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Properties

Label 304.7.e.a.113.1
Level $304$
Weight $7$
Character 304.113
Self dual yes
Analytic conductor $69.936$
Analytic rank $0$
Dimension $1$
CM discriminant -19
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,7,Mod(113,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.113");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 304.e (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: \(69.9364414204\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 113.1
Character \(\chi\) \(=\) 304.113

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-54.0000 q^{5} -610.000 q^{7} +729.000 q^{9} +O(q^{10})\) \(q-54.0000 q^{5} -610.000 q^{7} +729.000 q^{9} +1062.00 q^{11} -9630.00 q^{17} +6859.00 q^{19} -20610.0 q^{23} -12709.0 q^{25} +32940.0 q^{35} +142630. q^{43} -39366.0 q^{45} +75150.0 q^{47} +254451. q^{49} -57348.0 q^{55} -57062.0 q^{61} -444690. q^{63} +384050. q^{73} -647820. q^{77} +531441. q^{81} +1.13103e6 q^{83} +520020. q^{85} -370386. q^{95} +774198. q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) −54.0000 −0.432000 −0.216000 − 0.976393i \(-0.569301\pi\)
−0.216000 + 0.976393i \(0.569301\pi\)
\(6\) 0 0
\(7\) −610.000 −1.77843 −0.889213 − 0.457494i \(-0.848747\pi\)
−0.889213 + 0.457494i \(0.848747\pi\)
\(8\) 0 0
\(9\) 729.000 1.00000
\(10\) 0 0
\(11\) 1062.00 0.797896 0.398948 − 0.916973i \(-0.369375\pi\)
0.398948 + 0.916973i \(0.369375\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −9630.00 −1.96011 −0.980053 − 0.198737i \(-0.936316\pi\)
−0.980053 + 0.198737i \(0.936316\pi\)
\(18\) 0 0
\(19\) 6859.00 1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −20610.0 −1.69393 −0.846963 − 0.531652i \(-0.821572\pi\)
−0.846963 + 0.531652i \(0.821572\pi\)
\(24\) 0 0
\(25\) −12709.0 −0.813376
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 32940.0 0.768280
\(36\) 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
\(43\) 142630. 1.79393 0.896965 − 0.442101i \(-0.145767\pi\)
0.896965 + 0.442101i \(0.145767\pi\)
\(44\) 0 0
\(45\) −39366.0 −0.432000
\(46\) 0 0
\(47\) 75150.0 0.723828 0.361914 − 0.932211i \(-0.382123\pi\)
0.361914 + 0.932211i \(0.382123\pi\)
\(48\) 0 0
\(49\) 254451. 2.16280
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −57348.0 −0.344691
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −57062.0 −0.251395 −0.125698 − 0.992069i \(-0.540117\pi\)
−0.125698 + 0.992069i \(0.540117\pi\)
\(62\) 0 0
\(63\) −444690. −1.77843
\(64\) 0 0
\(65\) 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
\(73\) 384050. 0.987232 0.493616 − 0.869680i \(-0.335675\pi\)
0.493616 + 0.869680i \(0.335675\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −647820. −1.41900
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 531441. 1.00000
\(82\) 0 0
\(83\) 1.13103e6 1.97806 0.989031 − 0.147709i \(-0.0471899\pi\)
0.989031 + 0.147709i \(0.0471899\pi\)
\(84\) 0 0
\(85\) 520020. 0.846766
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −370386. −0.432000
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 774198. 0.797896
\(100\) 0 0
\(101\) 2.06030e6 1.99970 0.999852 − 0.0171767i \(-0.00546777\pi\)
0.999852 + 0.0171767i \(0.00546777\pi\)
\(102\) 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
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 1.11294e6 0.731776
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.87430e6 3.48590
\(120\) 0 0
\(121\) −643717. −0.363361
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.53004e6 0.783378
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.30228e6 −0.579283 −0.289642 − 0.957135i \(-0.593536\pi\)
−0.289642 + 0.957135i \(0.593536\pi\)
\(132\) 0 0
\(133\) −4.18399e6 −1.77843
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.22309e6 0.864560 0.432280 − 0.901739i \(-0.357709\pi\)
0.432280 + 0.901739i \(0.357709\pi\)
\(138\) 0 0
\(139\) −3.77334e6 −1.40502 −0.702508 − 0.711676i \(-0.747937\pi\)
