Properties

Label 320.6.f.a.289.4
Level 320
Weight 6
Character 320.289
Analytic conductor 51.323
Analytic rank 0
Dimension 4
CM discriminant -40
Inner twists 8

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 289.4
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 320.289
Dual form 320.6.f.a.289.3

$q$-expansion

\(f(q)\) \(=\) \(q+55.9017 q^{5} +245.967i q^{7} -243.000 q^{9} +O(q^{10})\) \(q+55.9017 q^{5} +245.967i q^{7} -243.000 q^{9} -802.000i q^{11} -245.967 q^{13} -1914.00i q^{19} -3331.74i q^{23} +3125.00 q^{25} +13750.0i q^{35} +15853.7 q^{37} +15202.0 q^{41} -13584.1 q^{45} +15361.8i q^{47} -43693.0 q^{49} +16837.6 q^{53} -44833.2i q^{55} -27986.0i q^{59} -59770.1i q^{63} -13750.0 q^{65} +197266. q^{77} +59049.0 q^{81} +128786. q^{89} -60500.0i q^{91} -106996. i q^{95} +194886. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 972q^{9} + O(q^{10}) \) \( 4q - 972q^{9} + 12500q^{25} + 60808q^{41} - 174772q^{49} - 55000q^{65} + 236196q^{81} + 515144q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 55.9017 1.00000
\(6\) 0 0
\(7\) 245.967i 1.89729i 0.316350 + 0.948643i \(0.397543\pi\)
−0.316350 + 0.948643i \(0.602457\pi\)
\(8\) 0 0
\(9\) −243.000 −1.00000
\(10\) 0 0
\(11\) − 802.000i − 1.99845i −0.0393993 0.999224i \(-0.512544\pi\)
0.0393993 0.999224i \(-0.487456\pi\)
\(12\) 0 0
\(13\) −245.967 −0.403663 −0.201832 0.979420i \(-0.564689\pi\)
−0.201832 + 0.979420i \(0.564689\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) − 1914.00i − 1.21635i −0.793804 0.608174i \(-0.791902\pi\)
0.793804 0.608174i \(-0.208098\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 3331.74i − 1.31326i −0.754212 0.656631i \(-0.771981\pi\)
0.754212 0.656631i \(-0.228019\pi\)
\(24\) 0 0
\(25\) 3125.00 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 13750.0i 1.89729i
\(36\) 0 0
\(37\) 15853.7 1.90382 0.951912 0.306371i \(-0.0991147\pi\)
0.951912 + 0.306371i \(0.0991147\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 15202.0 1.41235 0.706173 0.708039i \(-0.250420\pi\)
0.706173 + 0.708039i \(0.250420\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −13584.1 −1.00000
\(46\) 0 0
\(47\) 15361.8i 1.01437i 0.861837 + 0.507186i \(0.169314\pi\)
−0.861837 + 0.507186i \(0.830686\pi\)
\(48\) 0 0
\(49\) −43693.0 −2.59969
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 16837.6 0.823361 0.411681 0.911328i \(-0.364942\pi\)
0.411681 + 0.911328i \(0.364942\pi\)
\(54\) 0 0
\(55\) − 44833.2i − 1.99845i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 27986.0i − 1.04667i −0.852126 0.523336i \(-0.824687\pi\)
0.852126 0.523336i \(-0.175313\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) − 59770.1i − 1.89729i
\(64\) 0 0
\(65\) −13750.0 −0.403663
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 197266. 3.79162
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 59049.0 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 128786. 1.72343 0.861715 0.507393i \(-0.169391\pi\)
0.861715 + 0.507393i \(0.169391\pi\)
\(90\) 0 0
\(91\) − 60500.0i − 0.765864i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 106996.i − 1.21635i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 194886.i 1.99845i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) − 192950.i − 1.79206i −0.443994 0.896030i \(-0.646439\pi\)
0.443994 0.896030i \(-0.353561\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) − 186250.i − 1.31326i
\(116\) 0 0
\(117\) 59770.1 0.403663
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −482153. −2.99379
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 174693. 1.00000
\(126\) 0 0
\(127\) − 293931.i − 1.61710i −0.588429 0.808549i \(-0.700253\pi\)
0.588429 0.808549i \(-0.299747\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 66902.0i 0.340613i 0.985391 + 0.170306i \(0.0544757\pi\)
−0.985391 + 0.170306i \(0.945524\pi\)
\(132\) 0 0
\(133\) 470782. 2.30776
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) − 16786.0i − 0.0736903i −0.999321 0.0368451i \(-0.988269\pi\)
