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Properties

Label 403.3.b.a.402.1
Level $403$
Weight $3$
Character 403.402
Self dual yes
Analytic conductor $10.981$
Analytic rank $0$
Dimension $1$
CM discriminant -403
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [403,3,Mod(402,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.402");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 403.b (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: \(10.9809546537\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 402.1
Character \(\chi\) \(=\) 403.402

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{4} +9.00000 q^{9} +O(q^{10})\) \(q+4.00000 q^{4} +9.00000 q^{9} -9.00000 q^{11} +13.0000 q^{13} +16.0000 q^{16} +25.0000 q^{25} -31.0000 q^{31} +36.0000 q^{36} +43.0000 q^{37} -36.0000 q^{44} +49.0000 q^{49} +52.0000 q^{52} +64.0000 q^{64} -133.000 q^{73} +81.0000 q^{81} +42.0000 q^{83} +147.000 q^{89} -81.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\) \(249\) \(313\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 4.00000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) −9.00000 −0.818182 −0.409091 − 0.912494i \(-0.634154\pi\)
−0.409091 + 0.912494i \(0.634154\pi\)
\(12\) 0 0
\(13\) 13.0000 1.00000
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −31.0000 −1.00000
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 36.0000 1.00000
\(37\) 43.0000 1.16216 0.581081 − 0.813846i \(-0.302630\pi\)
0.581081 + 0.813846i \(0.302630\pi\)
\(38\) 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 \(\pi\)
\(44\) −36.0000 −0.818182
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 52.0000 1.00000
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −133.000 −1.82192 −0.910959 − 0.412497i \(-0.864657\pi\)
−0.910959 + 0.412497i \(0.864657\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 42.0000 0.506024 0.253012 − 0.967463i \(-0.418579\pi\)
0.253012 + 0.967463i \(0.418579\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 147.000 1.65169 0.825843 − 0.563901i \(-0.190700\pi\)
0.825843 + 0.563901i \(0.190700\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) −81.0000 −0.818182
\(100\) 100.000 1.00000
\(101\) −201.000 −1.99010 −0.995050 − 0.0993805i \(-0.968314\pi\)
−0.995050 + 0.0993805i \(0.968314\pi\)
\(102\) 0 0
\(103\) −197.000 −1.91262 −0.956311 − 0.292352i \(-0.905562\pi\)
−0.956311 + 0.292352i \(0.905562\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −189.000 −1.76636 −0.883178 − 0.469039i \(-0.844600\pi\)
−0.883178 + 0.469039i \(0.844600\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −177.000 −1.56637 −0.783186 − 0.621788i \(-0.786407\pi\)
−0.783186 + 0.621788i \(0.786407\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 117.000 1.00000
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −40.0000 −0.330579
\(122\) 0 0
\(123\) 0 0
\(124\) −124.000 −1.00000
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −141.000 −1.07634 −0.538168 − 0.842838i \(-0.680883\pi\)
−0.538168 + 0.842838i \(0.680883\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −222.000 −1.62044 −0.810219 − 0.586127i \(-0.800652\pi\)
−0.810219 + 0.586127i \(0.800652\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −117.000 −0.818182
\(144\) 144.000 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 172.000 1.16216
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 23.0000 0.152318 0.0761589 − 0.997096i \(-0.475734\pi\)
0.0761589 + 0.997096i \(0.475734\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −89.0000 −0.566879 −0.283439 − 0.958990i \(-0.591476\pi\)
−0.283439 + 0.958990i \(0.591476\pi\)
\(158\) 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
\(167\) 303.000 1.81437 0.907186 − 0.420731i \(-0.138226\pi\)
0.907186 + 0.420731i \(0.138226\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −57.0000 −0.329480 −0.164740 − 0.986337i \(-0.552678\pi\)
−0.164740 + 0.986337i \(0.552678\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −144.000 −0.818182
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 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
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −21.0000 −0.109948 −0.0549738 − 0.998488i \(-0.517508\pi\)
−0.0549738 + 0.998488i \(0.517508\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 196.000 1.00000
\(197\) −381.000 −1.93401 −0.967005 − 0.254757i \(-0.918004\pi\)
−0.967005 + 0.254757i \(0.918004\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 208.000 1.00000
\(209\) 0 0
\(210\) 0 0
\(211\) 19.0000 0.0900474 0.0450237 − 0.998986i \(-0.485664\pi\)