−0.702508 + 0.711676i \(0.747937\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.24350e6 1.88742 0.943711 − 0.330770i \(-0.107308\pi\)
0.943711 + 0.330770i \(0.107308\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −7.02027e6 −1.96011
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −738470. −0.190824 −0.0954122 − 0.995438i \(-0.530417\pi\)
−0.0954122 + 0.995438i \(0.530417\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.25721e7 3.01252
\(162\) 0 0
\(163\) 4.30175e6 0.993304 0.496652 − 0.867950i \(-0.334562\pi\)
0.496652 + 0.867950i \(0.334562\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 4.82681e6 1.00000
\(170\) 0 0
\(171\) 5.00021e6 1.00000
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 7.75249e6 1.44653
\(176\) 0 0
\(177\) 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
\(187\) −1.02271e7 −1.56396
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.37384e6 −1.34529 −0.672647 − 0.739963i \(-0.734843\pi\)
−0.672647 + 0.739963i \(0.734843\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −9.74943e6 −1.27521 −0.637603 − 0.770365i \(-0.720074\pi\)
−0.637603 + 0.770365i \(0.720074\pi\)
\(198\) 0 0
\(199\) 1.52712e7 1.93782 0.968911 − 0.247410i \(-0.0795793\pi\)
0.968911 + 0.247410i \(0.0795793\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.50247e7 −1.69393
\(208\) 0 0
\(209\) 7.28426e6 0.797896
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −7.70202e6 −0.774978
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −9.26486e6 −0.813376
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 2.66958e6 0.222298 0.111149 − 0.993804i \(-0.464547\pi\)
0.111149 + 0.993804i \(0.464547\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.48115e7 −1.96148 −0.980742 − 0.195308i \(-0.937429\pi\)
−0.980742 + 0.195308i \(0.937429\pi\)
\(234\) 0 0
\(235\) −4.05810e6 −0.312694
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.53322e7 1.12308 0.561541 − 0.827449i \(-0.310209\pi\)
0.561541 + 0.827449i \(0.310209\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.37404e7 −0.934329
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.08340e6 0.321464 0.160732 − 0.986998i \(-0.448614\pi\)
0.160732 + 0.986998i \(0.448614\pi\)
\(252\) 0 0
\(253\) −2.18878e7 −1.35158
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.76102e7 −0.968049 −0.484024 − 0.875054i \(-0.660825\pi\)
−0.484024 + 0.875054i \(0.660825\pi\)
\(264\) 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
\(271\) −2.84226e7 −1.42809 −0.714045 − 0.700100i \(-0.753139\pi\)
−0.714045 + 0.700100i \(0.753139\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.34970e7 −0.648990
\(276\) 0 0
\(277\) 2.99803e7 1.41058 0.705289 − 0.708920i \(-0.250817\pi\)
0.705289 + 0.708920i \(0.250817\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 3.32756e7 1.46814 0.734068 − 0.679076i \(-0.237619\pi\)
0.734068 + 0.679076i \(0.237619\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.85993e7 2.84201
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 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
\(301\) −8.70043e7 −3.19037
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.08135e6 0.108603
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.42879e7 −1.47233 −0.736164 − 0.676804i \(-0.763365\pi\)
−0.736164 + 0.676804i \(0.763365\pi\)
\(312\) 0 0
\(313\) −3.19739e7 −1.04271 −0.521353 − 0.853341i \(-0.674573\pi\)
−0.521353 + 0.853341i \(0.674573\pi\)
\(314\) 0 0
\(315\) 2.40133e7 0.768280
\(316\) 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
\(323\) −6.60522e7 −1.96011
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.58415e7 −1.28727
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −8.34492e7 −2.06795
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.37186e7 −1.52503 −0.762515 − 0.646971i \(-0.776035\pi\)
−0.762515 + 0.646971i \(0.776035\pi\)
\(348\) 0 0
\(349\) −4.62042e7 −1.08694 −0.543469 − 0.839429i \(-0.682890\pi\)
