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Properties

Label 9.94.a.a.1.1
Level $9$
Weight $94$
Character 9.1
Self dual yes
Analytic conductor $492.953$
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] = [9,94,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, 94, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 94);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 94 \)
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: \(492.952887545\)
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)
 
\(f(q)\) \(=\) \(q-9.90352e27 q^{4} +3.64618e39 q^{7} +O(q^{10})\) \(q-9.90352e27 q^{4} +3.64618e39 q^{7} +1.06025e52 q^{13} +9.80797e55 q^{16} +5.52916e59 q^{19} -1.00974e65 q^{25} -3.61101e67 q^{28} -1.68448e69 q^{31} -1.63529e73 q^{37} -1.75311e76 q^{43} +9.36715e78 q^{49} -1.05002e80 q^{52} +6.66175e82 q^{61} -9.71334e83 q^{64} -9.11409e84 q^{67} -8.82009e86 q^{73} -5.47582e87 q^{76} -2.68500e88 q^{79} +3.86587e91 q^{91} -4.62654e92 q^{97} +O(q^{100})\)

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
\(4\) −9.90352e27 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 3.64618e39 1.83984 0.919919 − 0.392108i \(-0.128254\pi\)
0.919919 + 0.392108i \(0.128254\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.06025e52 1.68672 0.843359 − 0.537350i \(-0.180575\pi\)
0.843359 + 0.537350i \(0.180575\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 9.80797e55 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 5.52916e59 1.90817 0.954084 − 0.299538i \(-0.0968326\pi\)
0.954084 + 0.299538i \(0.0968326\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00974e65 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −3.61101e67 −1.83984
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −1.68448e69 −0.755346 −0.377673 − 0.925939i \(-0.623276\pi\)
−0.377673 + 0.925939i \(0.623276\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.63529e73 −1.95982 −0.979909 − 0.199447i \(-0.936085\pi\)
−0.979909 + 0.199447i \(0.936085\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −1.75311e76 −1.93876 −0.969382 − 0.245556i \(-0.921030\pi\)
−0.969382 + 0.245556i \(0.921030\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 9.36715e78 2.38501
\(50\) 0 0
\(51\) 0 0
\(52\) −1.05002e80 −1.68672
\(53\) 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
\(61\) 6.66175e82 0.639368 0.319684 − 0.947524i \(-0.396423\pi\)
0.319684 + 0.947524i \(0.396423\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −9.71334e83 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −9.11409e84 −1.11490 −0.557449 − 0.830211i \(-0.688220\pi\)
−0.557449 + 0.830211i \(0.688220\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −8.82009e86 −1.99968 −0.999838 − 0.0179739i \(-0.994278\pi\)
−0.999838 + 0.0179739i \(0.994278\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −5.47582e87 −1.90817
\(77\) 0 0
\(78\) 0 0
\(79\) −2.68500e88 −1.54627 −0.773134 − 0.634242i \(-0.781312\pi\)
−0.773134 + 0.634242i \(0.781312\pi\)
\(80\) 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
\(91\) 3.86587e91 3.10329
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.62654e92 −1.90709 −0.953544 − 0.301255i \(-0.902595\pi\)
−0.953544 + 0.301255i \(0.902595\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e93 1.00000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −7.67595e93 −1.94178 −0.970892 − 0.239516i \(-0.923011\pi\)
−0.970892 + 0.239516i \(0.923011\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 5.30529e94 0.964667 0.482333 − 0.875988i \(-0.339789\pi\)
0.482333 + 0.875988i \(0.339789\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.57617e95 1.83984
\(113\) 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
\(121\) −7.07163e96 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 1.66822e97 0.755346
\(125\) 0 0
\(126\) 0 0
\(127\) 1.11936e98 1.66762 0.833808 − 0.552055i \(-0.186156\pi\)
0.833808 + 0.552055i \(0.186156\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 2.01603e99 3.51072
