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Properties

Label 315.5.e.a.244.1
Level $315$
Weight $5$
Character 315.244
Self dual yes
Analytic conductor $32.562$
Analytic rank $0$
Dimension $1$
CM discriminant -35
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] = [315,5,Mod(244,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("315.244");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 315.e (of order \(2\), degree \(1\), 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: \(32.5615383714\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 244.1
Character \(\chi\) \(=\) 315.244

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{4} -25.0000 q^{5} +49.0000 q^{7} +O(q^{10})\) \(q+16.0000 q^{4} -25.0000 q^{5} +49.0000 q^{7} +73.0000 q^{11} +23.0000 q^{13} +256.000 q^{16} -263.000 q^{17} -400.000 q^{20} +625.000 q^{25} +784.000 q^{28} +1153.00 q^{29} -1225.00 q^{35} +1168.00 q^{44} +3457.00 q^{47} +2401.00 q^{49} +368.000 q^{52} -1825.00 q^{55} +4096.00 q^{64} -575.000 q^{65} -4208.00 q^{68} +10078.0 q^{71} -9502.00 q^{73} +3577.00 q^{77} +12167.0 q^{79} -6400.00 q^{80} +6382.00 q^{83} +6575.00 q^{85} +1127.00 q^{91} +3383.00 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0
\(4\) 16.0000 1.00000
\(5\) −25.0000 −1.00000
\(6\) 0 0
\(7\) 49.0000 1.00000
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 73.0000 0.603306 0.301653 − 0.953418i \(-0.402462\pi\)
0.301653 + 0.953418i \(0.402462\pi\)
\(12\) 0 0
\(13\) 23.0000 0.136095 0.0680473 − 0.997682i \(-0.478323\pi\)
0.0680473 + 0.997682i \(0.478323\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 256.000 1.00000
\(17\) −263.000 −0.910035 −0.455017 − 0.890483i \(-0.650367\pi\)
−0.455017 + 0.890483i \(0.650367\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −400.000 −1.00000
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 784.000 1.00000
\(29\) 1153.00 1.37099 0.685493 − 0.728079i \(-0.259587\pi\)
0.685493 + 0.728079i \(0.259587\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1225.00 −1.00000
\(36\) 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 \(\pi\)
\(44\) 1168.00 0.603306
\(45\) 0 0
\(46\) 0 0
\(47\) 3457.00 1.56496 0.782481 − 0.622675i \(-0.213954\pi\)
0.782481 + 0.622675i \(0.213954\pi\)
\(48\) 0 0
\(49\) 2401.00 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 368.000 0.136095
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −1825.00 −0.603306
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4096.00 1.00000
\(65\) −575.000 −0.136095
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −4208.00 −0.910035
\(69\) 0 0
\(70\) 0 0
\(71\) 10078.0 1.99921 0.999603 − 0.0281662i \(-0.00896677\pi\)
0.999603 + 0.0281662i \(0.00896677\pi\)
\(72\) 0 0
\(73\) −9502.00 −1.78307 −0.891537 − 0.452948i \(-0.850372\pi\)
−0.891537 + 0.452948i \(0.850372\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3577.00 0.603306
\(78\) 0 0
\(79\) 12167.0 1.94953 0.974764 − 0.223239i \(-0.0716632\pi\)
0.974764 + 0.223239i \(0.0716632\pi\)
\(80\) −6400.00 −1.00000
\(81\) 0 0
\(82\) 0 0
\(83\) 6382.00 0.926404 0.463202 − 0.886253i \(-0.346700\pi\)
0.463202 + 0.886253i \(0.346700\pi\)
\(84\) 0 0
\(85\) 6575.00 0.910035
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 1127.00 0.136095
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3383.00 0.359549 0.179775 − 0.983708i \(-0.442463\pi\)
0.179775 + 0.983708i \(0.442463\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 10000.0 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 18383.0 1.73277 0.866387 − 0.499373i \(-0.166436\pi\)
0.866387 + 0.499373i \(0.166436\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −14353.0 −1.20806 −0.604032 − 0.796960i \(-0.706440\pi\)
−0.604032 + 0.796960i \(0.706440\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 12544.0 1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 18448.0 1.37099
\(117\) 0 0
\(118\) 0 0
\(119\) −12887.0 −0.910035
\(120\) 0 0
\(121\) −9312.00 −0.636022
\(122\) 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
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −19600.0 −1.00000
\(141\) 0 0
\(142\) 0 0
\(143\) 1679.00 0.0821067
\(144\) 0 0
\(145\) −28825.0 −1.37099