−0.543469 + 0.839429i \(0.682890\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5.80008e7 1.31859 0.659295 − 0.751885i \(-0.270855\pi\)
0.659295 + 0.751885i \(0.270855\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.96053e7 1.72052 0.860258 − 0.509858i \(-0.170302\pi\)
0.860258 + 0.509858i \(0.170302\pi\)
\(360\) 0 0
\(361\) 4.70459e7 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.07387e7 −0.426484
\(366\) 0 0
\(367\) 2.00784e7 0.406191 0.203095 − 0.979159i \(-0.434900\pi\)
0.203095 + 0.979159i \(0.434900\pi\)
\(368\) 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
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 3.49823e7 0.613008
\(386\) 0 0
\(387\) 1.03977e8 1.79393
\(388\) 0 0
\(389\) 6.58748e7 1.11910 0.559552 − 0.828795i \(-0.310973\pi\)
0.559552 + 0.828795i \(0.310973\pi\)
\(390\) 0 0
\(391\) 1.98474e8 3.32027
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −6.12375e7 −0.978692 −0.489346 − 0.872090i \(-0.662764\pi\)
−0.489346 + 0.872090i \(0.662764\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −2.86978e7 −0.432000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.10756e7 −0.854523
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.11183e7 −0.558976 −0.279488 − 0.960149i \(-0.590165\pi\)
−0.279488 + 0.960149i \(0.590165\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 5.47844e7 0.723828
\(424\) 0 0
\(425\) 1.22388e8 1.59430
\(426\) 0 0
\(427\) 3.48078e7 0.447088
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.41364e8 −1.69393
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.85495e8 2.16280
\(442\) 0 0
\(443\) −2.64025e7 −0.303692 −0.151846 − 0.988404i \(-0.548522\pi\)
−0.151846 + 0.988404i \(0.548522\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.47451e8 1.54490 0.772449 − 0.635077i \(-0.219032\pi\)
0.772449 + 0.635077i \(0.219032\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.95676e8 −1.99726 −0.998631 − 0.0523154i \(-0.983340\pi\)
−0.998631 + 0.0523154i \(0.983340\pi\)
\(462\) 0 0
\(463\) 5.51769e7 0.555923 0.277961 − 0.960592i \(-0.410341\pi\)
0.277961 + 0.960592i \(0.410341\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.67407e8 −1.64370 −0.821849 − 0.569706i \(-0.807057\pi\)
−0.821849 + 0.569706i \(0.807057\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.51473e8 1.43137
\(474\) 0 0
\(475\) −8.71710e7 −0.813376
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.87497e8 1.70603 0.853015 − 0.521886i \(-0.174771\pi\)
0.853015 + 0.521886i \(0.174771\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.09684e8 0.926610 0.463305 − 0.886199i \(-0.346663\pi\)
0.463305 + 0.886199i \(0.346663\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −4.18067e7 −0.344691
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.47627e8 1.99295 0.996475 − 0.0838961i \(-0.0267364\pi\)
0.996475 + 0.0838961i \(0.0267364\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −9.84619e7 −0.773685 −0.386843 − 0.922146i \(-0.626434\pi\)
−0.386843 + 0.922146i \(0.626434\pi\)
\(504\) 0 0
\(505\) −1.11256e8 −0.863873
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −2.34270e8 −1.75572
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 7.98093e7 0.577540
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 2.76736e8 1.86939
\(530\) 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
\(539\) 2.70227e8 1.72569
\(540\) 0 0
\(541\) 3.05822e8 1.93142 0.965709 − 0.259626i \(-0.0835991\pi\)
0.965709 + 0.259626i \(0.0835991\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) −4.15982e7 −0.251395
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.93764e8 1.69994 0.849970 − 0.526831i \(-0.176620\pi\)
0.849970 + 0.526831i \(0.176620\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −3.24179e8 −1.77843
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 3.51908e8 1.89026 0.945130 − 0.326696i \(-0.105935\pi\)
0.945130 + 0.326696i \(0.105935\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.61932e8 1.37780
\(576\) 0 0
\(577\) −3.57513e8 −1.86107 −0.930537 − 0.366197i \(-0.880660\pi\)
