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Properties

Label 9.70.a.a.1.1
Level $9$
Weight $70$
Character 9.1
Self dual yes
Analytic conductor $271.363$
Analytic rank $1$
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] = [9,70,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, 70, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 70);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 70 \)
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: \(271.363457963\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \(q-5.90296e20 q^{4} -1.50237e29 q^{7} +O(q^{10})\) \(q-5.90296e20 q^{4} -1.50237e29 q^{7} +2.78287e38 q^{13} +3.48449e41 q^{16} -2.61033e44 q^{19} -1.69407e48 q^{25} +8.86842e49 q^{28} -1.23225e51 q^{31} +1.44188e54 q^{37} +4.34268e56 q^{43} +2.07060e57 q^{49} -1.64272e59 q^{52} +7.51818e61 q^{61} -2.05688e62 q^{64} +1.08612e63 q^{67} +2.88890e64 q^{73} +1.54087e65 q^{76} -2.79932e64 q^{79} -4.18090e67 q^{91} -6.35708e68 q^{97} +O(q^{100})\)

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
\(4\) −5.90296e20 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −1.50237e29 −1.04929 −0.524643 − 0.851322i \(-0.675801\pi\)
−0.524643 + 0.851322i \(0.675801\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 2.78287e38 1.03145 0.515724 − 0.856755i \(-0.327523\pi\)
0.515724 + 0.856755i \(0.327523\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.48449e41 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −2.61033e44 −1.99387 −0.996934 − 0.0782418i \(-0.975069\pi\)
−0.996934 + 0.0782418i \(0.975069\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.69407e48 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 8.86842e49 1.04929
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −1.23225e51 −0.435230 −0.217615 − 0.976035i \(-0.569828\pi\)
−0.217615 + 0.976035i \(0.569828\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.44188e54 1.13755 0.568773 − 0.822494i \(-0.307418\pi\)
0.568773 + 0.822494i \(0.307418\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 4.34268e56 1.91909 0.959546 − 0.281552i \(-0.0908490\pi\)
0.959546 + 0.281552i \(0.0908490\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 2.07060e57 0.101002
\(50\) 0 0
\(51\) 0 0
\(52\) −1.64272e59 −1.03145
\(53\) 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
\(61\) 7.51818e61 1.91529 0.957643 − 0.287958i \(-0.0929763\pi\)
0.957643 + 0.287958i \(0.0929763\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −2.05688e62 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.08612e63 1.08717 0.543586 − 0.839354i \(-0.317066\pi\)
0.543586 + 0.839354i \(0.317066\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.88890e64 1.50001 0.750004 − 0.661433i \(-0.230051\pi\)
0.750004 + 0.661433i \(0.230051\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 1.54087e65 1.99387
\(77\) 0 0
\(78\) 0 0
\(79\) −2.79932e64 −0.0952616 −0.0476308 − 0.998865i \(-0.515167\pi\)
−0.0476308 + 0.998865i \(0.515167\pi\)
\(80\) 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
\(91\) −4.18090e67 −1.08228
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.35708e68 −1.81816 −0.909082 − 0.416618i \(-0.863215\pi\)
−0.909082 + 0.416618i \(0.863215\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e69 1.00000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 4.12239e69 1.48684 0.743421 − 0.668824i \(-0.233202\pi\)
0.743421 + 0.668824i \(0.233202\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 1.28924e70 0.659358 0.329679 − 0.944093i \(-0.393059\pi\)
0.329679 + 0.944093i \(0.393059\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −5.23499e70 −1.04929
\(113\) 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
\(121\) −7.17952e71 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 7.27389e71 0.435230
\(125\) 0 0
\(126\) 0 0
\(127\) −6.03351e71 −0.158249 −0.0791246 − 0.996865i \(-0.525213\pi\)