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −2.15297e99 −0.481780 −0.240890 − 0.970552i \(-0.577439\pi\)
−0.240890 + 0.970552i \(0.577439\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 1.61952e101 1.95982
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −2.02878e101 −0.965620 −0.482810 − 0.875725i \(-0.660384\pi\)
−0.482810 + 0.875725i \(0.660384\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.54375e102 −1.97761 −0.988806 − 0.149205i \(-0.952329\pi\)
−0.988806 + 0.149205i \(0.952329\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.43952e103 −1.95657 −0.978284 − 0.207269i \(-0.933542\pi\)
−0.978284 + 0.207269i \(0.933542\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 7.29009e103 1.84502
\(170\) 0 0
\(171\) 0 0
\(172\) 1.73619e104 1.93876
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −3.68171e104 −1.83984
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 1.17196e105 1.22140 0.610701 − 0.791862i \(-0.290888\pi\)
0.610701 + 0.791862i \(0.290888\pi\)
\(182\) 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
\(193\) 1.53751e106 0.809848 0.404924 − 0.914350i \(-0.367298\pi\)
0.404924 + 0.914350i \(0.367298\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −9.27677e106 −2.38501
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −6.88599e106 −0.873571 −0.436786 − 0.899566i \(-0.643883\pi\)
−0.436786 + 0.899566i \(0.643883\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1.03989e108 1.68672
\(209\) 0 0
\(210\) 0 0
\(211\) 1.77977e108 1.48330 0.741651 − 0.670786i \(-0.234043\pi\)
0.741651 + 0.670786i \(0.234043\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6.14191e108 −1.38972
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2.03936e108 0.129813 0.0649064 − 0.997891i \(-0.479325\pi\)
0.0649064 + 0.997891i \(0.479325\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −3.03581e109 −0.562245 −0.281122 − 0.959672i \(-0.590707\pi\)
−0.281122 + 0.959672i \(0.590707\pi\)
\(230\) 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
\(241\) −5.34431e110 −0.920659 −0.460330 − 0.887748i \(-0.652269\pi\)
−0.460330 + 0.887748i \(0.652269\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −6.59748e110 −0.639368
\(245\) 0 0
\(246\) 0 0
\(247\) 5.86230e111 3.21854
\(248\) 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
\(256\) 9.61963e111 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −5.96258e112 −3.60575
\(260\) 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
\(268\) 9.02616e112 1.11490
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −8.45629e112 −0.622455 −0.311227 − 0.950335i \(-0.600740\pi\)
−0.311227 + 0.950335i \(0.600740\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.47283e113 1.98690 0.993449 − 0.114272i \(-0.0364535\pi\)
0.993449 + 0.114272i \(0.0364535\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −1.21972e114 −1.19726 −0.598632 − 0.801024i \(-0.704289\pi\)
−0.598632 + 0.801024i \(0.704289\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.70240e114 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 8.73499e114 1.99968
\(293\) 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
\(301\) −6.39215e115 −3.56701
\(302\) 0 0
\(303\) 0 0
\(304\) 5.42298e115 1.90817
\(305\) 0 0
\(306\) 0 0
\(307\) −1.01929e115 −0.227177 −0.113588 − 0.993528i \(-0.536234\pi\)
−0.113588 + 0.993528i \(0.536234\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 9.20851e115 0.834406 0.417203 − 0.908813i \(-0.363010\pi\)
0.417203 + 0.908813i \(0.363010\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 2.65910e116 1.54627
\(317\) 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
\(325\) −1.07058e117 −1.68672
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.64206e117 −1.10507 −0.552533 − 0.833491i \(-0.686338\pi\)
−0.552533 + 0.833491i \(0.686338\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9.43871e116 0.275500 0.137750 − 0.990467i \(-0.456013\pi\)