0.999321 0.0368451i \(-0.0117308\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 197266.i 0.806700i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −430555. −1.39405 −0.697027 0.717045i \(-0.745494\pi\)
−0.697027 + 0.717045i \(0.745494\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 819500. 2.49163
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 550721.i − 1.52806i −0.645180 0.764030i \(-0.723218\pi\)
0.645180 0.764030i \(-0.276782\pi\)
\(168\) 0 0
\(169\) −310793. −0.837056
\(170\) 0 0
\(171\) 465102.i 1.21635i
\(172\) 0 0
\(173\) −353455. −0.897882 −0.448941 0.893561i \(-0.648199\pi\)
−0.448941 + 0.893561i \(0.648199\pi\)
\(174\) 0 0
\(175\) 768648.i 1.89729i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 316186.i − 0.737582i −0.929512 0.368791i \(-0.879772\pi\)
0.929512 0.368791i \(-0.120228\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 886250. 1.90382
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.08349e6 −1.98911 −0.994553 0.104230i \(-0.966762\pi\)
−0.994553 + 0.104230i \(0.966762\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 849818. 1.41235
\(206\) 0 0
\(207\) 809613.i 1.31326i
\(208\) 0 0
\(209\) −1.53503e6 −2.43081
\(210\) 0 0
\(211\) − 1.18270e6i − 1.82881i −0.404805 0.914403i \(-0.632660\pi\)
0.404805 0.914403i \(-0.367340\pi\)
\(212\) 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
\(223\) 1.31604e6i 1.77217i 0.463520 + 0.886087i \(0.346586\pi\)
−0.463520 + 0.886087i \(0.653414\pi\)
\(224\) 0 0
\(225\) −759375. −1.00000
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 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
\(235\) 858750.i 1.01437i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 89298.0 0.0990374 0.0495187 0.998773i \(-0.484231\pi\)
0.0495187 + 0.998773i \(0.484231\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.44251e6 −2.59969
\(246\) 0 0
\(247\) 470782.i 0.490995i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 620002.i − 0.621168i −0.950546 0.310584i \(-0.899475\pi\)
0.950546 0.310584i \(-0.100525\pi\)
\(252\) 0 0
\(253\) −2.67206e6 −2.62449
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 3.89950e6i 3.61210i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 549245.i 0.489640i 0.969569 + 0.244820i \(0.0787289\pi\)
−0.969569 + 0.244820i \(0.921271\pi\)
\(264\) 0 0
\(265\) 941250. 0.823361
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 2.50625e6i − 1.99845i
\(276\) 0 0
\(277\) 414947. 0.324933 0.162466 0.986714i \(-0.448055\pi\)
0.162466 + 0.986714i \(0.448055\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.12110e6 0.846989 0.423495 0.905899i \(-0.360803\pi\)
0.423495 + 0.905899i \(0.360803\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.73920e6i 2.67962i
\(288\) 0 0
\(289\) 1.41986e6 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.79040e6 1.21837 0.609187 0.793027i \(-0.291496\pi\)
0.609187 + 0.793027i \(0.291496\pi\)
\(294\) 0 0
\(295\) − 1.56446e6i − 1.04667i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 819500.i 0.530116i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) − 3.34125e6i − 1.89729i
\(316\) 0 0
\(317\) 2.83169e6 1.58270 0.791348 0.611366i \(-0.209380\pi\)
0.791348 + 0.611366i \(0.209380\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −768648. −0.403663
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.77850e6 −1.92455
\(330\) 0 0
\(331\) − 2.09810e6i − 1.05258i −0.850305 0.526291i \(-0.823582\pi\)
0.850305 0.526291i \(-0.176418\pi\)
\(332\) 0 0
\(333\) −3.85245e6 −1.90382
\(334\) 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
\(341\) 0 0
\(342\) 0 0
\(343\) − 6.61308e6i − 3.03507i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −1.18730e6 −0.479503
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 132934.i − 0.0515195i −0.999668 0.0257598i \(-0.991800\pi\)
0.999668 0.0257598i \(-0.00820049\pi\)
\(368\) 0 0
\(369\) −3.69409e6 −1.41235
\(370\) 0 0
\(371\) 4.14150e6i 1.56215i
\(372\) 0 0