−0.0791246 + 0.996865i \(0.525213\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 3.92168e73 2.09214
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −1.59749e74 −1.85964 −0.929818 − 0.368020i \(-0.880036\pi\)
−0.929818 + 0.368020i \(0.880036\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −8.51135e74 −1.13755
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 2.86990e75 1.91939 0.959693 − 0.281051i \(-0.0906829\pi\)
0.959693 + 0.281051i \(0.0906829\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −6.92883e74 −0.120815 −0.0604077 − 0.998174i \(-0.519240\pi\)
−0.0604077 + 0.998174i \(0.519240\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.20932e74 0.00578187 0.00289094 − 0.999996i \(-0.499080\pi\)
0.00289094 + 0.999996i \(0.499080\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 4.65035e75 0.0638843
\(170\) 0 0
\(171\) 0 0
\(172\) −2.56346e77 −1.91909
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 2.54511e77 1.04929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 9.88130e77 1.27322 0.636610 − 0.771186i \(-0.280336\pi\)
0.636610 + 0.771186i \(0.280336\pi\)
\(182\) 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
\(193\) −1.37425e79 −1.93348 −0.966739 − 0.255765i \(-0.917673\pi\)
−0.966739 + 0.255765i \(0.917673\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.22226e78 −0.101002
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 2.43908e79 1.19342 0.596711 − 0.802456i \(-0.296474\pi\)
0.596711 + 0.802456i \(0.296474\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 9.69689e79 1.03145
\(209\) 0 0
\(210\) 0 0
\(211\) −1.73030e80 −1.12299 −0.561496 − 0.827480i \(-0.689774\pi\)
−0.561496 + 0.827480i \(0.689774\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.85129e80 0.456681
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.70221e81 −1.63869 −0.819344 − 0.573302i \(-0.805662\pi\)
−0.819344 + 0.573302i \(0.805662\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 2.00580e81 0.772616 0.386308 − 0.922370i \(-0.373750\pi\)
0.386308 + 0.922370i \(0.373750\pi\)
\(230\) 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
\(241\) 2.29424e82 1.51723 0.758614 − 0.651540i \(-0.225877\pi\)
0.758614 + 0.651540i \(0.225877\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −4.43795e82 −1.91529
\(245\) 0 0
\(246\) 0 0
\(247\) −7.26421e82 −2.05657
\(248\) 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
\(256\) 1.21417e83 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −2.16623e83 −1.19361
\(260\) 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
\(268\) −6.41133e83 −1.08717
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −1.49826e84 −1.73040 −0.865198 − 0.501431i \(-0.832807\pi\)
−0.865198 + 0.501431i \(0.832807\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.05959e84 −1.66001 −0.830007 − 0.557753i \(-0.811664\pi\)
−0.830007 + 0.557753i \(0.811664\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −7.31869e84 −1.89584 −0.947920 − 0.318509i \(-0.896818\pi\)
−0.947920 + 0.318509i \(0.896818\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −7.96115e84 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −1.70530e85 −1.50001
\(293\) 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
\(301\) −6.52430e85 −2.01368
\(302\) 0 0
\(303\) 0 0
\(304\) −9.09568e85 −1.99387
\(305\) 0 0
\(306\) 0 0
\(307\) −1.23381e86 −1.92741 −0.963707 − 0.266961i \(-0.913980\pi\)
−0.963707 + 0.266961i \(0.913980\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −1.83542e86 −1.47047 −0.735234 − 0.677813i \(-0.762928\pi\)
−0.735234 + 0.677813i \(0.762928\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.65243e85 0.0952616
\(317\) 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
\(325\) −4.71436e86 −1.03145
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.62793e87 −1.89483 −0.947416 − 0.320004i \(-0.896316\pi\)
−0.947416 + 0.320004i \(0.896316\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3.78751e84 −0.00237204 −0.00118602 − 0.999999i \(-0.500378\pi\)
