.865376 − 0.501124i \(-0.832920\pi\)
−0.865376 + 0.501124i \(0.832920\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 204764. 1.26079
\(404\) 0 0
\(405\) 164025. 1.00000
\(406\) 0 0
\(407\) −376516. −2.27297
\(408\) 0 0
\(409\) −220318. −1.31705 −0.658527 − 0.752557i \(-0.728820\pi\)
−0.658527 + 0.752557i \(0.728820\pi\)
\(410\) 0 0
\(411\) −298962. −1.76983
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 236878. 1.34926 0.674632 − 0.738155i \(-0.264303\pi\)
0.674632 + 0.738155i \(0.264303\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 264222. 1.47669
\(424\) 0 0
\(425\) 61250.0 0.339100
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 304164. 1.65270
\(430\) 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
\(435\) 53550.0 0.282996
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 344638. 1.78827 0.894137 − 0.447793i \(-0.147790\pi\)
0.894137 + 0.447793i \(0.147790\pi\)
\(440\) 0 0
\(441\) 194481. 1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 32382.0 0.162065
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −112302. −0.547257
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) −71442.0 −0.339100
\(460\) 0 0
\(461\) −421678. −1.98417 −0.992085 − 0.125564i \(-0.959926\pi\)
−0.992085 + 0.125564i \(0.959926\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 324450. 1.50052
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −232002. −1.04580
\(472\) 0 0
\(473\) −423164. −1.89141
\(474\) 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
\(481\) 224644. 0.970967
\(482\) 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
\(489\) 409122. 1.71094
\(490\) 0 0
\(491\) −443282. −1.83873 −0.919363 − 0.393410i \(-0.871295\pi\)
−0.919363 + 0.393410i \(0.871295\pi\)
\(492\) 0 0
\(493\) −23324.0 −0.0959642
\(494\) 0 0
\(495\) 481950. 1.96694
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) −120078. −0.478397
\(502\) 0 0
\(503\) 261982. 1.03547 0.517733 − 0.855543i \(-0.326776\pi\)
0.517733 + 0.855543i \(0.326776\pi\)
\(504\) 0 0
\(505\) 318050. 1.24713
\(506\) 0 0
\(507\) 75573.0 0.294002
\(508\) 0 0
\(509\) −249838. −0.964324 −0.482162 − 0.876082i \(-0.660148\pi\)
−0.482162 + 0.876082i \(0.660148\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 776356. 2.90456
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 382222. 1.39737 0.698686 − 0.715428i \(-0.253768\pi\)
0.698686 + 0.715428i \(0.253768\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −141316. −0.508827
\(528\) 0 0
\(529\) 463203. 1.65524
\(530\) 0 0
\(531\) −214002. −0.758977
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −395262. −1.37068
\(538\) 0 0
\(539\) 571438. 1.96694
\(540\) 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
\(547\) −273938. −0.915541 −0.457770 − 0.889070i \(-0.651352\pi\)
−0.457770 + 0.889070i \(0.651352\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 355950. 1.15559
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 252476. 0.807972
\(560\) 0 0
\(561\) −209916. −0.666991
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 50450.0 0.158039
\(566\) 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
\(575\) 538750. 1.62949
\(576\) 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
\(585\) −287550. −0.840237
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 564578. 1.60552 0.802758 − 0.596305i \(-0.203365\pi\)
0.802758 + 0.596305i \(0.203365\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −17262.0 −0.0484331
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −206878. −0.572750 −0.286375 − 0.958118i \(-0.592450\pi\)
−0.286375 + 0.958118i \(0.592450\pi\)
\(602\) 0 0
\(603\) 672462. 1.84941
\(604\) 0 0
\(605\) 1.05008e6 2.86886
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −463204. −1.24077
\(612\) 0 0
\(613\) −700462. −1.86408 −0.932038 − 0.362360i \(-0.881971\pi\)
−0.932038 + 0.362360i \(0.881971\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −690622. −1.81414 −0.907068 − 0.420983i \(-0.861685\pi\)
−0.907068 + 0.420983i \(0.861685\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) −628398. −1.62949