\(373\) 5.37365e6 1.99985 0.999925 0.0122483i \(-0.00389886\pi\)
0.999925 + 0.0122483i \(0.00389886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 4.62669e6i 1.65452i 0.561819 + 0.827260i \(0.310102\pi\)
−0.561819 + 0.827260i \(0.689898\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.65025e6i 1.96821i 0.177593 + 0.984104i \(0.443169\pi\)
−0.177593 + 0.984104i \(0.556831\pi\)
\(384\) 0 0
\(385\) 1.10275e7 3.79162
\(386\) 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
\(396\) 0 0
\(397\) 3.34849e6 1.06628 0.533142 0.846026i \(-0.321011\pi\)
0.533142 + 0.846026i \(0.321011\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 231002. 0.0717389 0.0358695 0.999356i \(-0.488580\pi\)
0.0358695 + 0.999356i \(0.488580\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.30094e6 1.00000
\(406\) 0 0
\(407\) − 1.27147e7i − 3.80469i
\(408\) 0 0
\(409\) −6.63601e6 −1.96155 −0.980774 0.195146i \(-0.937482\pi\)
−0.980774 + 0.195146i \(0.937482\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.88365e6 1.98584
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.11741e6i 0.589211i 0.955619 + 0.294605i \(0.0951882\pi\)
−0.955619 + 0.294605i \(0.904812\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) − 3.73291e6i − 1.01437i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.37695e6 −1.59739
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 1.06174e7 2.59969
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 7.19936e6 1.72343
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8.52989e6 −1.99677 −0.998383 0.0568371i \(-0.981898\pi\)
−0.998383 + 0.0568371i \(0.981898\pi\)
\(450\) 0 0
\(451\) − 1.21920e7i − 2.82250i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 3.38205e6i − 0.765864i
\(456\) 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
\(463\) − 3.67487e6i − 0.796689i −0.917236 0.398345i \(-0.869585\pi\)
0.917236 0.398345i \(-0.130415\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) − 5.98125e6i − 1.21635i
\(476\) 0 0
\(477\) −4.09153e6 −0.823361
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −3.89950e6 −0.768504
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 7.51491e6i − 1.43582i −0.696134 0.717912i \(-0.745098\pi\)
0.696134 0.717912i \(-0.254902\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 2.39120e6i − 0.447623i −0.974632 0.223812i \(-0.928150\pi\)
0.974632 0.223812i \(-0.0718500\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 1.08945e7i 1.99845i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 6.93889e6i 1.24749i 0.781626 + 0.623747i \(0.214390\pi\)
−0.781626 + 0.623747i \(0.785610\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 7.44027e6i 1.31120i 0.755109 + 0.655600i \(0.227584\pi\)
−0.755109 + 0.655600i \(0.772416\pi\)
\(504\) 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
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 1.07863e7i − 1.79206i
\(516\) 0 0
\(517\) 1.23202e7 2.02717
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.21326e7 1.95821 0.979106 0.203351i \(-0.0651833\pi\)
0.979106 + 0.203351i \(0.0651833\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −4.66416e6 −0.724659
\(530\) 0 0
\(531\) 6.80060e6i 1.04667i
\(532\) 0 0
\(533\) −3.73920e6 −0.570112
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.50418e7i 5.19534i
\(540\) 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
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.45212e7 −1.98319 −0.991594 0.129387i \(-0.958699\pi\)
−0.991594 + 0.129387i \(0.958699\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.45241e7i 1.89729i
\(568\) 0 0
\(569\) −3.50709e6 −0.454115 −0.227057 0.973881i \(-0.572911\pi\)
−0.227057 + 0.973881i \(0.572911\pi\)
\(570\) 0 0
\(571\) − 1.45829e7i − 1.87177i −0.352299 0.935887i \(-0.614600\pi\)
0.352299 0.935887i \(-0.385400\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 1.04117e7i − 1.31326i
\(576\) 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
\(583\) − 1.35037e7i − 1.64544i