−0.00118602 + 0.999999i \(0.500378\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 2.76885e87 0.943306
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 1.04605e88 1.95918 0.979590 − 0.201005i \(-0.0644206\pi\)
0.979590 + 0.201005i \(0.0644206\pi\)
\(350\) 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
\(361\) 5.09988e88 2.97551
\(362\) 0 0
\(363\) 0 0
\(364\) 2.46797e88 1.08228
\(365\) 0 0
\(366\) 0 0
\(367\) −4.86529e88 −1.60742 −0.803711 − 0.595020i \(-0.797144\pi\)
−0.803711 + 0.595020i \(0.797144\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.05123e89 1.98492 0.992459 − 0.122580i \(-0.0391167\pi\)
0.992459 + 0.122580i \(0.0391167\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.81848e89 1.97996 0.989978 − 0.141223i \(-0.0451033\pi\)
0.989978 + 0.141223i \(0.0451033\pi\)
\(380\) 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
\(388\) 3.75256e89 1.81816
\(389\) 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
\(397\) −8.98787e89 −1.97417 −0.987085 − 0.160198i \(-0.948787\pi\)
−0.987085 + 0.160198i \(0.948787\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −5.90296e89 −1.00000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −3.42918e89 −0.448917
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.36100e90 1.07005 0.535024 − 0.844837i \(-0.320302\pi\)
0.535024 + 0.844837i \(0.320302\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.43343e90 −1.48684
\(413\) 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
\(421\) −6.71945e90 −1.94806 −0.974028 − 0.226429i \(-0.927295\pi\)
−0.974028 + 0.226429i \(0.927295\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.12951e91 −2.00968
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −3.17823e90 −0.349424 −0.174712 − 0.984620i \(-0.555899\pi\)
−0.174712 + 0.984620i \(0.555899\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −7.61036e90 −0.659358
\(437\) 0 0
\(438\) 0 0
\(439\) −6.56119e90 −0.448700 −0.224350 − 0.974509i \(-0.572026\pi\)
−0.224350 + 0.974509i \(0.572026\pi\)
\(440\) 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
\(448\) 3.09019e91 1.04929
\(449\) 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
\(457\) 6.42431e91 1.09829 0.549146 − 0.835727i \(-0.314953\pi\)
0.549146 + 0.835727i \(0.314953\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 1.71469e92 1.86913 0.934567 − 0.355788i \(-0.115787\pi\)
0.934567 + 0.355788i \(0.115787\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −1.63176e92 −1.14075
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.42208e92 1.99387
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 4.01256e92 1.17332
\(482\) 0 0
\(483\) 0 0
\(484\) 4.23804e92 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 6.82800e92 1.30180 0.650900 − 0.759163i \(-0.274392\pi\)
0.650900 + 0.759163i \(0.274392\pi\)
\(488\) 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
\(496\) −4.29375e92 −0.435230
\(497\) 0 0
\(498\) 0 0
\(499\) −2.41377e93 −1.98713 −0.993566 − 0.113258i \(-0.963871\pi\)
−0.993566 + 0.113258i \(0.963871\pi\)
\(500\) 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
\(508\) 3.56156e92 0.158249
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −4.34019e93 −1.57394
\(512\) 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
\(523\) −5.18032e93 −0.843428 −0.421714 − 0.906729i \(-0.638571\pi\)
−0.421714 + 0.906729i \(0.638571\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −9.10376e93 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −2.31495e94 −2.09214
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.48469e94 1.25883 0.629413 − 0.777071i \(-0.283295\pi\)
0.629413 + 0.777071i \(0.283295\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −5.74919e94 −1.99087 −0.995437 − 0.0954250i \(-0.969579\pi\)
−0.995437 + 0.0954250i \(0.969579\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 4.20561e93 0.0999567