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 390625. 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −155036. −0.391860
\(630\) 0 0
\(631\) 763198. 1.91681 0.958404 − 0.285416i \(-0.0921317\pi\)
0.958404 + 0.285416i \(0.0921317\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −340942. −0.840237
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −796178. −1.92570 −0.962849 − 0.270040i \(-0.912963\pi\)
−0.962849 + 0.270040i \(0.912963\pi\)
\(644\) 0 0
\(645\) 400050. 0.961601
\(646\) 0 0
\(647\) 268702. 0.641893 0.320946 − 0.947097i \(-0.395999\pi\)
0.320946 + 0.947097i \(0.395999\pi\)
\(648\) 0 0
\(649\) −628796. −1.49286
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −558050. −1.30074
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −345842. −0.796355 −0.398178 − 0.917308i \(-0.630357\pi\)
−0.398178 + 0.917308i \(0.630357\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 125244. 0.284925
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −205156. −0.461140
\(668\) 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
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 830450. 1.76983
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 927918. 1.89913
\(700\) 0 0
\(701\) 53522.0 0.108917 0.0544586 − 0.998516i \(-0.482657\pi\)
0.0544586 + 0.998516i \(0.482657\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −733950. −1.47669
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 894078. 1.76863
\(712\) 0 0
\(713\) −1.24300e6 −2.44508
\(714\) 0 0
\(715\) −844900. −1.65270
\(716\) 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
\(723\) 1.04542e6 1.99993
\(724\) 0 0
\(725\) −148750. −0.282996
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 531441. 1.00000
\(730\) 0 0
\(731\) −174244. −0.326079
\(732\) 0 0
\(733\) −830542. −1.54580 −0.772901 − 0.634527i \(-0.781195\pi\)
−0.772901 + 0.634527i \(0.781195\pi\)
\(734\) 0 0
\(735\) −540225. −1.00000
\(736\) 0 0
\(737\) 1.97588e6 3.63768
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.01002e6 −1.82958 −0.914790 − 0.403929i \(-0.867644\pi\)
−0.914790 + 0.403929i \(0.867644\pi\)
\(744\) 0 0
\(745\) −89950.0 −0.162065
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.11600e6 −1.97872 −0.989362 − 0.145476i \(-0.953529\pi\)
−0.989362 + 0.145476i \(0.953529\pi\)
\(752\) 0 0
\(753\) 922338. 1.62667
\(754\) 0 0
\(755\) 311950. 0.547257
\(756\) 0 0
\(757\) −881902. −1.53896 −0.769482 − 0.638668i \(-0.779486\pi\)
−0.769482 + 0.638668i \(0.779486\pi\)
\(758\) 0 0
\(759\) −1.84640e6 −3.20511
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 198450. 0.339100
\(766\) 0 0
\(767\) 375164. 0.637721
\(768\) 0 0
\(769\) −1.03680e6 −1.75324 −0.876620 − 0.481183i \(-0.840207\pi\)
−0.876620 + 0.481183i \(0.840207\pi\)
\(770\) 0 0
\(771\) −666162. −1.12065
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −901250. −1.50052
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 173502. 0.282996
\(784\) 0 0
\(785\) 644450. 1.04580
\(786\) 0 0
\(787\) 266542. 0.430344 0.215172 − 0.976576i \(-0.430969\pi\)
0.215172 + 0.976576i \(0.430969\pi\)
\(788\) 0 0
\(789\) −1.24328e6 −1.99717
\(790\) 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
\(799\) 319676. 0.500745
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −455778. −0.699852
\(808\) 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
\(813\) 1.28306e6 1.94118
\(814\) 0 0
\(815\) −1.13645e6 −1.71094
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −742798. −1.10201 −0.551004 − 0.834503i \(-0.685755\pi\)
−0.551004 + 0.834503i \(0.685755\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) −1.33875e6 −1.96694
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) −409122. −0.592449
\(832\) 0 0
\(833\) 235298. 0.339100
\(834\) 0 0
\(835\) 333550. 0.478397
\(836\) 0 0
\(837\) 1.05122e6 1.50052
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −650637. −0.919913
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −209925. −0.294002
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 41922.0 0.0581603
\(850\) 0 0
\(851\) −1.36368e6 −1.88302
\(852\) 0 0
