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Properties

Label 9.76.a.a.1.1
Level $9$
Weight $76$
Character 9.1
Self dual yes
Analytic conductor $320.606$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9,76,Mod(1,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 76, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 76);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 76 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

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: \(320.605553540\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.77789e22 q^{4} +9.74884e31 q^{7} +O(q^{10})\) \(q-3.77789e22 q^{4} +9.74884e31 q^{7} -9.38529e41 q^{13} +1.42725e45 q^{16} +1.29171e48 q^{19} -2.64698e52 q^{25} -3.68301e54 q^{28} +1.10940e56 q^{31} -1.42572e58 q^{37} +3.14159e61 q^{43} +7.09212e63 q^{49} +3.54566e64 q^{52} -1.77438e67 q^{61} -5.39199e67 q^{64} +5.77008e68 q^{67} +3.54520e69 q^{73} -4.87993e70 q^{76} -2.80272e71 q^{79} -9.14957e73 q^{91} -2.47825e74 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −3.77789e22 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 9.74884e31 1.98507 0.992536 − 0.121949i \(-0.0389144\pi\)
0.992536 + 0.121949i \(0.0389144\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −9.38529e41 −1.58333 −0.791666 − 0.610954i \(-0.790786\pi\)
−0.791666 + 0.610954i \(0.790786\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.42725e45 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 1.29171e48 1.43848 0.719241 − 0.694761i \(-0.244490\pi\)
0.719241 + 0.694761i \(0.244490\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −2.64698e52 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −3.68301e54 −1.98507
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.10940e56 1.31529 0.657647 − 0.753326i \(-0.271552\pi\)
0.657647 + 0.753326i \(0.271552\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.42572e58 −0.222060 −0.111030 − 0.993817i \(-0.535415\pi\)
−0.111030 + 0.993817i \(0.535415\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 3.14159e61 1.74616 0.873078 − 0.487581i \(-0.162121\pi\)
0.873078 + 0.487581i \(0.162121\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.09212e63 2.94051
\(50\) 0 0
\(51\) 0 0
\(52\) 3.54566e64 1.58333
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.77438e67 −1.99149 −0.995745 − 0.0921488i \(-0.970626\pi\)
−0.995745 + 0.0921488i \(0.970626\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −5.39199e67 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 5.77008e68 1.92033 0.960166 − 0.279430i \(-0.0901453\pi\)
0.960166 + 0.279430i \(0.0901453\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 3.54520e69 0.473187 0.236594 − 0.971609i \(-0.423969\pi\)
0.236594 + 0.971609i \(0.423969\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −4.87993e70 −1.43848
\(77\) 0 0
\(78\) 0 0
\(79\) −2.80272e71 −1.93448 −0.967238 − 0.253869i \(-0.918297\pi\)
−0.967238 + 0.253869i \(0.918297\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 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
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −9.14957e73 −3.14303
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.47825e74 −0.776613 −0.388307 − 0.921530i \(-0.626940\pi\)
−0.388307 + 0.921530i \(0.626940\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e75 1.00000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 5.79513e75 1.91279 0.956395 − 0.292075i \(-0.0943458\pi\)
0.956395 + 0.292075i \(0.0943458\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −4.00311e76 −1.58090 −0.790449 − 0.612528i \(-0.790153\pi\)
−0.790449 + 0.612528i \(0.790153\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.39140e77 1.98507
\(113\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.27190e78 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −4.19118e78 −1.31529
\(125\) 0 0
\(126\) 0 0
\(127\) −2.90520e78 −0.371995 −0.185997 − 0.982550i \(-0.559552\pi\)
−0.185997 + 0.982550i \(0.559552\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 1.25927e80 2.85549
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −2.66039e80 −1.15316 −0.576581 − 0.817040i \(-0.695613\pi\)
−0.576581 + 0.817040i \(0.695613\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 5.38624e80 0.222060
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −3.09177e81 −0.600582 −0.300291 − 0.953848i \(-0.597084\pi\)
−0.300291 + 0.953848i \(0.597084\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.03191e82 −0.915519 −0.457759 − 0.889076i \(-0.651348\pi\)
−0.457759 + 0.889076i \(0.651348\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.70891e83 1.88662 0.943309 − 0.331917i \(-0.107695\pi\)
0.943309 + 0.331917i \(0.107695\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 5.29477e83 1.50694
\(170\) 0 0
\(171\) 0 0
\(172\) −1.18686e84 −1.74616
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −2.58050e84 −1.98507