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −24242.0 −1.09193 −0.545966 − 0.837807i \(-0.683837\pi\)
−0.545966 + 0.837807i \(0.683837\pi\)
\(150\) 0 0
\(151\) −45433.0 −1.99259 −0.996294 − 0.0860129i \(-0.972587\pi\)
−0.996294 + 0.0860129i \(0.972587\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −31342.0 −1.27153 −0.635766 − 0.771882i \(-0.719316\pi\)
−0.635766 + 0.771882i \(0.719316\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 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
\(167\) −17663.0 −0.633332 −0.316666 − 0.948537i \(-0.602563\pi\)
−0.316666 + 0.948537i \(0.602563\pi\)
\(168\) 0 0
\(169\) −28032.0 −0.981478
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11017.0 0.368105 0.184052 − 0.982916i \(-0.441078\pi\)
0.184052 + 0.982916i \(0.441078\pi\)
\(174\) 0 0
\(175\) 30625.0 1.00000
\(176\) 18688.0 0.603306
\(177\) 0 0
\(178\) 0 0
\(179\) 16558.0 0.516775 0.258388 − 0.966041i \(-0.416809\pi\)
0.258388 + 0.966041i \(0.416809\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −19199.0 −0.549029
\(188\) 55312.0 1.56496
\(189\) 0 0
\(190\) 0 0
\(191\) −47447.0 −1.30059 −0.650297 − 0.759680i \(-0.725356\pi\)
−0.650297 + 0.759680i \(0.725356\pi\)
\(192\) 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
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 56497.0 1.37099
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 5888.00 0.136095
\(209\) 0 0
\(210\) 0 0
\(211\) −77593.0 −1.74284 −0.871420 − 0.490537i \(-0.836801\pi\)
−0.871420 + 0.490537i \(0.836801\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −29200.0 −0.603306
\(221\) −6049.00 −0.123851
\(222\) 0 0
\(223\) 61343.0 1.23355 0.616773 − 0.787141i \(-0.288440\pi\)
0.616773 + 0.787141i \(0.288440\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −49823.0 −0.966892 −0.483446 − 0.875374i \(-0.660615\pi\)
−0.483446 + 0.875374i \(0.660615\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −86425.0 −1.56496
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −43367.0 −0.759213 −0.379606 − 0.925148i \(-0.623941\pi\)
−0.379606 + 0.925148i \(0.623941\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −60025.0 −1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 65536.0 1.00000
\(257\) −111938. −1.69477 −0.847386 − 0.530977i \(-0.821825\pi\)
−0.847386 + 0.530977i \(0.821825\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −9200.00 −0.136095
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −67328.0 −0.910035
\(273\) 0 0
\(274\) 0 0
\(275\) 45625.0 0.603306
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −119807. −1.51729 −0.758647 − 0.651502i \(-0.774139\pi\)
−0.758647 + 0.651502i \(0.774139\pi\)
\(282\) 0 0
\(283\) 152303. 1.90167 0.950836 − 0.309695i \(-0.100227\pi\)
0.950836 + 0.309695i \(0.100227\pi\)
\(284\) 161248. 1.99921
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −14352.0 −0.171837
\(290\) 0 0
\(291\) 0 0
\(292\) −152032. −1.78307
\(293\) 171337. 1.99579 0.997897 − 0.0648123i \(-0.0206449\pi\)
0.997897 + 0.0648123i \(0.0206449\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 135263. 1.43517 0.717583 − 0.696473i \(-0.245248\pi\)
0.717583 + 0.696473i \(0.245248\pi\)
\(308\) 57232.0 0.603306
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −147097. −1.50146 −0.750732 − 0.660606i \(-0.770299\pi\)
−0.750732 + 0.660606i \(0.770299\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 194672. 1.94953
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 84169.0 0.827124
\(320\) −102400. −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 14375.0 0.136095
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 169393. 1.56496
\(330\) 0 0
\(331\) 138482. 1.26397 0.631986 − 0.774980i \(-0.282240\pi\)
0.631986 + 0.774980i \(0.282240\pi\)
\(332\) 102112. 0.926404
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 105200. 0.910035
\(341\) 0 0
\(342\) 0 0
\(343\) 117649. 1.00000
\(344\) 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
\(353\) 229897. 1.84495 0.922473 − 0.386060i \(-0.126164\pi\)
0.922473 + 0.386060i \(0.126164\pi\)
\(354\) 0 0
\(355\) −251950. −1.99921
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −76322.0 −0.592190 −0.296095 − 0.955159i \(-0.595684\pi\)
−0.296095 + 0.955159i \(0.595684\pi\)