0.137750 + 0.990467i \(0.456013\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.98339e118 2.54819
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −3.19859e118 −1.83474 −0.917372 − 0.398030i \(-0.869694\pi\)
−0.917372 + 0.398030i \(0.869694\pi\)
\(350\) 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
\(361\) 2.21754e119 2.64111
\(362\) 0 0
\(363\) 0 0
\(364\) −3.82857e119 −3.10329
\(365\) 0 0
\(366\) 0 0
\(367\) 1.07518e119 0.594988 0.297494 − 0.954724i \(-0.403849\pi\)
0.297494 + 0.954724i \(0.403849\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3.73093e119 −0.971306 −0.485653 − 0.874152i \(-0.661418\pi\)
−0.485653 + 0.874152i \(0.661418\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 6.22158e119 0.771219 0.385610 − 0.922662i \(-0.373991\pi\)
0.385610 + 0.922662i \(0.373991\pi\)
\(380\) 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
\(388\) 4.58190e120 1.90709
\(389\) 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
\(397\) −1.00855e121 −1.44523 −0.722616 − 0.691250i \(-0.757061\pi\)
−0.722616 + 0.691250i \(0.757061\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −9.90352e120 −1.00000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −1.78597e121 −1.27406
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.51652e121 0.902958 0.451479 − 0.892282i \(-0.350897\pi\)
0.451479 + 0.892282i \(0.350897\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 7.60189e121 1.94178
\(413\) 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
\(421\) −1.51308e122 −1.41496 −0.707480 − 0.706734i \(-0.750168\pi\)
−0.707480 + 0.706734i \(0.750168\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.42900e122 1.17633
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −7.67258e122 −1.94202 −0.971011 − 0.239035i \(-0.923169\pi\)
−0.971011 + 0.239035i \(0.923169\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −5.25411e122 −0.964667
\(437\) 0 0
\(438\) 0 0
\(439\) 1.30782e123 1.74562 0.872809 − 0.488061i \(-0.162296\pi\)
0.872809 + 0.488061i \(0.162296\pi\)
\(440\) 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
\(448\) −3.54166e123 −1.83984
\(449\) 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
\(457\) −3.49400e122 −0.0719813 −0.0359906 − 0.999352i \(-0.511459\pi\)
−0.0359906 + 0.999352i \(0.511459\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 1.46706e124 1.64790 0.823952 − 0.566660i \(-0.191765\pi\)
0.823952 + 0.566660i \(0.191765\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −3.32317e124 −2.05123
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −5.58303e124 −1.90817
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −1.73382e125 −3.30566
\(482\) 0 0
\(483\) 0 0
\(484\) 7.00341e124 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 9.71361e124 1.04060 0.520300 − 0.853984i \(-0.325820\pi\)
0.520300 + 0.853984i \(0.325820\pi\)
\(488\) 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
\(496\) −1.65213e125 −0.755346
\(497\) 0 0
\(498\) 0 0
\(499\) −5.27246e125 −1.82112 −0.910559 − 0.413379i \(-0.864348\pi\)
−0.910559 + 0.413379i \(0.864348\pi\)
\(500\) 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
\(508\) −1.10856e126 −1.66762
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −3.21597e126 −3.67908
\(512\) 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
\(523\) −3.08295e126 −1.19850 −0.599249 − 0.800562i \(-0.704534\pi\)
−0.599249 + 0.800562i \(0.704534\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −4.37209e126 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −1.99658e127 −3.51072
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5.76921e126 −0.464990 −0.232495 − 0.972598i \(-0.574689\pi\)
−0.232495 + 0.972598i \(0.574689\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.78815e127 −1.82817 −0.914085 − 0.405524i \(-0.867089\pi\)
−0.914085 + 0.405524i \(0.867089\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −9.79001e127 −2.84488