−0.930537 + 0.366197i \(0.880660\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.89928e8 −3.51784
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.60908e8 1.28995 0.644975 − 0.764204i \(-0.276868\pi\)
0.644975 + 0.764204i \(0.276868\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.16214e7 0.151641 0.0758206 − 0.997121i \(-0.475842\pi\)
0.0758206 + 0.997121i \(0.475842\pi\)
\(594\) 0 0
\(595\) −3.17212e8 −1.50591
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.47607e7 0.156972
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −3.06883e8 −1.33227 −0.666134 − 0.745832i \(-0.732052\pi\)
−0.666134 + 0.745832i \(0.732052\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.35173e6 0.0355567 0.0177783 − 0.999842i \(-0.494341\pi\)
0.0177783 + 0.999842i \(0.494341\pi\)
\(618\) 0 0
\(619\) −4.70840e8 −1.98519 −0.992594 − 0.121480i \(-0.961236\pi\)
−0.992594 + 0.121480i \(0.961236\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.15956e8 0.474957
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.23521e8 −0.491647 −0.245824 − 0.969315i \(-0.579058\pi\)
−0.245824 + 0.969315i \(0.579058\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −3.20897e6 −0.0120707 −0.00603535 − 0.999982i \(-0.501921\pi\)
−0.00603535 + 0.999982i \(0.501921\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.47031e8 −0.912092 −0.456046 − 0.889956i \(-0.650735\pi\)
−0.456046 + 0.889956i \(0.650735\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4.26976e8 −1.53343 −0.766714 − 0.641988i \(-0.778110\pi\)
−0.766714 + 0.641988i \(0.778110\pi\)
\(654\) 0 0
\(655\) 7.03232e7 0.250250
\(656\) 0 0
\(657\) 2.79972e8 0.987232
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.25935e8 0.768280
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.05998e7 −0.200588
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 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
\(685\) −1.20047e8 −0.373490
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −2.21024e8 −0.669892 −0.334946 − 0.942237i \(-0.608718\pi\)
−0.334946 + 0.942237i \(0.608718\pi\)
\(692\) 0 0
\(693\) −4.72261e8 −1.41900
\(694\) 0 0
\(695\) 2.03760e8 0.606967
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.94922e8 −0.856156 −0.428078 − 0.903742i \(-0.640809\pi\)
−0.428078 + 0.903742i \(0.640809\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.25678e9 −3.55633
\(708\) 0 0
\(709\) −3.01929e8 −0.847161 −0.423580 − 0.905859i \(-0.639227\pi\)
−0.423580 + 0.905859i \(0.639227\pi\)
\(710\) 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
\(719\) 6.00444e8 1.61542 0.807711 − 0.589579i \(-0.200706\pi\)
0.807711 + 0.589579i \(0.200706\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.34161e8 −0.349158 −0.174579 − 0.984643i \(-0.555856\pi\)
−0.174579 + 0.984643i \(0.555856\pi\)
\(728\) 0 0
\(729\) 3.87420e8 1.00000
\(730\) 0 0
\(731\) −1.37353e9 −3.51629
\(732\) 0 0
\(733\) −1.31191e6 −0.00333113 −0.00166557 − 0.999999i \(-0.500530\pi\)
−0.00166557 + 0.999999i \(0.500530\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 7.32517e8 1.81503 0.907517 − 0.420016i \(-0.137976\pi\)
0.907517 + 0.420016i \(0.137976\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −3.37149e8 −0.815367
\(746\) 0 0
\(747\) 8.24521e8 1.97806
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.65794e8 1.99585 0.997923 − 0.0644255i \(-0.0205215\pi\)
0.997923 + 0.0644255i \(0.0205215\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 7.84034e8 1.77902 0.889510 − 0.456915i \(-0.151046\pi\)
0.889510 + 0.456915i \(0.151046\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 3.79095e8 0.846766
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −6.84693e8 −1.50563 −0.752813 − 0.658235i \(-0.771303\pi\)
−0.752813 + 0.658235i \(0.771303\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.98774e7 0.0824361
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 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