\(360\) 0 0
\(361\) 130321. 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 18032.0 0.136095
\(365\) 237550. 1.78307
\(366\) 0 0
\(367\) −116497. −0.864933 −0.432467 − 0.901650i \(-0.642357\pi\)
−0.432467 + 0.901650i \(0.642357\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 26519.0 0.186584
\(378\) 0 0
\(379\) −35278.0 −0.245598 −0.122799 − 0.992432i \(-0.539187\pi\)
−0.122799 + 0.992432i \(0.539187\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 29182.0 0.198938 0.0994689 − 0.995041i \(-0.468286\pi\)
0.0994689 + 0.995041i \(0.468286\pi\)
\(384\) 0 0
\(385\) −89425.0 −0.603306
\(386\) 0 0
\(387\) 0 0
\(388\) 54128.0 0.359549
\(389\) −249407. −1.64820 −0.824099 − 0.566446i \(-0.808318\pi\)
−0.824099 + 0.566446i \(0.808318\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −304175. −1.94953
\(396\) 0 0
\(397\) −163897. −1.03990 −0.519948 − 0.854198i \(-0.674049\pi\)
−0.519948 + 0.854198i \(0.674049\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 160000. 1.00000
\(401\) 316273. 1.96686 0.983430 − 0.181289i \(-0.0580270\pi\)
0.983430 + 0.181289i \(0.0580270\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 294128. 1.73277
\(413\) 0 0
\(414\) 0 0
\(415\) −159550. −0.926404
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −76753.0 −0.433043 −0.216522 − 0.976278i \(-0.569471\pi\)
−0.216522 + 0.976278i \(0.569471\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −164375. −0.910035
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −356087. −1.91691 −0.958455 − 0.285245i \(-0.907925\pi\)
−0.958455 + 0.285245i \(0.907925\pi\)
\(432\) 0 0
\(433\) 193538. 1.03226 0.516132 − 0.856509i \(-0.327372\pi\)
0.516132 + 0.856509i \(0.327372\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −229648. −1.20806
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 200704. 1.00000
\(449\) −264287. −1.31094 −0.655470 − 0.755221i \(-0.727530\pi\)
−0.655470 + 0.755221i \(0.727530\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −28175.0 −0.136095
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 295168. 1.37099
\(465\) 0 0
\(466\) 0 0
\(467\) −322463. −1.47858 −0.739292 − 0.673385i \(-0.764840\pi\)
−0.739292 + 0.673385i \(0.764840\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −206192. −0.910035
\(477\) 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
\(484\) −148992. −0.636022
\(485\) −84575.0 −0.359549
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 470713. 1.95251 0.976255 − 0.216625i \(-0.0695049\pi\)
0.976255 + 0.216625i \(0.0695049\pi\)
\(492\) 0 0
\(493\) −303239. −1.24765
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 493822. 1.99921
\(498\) 0 0
\(499\) −31513.0 −0.126558 −0.0632789 − 0.997996i \(-0.520156\pi\)
−0.0632789 + 0.997996i \(0.520156\pi\)
\(500\) −250000. −1.00000
\(501\) 0 0
\(502\) 0 0
\(503\) 313297. 1.23828 0.619142 − 0.785279i \(-0.287481\pi\)
0.619142 + 0.785279i \(0.287481\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −465598. −1.78307
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −459575. −1.73277
\(516\) 0 0
\(517\) 252361. 0.944150
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −440782. −1.61146 −0.805732 − 0.592281i \(-0.798228\pi\)
−0.805732 + 0.592281i \(0.798228\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 279841. 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 175273. 0.603306
\(540\) 0 0
\(541\) 2927.00 0.0100006 0.00500032 − 0.999987i \(-0.498408\pi\)
0.00500032 + 0.999987i \(0.498408\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 358825. 1.20806
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 596183. 1.94953
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −313600. −1.00000
\(561\) 0 0
\(562\) 0 0
\(563\) −129938. −0.409939 −0.204970 − 0.978768i \(-0.565710\pi\)
−0.204970 + 0.978768i \(0.565710\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −566882. −1.75093 −0.875464 − 0.483284i \(-0.839444\pi\)
−0.875464 + 0.483284i \(0.839444\pi\)
\(570\) 0 0
\(571\) −638158. −1.95729 −0.978647 − 0.205549i \(-0.934102\pi\)
−0.978647 + 0.205549i \(0.934102\pi\)
\(572\) 26864.0 0.0821067
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −665017. −1.99747 −0.998737 − 0.0502441i \(-0.984000\pi\)