\(554\) 0 0
\(555\) 0 0
\(556\) 2.13219e127 0.481780
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −1.85873e128 −3.27015
\(560\) 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
\(571\) −9.96515e127 −0.652987 −0.326494 − 0.945199i \(-0.605867\pi\)
−0.326494 + 0.945199i \(0.605867\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.40757e128 0.970295 0.485148 − 0.874432i \(-0.338766\pi\)
0.485148 + 0.874432i \(0.338766\pi\)
\(578\) 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
\(589\) −9.31374e128 −1.44133
\(590\) 0 0
\(591\) 0 0
\(592\) −1.60389e129 −1.95982
\(593\) 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
\(601\) 3.26700e129 1.97918 0.989592 − 0.143900i \(-0.0459642\pi\)
0.989592 + 0.143900i \(0.0459642\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2.00920e129 0.965620
\(605\) 0 0
\(606\) 0 0
\(607\) 4.82486e129 1.84166 0.920832 − 0.389959i \(-0.127511\pi\)
0.920832 + 0.389959i \(0.127511\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −5.10261e128 −0.123277 −0.0616383 − 0.998099i \(-0.519633\pi\)
−0.0616383 + 0.998099i \(0.519633\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 1.10263e130 1.69362 0.846810 − 0.531896i \(-0.178520\pi\)
0.846810 + 0.531896i \(0.178520\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.01958e130 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 2.51921e130 1.97761
\(629\) 0 0
\(630\) 0 0
\(631\) −3.05553e130 −1.92186 −0.960928 − 0.276799i \(-0.910726\pi\)
−0.960928 + 0.276799i \(0.910726\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 9.93152e130 4.02284
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −2.33808e130 −0.612418 −0.306209 − 0.951964i \(-0.599061\pi\)
−0.306209 + 0.951964i \(0.599061\pi\)
\(644\) 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
\(652\) 1.42563e131 1.95657
\(653\) 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
\(661\) 2.56985e131 1.86440 0.932201 − 0.361941i \(-0.117886\pi\)
0.932201 + 0.361941i \(0.117886\pi\)
\(662\) 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
\(673\) 6.36373e131 1.99992 0.999958 − 0.00921690i \(-0.00293387\pi\)
0.999958 + 0.00921690i \(0.00293387\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −7.21975e131 −1.84502
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) −1.68692e132 −3.50873
\(680\) 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
\(688\) −1.71944e132 −1.93876
\(689\) 0 0
\(690\) 0 0
\(691\) 2.16624e132 1.99516 0.997581 − 0.0695137i \(-0.0221448\pi\)
0.997581 + 0.0695137i \(0.0221448\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 3.64618e132 1.83984
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −9.04180e132 −3.73966
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.77770e132 −1.88813 −0.944066 − 0.329758i \(-0.893033\pi\)
−0.944066 + 0.329758i \(0.893033\pi\)
\(710\) 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
\(721\) −2.79879e133 −3.57257
\(722\) 0 0
\(723\) 0 0
\(724\) −1.16065e133 −1.22140
\(725\) 0 0
\(726\) 0 0
\(727\) 1.63534e133 1.41990 0.709950 − 0.704252i \(-0.248718\pi\)
0.709950 + 0.704252i \(0.248718\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.64798e133 0.976376 0.488188 − 0.872739i \(-0.337658\pi\)
0.488188 + 0.872739i \(0.337658\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −2.34088e133 −0.949321 −0.474661 − 0.880169i \(-0.657429\pi\)
−0.474661 + 0.880169i \(0.657429\pi\)
\(740\) 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
\(751\) −5.94464e132 −0.113990 −0.0569952 − 0.998374i \(-0.518152\pi\)
−0.0569952 + 0.998374i \(0.518152\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.47565e134 1.95446 0.977228 − 0.212194i \(-0.0680610\pi\)
0.977228 + 0.212194i \(0.0680610\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 1.93441e134 1.77483
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −6.62666e133 −0.422396 −0.211198 − 0.977443i \(-0.567736\pi\)