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) −7.23694e8 −1.41878
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.07861e8 0.787709
\(804\) 0 0
\(805\) −6.78893e8 −1.30141
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −9.14710e8 −1.72758 −0.863790 − 0.503853i \(-0.831915\pi\)
−0.863790 + 0.503853i \(0.831915\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.32294e8 −0.429107
\(816\) 0 0
\(817\) 9.78299e8 1.79393
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.70493e8 −1.39232 −0.696160 − 0.717886i \(-0.745110\pi\)
−0.696160 + 0.717886i \(0.745110\pi\)
\(822\) 0 0
\(823\) 6.52977e8 1.17138 0.585691 − 0.810534i \(-0.300823\pi\)
0.585691 + 0.810534i \(0.300823\pi\)
\(824\) 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
\(832\) 0 0
\(833\) −2.45036e9 −4.23931
\(834\) 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
\(841\) 5.94823e8 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.60648e8 −0.432000
\(846\) 0 0
\(847\) 3.92667e8 0.646211
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.15558e9 1.86189 0.930947 − 0.365156i \(-0.118984\pi\)
0.930947 + 0.365156i \(0.118984\pi\)
\(854\) 0 0
\(855\) −2.70011e8 −0.432000
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 2.30012e7 0.0362886 0.0181443 − 0.999835i \(-0.494224\pi\)
0.0181443 + 0.999835i \(0.494224\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 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
\(875\) −9.33322e8 −1.39318
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.09554e9 1.60214 0.801071 − 0.598569i \(-0.204264\pi\)
0.801071 + 0.598569i \(0.204264\pi\)
\(882\) 0 0
\(883\) 1.37094e9 1.99130 0.995648 − 0.0931960i \(-0.0297083\pi\)
0.995648 + 0.0931960i \(0.0297083\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 5.64390e8 0.797896
\(892\) 0 0
\(893\) 5.15454e8 0.723828
\(894\) 0 0
\(895\) 0 0
\(896\) 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 \(\pi\)
\(908\) 0 0
\(909\) 1.50196e9 1.99970
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 1.20115e9 1.57829
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.94392e8 1.03021
\(918\) 0 0
\(919\) −1.00603e9 −1.29618 −0.648091 − 0.761563i \(-0.724432\pi\)
−0.648091 + 0.761563i \(0.724432\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.39761e9 −1.74316 −0.871582 − 0.490250i \(-0.836905\pi\)
−0.871582 + 0.490250i \(0.836905\pi\)
\(930\) 0 0
\(931\) 1.74528e9 2.16280
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.52261e8 0.675631
\(936\) 0 0
\(937\) −8.44763e8 −1.02687 −0.513436 − 0.858128i \(-0.671628\pi\)
−0.513436 + 0.858128i \(0.671628\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.20501e9 1.41886 0.709429 − 0.704777i \(-0.248953\pi\)
0.709429 + 0.704777i \(0.248953\pi\)
\(948\) 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
\(955\) 5.06187e8 0.581167
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.35608e9 −1.53756
\(960\) 0 0
\(961\) 8.87504e8 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 7.13911e8 0.789523 0.394761 − 0.918784i \(-0.370827\pi\)
0.394761 + 0.918784i \(0.370827\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 2.30174e9 2.49872
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 5.26469e8 0.550889
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.93960e9 −3.03879
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −8.24645e8 −0.837139
\(996\) 0 0
\(997\) 1.81423e9 1.83066 0.915329 − 0.402707i \(-0.131931\pi\)
0.915329 + 0.402707i \(0.131931\pi\)
\(998\) 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 304.7.e.a.113.1 1
4.3 odd 2 19.7.b.a.18.1 ✓ 1
12.11 even 2 171.7.c.a.37.1 1
19.18 odd 2 CM 304.7.e.a.113.1 1
76.75 even 2 19.7.b.a.18.1 ✓ 1
228.227 odd 2 171.7.c.a.37.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
19.7.b.a.18.1 ✓ 1 4.3 odd 2
19.7.b.a.18.1 ✓ 1 76.75 even 2
171.7.c.a.37.1 1 12.11 even 2
171.7.c.a.37.1 1 228.227 odd 2
304.7.e.a.113.1 1 1.1 even 1 trivial
304.7.e.a.113.1 1 19.18 odd 2 CM