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 7.14631e84 1.55287 0.776435 − 0.630197i \(-0.217026\pi\)
0.776435 + 0.630197i \(0.217026\pi\)
\(182\) 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
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 7.87587e85 1.54134 0.770671 − 0.637233i \(-0.219921\pi\)
0.770671 + 0.637233i \(0.219921\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.67933e86 −2.94051
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 3.16151e86 1.96293 0.981463 − 0.191652i \(-0.0613846\pi\)
0.981463 + 0.191652i \(0.0613846\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.33951e87 −1.58333
\(209\) 0 0
\(210\) 0 0
\(211\) −6.83220e86 −0.472029 −0.236014 − 0.971750i \(-0.575841\pi\)
−0.236014 + 0.971750i \(0.575841\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.08153e88 2.61095
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.12756e87 −0.0978829 −0.0489414 − 0.998802i \(-0.515585\pi\)
−0.0489414 + 0.998802i \(0.515585\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −6.07647e88 −1.94904 −0.974519 − 0.224304i \(-0.927989\pi\)
−0.974519 + 0.224304i \(0.927989\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 2.12648e89 1.00467 0.502334 − 0.864674i \(-0.332475\pi\)
0.502334 + 0.864674i \(0.332475\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 6.70342e89 1.99149
\(245\) 0 0
\(246\) 0 0
\(247\) −1.21231e90 −2.27759
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 2.03704e90 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −1.38992e90 −0.440806
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −2.17987e91 −1.92033
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 2.61927e91 1.51996 0.759978 − 0.649949i \(-0.225210\pi\)
0.759978 + 0.649949i \(0.225210\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.45400e91 1.90282 0.951411 − 0.307924i \(-0.0996342\pi\)
0.951411 + 0.307924i \(0.0996342\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −1.90146e90 −0.0217318 −0.0108659 − 0.999941i \(-0.503459\pi\)
−0.0108659 + 0.999941i \(0.503459\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.92163e92 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −1.33934e92 −0.473187
\(293\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 3.06269e93 3.46625
\(302\) 0 0
\(303\) 0 0
\(304\) 1.84359e93 1.43848
\(305\) 0 0
\(306\) 0 0
\(307\) −3.12489e93 −1.68712 −0.843559 − 0.537036i \(-0.819544\pi\)
−0.843559 + 0.537036i \(0.819544\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 4.46014e92 0.116530 0.0582648 − 0.998301i \(-0.481443\pi\)
0.0582648 + 0.998301i \(0.481443\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.05884e94 1.93448
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 2.48427e94 1.58333
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5.49619e94 1.76405 0.882027 − 0.471199i \(-0.156179\pi\)
0.882027 + 0.471199i \(0.156179\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.94321e94 1.46343 0.731715 − 0.681610i \(-0.238720\pi\)
0.731715 + 0.681610i \(0.238720\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 4.56271e95 3.85206
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 2.29971e95 1.01325 0.506627 − 0.862165i \(-0.330892\pi\)
0.506627 + 0.862165i \(0.330892\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 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
\(361\) 8.62166e95 1.06923
\(362\) 0 0
\(363\) 0 0
\(364\) 3.45661e96 3.14303
\(365\) 0 0
\(366\) 0 0
\(367\) 2.99217e96 1.99990 0.999952 − 0.00975168i \(-0.00310411\pi\)
0.999952 + 0.00975168i \(0.00310411\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 3.15635e96 1.14843 0.574213 − 0.818706i \(-0.305308\pi\)
0.574213 + 0.818706i \(0.305308\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.86364e96 −0.372728 −0.186364 − 0.982481i \(-0.559670\pi\)
−0.186364 + 0.982481i \(0.559670\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 9.36256e96 0.776613
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.42323e97 1.90377 0.951883 − 0.306460i \(-0.0991447\pi\)
0.951883 + 0.306460i \(0.0991447\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −3.77789e97 −1.00000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −1.04120e98 −2.08255
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 8.02302e97 0.921964 0.460982 − 0.887410i \(-0.347497\pi\)
0.460982 + 0.887410i \(0.347497\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.18934e98 −1.91279
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −5.14605e98 −1.99938 −0.999690 − 0.0249076i \(-0.992071\pi\)
−0.999690 + 0.0249076i \(0.992071\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.72982e99 −3.95325
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 6.61004e98 0.895176 0.447588 − 0.894240i \(-0.352283\pi\)
0.447588 + 0.894240i \(0.352283\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.51233e99 1.58090
\(437\) 0 0