\(554\) 0 0
\(555\) 0 0
\(556\) 9.42992e94 1.85964
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 1.20851e95 1.97944
\(560\) 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
\(571\) 2.54078e95 1.99996 0.999978 − 0.00659271i \(-0.00209854\pi\)
0.999978 + 0.00659271i \(0.00209854\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −8.38715e94 −0.460308 −0.230154 − 0.973154i \(-0.573923\pi\)
−0.230154 + 0.973154i \(0.573923\pi\)
\(578\) 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
\(589\) 3.21657e95 0.867791
\(590\) 0 0
\(591\) 0 0
\(592\) 5.02421e95 1.13755
\(593\) 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
\(601\) 1.13358e96 1.52504 0.762520 − 0.646965i \(-0.223962\pi\)
0.762520 + 0.646965i \(0.223962\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.69409e96 −1.91939
\(605\) 0 0
\(606\) 0 0
\(607\) −1.11990e96 −1.06947 −0.534735 − 0.845020i \(-0.679589\pi\)
−0.534735 + 0.845020i \(0.679589\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −2.92388e96 −1.98872 −0.994361 − 0.106052i \(-0.966179\pi\)
−0.994361 + 0.106052i \(0.966179\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −3.61907e96 −1.75902 −0.879508 − 0.475885i \(-0.842128\pi\)
−0.879508 + 0.475885i \(0.842128\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.86986e96 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 4.09006e95 0.120815
\(629\) 0 0
\(630\) 0 0
\(631\) −7.82593e96 −1.96121 −0.980603 − 0.196005i \(-0.937203\pi\)
−0.980603 + 0.196005i \(0.937203\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5.76220e95 0.104178
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 1.97491e96 0.258385 0.129193 − 0.991620i \(-0.458761\pi\)
0.129193 + 0.991620i \(0.458761\pi\)
\(644\) 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
\(652\) −7.13856e94 −0.00578187
\(653\) 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
\(661\) 2.92988e97 1.47877 0.739384 − 0.673284i \(-0.235117\pi\)
0.739384 + 0.673284i \(0.235117\pi\)
\(662\) 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
\(673\) 7.23048e97 1.96177 0.980884 − 0.194595i \(-0.0623394\pi\)
0.980884 + 0.194595i \(0.0623394\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −2.74508e96 −0.0638843
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 9.55068e97 1.90777
\(680\) 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
\(688\) 1.51320e98 1.91909
\(689\) 0 0
\(690\) 0 0
\(691\) −1.64591e98 −1.79644 −0.898221 − 0.439545i \(-0.855140\pi\)
−0.898221 + 0.439545i \(0.855140\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.50237e98 −1.04929
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −3.76378e98 −2.26812
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.98698e98 1.79205 0.896023 − 0.444008i \(-0.146444\pi\)
0.896023 + 0.444008i \(0.146444\pi\)
\(710\) 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
\(721\) −6.19335e98 −1.56012
\(722\) 0 0
\(723\) 0 0
\(724\) −5.83289e98 −1.27322
\(725\) 0 0
\(726\) 0 0
\(727\) 1.30373e98 0.246746 0.123373 − 0.992360i \(-0.460629\pi\)
0.123373 + 0.992360i \(0.460629\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −6.47639e98 −0.923093 −0.461547 − 0.887116i \(-0.652705\pi\)
−0.461547 + 0.887116i \(0.652705\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.82532e99 −1.96384 −0.981922 − 0.189285i \(-0.939383\pi\)
−0.981922 + 0.189285i \(0.939383\pi\)
\(740\) 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
\(751\) 3.17948e99 1.96236 0.981179 − 0.193102i \(-0.0618547\pi\)
0.981179 + 0.193102i \(0.0618547\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.41275e99 1.13163 0.565816 − 0.824532i \(-0.308561\pi\)
0.565816 + 0.824532i \(0.308561\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −1.93692e99 −0.691855
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 4.45657e99 1.21490 0.607452 − 0.794356i \(-0.292192\pi\)