−0.211198 + 0.977443i \(0.567736\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.52268e134 −0.809848
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 1.70089e134 0.755346
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 9.18727e134 2.38501
\(785\) 0 0
\(786\) 0 0
\(787\) 7.90336e134 1.71786 0.858928 − 0.512096i \(-0.171131\pi\)
0.858928 + 0.512096i \(0.171131\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 7.06313e134 1.07843
\(794\) 0 0
\(795\) 0 0
\(796\) 6.81955e134 0.873571
\(797\) 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
\(811\) −7.90800e134 −0.425204 −0.212602 − 0.977139i \(-0.568194\pi\)
−0.212602 + 0.977139i \(0.568194\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −9.69321e135 −3.69949
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 7.11521e135 1.93239 0.966196 − 0.257807i \(-0.0829999\pi\)
0.966196 + 0.257807i \(0.0829999\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 9.74072e135 1.88715 0.943573 − 0.331164i \(-0.107441\pi\)
0.943573 + 0.331164i \(0.107441\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.02986e136 −1.68672
\(833\) 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
\(841\) −1.00696e136 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −1.76260e136 −1.48330
\(845\) 0 0
\(846\) 0 0
\(847\) −2.57845e136 −1.83984
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −2.37459e135 −0.122028 −0.0610139 − 0.998137i \(-0.519433\pi\)
−0.0610139 + 0.998137i \(0.519433\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −5.01121e136 −1.85892 −0.929459 − 0.368926i \(-0.879726\pi\)
−0.929459 + 0.368926i \(0.879726\pi\)
\(860\) 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
\(868\) 6.08265e136 1.38972
\(869\) 0 0
\(870\) 0 0
\(871\) −9.66322e136 −1.88052
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.38795e136 −0.620556 −0.310278 − 0.950646i \(-0.600422\pi\)
−0.310278 + 0.950646i \(0.600422\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −1.83601e135 −0.0189105 −0.00945523 − 0.999955i \(-0.503010\pi\)
−0.00945523 + 0.999955i \(0.503010\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 4.08139e137 3.06814
\(890\) 0 0
\(891\) 0 0
\(892\) −2.01968e136 −0.129813
\(893\) 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
\(907\) 6.75723e137 2.00000 1.00000 0.000684466i \(-0.000217872\pi\)
1.00000 0.000684466i \(0.000217872\pi\)
\(908\) 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
\(916\) 3.00653e137 0.562245
\(917\) 0 0
\(918\) 0 0
\(919\) −1.24425e138 −1.99864 −0.999322 − 0.0368250i \(-0.988276\pi\)
−0.999322 + 0.0368250i \(0.988276\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.65122e138 1.95982
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 5.17925e138 4.55099
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.50121e137 0.0978474 0.0489237 − 0.998803i \(-0.484421\pi\)
0.0489237 + 0.998803i \(0.484421\pi\)
\(938\) 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
\(949\) −9.35150e138 −3.37289
\(950\) 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
\(961\) −2.13576e138 −0.429452
\(962\) 0 0
\(963\) 0 0
\(964\) 5.29275e138 0.920659
\(965\) 0 0
\(966\) 0 0
\(967\) −1.09444e139 −1.64763 −0.823816 − 0.566857i \(-0.808159\pi\)
−0.823816 + 0.566857i \(0.808159\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −7.85011e138 −0.886398
\(974\) 0 0
\(975\) 0 0
\(976\) 6.53383e138 0.639368
\(977\) 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
\(988\) −5.80574e139 −3.21854
\(989\) 0 0
\(990\) 0 0
\(991\) 1.20337e139 0.579397 0.289698 − 0.957118i \(-0.406445\pi\)
0.289698 + 0.957118i \(0.406445\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −4.37244e139 −1.59001 −0.795004 − 0.606604i \(-0.792531\pi\)
−0.795004 + 0.606604i \(0.792531\pi\)
\(998\) 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 9.94.a.a.1.1 ✓ 1
3.2 odd 2 CM 9.94.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.94.a.a.1.1 ✓ 1 1.1 even 1 trivial
9.94.a.a.1.1 ✓ 1 3.2 odd 2 CM