0.0450237 + 0.998986i \(0.485664\pi\)
\(212\) 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
\(223\) −329.000 −1.47534 −0.737668 − 0.675163i \(-0.764073\pi\)
−0.737668 + 0.675163i \(0.764073\pi\)
\(224\) 0 0
\(225\) 225.000 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 179.000 0.781659 0.390830 − 0.920463i \(-0.372188\pi\)
0.390830 + 0.920463i \(0.372188\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 63.0000 0.270386 0.135193 − 0.990819i \(-0.456835\pi\)
0.135193 + 0.990819i \(0.456835\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 354.000 1.48117 0.740586 − 0.671962i \(-0.234548\pi\)
0.740586 + 0.671962i \(0.234548\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.0580913 −0.0290456 − 0.999578i \(-0.509247\pi\)
−0.0290456 + 0.999578i \(0.509247\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 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
\(256\) 256.000 1.00000
\(257\) 111.000 0.431907 0.215953 − 0.976404i \(-0.430714\pi\)
0.215953 + 0.976404i \(0.430714\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 511.000 1.88561 0.942804 − 0.333346i \(-0.108178\pi\)
0.942804 + 0.333346i \(0.108178\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −225.000 −0.818182
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −279.000 −1.00000
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 163.000 0.575972 0.287986 − 0.957635i \(-0.407014\pi\)
0.287986 + 0.957635i \(0.407014\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −532.000 −1.82192
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 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
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 219.000 0.704180 0.352090 − 0.935966i \(-0.385471\pi\)
0.352090 + 0.935966i \(0.385471\pi\)
\(312\) 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
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 324.000 1.00000
\(325\) 325.000 1.00000
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −454.000 −1.37160 −0.685801 − 0.727789i \(-0.740548\pi\)
−0.685801 + 0.727789i \(0.740548\pi\)
\(332\) 168.000 0.506024
\(333\) 387.000 1.16216
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 279.000 0.818182
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −69.0000 −0.195467 −0.0977337 − 0.995213i \(-0.531159\pi\)
−0.0977337 + 0.995213i \(0.531159\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 588.000 1.65169
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 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
\(373\) 343.000 0.919571 0.459786 − 0.888030i \(-0.347926\pi\)
0.459786 + 0.888030i \(0.347926\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −753.000 −1.96606 −0.983029 − 0.183452i \(-0.941273\pi\)
−0.983029 + 0.183452i \(0.941273\pi\)
\(384\) 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
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −324.000 −0.818182
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 400.000 1.00000
\(401\) 771.000 1.92269 0.961347 − 0.275341i \(-0.0887907\pi\)
0.961347 + 0.275341i \(0.0887907\pi\)
\(402\) 0 0
\(403\) −403.000 −1.00000
\(404\) −804.000 −1.99010
\(405\) 0 0
\(406\) 0 0
\(407\) −387.000 −0.950860
\(408\) 0 0
\(409\) −701.000 −1.71394 −0.856968 − 0.515369i \(-0.827655\pi\)
−0.856968 + 0.515369i \(0.827655\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −788.000 −1.91262
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −774.000 −1.84726 −0.923628 − 0.383291i \(-0.874791\pi\)
−0.923628 + 0.383291i \(0.874791\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −756.000 −1.76636
\(429\) 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
\(437\) 0 0
\(438\) 0 0
\(439\) −734.000 −1.67198 −0.835991 − 0.548743i \(-0.815106\pi\)
−0.835991 + 0.548743i \(0.815106\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) 483.000 1.09029 0.545147 − 0.838341i \(-0.316474\pi\)
0.545147 + 0.838341i \(0.316474\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 402.000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −708.000 −1.56637
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 139.000 0.304158 0.152079 − 0.988368i \(-0.451403\pi\)
0.152079 + 0.988368i \(0.451403\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −597.000 −1.29501 −0.647505 − 0.762061i \(-0.724188\pi\)
−0.647505 + 0.762061i \(0.724188\pi\)
\(462\) 0 0
\(463\) 647.000 1.39741 0.698704 − 0.715411i \(-0.253760\pi\)
0.698704 + 0.715411i \(0.253760\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −678.000 −1.45182 −0.725910 − 0.687790i \(-0.758581\pi\)