\(853\) −1.39070e6 −1.91133 −0.955666 − 0.294454i \(-0.904862\pi\)
−0.955666 + 0.294454i \(0.904862\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −90622.0 −0.123388 −0.0616939 − 0.998095i \(-0.519650\pi\)
−0.0616939 + 0.998095i \(0.519650\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 15742.0 0.0211368 0.0105684 − 0.999944i \(-0.496636\pi\)
0.0105684 + 0.999944i \(0.496636\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 665253. 0.885011
\(868\) 0 0
\(869\) 2.62704e6 3.47879
\(870\) 0 0
\(871\) −1.17888e6 −1.55394
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.39954e6 1.81964 0.909820 − 0.415003i \(-0.136219\pi\)
0.909820 + 0.415003i \(0.136219\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 928942. 1.19143 0.595713 − 0.803197i \(-0.296870\pi\)
0.595713 + 0.803197i \(0.296870\pi\)
\(884\) 0 0
\(885\) 594450. 0.758977
\(886\) 0 0
\(887\) 1.49846e6 1.90458 0.952288 − 0.305200i \(-0.0987232\pi\)
0.952288 + 0.305200i \(0.0987232\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.56152e6 1.96694
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 1.09795e6 1.37068
\(896\) 0 0
\(897\) 1.10164e6 1.36916
\(898\) 0 0
\(899\) 343196. 0.424642
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.09042e6 −1.32550 −0.662748 − 0.748842i \(-0.730610\pi\)
−0.662748 + 0.748842i \(0.730610\pi\)
\(908\) 0 0
\(909\) 1.03048e6 1.24713
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 338878. 0.401248 0.200624 − 0.979668i \(-0.435703\pi\)
0.200624 + 0.979668i \(0.435703\pi\)
\(920\) 0 0
\(921\) −1.69040e6 −1.99283
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −988750. −1.15559
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 583100. 0.666991
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 473042. 0.534220 0.267110 − 0.963666i \(-0.413931\pi\)
0.267110 + 0.963666i \(0.413931\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.65158e6 −1.81850 −0.909252 − 0.416246i \(-0.863346\pi\)
−0.909252 + 0.416246i \(0.863346\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 509796. 0.556638
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.15584e6 1.25156
\(962\) 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
\(971\) 960238. 1.01845 0.509226 − 0.860633i \(-0.329932\pi\)
0.509226 + 0.860633i \(0.329932\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 798750. 0.840237
\(976\) 0 0
\(977\) −936862. −0.981491 −0.490746 − 0.871303i \(-0.663276\pi\)
−0.490746 + 0.871303i \(0.663276\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.70026e6 −1.75958 −0.879788 − 0.475367i \(-0.842315\pi\)
−0.879788 + 0.475367i \(0.842315\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.53264e6 −1.56692
\(990\) 0 0
\(991\) −1.50288e6 −1.53030 −0.765152 − 0.643850i \(-0.777336\pi\)
−0.765152 + 0.643850i \(0.777336\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 47950.0 0.0484331
\(996\) 0 0
\(997\) −1.98686e6 −1.99884 −0.999419 − 0.0340973i \(-0.989144\pi\)
−0.999419 + 0.0340973i \(0.989144\pi\)
\(998\) 0 0
\(999\) 1.15328e6 1.15559
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 480.5.i.b.209.1 1
3.2 odd 2 480.5.i.a.209.1 1
4.3 odd 2 120.5.i.d.29.1 yes 1
5.4 even 2 480.5.i.d.209.1 1
8.3 odd 2 120.5.i.c.29.1 yes 1
8.5 even 2 480.5.i.c.209.1 1
12.11 even 2 120.5.i.b.29.1 yes 1
15.14 odd 2 480.5.i.c.209.1 1
20.19 odd 2 120.5.i.a.29.1 ✓ 1
24.5 odd 2 480.5.i.d.209.1 1
24.11 even 2 120.5.i.a.29.1 ✓ 1
40.19 odd 2 120.5.i.b.29.1 yes 1
40.29 even 2 480.5.i.a.209.1 1
60.59 even 2 120.5.i.c.29.1 yes 1
120.29 odd 2 CM 480.5.i.b.209.1 1
120.59 even 2 120.5.i.d.29.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.5.i.a.29.1 ✓ 1 20.19 odd 2
120.5.i.a.29.1 ✓ 1 24.11 even 2
120.5.i.b.29.1 yes 1 12.11 even 2
120.5.i.b.29.1 yes 1 40.19 odd 2
120.5.i.c.29.1 yes 1 8.3 odd 2
120.5.i.c.29.1 yes 1 60.59 even 2
120.5.i.d.29.1 yes 1 4.3 odd 2
120.5.i.d.29.1 yes 1 120.59 even 2
480.5.i.a.209.1 1 3.2 odd 2
480.5.i.a.209.1 1 40.29 even 2
480.5.i.b.209.1 1 1.1 even 1 trivial
480.5.i.b.209.1 1 120.29 odd 2 CM
480.5.i.c.209.1 1 8.5 even 2
480.5.i.c.209.1 1 15.14 odd 2
480.5.i.d.209.1 1 5.4 even 2
480.5.i.d.209.1 1 24.5 odd 2