\(584\) 0 0
\(585\) 3.34125e6 0.403663
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −1.60910e7 −1.81718 −0.908588 0.417694i \(-0.862838\pi\)
−0.908588 + 0.417694i \(0.862838\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.69532e7 −2.99379
\(606\) 0 0
\(607\) 1.74118e7i 1.91810i 0.283233 + 0.959051i \(0.408593\pi\)
−0.283233 + 0.959051i \(0.591407\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 3.77850e6i − 0.409465i
\(612\) 0 0
\(613\) −1.56408e7 −1.68116 −0.840579 0.541689i \(-0.817785\pi\)
−0.840579 + 0.541689i \(0.817785\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) − 7.16491e6i − 0.751596i −0.926702 0.375798i \(-0.877369\pi\)
0.926702 0.375798i \(-0.122631\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.16772e7i 3.26984i
\(624\) 0 0
\(625\) 9.76562e6 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 1.64313e7i − 1.61710i
\(636\) 0 0
\(637\) 1.07471e7 1.04940
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.08320e6 0.392515 0.196257 0.980552i \(-0.437121\pi\)
0.196257 + 0.980552i \(0.437121\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.10606e7i 1.97793i 0.148160 + 0.988963i \(0.452665\pi\)
−0.148160 + 0.988963i \(0.547335\pi\)
\(648\) 0 0
\(649\) −2.24448e7 −2.09172
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.24455e6 0.848405 0.424202 0.905567i \(-0.360555\pi\)
0.424202 + 0.905567i \(0.360555\pi\)
\(654\) 0 0
\(655\) 3.73994e6i 0.340613i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1.04250e7i − 0.935108i −0.883964 0.467554i \(-0.845135\pi\)
0.883964 0.467554i \(-0.154865\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.63175e7 2.30776
\(666\) 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
\(675\) 0 0
\(676\) 0 0
\(677\) 2.14754e6 0.180082 0.0900409 0.995938i \(-0.471300\pi\)
0.0900409 + 0.995938i \(0.471300\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.14150e6 −0.332361
\(690\) 0 0
\(691\) 2.50118e7i 1.99274i 0.0851509 + 0.996368i \(0.472863\pi\)
−0.0851509 + 0.996368i \(0.527137\pi\)
\(692\) 0 0
\(693\) −4.79356e7 −3.79162
\(694\) 0 0
\(695\) − 938366.i − 0.0736903i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) − 3.03440e7i − 2.31571i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 1.10275e7i 0.806700i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 4.74595e7 3.40005
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 1.14949e7i − 0.806623i −0.915063 0.403311i \(-0.867859\pi\)
0.915063 0.403311i \(-0.132141\pi\)
\(728\) 0 0
\(729\) −1.43489e7 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.60595e7 1.10401 0.552003 0.833842i \(-0.313864\pi\)
0.552003 + 0.833842i \(0.313864\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 2.47882e7i 1.66968i 0.550490 + 0.834842i \(0.314441\pi\)
−0.550490 + 0.834842i \(0.685559\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.55041e7i 1.69488i 0.530893 + 0.847439i \(0.321857\pi\)
−0.530893 + 0.847439i \(0.678143\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −3.64217e6 −0.231005 −0.115502 0.993307i \(-0.536848\pi\)
−0.115502 + 0.993307i \(0.536848\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.83413e7 −1.77402 −0.887009 0.461752i \(-0.847221\pi\)
−0.887009 + 0.461752i \(0.847221\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.88365e6i 0.422503i
\(768\) 0 0
\(769\) −1.33834e7 −0.816114 −0.408057 0.912956i \(-0.633794\pi\)
−0.408057 + 0.912956i \(0.633794\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.91839e7 1.75669 0.878345 0.478028i \(-0.158648\pi\)
0.878345 + 0.478028i \(0.158648\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 2.90966e7i − 1.71790i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.40688e7 −1.39405
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 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
\(797\) −2.63735e7 −1.47069 −0.735347 0.677691i \(-0.762981\pi\)
−0.735347 + 0.677691i \(0.762981\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −3.12950e7 −1.72343