−0.725910 + 0.687790i \(0.758581\pi\)
\(468\) 468.000 1.00000
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 559.000 1.16216
\(482\) 0 0
\(483\) 0 0
\(484\) −160.000 −0.330579
\(485\) 0 0
\(486\) 0 0
\(487\) −142.000 −0.291581 −0.145791 − 0.989315i \(-0.546573\pi\)
−0.145791 + 0.989315i \(0.546573\pi\)
\(488\) 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
\(496\) −496.000 −1.00000
\(497\) 0 0
\(498\) 0 0
\(499\) 874.000 1.75150 0.875752 − 0.482762i \(-0.160366\pi\)
0.875752 + 0.482762i \(0.160366\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −606.000 −1.20477 −0.602386 − 0.798205i \(-0.705783\pi\)
−0.602386 + 0.798205i \(0.705783\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −966.000 −1.89784 −0.948919 − 0.315518i \(-0.897822\pi\)
−0.948919 + 0.315518i \(0.897822\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 639.000 1.22649 0.613244 − 0.789894i \(-0.289864\pi\)
0.613244 + 0.789894i \(0.289864\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −564.000 −1.07634
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 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
\(539\) −441.000 −0.818182
\(540\) 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
\(547\) −518.000 −0.946984 −0.473492 − 0.880798i \(-0.657007\pi\)
−0.473492 + 0.880798i \(0.657007\pi\)
\(548\) −888.000 −1.62044
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1083.00 1.94434 0.972172 − 0.234267i \(-0.0752689\pi\)
0.972172 + 0.234267i \(0.0752689\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 723.000 1.28419 0.642096 − 0.766624i \(-0.278065\pi\)
0.642096 + 0.766624i \(0.278065\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 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 \(\pi\)
\(572\) −468.000 −0.818182
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 576.000 1.00000
\(577\) 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
\(587\) 399.000 0.679727 0.339864 − 0.940475i \(-0.389619\pi\)
0.339864 + 0.940475i \(0.389619\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 688.000 1.16216
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −414.000 −0.691152 −0.345576 − 0.938391i \(-0.612316\pi\)
−0.345576 + 0.938391i \(0.612316\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 92.0000 0.152318
\(605\) 0 0
\(606\) 0 0
\(607\) 811.000 1.33608 0.668040 − 0.744126i \(-0.267134\pi\)
0.668040 + 0.744126i \(0.267134\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −758.000 −1.23654 −0.618271 − 0.785965i \(-0.712166\pi\)
−0.618271 + 0.785965i \(0.712166\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 959.000 1.54927 0.774637 − 0.632407i \(-0.217933\pi\)
0.774637 + 0.632407i \(0.217933\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −356.000 −0.566879
\(629\) 0 0
\(630\) 0 0
\(631\) −1249.00 −1.97940 −0.989699 − 0.143165i \(-0.954272\pi\)
−0.989699 + 0.143165i \(0.954272\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 637.000 1.00000
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −233.000 −0.362364 −0.181182 − 0.983450i \(-0.557992\pi\)
−0.181182 + 0.983450i \(0.557992\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 903.000 1.38285 0.691424 − 0.722449i \(-0.256984\pi\)
0.691424 + 0.722449i \(0.256984\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1197.00 −1.82192
\(658\) 0 0
\(659\) −294.000 −0.446131 −0.223065 − 0.974804i \(-0.571606\pi\)
−0.223065 + 0.974804i \(0.571606\pi\)
\(660\) 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
\(668\) 1212.00 1.81437
\(669\) 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
\(676\) 676.000 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 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 \(\pi\)
\(692\) −228.000 −0.329480
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 999.000 1.42511 0.712553 − 0.701618i \(-0.247539\pi\)
0.712553 + 0.701618i \(0.247539\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −576.000 −0.818182
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1093.00 −1.54161 −0.770804 − 0.637072i \(-0.780145\pi\)
−0.770804 + 0.637072i \(0.780145\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 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
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −158.000 −0.217331 −0.108666 − 0.994078i \(-0.534658\pi\)
−0.108666 + 0.994078i \(0.534658\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1447.00 1.95805 0.979026 − 0.203737i \(-0.0653087\pi\)
0.979026 + 0.203737i \(0.0653087\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 711.000 0.956931 0.478466 − 0.878106i \(-0.341193\pi\)