0.607452 + 0.794356i \(0.292192\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.11215e99 1.93348
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 2.08750e99 0.435230
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 7.21498e98 0.101002
\(785\) 0 0
\(786\) 0 0
\(787\) −7.08289e99 −0.869125 −0.434563 − 0.900642i \(-0.643097\pi\)
−0.434563 + 0.900642i \(0.643097\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.09221e100 1.97552
\(794\) 0 0
\(795\) 0 0
\(796\) −1.43978e100 −1.19342
\(797\) 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
\(811\) 4.57416e100 1.99109 0.995547 − 0.0942641i \(-0.0300498\pi\)
0.995547 + 0.0942641i \(0.0300498\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.13358e101 −3.82642
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −1.97960e100 −0.519139 −0.259570 − 0.965724i \(-0.583581\pi\)
−0.259570 + 0.965724i \(0.583581\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −8.33059e100 −1.70037 −0.850187 − 0.526481i \(-0.823511\pi\)
−0.850187 + 0.526481i \(0.823511\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −5.72403e100 −1.03145
\(833\) 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
\(841\) −8.04381e100 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 1.02139e101 1.12299
\(845\) 0 0
\(846\) 0 0
\(847\) 1.07863e101 1.04929
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 8.97694e100 0.684521 0.342260 − 0.939605i \(-0.388807\pi\)
0.342260 + 0.939605i \(0.388807\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −9.13612e100 −0.547013 −0.273507 − 0.961870i \(-0.588184\pi\)
−0.273507 + 0.961870i \(0.588184\pi\)
\(860\) 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
\(868\) −1.09281e101 −0.456681
\(869\) 0 0
\(870\) 0 0
\(871\) 3.02254e101 1.12136
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 6.66390e101 1.95093 0.975466 − 0.220150i \(-0.0706546\pi\)
0.975466 + 0.220150i \(0.0706546\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 9.34719e100 0.216290 0.108145 − 0.994135i \(-0.465509\pi\)
0.108145 + 0.994135i \(0.465509\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 9.06455e100 0.166049
\(890\) 0 0
\(891\) 0 0
\(892\) 1.00481e102 1.63869
\(893\) 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
\(907\) 3.31226e101 0.303859 0.151930 − 0.988391i \(-0.451451\pi\)
0.151930 + 0.988391i \(0.451451\pi\)
\(908\) 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
\(916\) −1.18402e102 −0.772616
\(917\) 0 0
\(918\) 0 0
\(919\) −3.11476e102 −1.81568 −0.907839 − 0.419319i \(-0.862269\pi\)
−0.907839 + 0.419319i \(0.862269\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −2.44264e102 −1.13755
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) −5.40495e101 −0.201385
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −6.11804e102 −1.82640 −0.913201 − 0.407509i \(-0.866397\pi\)
−0.913201 + 0.407509i \(0.866397\pi\)
\(938\) 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
\(949\) 8.03943e102 1.54718
\(950\) 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
\(961\) −6.49756e102 −0.810575
\(962\) 0 0
\(963\) 0 0
\(964\) −1.35428e103 −1.51723
\(965\) 0 0
\(966\) 0 0
\(967\) −1.12070e103 −1.12791 −0.563955 − 0.825806i \(-0.690721\pi\)
−0.563955 + 0.825806i \(0.690721\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 2.40002e103 1.95129
\(974\) 0 0
\(975\) 0 0
\(976\) 2.61970e103 1.91529
\(977\) 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
\(988\) 4.28804e103 2.05657
\(989\) 0 0
\(990\) 0 0
\(991\) 7.12054e101 0.0307590 0.0153795 − 0.999882i \(-0.495104\pi\)
0.0153795 + 0.999882i \(0.495104\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 5.70029e103 1.99947 0.999733 − 0.0231264i \(-0.00736203\pi\)
0.999733 + 0.0231264i \(0.00736203\pi\)
\(998\) 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 9.70.a.a.1.1 ✓ 1
3.2 odd 2 CM 9.70.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.70.a.a.1.1 ✓ 1 1.1 even 1 trivial
9.70.a.a.1.1 ✓ 1 3.2 odd 2 CM