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 4.58114e7 2.49163
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.58690e7 1.38966 0.694829 0.719175i \(-0.255480\pi\)
0.694829 + 0.719175i \(0.255480\pi\)
\(810\) 0 0
\(811\) − 6.54480e6i − 0.349417i −0.984620 0.174709i \(-0.944102\pi\)
0.984620 0.174709i \(-0.0558984\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 1.47015e7i 0.765864i
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 9.84056e6i 0.506431i 0.967410 + 0.253215i \(0.0814881\pi\)
−0.967410 + 0.253215i \(0.918512\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 3.07862e7i − 1.52806i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 2.05111e7 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.73739e7 −0.837056
\(846\) 0 0
\(847\) − 1.18594e8i − 5.68007i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 5.28205e7i − 2.50022i
\(852\) 0 0
\(853\) 3.76003e7 1.76937 0.884685 0.466189i \(-0.154373\pi\)
0.884685 + 0.466189i \(0.154373\pi\)
\(854\) 0 0
\(855\) 2.60000e7i 1.21635i
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 4.04040e7i 1.86828i 0.356909 + 0.934139i \(0.383831\pi\)
−0.356909 + 0.934139i \(0.616169\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 3.27632e7i − 1.49748i −0.662866 0.748738i \(-0.730660\pi\)
0.662866 0.748738i \(-0.269340\pi\)
\(864\) 0 0
\(865\) −1.97588e7 −0.897882
\(866\) 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
\(875\) 4.29688e7i 1.89729i
\(876\) 0 0
\(877\) 2.60890e7 1.14540 0.572702 0.819763i \(-0.305895\pi\)
0.572702 + 0.819763i \(0.305895\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2.19751e7 0.953874 0.476937 0.878937i \(-0.341747\pi\)
0.476937 + 0.878937i \(0.341747\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1.71978e7i − 0.733946i −0.930232 0.366973i \(-0.880394\pi\)
0.930232 0.366973i \(-0.119606\pi\)
\(888\) 0 0
\(889\) 7.22975e7 3.06810
\(890\) 0 0
\(891\) − 4.73573e7i − 1.99845i
\(892\) 0 0
\(893\) 2.94025e7 1.23383
\(894\) 0 0
\(895\) − 1.76753e7i − 0.737582i
\(896\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.64557e7 −0.646239
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 4.95429e7 1.90382
\(926\) 0 0
\(927\) 4.68869e7i 1.79206i
\(928\) 0 0
\(929\) 9.58531e6 0.364391 0.182195 0.983262i \(-0.441680\pi\)
0.182195 + 0.983262i \(0.441680\pi\)
\(930\) 0 0
\(931\) 8.36284e7i 3.16213i
\(932\) 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
\(943\) − 5.06491e7i − 1.85478i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.86292e7 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 2.80976e7i 0.966280i 0.875543 + 0.483140i \(0.160504\pi\)
−0.875543 + 0.483140i \(0.839496\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 5.13311e7i 1.74716i 0.486681 + 0.873580i \(0.338207\pi\)
−0.486681 + 0.873580i \(0.661793\pi\)
\(972\) 0 0
\(973\) 4.12881e6 0.139811
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) − 1.03286e8i − 3.44418i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.03844e7i 0.672844i 0.941711 + 0.336422i \(0.109217\pi\)
−0.941711 + 0.336422i \(0.890783\pi\)
\(984\) 0 0
\(985\) −6.05687e7 −1.98911
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 6.08649e7 1.93923 0.969614 0.244639i \(-0.0786694\pi\)
0.969614 + 0.244639i \(0.0786694\pi\)
\(998\) 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 320.6.f.a.289.4 yes 4
4.3 odd 2 inner 320.6.f.a.289.3 yes 4
5.4 even 2 inner 320.6.f.a.289.1 4
8.3 odd 2 inner 320.6.f.a.289.1 4
8.5 even 2 inner 320.6.f.a.289.2 yes 4
20.19 odd 2 inner 320.6.f.a.289.2 yes 4
40.19 odd 2 CM 320.6.f.a.289.4 yes 4
40.29 even 2 inner 320.6.f.a.289.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.6.f.a.289.1 4 5.4 even 2 inner
320.6.f.a.289.1 4 8.3 odd 2 inner
320.6.f.a.289.2 yes 4 8.5 even 2 inner
320.6.f.a.289.2 yes 4 20.19 odd 2 inner
320.6.f.a.289.3 yes 4 4.3 odd 2 inner
320.6.f.a.289.3 yes 4 40.29 even 2 inner
320.6.f.a.289.4 yes 4 1.1 even 1 trivial
320.6.f.a.289.4 yes 4 40.19 odd 2 CM