0.478466 + 0.878106i \(0.341193\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 378.000 0.506024
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1099.00 1.46338 0.731691 − 0.681636i \(-0.238731\pi\)
0.731691 + 0.681636i \(0.238731\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1026.00 1.34823 0.674113 − 0.738628i \(-0.264526\pi\)
0.674113 + 0.738628i \(0.264526\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −84.0000 −0.109948
\(765\) 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
\(773\) 27.0000 0.0349288 0.0174644 − 0.999847i \(-0.494441\pi\)
0.0174644 + 0.999847i \(0.494441\pi\)
\(774\) 0 0
\(775\) −775.000 −1.00000
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 784.000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −937.000 −1.19060 −0.595299 − 0.803505i \(-0.702966\pi\)
−0.595299 + 0.803505i \(0.702966\pi\)
\(788\) −1524.00 −1.93401
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 1323.00 1.65169
\(802\) 0 0
\(803\) 1197.00 1.49066
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −342.000 −0.416565 −0.208283 − 0.978069i \(-0.566787\pi\)
−0.208283 + 0.978069i \(0.566787\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1446.00 −1.74849 −0.874244 − 0.485486i \(-0.838643\pi\)
−0.874244 + 0.485486i \(0.838643\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 832.000 1.00000
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 76.0000 0.0900474
\(845\) 0 0
\(846\) 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
\(855\) 0 0
\(856\) 0 0
\(857\) 1311.00 1.52975 0.764877 − 0.644176i \(-0.222799\pi\)
0.764877 + 0.644176i \(0.222799\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1602.00 1.85632 0.928158 − 0.372187i \(-0.121392\pi\)
0.928158 + 0.372187i \(0.121392\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 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
\(887\) 162.000 0.182638 0.0913191 − 0.995822i \(-0.470892\pi\)
0.0913191 + 0.995822i \(0.470892\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −729.000 −0.818182
\(892\) −1316.00 −1.47534
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 900.000 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1813.00 −1.99890 −0.999449 − 0.0331999i \(-0.989430\pi\)
−0.999449 + 0.0331999i \(0.989430\pi\)
\(908\) 0 0
\(909\) −1809.00 −1.99010
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −378.000 −0.414020
\(914\) 0 0
\(915\) 0 0
\(916\) 716.000 0.781659
\(917\) 0 0
\(918\) 0 0
\(919\) −1789.00 −1.94668 −0.973341 − 0.229365i \(-0.926335\pi\)
−0.973341 + 0.229365i \(0.926335\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1075.00 1.16216
\(926\) 0 0
\(927\) −1773.00 −1.91262
\(928\) 0 0
\(929\) 339.000 0.364909 0.182454 − 0.983214i \(-0.441596\pi\)
0.182454 + 0.983214i \(0.441596\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 252.000 0.270386
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1753.00 −1.87086 −0.935432 − 0.353506i \(-0.884989\pi\)
−0.935432 + 0.353506i \(0.884989\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1869.00 −1.98618 −0.993092 − 0.117334i \(-0.962565\pi\)
−0.993092 + 0.117334i \(0.962565\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1863.00 1.96727 0.983633 − 0.180186i \(-0.0576700\pi\)
0.983633 + 0.180186i \(0.0576700\pi\)
\(948\) 0 0
\(949\) −1729.00 −1.82192
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1416.00 1.48117
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 0 0
\(963\) −1701.00 −1.76636
\(964\) −56.0000 −0.0580913
\(965\) 0 0
\(966\) 0 0
\(967\) −1817.00 −1.87901 −0.939504 − 0.342539i \(-0.888713\pi\)
−0.939504 + 0.342539i \(0.888713\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1539.00 1.58496 0.792482 − 0.609895i \(-0.208789\pi\)
0.792482 + 0.609895i \(0.208789\pi\)
\(972\) 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
\(979\) −1323.00 −1.35138
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1134.00 −1.15361 −0.576806 − 0.816881i \(-0.695701\pi\)
−0.576806 + 0.816881i \(0.695701\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1633.00 −1.63791 −0.818957 − 0.573855i \(-0.805447\pi\)
−0.818957 + 0.573855i \(0.805447\pi\)
\(998\) 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 403.3.b.a.402.1 ✓ 1
13.12 even 2 403.3.b.b.402.1 yes 1
31.30 odd 2 403.3.b.b.402.1 yes 1
403.402 odd 2 CM 403.3.b.a.402.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
403.3.b.a.402.1 ✓ 1 1.1 even 1 trivial
403.3.b.a.402.1 ✓ 1 403.402 odd 2 CM
403.3.b.b.402.1 yes 1 13.12 even 2
403.3.b.b.402.1 yes 1 31.30 odd 2