−0.998737 + 0.0502441i \(0.984000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −461200. −1.37099
\(581\) 312718. 0.926404
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −507698. −1.47343 −0.736715 − 0.676204i \(-0.763624\pi\)
−0.736715 + 0.676204i \(0.763624\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 320137. 0.910388 0.455194 − 0.890392i \(-0.349570\pi\)
0.455194 + 0.890392i \(0.349570\pi\)
\(594\) 0 0
\(595\) 322175. 0.910035
\(596\) −387872. −1.09193
\(597\) 0 0
\(598\) 0 0
\(599\) 613273. 1.70923 0.854614 − 0.519263i \(-0.173794\pi\)
0.854614 + 0.519263i \(0.173794\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −726928. −1.99259
\(605\) 232800. 0.636022
\(606\) 0 0
\(607\) −82417.0 −0.223686 −0.111843 − 0.993726i \(-0.535675\pi\)
−0.111843 + 0.993726i \(0.535675\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 79511.0 0.212983
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 390625. 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −501472. −1.27153
\(629\) 0 0
\(630\) 0 0
\(631\) 100487. 0.252378 0.126189 − 0.992006i \(-0.459725\pi\)
0.126189 + 0.992006i \(0.459725\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 55223.0 0.136095
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −96002.0 −0.233649 −0.116825 − 0.993153i \(-0.537272\pi\)
−0.116825 + 0.993153i \(0.537272\pi\)
\(642\) 0 0
\(643\) 713183. 1.72496 0.862480 − 0.506091i \(-0.168910\pi\)
0.862480 + 0.506091i \(0.168910\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −111458. −0.266258 −0.133129 − 0.991099i \(-0.542502\pi\)
−0.133129 + 0.991099i \(0.542502\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −777527. −1.79038 −0.895189 − 0.445687i \(-0.852959\pi\)
−0.895189 + 0.445687i \(0.852959\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −282608. −0.633332
\(669\) 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
\(676\) −448512. −0.981478
\(677\) −750023. −1.63643 −0.818215 − 0.574913i \(-0.805036\pi\)
−0.818215 + 0.574913i \(0.805036\pi\)
\(678\) 0 0
\(679\) 165767. 0.359549
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 176272. 0.368105
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 490000. 1.00000
\(701\) 884833. 1.80063 0.900317 − 0.435235i \(-0.143335\pi\)
0.900317 + 0.435235i \(0.143335\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 299008. 0.603306
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.00253e6 1.99436 0.997180 − 0.0750454i \(-0.0239102\pi\)
0.997180 + 0.0750454i \(0.0239102\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −41975.0 −0.0821067
\(716\) 264928. 0.516775
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 900767. 1.73277
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 720625. 1.37099
\(726\) 0 0
\(727\) 976418. 1.84743 0.923713 − 0.383086i \(-0.125139\pi\)
0.923713 + 0.383086i \(0.125139\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 51143.0 0.0951871 0.0475936 − 0.998867i \(-0.484845\pi\)
0.0475936 + 0.998867i \(0.484845\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 749207. 1.37187 0.685935 − 0.727663i \(-0.259393\pi\)
0.685935 + 0.727663i \(0.259393\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 606050. 1.09193
\(746\) 0 0
\(747\) 0 0
\(748\) −307184. −0.549029
\(749\) 0 0
\(750\) 0 0
\(751\) 648887. 1.15051 0.575253 − 0.817975i \(-0.304903\pi\)
0.575253 + 0.817975i \(0.304903\pi\)
\(752\) 884992. 1.56496
\(753\) 0 0
\(754\) 0 0
\(755\) 1.13582e6 1.99259
\(756\) 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
\(763\) −703297. −1.20806
\(764\) −759152. −1.30059
\(765\) 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
\(773\) −1.17962e6 −1.97417 −0.987084 − 0.160202i \(-0.948786\pi\)
−0.987084 + 0.160202i \(0.948786\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 735694. 1.20613
\(782\) 0 0
\(783\) 0 0
\(784\) 614656. 1.00000
\(785\) 783550. 1.27153
\(786\) 0 0
\(787\) −1.03714e6 −1.67451 −0.837253 − 0.546816i \(-0.815840\pi\)
−0.837253 + 0.546816i \(0.815840\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.07354e6 −1.69006 −0.845031 − 0.534717i \(-0.820418\pi\)
−0.845031 + 0.534717i \(0.820418\pi\)
\(798\) 0 0
\(799\) −909191. −1.42417
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −693646. −1.07574