\(438\) 0 0
\(439\) −2.46300e99 −1.99088 −0.995441 − 0.0953799i \(-0.969593\pi\)
−0.995441 + 0.0953799i \(0.969593\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −5.25656e99 −1.98507
\(449\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.01632e100 −1.82042 −0.910210 − 0.414148i \(-0.864080\pi\)
−0.910210 + 0.414148i \(0.864080\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 1.80745e100 1.98508 0.992542 − 0.121902i \(-0.0388994\pi\)
0.992542 + 0.121902i \(0.0388994\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 5.62515e100 3.81200
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −3.41912e100 −1.43848
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 1.33808e100 0.351595
\(482\) 0 0
\(483\) 0 0
\(484\) 4.80508e100 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 1.12035e101 1.84935 0.924673 − 0.380763i \(-0.124339\pi\)
0.924673 + 0.380763i \(0.124339\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.58338e101 1.31529
\(497\) 0 0
\(498\) 0 0
\(499\) 5.75021e100 0.380990 0.190495 − 0.981688i \(-0.438991\pi\)
0.190495 + 0.981688i \(0.438991\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 1.09755e101 0.371995
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 3.45615e101 0.939311
\(512\) 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
\(521\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −1.02039e102 −1.16132 −0.580659 − 0.814147i \(-0.697205\pi\)
−0.580659 + 0.814147i \(0.697205\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.34768e102 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −4.75737e102 −2.85549
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.34052e102 −0.428921 −0.214460 − 0.976733i \(-0.568799\pi\)
−0.214460 + 0.976733i \(0.568799\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 9.44762e102 1.99893 0.999465 − 0.0327042i \(-0.0104119\pi\)
0.999465 + 0.0327042i \(0.0104119\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −2.73233e103 −3.84008
\(554\) 0 0
\(555\) 0 0
\(556\) 1.00507e103 1.15316
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −2.94848e103 −2.76474
\(560\) 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
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 2.14607e103 0.907381 0.453691 − 0.891159i \(-0.350107\pi\)
0.453691 + 0.891159i \(0.350107\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 6.99249e103 1.99774 0.998870 − 0.0475261i \(-0.0151337\pi\)
0.998870 + 0.0475261i \(0.0151337\pi\)
\(578\) 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
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 1.43301e104 1.89203
\(590\) 0 0
\(591\) 0 0
\(592\) −2.03486e103 −0.222060
\(593\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 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
\(601\) 2.84501e104 1.76316 0.881580 − 0.472035i \(-0.156480\pi\)
0.881580 + 0.472035i \(0.156480\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.16804e104 0.600582
\(605\) 0 0
\(606\) 0 0
\(607\) 3.26202e103 0.139287 0.0696436 − 0.997572i \(-0.477814\pi\)
0.0696436 + 0.997572i \(0.477814\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 2.98856e104 0.882460 0.441230 − 0.897394i \(-0.354542\pi\)
0.441230 + 0.897394i \(0.354542\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −8.30167e104 −1.70124 −0.850621 − 0.525779i \(-0.823774\pi\)
−0.850621 + 0.525779i \(0.823774\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 7.00649e104 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 7.67634e104 0.915519
\(629\) 0 0
\(630\) 0 0
\(631\) 1.99900e105 1.99394 0.996972 − 0.0777677i \(-0.0247792\pi\)
0.996972 + 0.0777677i \(0.0247792\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −6.65616e105 −4.65581
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −3.32995e105 −1.63880 −0.819402 − 0.573220i \(-0.805694\pi\)
−0.819402 + 0.573220i \(0.805694\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −6.45608e105 −1.88662
\(653\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 5.09491e105 0.890394 0.445197 − 0.895433i \(-0.353134\pi\)
0.445197 + 0.895433i \(0.353134\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 3.23733e105 0.288153 0.144076 − 0.989567i \(-0.453979\pi\)
0.144076 + 0.989567i \(0.453979\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −2.00031e106 −1.50694
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) −2.41600e106 −1.54163
\(680\) 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
\(688\) 4.48383e106 1.74616
\(689\) 0 0
\(690\) 0 0
\(691\) 5.41344e106 1.79080 0.895398 − 0.445266i \(-0.146891\pi\)
0.895398 + 0.445266i \(0.146891\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 9.74884e106 1.98507
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −1.84162e106 −0.319430
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 7.93973e106 1.00132 0.500659 − 0.865645i \(-0.333091\pi\)
0.500659 + 0.865645i \(0.333091\pi\)