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.29955e6 1.98562 0.992812 − 0.119685i \(-0.0381886\pi\)
0.992812 + 0.119685i \(0.0381886\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 903952. 1.37099
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.23437e6 −1.83129 −0.915647 − 0.401984i \(-0.868321\pi\)
−0.915647 + 0.401984i \(0.868321\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 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
\(831\) 0 0
\(832\) 94208.0 0.136095
\(833\) −631463. −0.910035
\(834\) 0 0
\(835\) 441575. 0.633332
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 622128. 0.879605
\(842\) 0 0
\(843\) 0 0
\(844\) −1.24149e6 −1.74284
\(845\) 700800. 0.981478
\(846\) 0 0
\(847\) −456288. −0.636022
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.44782e6 −1.98984 −0.994918 − 0.100692i \(-0.967894\pi\)
−0.994918 + 0.100692i \(0.967894\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 970462. 1.32135 0.660674 − 0.750673i \(-0.270271\pi\)
0.660674 + 0.750673i \(0.270271\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −275425. −0.368105
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 888191. 1.17616
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −765625. −1.00000
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −467200. −0.603306
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −96784.0 −0.123851
\(885\) 0 0
\(886\) 0 0
\(887\) 1.32950e6 1.68983 0.844913 − 0.534904i \(-0.179652\pi\)
0.844913 + 0.534904i \(0.179652\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 981488. 1.23355
\(893\) 0 0
\(894\) 0 0
\(895\) −413950. −0.516775
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) −797168. −0.966892
\(909\) 0 0
\(910\) 0 0
\(911\) −1.15584e6 −1.39271 −0.696357 − 0.717696i \(-0.745197\pi\)
−0.696357 + 0.717696i \(0.745197\pi\)
\(912\) 0 0
\(913\) 465886. 0.558905
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −695113. −0.823047 −0.411523 − 0.911399i \(-0.635003\pi\)
−0.411523 + 0.911399i \(0.635003\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 231794. 0.272081
\(924\) 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
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 479975. 0.549029
\(936\) 0 0
\(937\) 1.58930e6 1.81020 0.905102 − 0.425195i \(-0.139794\pi\)
0.905102 + 0.425195i \(0.139794\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −1.38280e6 −1.56496
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −218546. −0.242667
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 1.18618e6 1.30059
\(956\) −693872. −0.759213
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 923521. 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −960400. −1.00000
\(981\) 0 0
\(982\) 0 0
\(983\) 675937. 0.699518 0.349759 − 0.936840i \(-0.386263\pi\)
0.349759 + 0.936840i \(0.386263\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.44288e6 −1.46920 −0.734602 − 0.678498i \(-0.762631\pi\)
−0.734602 + 0.678498i \(0.762631\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 737783. 0.742230 0.371115 − 0.928587i \(-0.378976\pi\)
0.371115 + 0.928587i \(0.378976\pi\)
\(998\) 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 315.5.e.a.244.1 1
3.2 odd 2 35.5.c.a.34.1 ✓ 1
5.4 even 2 315.5.e.b.244.1 1
7.6 odd 2 315.5.e.b.244.1 1
12.11 even 2 560.5.p.b.209.1 1
15.2 even 4 175.5.d.c.76.2 2
15.8 even 4 175.5.d.c.76.1 2
15.14 odd 2 35.5.c.b.34.1 yes 1
21.20 even 2 35.5.c.b.34.1 yes 1
35.34 odd 2 CM 315.5.e.a.244.1 1
60.59 even 2 560.5.p.a.209.1 1
84.83 odd 2 560.5.p.a.209.1 1
105.62 odd 4 175.5.d.c.76.1 2
105.83 odd 4 175.5.d.c.76.2 2
105.104 even 2 35.5.c.a.34.1 ✓ 1
420.419 odd 2 560.5.p.b.209.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.c.a.34.1 ✓ 1 3.2 odd 2
35.5.c.a.34.1 ✓ 1 105.104 even 2
35.5.c.b.34.1 yes 1 15.14 odd 2
35.5.c.b.34.1 yes 1 21.20 even 2
175.5.d.c.76.1 2 15.8 even 4
175.5.d.c.76.1 2 105.62 odd 4
175.5.d.c.76.2 2 15.2 even 4
175.5.d.c.76.2 2 105.83 odd 4
315.5.e.a.244.1 1 1.1 even 1 trivial
315.5.e.a.244.1 1 35.34 odd 2 CM
315.5.e.b.244.1 1 5.4 even 2
315.5.e.b.244.1 1 7.6 odd 2
560.5.p.a.209.1 1 60.59 even 2
560.5.p.a.209.1 1 84.83 odd 2
560.5.p.b.209.1 1 12.11 even 2
560.5.p.b.209.1 1 420.419 odd 2