\(710\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 5.64958e107 3.79703
\(722\) 0 0
\(723\) 0 0
\(724\) −2.69980e107 −1.55287
\(725\) 0 0
\(726\) 0 0
\(727\) −2.55571e107 −1.25884 −0.629421 − 0.777065i \(-0.716708\pi\)
−0.629421 + 0.777065i \(0.716708\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1.62304e107 −0.587393 −0.293697 − 0.955899i \(-0.594886\pi\)
−0.293697 + 0.955899i \(0.594886\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −3.00343e107 −0.800666 −0.400333 − 0.916370i \(-0.631106\pi\)
−0.400333 + 0.916370i \(0.631106\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −7.20567e107 −1.04997 −0.524985 − 0.851112i \(-0.675929\pi\)
−0.524985 + 0.851112i \(0.675929\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.31442e108 1.42114 0.710572 − 0.703624i \(-0.248436\pi\)
0.710572 + 0.703624i \(0.248436\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −3.90257e108 −3.13820
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1.41031e108 −0.845426 −0.422713 − 0.906264i \(-0.638922\pi\)
−0.422713 + 0.906264i \(0.638922\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.97542e108 −1.54134
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −2.93655e108 −1.31529
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.01222e109 2.94051
\(785\) 0 0
\(786\) 0 0
\(787\) −6.25966e107 −0.157579 −0.0787895 − 0.996891i \(-0.525106\pi\)
−0.0787895 + 0.996891i \(0.525106\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.66531e109 3.15319
\(794\) 0 0
\(795\) 0 0
\(796\) −1.19438e109 −1.96293
\(797\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 8.98707e108 0.733392 0.366696 − 0.930341i \(-0.380489\pi\)
0.366696 + 0.930341i \(0.380489\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4.05802e109 2.51181
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −1.09685e109 −0.516006 −0.258003 − 0.966144i \(-0.583064\pi\)
−0.258003 + 0.966144i \(0.583064\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 1.57047e109 0.562645 0.281323 − 0.959613i \(-0.409227\pi\)
0.281323 + 0.959613i \(0.409227\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.06054e109 1.58333
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −4.78465e109 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 2.58113e109 0.472029
\(845\) 0 0
\(846\) 0 0
\(847\) −1.23995e110 −1.98507
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.42218e110 1.74730 0.873649 − 0.486557i \(-0.161747\pi\)
0.873649 + 0.486557i \(0.161747\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −1.85193e110 −1.74937 −0.874684 − 0.484694i \(-0.838931\pi\)
−0.874684 + 0.484694i \(0.838931\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) −4.08591e110 −2.61095
\(869\) 0 0
\(870\) 0 0
\(871\) −5.41538e110 −3.04052
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 4.39522e110 1.90764 0.953819 − 0.300383i \(-0.0971146\pi\)
0.953819 + 0.300383i \(0.0971146\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 3.84148e110 1.29114 0.645568 − 0.763703i \(-0.276621\pi\)
0.645568 + 0.763703i \(0.276621\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −2.83224e110 −0.738437
\(890\) 0 0
\(891\) 0 0
\(892\) 4.25980e109 0.0978829
\(893\) 0 0
\(894\) 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
\(907\) −1.42185e111 −1.74816 −0.874079 − 0.485785i \(-0.838534\pi\)
−0.874079 + 0.485785i \(0.838534\pi\)
\(908\) 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
\(916\) 2.29562e111 1.94904
\(917\) 0 0
\(918\) 0 0
\(919\) 2.51602e111 1.88965 0.944827 − 0.327569i \(-0.106229\pi\)
0.944827 + 0.327569i \(0.106229\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 3.77386e110 0.222060
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 9.16095e111 4.22987
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −5.48906e111 −1.99188 −0.995940 − 0.0900192i \(-0.971307\pi\)
−0.995940 + 0.0900192i \(0.971307\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) −3.32727e111 −0.749212
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 5.19337e111 0.729998
\(962\) 0 0
\(963\) 0 0
\(964\) −8.03362e111 −1.00467
\(965\) 0 0
\(966\) 0 0
\(967\) 9.81810e111 1.09278 0.546392 − 0.837529i \(-0.316001\pi\)
0.546392 + 0.837529i \(0.316001\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −2.59357e112 −2.28911
\(974\) 0 0
\(975\) 0 0
\(976\) −2.53248e112 −1.99149
\(977\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 4.57996e112 2.27759
\(989\) 0 0
\(990\) 0 0
\(991\) −4.48974e112 −1.99278 −0.996389 − 0.0849106i \(-0.972940\pi\)
−0.996389 + 0.0849106i \(0.972940\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −3.37363e112 −1.19407 −0.597034 − 0.802216i \(-0.703654\pi\)
−0.597034 + 0.802216i \(0.703654\pi\)
\(998\) 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 9.76.a.a.1.1 ✓ 1
3.2 odd 2 CM 9.76.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.76.a.a.1.1 ✓ 1 1.1 even 1 trivial
9.76.a.a.1.1 ✓ 1 3.2 odd 2 CM